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M . ‘ -- 7 . ~ ~I‘I'."‘I JI'm‘AI r . -~ ILllllllzllljljlllllljllllfllfllllllllllllll 035 - 1’11"“: it ~ LIBRARY Mir". ‘wan Scam U £th “at, This is to certify that the thesis entitled THE EFFECTS ON THE WEIGHING COEFFICIENTS OF ERROR OF MEASUREMENT IN THE RANDOM PREDICTORS OF A QUANTAL RESPONSE 'TECHNIQUE presented by Robert Alan Carr has been accepted towards fulfillment of the requirements for Ph.D Education degree in Md Major professor Date 5/2. ! 7? 0-7639 WWW 1978 9 Copyright by “- I C ’. I‘ 2 .1 uk l n"‘ I .1 ’l 3;- '23'.‘ u‘-‘ I .A -—- :0: ‘ 444.de 2&9“ THE EFFECTS ON THE WEIGHTING COEFFICIENTS - v ' OF ERRORS OF MEASUREMENT IN THE Hi IINDOM PREDICTORS OF A QUANTAL RESPONSE TECHNIQUE BY ‘ Robert Alan Carr _ it ., .. ans- “ I .‘Ih' A DISSERTATION Submitted to ‘ Michigan State University 4 1n partial fulfillment of the requirements for the degree of ‘ DOCTOR OF PHILOSOPHY '1’575fit of counseling, Personnel Services ‘and Edueatlonal Psychology 1978 s.‘- n~-~ s V—xp-n' nu.- .va‘n .I‘ p- o.- o- 5-.- _- \ 'l—N n ”.5. .‘~ ~~o~ - ’ _v A’ . "'-..-.o 1 u. . . ‘9‘ , lb.- «1- vs . — by. ‘. . . “-~ : "‘-h ‘-.~ . a n... V n. ‘O -. ' "a ; '-«. .Y :r “A A 0 '¢ - _ . \._ “a “’1 . . u p. u.‘ ‘ . 'x' y A '~. r.‘ .; ‘w b u p .‘p. “P. p. -,‘ e“ " '7‘ n x h" .‘-. ABSTRACT - THE EFFECTS ON THE WEIGHTING COEFFICIENTS OF ERRORS OF MEASUREMENT IN THE BY Robert Alan Carr The random predictor quantal response model is examined in r{§%%eirch. Quantal response models are qualitative data analysis The general situation addressed by quantal response models Irisearch concern the relationship between a single qualitative ii'iiquantal response) and one or more quantitative random pre- rlNiiiables. This relationship is expressed in a series of ‘swc$efficients. For each category of the criterion there is a gheifihting coefficients with one Weighting coefficient asso- fiith“each predictor variable. fifbblem is to describe a procedure for producing estimates h;5tinq'coefficients which would be produced if there were - igisurement in the random predictors. ifléocedure used describes: a quantal response model and efilents based on the assumed existence of error-free iia'éqmm' eesp’onse model and weighting ,. .n «v o . ' ' ‘ v v .- ".."U.vs. ,znl :7'“' '0‘”. érv- . ,- .h‘.‘ w we Robert Alan Carr énts based on the observed predictor counterparts, which con- Térrors of measurement, of these latent predictors; and two ‘dfi"Ement models which provide two possible relationships between it§o quantal response models. Then the value of a weighting co- 1%E§fcient based on the use of latent predictors is compared to the ,fie of the corresponding weighting coefficient based on the use of ,9“3Q:Iicab1e across the universe of situations which define the quantal o .05 e£::-onse models. Then a set of estimation procedures called analysis of co- -_£flgafiid'be used to derive estimates of the weighting coefficients which ._ i ‘ No generally applicable algebraic results of the effects of ‘fis of measurement were discovered which apply to all possible ran- «predictor quantal response models. Therefore, the two simplest ‘fiore examined in detail. For one predictor quantal response 0" ,..~ . u _ ‘.o~ v. ‘ - ~- ..- r r ’ h . ,voo' . . a; a :amy-Y-nz1 .-..'Il”" -. _y:. . .. “‘.‘c.‘llt ';~~‘...,~ -.~l..-..u ‘ a b . - A “h : “r,“ .0. .7 'Y‘n -."° ”-~¢ Robert Alan Carr ‘fland latent predictors. The derivation of these categories, Y with their descriptions and examples of situations, are ”vi in this research. SSE 0rfr'he analysis of covariance structures procedures as applied Vigwi.maximum-likelihood estimates of the weighting coefficients :gg the use of latent predictors are described. Since these pro- 1 “go POt lead to explicitly solvable estimates, a numerical ‘_;;tion procedure is needed to produce the estimates. This re- .“ ~b. --~ . ... C ‘ fl‘ 5. Ir‘ "Q ..n. ., "~ h “-0... no: a. V V. 1 I d . \ A ~v S '\ :‘erR t nun . a; . a o. - V “‘ -~...‘3‘ s‘.‘ .’ :n,“ A. ‘ nu . 9» :“rr. h .VIK. . ACKNOWLEDGEMENTS During the final stages of writing the thesis and preparing ‘the oral examination Dr. William Schmidt, my committee chairman, Sggarticularly helpful. I thank him for encouraging me to set an Vjis date earlier than I originally thought was feasible and for his 1ft review and return of large volumes of draft material which - the early completion date to come to fruition on schedule. Also, I appreciate the time and effort in reviewing the i which was spent by each member of my committee: Dr. Andrew ’ , Dr. Donald Freeman and Dr. Dennis Gilliland. The care, concern, support and understanding of my family E to keep me focused on the task over the nearly seven years it ‘3. to complete the degree. Thank you Mother and Dad Birchard and Dad Carr. grinning from handwritten drafts, especially just prior to ‘ “Mi-pod to make an early completion of the oral examination n, _ » Thank you dear. Without your support I never would have In (In (I) (I) II) ' uSection ‘1'8ection TABLE OF CONTENTS Introduction Errors of Measurement in Quantitative Data Analysis Models Errors of Measurement in Qualitative Data Analysis Models The Data Analysis Model to be Examined in this Research Presentation of the Problem for this Research The Random Predictor Quantal Response Model - An Introduction The Observed Random Predictor Quantal Response Model The Latent Random Predictor Quantal Response Model The Measurement Model Summary Introduction and Approach to the Problem One Predictor Models (p = 1) TWO Predictor Models (p = 2) Tho Category, TWO Predictor Models (J . 2: P ‘ 2) 'I. Simplify the Notation II. Derive Expressions for B x/B; and By/B; in Terms of Latent , and Error Parameters III. Presentation of the Approach to _r’ the Examination of 8 x/B I‘.sl.The Search for Categories of Dis- -tributions of 8 8/85 as a Func- tion of pfn iv . Page 13 16 18 32 36 45 47 51 54 55 55 57 66 70 Y H n‘ "' -_..o lav... u‘~.‘. . ‘5 oev v. Page V. The Search for Categories of Dis- tributions of B B; as a Func- tion of pan 95 VI. Additional Algebraic* Results Involving Both 8 xg/B and By/B; 115 Derivation 1 115 Derivation 2 117 Derivation 3 120 VII. JOint General Categories of Dis- * * t 'b t' f r1 u ions or Bx/Bg and By/Bn Together 127 J Category, Two Predictor Models (J 3_2, p = 2) 139 Summary 146 Introduction 156 Reformulation of the Observed Random Predictor Quantal Response Model 158 Identifiability of the Models for the Covariance Matrix and Vectors of Category Means of the Reformulated Observed Random Predictor Quantal Response Model 161 ‘Summary for Section C 183 Maximum Likelihood Estimation Pro- cedures Associated with the Reformu- lated Observed Random Predictor Quantal ' Response Model 185 Summary 193 Introduction 195 "The Computer Program (TQUANER) 197 1“ Tflo‘Examples 200 ’.+'Eaample‘l 205 Example 2 212 ifillfllly 219 (u (t- (I) (D ‘l ‘ T r... u . q n-p - ‘ _' 4 "be uvo 4 (ll LA) Ill '- . t- a "v ‘ p n- .- ~.. 9 ‘ .- :50dtion 3: Recommendations for Further Study .1“2 fippendices , - 3L3. Identification and Justification of the Existence “ of a Base Set of Weighting Coefficients in the Multiple Observed Predictor, Polychotomous Criterion Model Identification and Justification of the Existence of a Base Set of Weighting Coefficients in the Multiple Latent Predictor, Polychotomous Criterion Model £.-‘« 3;: 3 Appendices Development of the Property of Inter- changeability of x and Justification of the Need to Examine * J ‘ Bx/BE C. Summary * Examination of 8 /B for Special Case Situations: x g A. dn = 0 (i.e., d = d Only for Values of d 3_0 E = l/d is defined) d = de a o pEn = 0 pxx = pyy = 1 pxx = 1. pyy < 1 or pxx < 1. pyy = 1 Algebraic Examinations of Relationships Between Expressions Needed for Work in Appendix 8.4 2 Examination of = 1- - A; Q pgnpxx( pyy) ‘. dp£n(l-pxxpyy) + (l-pxx) as a Funct1on of 7flngn. Identification and Determination of Existence Conditions of the Roots of Q as a _Phnction of pE flationship of 9;:X) to l/d when d > 1 ).§nsti2n of the Arithmetic Sign of 83/85 BY/Bn ;_e l/(Z-pyy) to (Zpyy-l)/pyy .vi Page 221 229 231 233 235 236 238 239 243 244 245 245 249 256 269 271 275 . .-~ . '..u--.y .4» . u. 2 (I) U, , ‘3 I), A ‘ via 1. it. o ‘0- u..— o ‘9 ¢‘.. Comparison of p2?) to d In Interpretation of dpg as a Ratio of Two Slopes n ’ ' 9. Appendices ‘ ‘ :‘Examination of‘Identifiability of Model (4.4) for 2 Examination of Conditions for Satisfaction of * athe Counting Condition for Identifiability Examination of Identifiability for Two Models for Z A. Model (4.5) ‘3’. Model (4.7) Examination of Identifiability of Model (4.11) ‘ for 2 under the Constraint W2 = 0:1 Derivatives of F = 9:42] + tr{2-ls } where 2 P Page 277 282 291 294 296 299 300 303 307 317 cap '1‘ “1 LIST OF TABLES General Categories G.C. I (x) and G.C. I (y) Subcategory a) and a') General Categories G.C. I (x) and G.C. I (y) Subcategory b) and b') General Categories G.C. II (x) and G.C. II (y) General Categories G.C. III (x) and G.C. III (y) d - l (0 < pxx §_1/2-pyy) 44-» 1 (1/2-pW < on 1 1) Estimates of the Latent Parameters of A, O, V2 and 241) (i = 1,2) for Example 1 7 Parameter Values and Estimates of Weighting ;.056£ficients for Example 1 - Eitimates of the Latent Parameters of A, 0. W2 ‘.J§d 341) (i = 1,2,) for Example 2 ‘.:2Irameter values and Estimates of Weighting Page 106 108 110 111 113 114 208 210 215 217 ‘A‘ .a- in .‘2. - H .a "vh «‘1 "a. n ‘ .1 «.~ (A. f'! LIST OF FIGURES G.C. I subcategory a) rG.C. I subcategory b) G.C. II G.C. III d=1 and 0 8 while the corresponding regression coefficients 2 based on observed scores, 8' 1 and 85, have the relationship Bi < 8%. Although Cochran's example is based on a specific set of para- meter values it does illustrate the potential problem which can arise When the effects of errors of measurement are not considered. Wiley and Hornik (1973) provide a data example which illus- trates the potential for misinterpretation which exists when fallible measures are used in a regression analysis. The data came from a Study of communication processes conducted in Central America. Suf- ficient information was collected to provide estimates of the true regressian . e ‘1 . I‘ .un ..yvv .‘9 un‘ '...’6‘A .‘0 ”:Q nya r x; a . .rtuiv 'U. o I. “ ‘4‘ n5AY -::“" '9‘ w reliable. For HQ:- .1080, a la 0 g Q r; - :r '- s‘u..h~‘. .5 t: regress . V" " " .... 30 e .. . . _ .3. 'N "H . ‘ "b v‘ A :55: ZEVA‘ ev,’ "a 1 . In .M' ‘h . ,. cad...“ . e .. v ‘I.. 5‘51““ .I C‘u ‘ "‘-=. tne E e 952::- 5 ~. Se "“4: ‘7 '§ . l :36“ .. “\. ‘V' t -V‘ , tot: ’9‘; u e ‘.:’ w ~ 0 u 's 7 gm .. . ‘NA 6' ‘ regression coefficients. Each of two dependent variables were individually regressed on two fallible predictor variables. The two predictor variables were positively related to each other. One predictor was highly reliable while the other was considerably less reliable. Considering the estimated true relationship, for one dependent variable the more reliable predictor had the stronger relationship (i.e., a larger estimated true regression coefficient) and the less reliable predictor had virtually no relationship. In this situation the regression coefficients estimated solely from the observed scores with no consideration of errors of measurement did not differ greatly from the estimated true regression coefficients. For the second dependent variable the more reliable predictor had virtually no relationship (i.e., a true regression coefficient near zero) while the less reliable predictor had a very large rela— tionship. In this situation, however, the regression coefficients estimated solely from the observed scores with no consideration of errors of measurement differed markedly from the estimated true regression coefficients. Because of the positive relationship be- tween the predictors, not only did the errors of measurement attenuate the estimated relationship of the less reliable predictor (with the stronger true relationship) but also some of the relationship of this Predictor to the dependent variable is spuriously attributed to the more reliable predictor (with virtually no true relationship). That i3, when errors of measurement were not considered the predictor which had a high estimated true regression coefficient but low . o ‘ 9 . 5. “:r' . . Ianw“ 3:05: ha'. -_'e ether gear zero :Ee “' '<-‘ A..- v was ’65:: +4 ‘0' "ye. 5- ero v Err "die .. ‘9?" ~ v Lie..e.a 62:38 65 5 ‘9. g , .1 : ~ ”7 ability produced an observed regression coefficient which was . \-- 'mt half the size of the true regression coefficient. However, : 'v'fihgzgother predictor which had estimated true regression coefficient m zero but had high reliability and was positively correlated with ' [‘1 :the first predictor produced an observed regression coefficient which I'- V.“ relatively large . -;-. Therefore in this example for the second dependent variable . w ‘dhterpretations based on regression coefficients estimated solely 72:;er the use of observed scores with no consideration of errors of 3‘fleasurement in the predictors would lead to conclusions which are _. censiderably different from the conclusions based on an examination .~_~ (fifths estimated true regression coefficients. ’ gar . These two examples provide an indication of the adverse ffects of errors of measurements in regression analysis. Other re- “dredged the problems which can occur with other quantitative data . {c‘ ' pfll’ys‘is. models . u .1--. h The vn"'; “EC 5- :uavh . . Oh, gazes W. ..‘. y a; 5L “me. Uh b-J gvvn T30 “out I g.- Auc - ' ‘ 'I“~ Ibu&a5 . C 3;. ,_ h a“ ‘AEI.:‘V‘. . r a.._. ‘A v n , . .‘au ‘Vl ea: 3f the Eithes ‘ . \‘ ”I; fl~ Ub‘vll“ It? ‘\ . . ”\M‘. ‘ ‘ ‘e A.’ V U. I.“ ’x V‘ kfi‘ :h; n “‘i :z‘n ‘.°e v i Q a i I «_ . .‘v. Section C: Errors of measurement in qualitative data analysis models The problems associated with errors of measurement are not restricted solely to quantitative data analysis models. Bross (1954) examines the effects of errors in classification 1310 e variable in a 2 x 2 table. In this case samples from each of two populations are classified into one of two categories on a second dimension. The association of any unit with a particular population is assumed to be without error but the classification of that unit into one or the other of the two categories on the second dimension is subject to error. The interest in this case is in the proportion of units from one population which are classified into one category on the second dimension as compared to the proportion of units from the second population classified into the same category on the second dimension. In this case, if the proportions of false negatives from each of the two populations are equal and the proportions of false positives from each of the populations are also equal then the dif- ference in the proportions of units assigned to one category of the second dimension when using the observed proportions is an under— estimate of the difference based on the true proportions. Here the Type I errors of the test of significance will remain unchanged but the power of the test will be decreased. If, however, the assumptions of the equalitywaf false negatives and false positives across populations is not appropriate, the Type I errors of the test of significance are increased. Mote and Anderson (1965) work with units from a single popu- lation which are classified into one of several categories on some 56763; ~ A.ru..e~ ‘ u only v .. A. 'e I‘VDA . I‘ , .‘ “:5'Ay 'e-d‘»¢~. I ~ 1 \ “;?;~ "‘vo nu ' . I“ In. “fl “3"- o n--_’ _.r ’- ‘u'.. .E ' A ‘ 5A ~..: UV ‘r ‘u dimension. When errors of classification are present the estimates of the population proportions in any one category based on a random sample of units from the population are biased. And standard statistical tests where the null hypotheses specify particular popu- lation proportions or relationships among the population proportions will have increased Type I errors. Mote and Anderson (1965) discuss several special case situations where statistical tests can be con- structed which will have correct Type I errors. Each of these situa- tions requires some specialized information which may not be available in all cases. Assakul and Proctor (1967) examine the effects of misclassi- fication in the r X c contingency table on the standard x2 test. Here errors of classification in each of the two dimensions are con- sidered. If and only if the errors of misclassification in one dimension are independent of errors of misclassification in the second dimension then the null hypothesis for the x2 test of independence based on the true population proportions implies the null hypothesis based on the observed population proportions and vice-versa. Under this condition the Type I errors are unchanged but the power of the test in large samplesi£;never increased and nearly always reduced by misclassification. When the errors of misclassification are not independent, Assakul and Proctor show how to make an appropriate x2 test based on observed proportions when some very specialized information is available. In this case they follow the same procedure used by Mote and Anderson (1965) for one of their special case situations. . I . ‘ ,These three references indicate that errors of measurement ': co problems in quantitative data analysis models as well as 'jtdms data analysis models. A summary of the three references R l...‘ A U s ,‘U ‘ It. v The :5 :‘ge effe: 11:32 far 1’?- guaLLtative The sci-Liques c nation (; Enables. LEE values c 2::;:rer.:e c rie'. :1“; re 'g- one; I. ‘t.:ub....g CC I. . I ’.'~" ~-.‘:.._ “A A; 0 ‘Q (1' - a...“ h.“ ‘5' Ch. ‘V. Section D: The data analysis model to be examined in this research The research to be presented here extends the investigation of the effects of errors of measurement to a quantal response tech- nique for random predictor variables. Quantal responses models are qualitative data analysis models. The general situation which is addressed by quantal response techniques concerns the relationship between a single qualitative criterion (quantal response) and one or more quantitative predictor variables. The relationship of interest is the relationship between the values on the set of predictor variables and the probability of occurrence of a particular quantal response. In a quantal response model the relationship of interest is expressed in a series of weighting coefficients. For each category of the criterion variable I there is a set of weighting coefficients with one weighting coefficient associated with each predictor variable. The sign and relative size of the weighting coefficient give an indication of the type and strength of the relationship. These techniques can be employed as classification procedures. That is, for a particular subject whose classification on the criterion is not known but whose set of values on the predictor vari- ables is known, these techniques provide information about which of the categories of the criterion is most probable. For a general discussion about the classification of observations see Anderson (1958, chapter 6) and Tatsuoka (1974). Another use of quantal response techniques is to determine estimates of the relationship between the predictor variables and the 7.95 I n:v;l~l 0 tag...“ I s ‘ . .AAF'V A uCVAh.-~. . .— ' . A... V .H. - e4. ‘ I- . A "'“ vi 1 A'." v» “-) u..-“ e -._‘ I: I. [In 4 . "-3. 10 probability of occurrence of a category of the criterion. Estimates of the relationship, in the form of a weighting coefficient for each predictor variable, can be produced. It is this use of quantal response techniques that is of interest in this research. Quantal response techniques fall into one of two general types, each with a model and associated procedures for estimating the parameters of the model. The distinction between the two types of quantal response techniques depends on the type of relationship that is postulated be- tween the predictor variables and the probability of occurrence of levels of the criterion. Since McSweeney and Schmidt (1974) and Cornfield, Gordon and Smith (1960) both provide discussion about the two types of quantal response techniques, only a brief description will be given here. The first type of quantal response model assumes that, either by the sampling procedure and/or by the theoretical considera- tion of the location ofthepredictor variables late in the causal chain ending with the criterion and the indirect mediational rela- tionship of the predictors between other links in the causal chain and the criterion, a functional relationship between the predictors and the probability of occurrence of levels of the criterion seems reasonable. For this type of quantal response model the predictor variables are treated as fixed mathematical variables regardless of their method of selection. The second type of quantal response model becomes appropriate when a functional relationship between the predictors and the links :in th rs: likely we a dire: as u indire :se a s_t__t_i are re: tre 35.25. The .I 'I O. J T “-M beAC ’54 h! : I .21 3 d nib-p, '_ «. ; f £‘¢¢‘~¢u 5‘\ 12:6: of e: .5 n u I“ . . v- ~~ “es! in; ‘I. A N” wit J14 ."b ' ’Q . ‘> :‘oghl‘lto'v a "e ..,e p on“: F- . . . "U J‘ :‘n‘ UV ‘-v;|:Cuc. . "W; ‘ 4 ID. ‘ probability of occurrence of levels of the criterion does not seem to be a reasonable assumption. Because of sampling techniques and/or because the predictor variables "...can be thought of as intermediate links [in the causal chain] or as outcomes themselves then it is most likely true that the factors influencing the predictors will have a direct effect on the probability of [the criterion] as well as an indirect influence mediated through the predictors.“1 In this case a statistical relationship is assumed andthepmedictor variables are then treated as random variables rather than mathematical vari- ables. The first type of quantal response model has been employed in the biological sciences, particularly in assessments of drug potency. A simplistic prototypical experiment would involve the pre- determination of a fixed number of drug dosage levels. A preset number of experimental animals at each dosage level would be injected with the drug and their response on some criterion would be noted. The criterion might be dichotomous (e.g. alive or dead) or polychotomous (e.g. alive, moribund or dead). The important thing to note here is that the dosage level is experimentally controllable and the drug dosage level is expected to have a direct effect on the probability of survival or non-survival. The second type of quantal response model whichwillbe the focus of this research will generally be more appropriate for social 1McSweeney and Schmidt; "Quantal Response Techniques for Random Predictor Variables," AERA presentation, 1974. :er.ce aE S: ,e Cf « D. u... is f n f . c , . up. Q - V ~PA v...v . a: S q. “.3 s. 8x39: . “ 12 science applications. McSweeney and Schmidt (1974, pp. 5, 6) pose a hypothetical example of this second type of model. In the example, mastery of a learning task (with two levels, mastery and non-mastery) is the criterion of interest. Theprobability of mastery is to be expressed as a function of entry level knowledge of the student. "In this case, the data would be generated by classifying a random sample of subjects on the basis of their entry knowledge and their mastery. The choice of levels of entry knowledge of the subjects to be observed is not under the control of the experimenter and as such the number of subjects exhibiting xk units of entry knowledge is a random variable (usually taking on only the values zero and one) rather than an experimenter-imposed-constraint. Furthermore, it would be plausible, logically, to postulate the existence of other variables (e.g. motivation, need for achievenemt, interest in subject matter) that affect both [entry level knowledge] and mastery. Consequently the observed relationship between the predictor and the criterion could be a result of the direct relationship of each to other vari- ables."2 Thus in this case the predictor variables are expected to have a statistical, as opposed to a functional, relationship with the probability of occurrence of levels of the criterion. Therefore the predictor variables are treated as random variables. This second type of quantal response model which is the model of interest for this research will be called the Random Predictor Quantal Response Model to distinguish it from other uses of quantal response techniques not involved in this research. 21bid, p. 6. .;:-‘u- 1- ...-..a b- ‘- O ~1~° on-.. ‘5 . fi'V‘v O b...~ 'I. ‘o-.. ‘. and t" .4. CL; ’ce -.~ . ,- Q . f~:'-- \_. v.,~ 13 Section E: Presentation of the problem for this research In the presentation of the model it is clear that the Quantal Response model contains both qualitative and quantitative variables. The criterion is a qualitative variable with two or more categories which have no necessary ordered relationship. The predictor vari- ables are quantitative statistical variables which can conceivably assume any real value, positive, negative or zero. For this research the qualitative criterion variable will be assumed to be error-free. That is for any given unit the classification of that unit into one unique category of the criterion is accomplished without error. However, one or more of the predictor variables may be measured with error. Therefore the impetus for this research is provided by a situation such as the following. There is an interest in determining the relationship between the occurrence of some category of a qualitative criterion variable and the true values on one or more quantitative predictor variables where the Random Predictor Quantal Response Model is the model of choice. Since the relationship of interest involves the true values of the predictor variables, rather than the observed values, and since it is known for other statistical models (e.g. Linear Regression) that in the presence of errors of measurement the relationships estimated on the basis of observed scores do not always approximate well the relationships estimated on the basis of true scores, two general questions arise. . ' ' -6 n ' ”1'. »C J {reg 26.2319 on a :5 masxeze as: betwee: seas: nest H ‘ '1 1 $.22 ‘11; H L a :56 SCCIES . ‘ a. ‘. “Q ' n “‘e :‘Dq, M.,_-~ . L‘F‘aa. ‘ o~¢vc~es ’7- \III 1 .V:- ~ . u. ‘1 o '- 5'0 lit". . f'cn. 14 The first question is: How much variation is there in the estimated relationship based on observed scores of the predictor variables compared to the relationship based on the true scores of the predictor variables? The response to this question may vary de- pending on a variety of factors such as the extent to which errors of measurement are present in the predictor variables and the correla- tion between the predictor variables, among others. Since some difference in estimated relationships can be ex- pected based on research with other models and since true scores on the predictor variables are typically not directly measurable, the second question becomes: What estimation procedures can be developed which will provide the estimated relationship of interest based on true scores of the predictor variables? These two questions provide the direction for this research. The first question provides the direction for the first major area of the research. Area one involves determining the effects of various levels of errors of measurement on the weighting coefficients in the Random Predictor Quantal Response Model. The second question indicates the direction of the second major area of the research. Area two involves developing techniques to estimate the weighting coefficients which would result if the true score for each predictor Variable were available for use in the model. Chapter 2 will provide a detailed presentation of the Random Predictor Quantal Response Model for observed predictors and for true Predictors. In each case, the general model and two special cases "111 be presented along with various simplifying derivations and 'e'ates the i"... " on » Oiw‘. "c “4-6 .5 b. . 282385 SIZE :2 aoiel a: C .-O Q I .Z!::.(L‘G (p .5s;»::s cf "1 ‘i... he use: C‘F " 066‘ c 15 other interesting algebraic results. The measurement model which relates the true predictors with the associated observed predictors will be defined in this chapter. Chapters 3 and 4 will present the results of the research for the two major areas identified above; chapter 3 for Area one and chapter 4 for Area two. For chapter 4 an expanded measurement model will relate the observed predictors to the latent predictors. The task will then be to estimate latent parameters from the observed data. A set of pro- cedures often used where errors of measurement are incorporated in the model are termed Analysis of Covariance Structures (ANCOVST). Joreskog (1970), Wiley, Schmidt and Bramble (1973) present dis— cussions of ANCOVST procedures. Modifications of these procedures will be used in chapter 4. Chapter 5 will contain a brief description of a computer program, using the methods described in Chapter 4, which can produce estimates of the latent weighting coefficients. An illustration of the use of the computer program will also be provided. Chapter 6 will provide the summary of the results of both major areas of this research along with recommendations for further study. . . ,. no . A'- ‘PevoL-rl‘ It. .. 3313! U '. i C. , p . (t, a (3 :Prny. ‘fiav.: O . Q . IV“! 'A "‘~ '5‘ . '4 '- i"“ F‘ 9o m.-.“ VA; se‘ (I- O C. I 9‘ ‘ A. .- ‘1 ! .3». ~ ‘.._:‘ .- G‘h CHAPTER 2 Section A: The Random Predictor Quantal Response Model - An Introduction In this chapter the Random Predictor Quantal Response Model will be presented along with various algebraic derivations and results of interest. The weighting coefficients associated with each pre- dictor which provide an indication of the conditional relationship between a given predictor and the probability of classification into a particular category of the criterion will be identified. In fact, two Random Predictor Quantal Response Models will be presented. The first model to be presented (Section B) is based solely on the use of observed predictors with no consideration of errors of measurement. This model will be called the Observed Random Predictor Quantal Response Model. The weighting coefficients identified from this model will be called the observed weighting co— efficients. The second model to be presented (Section C) is based on the use of latent predictors. This model will be called the Latent Random Predictor Quantal Response Model. The weighting co- efficients identified from this model will be called the true weighting coefficients. Although the true weighting coefficients represent the relationship of interest between the predictors and the criterion, seldom if ever will there be available direct measure- ments of the latent predictors. Thus, there will not be available 16 est'jates ‘ 367.15 of t; (I) . ,_ a ‘ n ne'PV’" A .93Ca'e.‘ : . - v on— I “VA :»a: U ' t ' ‘A be. a I! .2...e .e .- “on..." =.i+"‘1-|M22 J 222 1:1 i(m-1)| ' °(i+1)1 _ m—l p-l = (:1) z (-1) 22 i=1 . 2 i 22 IMi(m-1)I ' °(i+1)1 matrix 222 and a(i+1)1 is the element in row (1 + 1) and . l‘of 2 which is also the 1th element of the vector £21. ' equations (2.8) and (2.9). HP” = (-1)2(-1)“"1 = (-1)"“1 Z 2 22%1 ' m-1)|=|M(m-1)1l = “‘11 1| from Note 4 above and Note 3 i"; "e b IOU. ‘ . u b. “7.: b v U. no . o ;;-opr now-U. .. ny;-u Abh ifi¥flfib 'U o a 1' incl] s 20 ~F .. '1 . .‘ “in. C." v‘ 29 Cm C11 Thus equation (2.5) becomes: C 33.3: -1 (k) 11E (k) (2 ) = u - g b u(k)] 15x 1 TH x1 1=21"‘ 8 ,i‘ -1-“here b1 1 (2 = 2,...,p) is the regression coefficient associated ‘ {yith predictor x£ when predictor X2 and the other p - 2 pre- thfféictors are regressed on the first predictor as in equation (2.6). ,Eé‘ In general for any predictor xq in category k, since any r; 'ggradictor can be put first in the ordering of the set of predictors: (2 -12 (k)) (Tg?x [u(k _ Z 2 “(t)] . 2=1bq° x /« lfq (k) (k) (k) (k) Note: u = b . + b . u +...+ b . _ u _ -— xq <10 qlxl qqlqu (k) u(k) +b +...+b . q <1+1u xq+ +1 q 9 "x13 (k) _ u(k) 1; "(In q-O xq =1 q.g xz 27k; ‘ r “2 , b(k) _ : q-O sents the intercept of the regression hyperplane of xq for is the constant in the regression equation above and ty k data. Therefore (213(k)) q=_c|:_ 2), the research to be presented in this chapter is based on two special cases of the general quantal response models. The two special cases of the general quantal response models to be examined are one predictor models (i.e., p = 1) in Section B and two predictor models (i.e., p = 2) in Section C. Most of the work on two predictor models will be done for models with a dichotomous criterion (J = 2). The two-predictor, polychotomous criterion model will be shown to repre- sent a simple extension of the two-predictor, dichotomous criterion model. 51 Section B: One Predictor Models (p = l) The first case to be examined is that with a polychotomous criterion (J 3 2) and one predictor (p = 1). For this case the Observed Random Predictor Quantal Response Model (2.2) becomes for some category k (k = 1,2,...,J) l (3.1) Pr{Y — klx} — Pk - J - + 1 + .2 exp{ (atk.j Bk-jX)} J=1 jfk where (k) 2 (j) 2 P. 1(u)-(u) _ _1. X X a - -Ln( ) - k-j pk 2 02 X and _ (k) (j) 2 Bk-j ("x 11x )/°x . (i) 2 . . with “X and ox, the mean and the variance, respectively, of the distribution of the single observed predictor X for category i. For this special case the general latent predictor model (2.19) becomes for that same category k identified above: * l (3.2) Pr{Y — le} — Pk — J * * l + .2 exp{-(0Lk.j + Bk_jT)} J=1 M where k 2 ' 2 * P. (u; )) -(ué3)) a ‘ -Ln(—J) - — k-j pk 2 02 T and * (k) (j) 2 8k°j - ( T “T )/°T for sinc Where 5e rvel the l. 52 with uél) and 0:, the mean and the variance, respectively, for the distribution of the single latent predictor T for category i. Using the expression for Bk'j from (3.1) and the expression * for Bk- from (3.2) for some j # k, j,k = 1,2,...,J (k) 11(j) 2 Bk.. = (ux )/0: =32- * (k) u(j) 2 Bk-j (uT )/0: OX . (k) _ (k) (j) _ (3') Since “X - “T and “X — “T from (2.24a). Thus for any category k and any weighting coefficients * Bk'j and Bk-j 2 Bk.. GT (3.3) 1.1.7:). (jaék. j.k=1.2.....J) B 0 xx koj X where pxx = oi/o: is the reliability of the observed predictor X. Therefore for all one predictor models the value of the ob- served weighting coefficient will be an underestimate of the value of the latent (true) weighting coefficient by a factor equal to the reliability of the predictor. The more reliable the predictor the closer the values of the observed weighting coefficient will be to the corresponding latent weighting coefficient. However, the values of the observed weighting coefficient will be identical to the values of the corresponding latent weighting coefficient only for a perfectly reliable predictor, i.e. pxx = l. * If Bk-j = O for some j # k, j,k = 1,2,...,J, this implies (k) (j) (k) (j) (k) (j) . t t - = - = - ha uT “T 0 but then “T “T “X ux requires (k) (j) _ - that “X - ”X - 0 and thus Bk-j — 0 and conversely. 53 Therefore in any one predictor model if Bk-j = 0 for any i j # k, j,k = 1,2,...,J then Bk-j = 0 and the observed weighting coefficient provides an exact estimate of the latent weighting co- efficient. 54 Section C: Two Predictor Models (p = 2) The polychotomous criterion (J 3_2) two-predictor (p = 2) models for both observed and latent predictors have the same appear- ance as the general case models given by (2.2) and (2.19). The specialization to two predictors is obvious only when the precise form of the veCtors and matrices of the models are examined. The parallel structure of the two models (2.2) and (2.19) allows the identifica- tion of the two predictor case to proceed for each model simultaneously. The identification of the vectors and matrices from the Observed Random Predictor Quantal Response Model (2.2) will be presented below on the left with the corresponding vectors and matrices from the Latent Random Predictor Quantal Response Model (2.19) on the right. The vectors of predictors become: r‘ o F- -1 x1 Tl X = 2 and T = 2 . LX T .4 C. .4 The vectors of predictor means for some category i (i = 1,2,...,J) become: 1 '- 1 F' (i) (i) uX1 lJTl (i) _ . (i) _ . Ex .. 11(3) and ET - p(g) - X L. .J L.T .J The matrices of predictor variances and covariances, assumed homogeneous across all categories become: 55 . 1 :2 . 1 X1 X1X2 T1 Tsz Z = 2 and ¢ = O' 0' 0‘ 0' 2 1 2 2 l 2 And the vectors of weighting coefficeints for some category k are: Bk-j(xl) Bk'j(T1) ékoj = j and Eko ' 3 Bk°j(X2) Bk-j(T2) where j # k, j,k = 1,2,...,J. The approach to this speical case (p 2) will proceed by first examining the simplest two predictor model. This simplest model involves two categories of the criterion (J = 2) and the two pre- dictors. Results for more complex models involving more than two categories of the criterion (J > 2) and two predictors will be shown to be simple extensions of the results for the simplest two predictor model. Two Category, Two Predictor Models (J = 2, p = 2) I. Simplify the notation In order to simplify the appearance of the algebra below several notational adjustments to the general models will be made. The two observed predictors will be denoted as x and y, i.e. ‘X' = [x y]. The two latent predictors will be denoted as g and n, i.e. ‘3' = [g n] where x = g + ex and y = n + ey, i.e. g = g + E. The two categories of the criterion will be identified by the numerals 56 O and 1, rather than 1 and 2, so that the category identification is consistent with the notation used for the dichotomous criterion model in both McSweeney and Schmidt (1974) and Cornfield, Gordon and Smith (1960). Also let p and po the unconditional probabilities of 1 occurrence of category 1 and 0 respectively be p1 = p and P0 = 1 ‘ P1 = Q- Since for the dichotomous case Pr{Y = llg} + Pr{Y = DIX} = l and Pr{Y = lIT} + Pr{Y = GIT} = 1, it will be sufficient to work with the expressions of the observed and latent predictor models associated with Pr{Y = lIX} and Pr{Y = llg} respectively. Associated with each of these model expressions is a single weighting coefficient, 51-0 and 51-0 each with two components. Since the proofs in Appendices A.l and A.2 indicated $1.0 = -§o-l and El-o = -§;.1, that is for each model there is only one distinct weighting coeffi- . * * Cient, let g = él-o and g = gl-o' Therefore the dichotomous criterion (J = 2), two predictor (p = 2) observed predictor model can be expressed as: l (3.4) Pr{Y = 1 X} = P = ' IN 1 1 + exp{-(al.o + g §)} where X = [x] y =_ g _1 (1)'-1(1)_ (ow-1(0) a... W.) 304x 2 Ex 2,. 2 1,. J and B _ X _ '1 (l) _ (o) E“ B ‘2 (Ex Ex ). Y and the latent predictor model can be expressed as: 57 * 1 (3.5) Pr{Y = 1|g} = P1 = * *0 1 + exp{-(a1. + g 3)} 0 where E = [E] n = _ g, _ l (1)' -l (1) _ (O)' -l (0) 01:0 Ln(p) 2£ET ¢ ET RT ¢ RT 3 and 8* * - £3, = E ”191;“ fig”). BY) * * II. Derive Expressions for Bx/Bg and B /8 in terms of latent and error parameters y n In order to study the effects of errors of measurement on the weighting coefficients the ratios Bx/B; and By/B; will be examined. The formulas presented previously for single predictor weighting coefficients will prove of limited usefulness for this task, therefore the first necessity will be to derive expressions and conditions for existence for Bx, By, B; and 8;. From these expressions the desired ratios can be formed. Consider first * B * _ E _ -l (l) (o) g * ¢ (ET RT ) 8T) 2 r— -p '7 0.: 0S -1 1 1. m o = n and ¢ = 2 O 0 o 02 l-p2 05 g n n5 n in _p ..Jfll ll. 0 0 2 L_ E n 0 .4 T) where pan is the correlation between the latent predictors E and n. Note: ¢-l exist only if 1 - pgn # 0 which implies pEn # :_l. 58 a (l) _ ”(0) Let a g 5 = (1) _ (o) = 5 5 ~T a RT ET “(1). _ r1(0) V) t. n n ..J Therefore r- o r- _ '1 r- o * 1 pg * Ba -1 1 "3' o o a: E = = ¢ ~T = 2 05 E n * l _ D 8n En _p L. .1 53 1__ a Logcn (,2 J L. U.) T) :5 _ an 0&1 2 o o * 1 O n E = 2 a 5 1 - on a a 0 _fl._ _§L_£fll C2 Gian L. n .4 and a a p (3.6a) 3* = ___-L_ _§. __ n an 5 1 - 02 02 Oion En E a a o p = l £[l-J-ggnjfora540 1 - 92 02 Onag E in 5 Thus 13* a 11(1) _ “(0) (3.6b) 8* = —-——§-—- 1 - d p where b* = —§-= 5 n C 1 _ 02 E in E 02 02 in E 5 a /o (u(1) - u(0))/0 - n n _ n 1 n and dg ‘ a /0 ‘ (1) (o) E E (“g - “g )/06 - . * (1) (o) 2 In this formulation bg = (uE - pg )/05 has the form of a latent weighting coefficient for g from a single predictor model (see (3.2) for an example). In this formulation dg represents the 59 ratio of category mean differences for the two latent predictors, where each mean difference is in standard units (i.e. divided by the standard deviation of the distribution of the latent predictors). Therefore, a large positive dg value indicates a larger standard unit mean difference between category 1 and category 0 for latent pre- dictor n than for latent predictor 5. Other values of d5 would carry appropriate corresponding interpretations in terms of ratios of standard unit mean differences between categories. And * a a D _ __1__. .1 E én (3.7a) 8 — _ n l - 02 02 Gian in n a a o p = 1 "—fl 1 - g n in for a # 0. 1 - 02 02 Cg a n En n * (3 7b) * br‘ 1 d in (1) (0) a _ a /O a» U Where b = “3': n n and d =-—§——§ . Since n 02 2 n a /0 n 0n T) n an/o * l d = n as defined above for B , d =-—- for d # O. E ag/og E n dg 6 Consider now the expressions for 8x and BY from the observed predictor model. B x _ -l (l) (o) B - B — 2 (11K uX ) Y f" "D ‘-‘ I_L__ xx 2 -1 1 2 o o 2 = o and 2 — 2 o x y x xy 1 _ p x 2 “Y yx O Y .jESL .l. o o 02 L. x y y .4 where p XY y. Note: 2-1 Let 3x = Thus 3 = E, = is the correlation between the observed predictors 2 exists onl if 1 - O which im lies y oxy 7‘ p pxy u __ (l) (o) " Ex Ex u ‘1 l = Z a = X l _ 2 XY r-a a p _ .3S...JL_3§L 2 0x0 02 Ox y xy a a p _X.- _§;_§X. O2 oxoy L. Y .4 60 (l) X I 'D XQ Q x and #11. Before proceeding further with finding expressions for 8x and 8y from the observed predictor model consider some of the im- portant relationships between expressions involving observed pre- dictor model parameters and the corresponding expressions involving latent predictor model parameters. (3.8) X (3.9a) pxx (3.9b) pyy O (1) Ex 2 E /o (o) ’ E 2 x 2 2 G O n/ Y = (1) x HT therefore therefore ~T a = Y 2 2 0x - Og/pxx 2 2 o = 0 Y n/pyy (o) 61 O O 0 xx. in gn r""" (3.10) p = = = = XY 2 2 2 2 l 2 pxxpyy 05” pxxpyy O O O O O O’ x Y 5 . _n__ 5 n Vpxx pyy l-e- p = 0 V0 xy En xprY ' * E sions of interest for the research which is to follow, expressions * Since the ratios Bx/B and By/Bn are the ultimate expres- (3.8) - (3.10) above will be used to express Bx and By in terms of parameters from the latent predictor model. The expression for Bx using parameters from the observed predictor model is: l .5. .x._§2. B = ' x 1 _ 02 02 0x0 xY X y (3.11a) l a a Ox 0 8 = - -5L 1 - -X--—§X- for a # O . x l _ D2 02 o ax x xy x y And using expressions (3.8) - (3.10) Bx becomes: a a - 0 V ° V B = l . g . 1 _ n 5/ pxx pgn pxxpyy x 2 2 l - p p o 0 V - a pan xx yy a/pxx n/ 0yy 5 * b o _ 5 xx _ and 62 (l) (o) d = an/On = (un "n )/on E a /o (1) _ u(o) ° E E (ug “E )/°g * Therefore using the ratio of (3.llb) and (3.6b). Bx/Bg will * exist if 85 f O, and (3.12) * 'k B B - 1 ‘ d D /L—_-———' l - d p 2 X/ E 1 _ 02 p p E Enpw 1 _ pg” E En En xx YY p2 Bx/8* = (1- pEn )p xx(1 dgpgnpyy) g (l — pgnp xx pyy)(1 - dEpEn) * This expression for Bx/BE (3.12) will exist if: 1.) pg“ # :.l (Needed for 4‘1 to exist.) Note: = therefore < there- oxy ognVoxxpyy (Oxyl ..lpgnl . -l . . if + 1 then + l and X will eXlSt. pEn # __ oxy ¢._ 2.) a 0 g f . (1) p(0) (l) (o) . . That is, a = - O‘=u . This is a “a “a 3 “a E “a needed only to guarantee the existence of the specific 'k formulation for BX/BE being used. A variation of the * expression Bx/Bg for a5 = 0 will be examined below. 3.) l - d 0 ==d 1 £°£n * apan * * This is needed to guarantee that 85 ¢ 0. The second * * requirement, ag # O, guarantees that b # 0 thus E * * Bg # 0. When BE = O, Bx will be examined briefly below. 63 Expressions for By can also be produced using arguments similar to those for Bx above: The expression for By using parameters from the observed predictor model is: l a axo p (3.13a) B =———-—1- 1-—Y—XX fora#0. y 2 2 o a Y 1 - p o x y xy y And using expressions (3.8) - (3.10) BY becomes: * b p n YY 3.13b = l-d ( ) By 1-020 0 [ nognoxx] En xx YY (l) (1) * a u - a /o where b =—"= TL ” and d =—i——§ with d =1— n 2 2 n a /o n d on 0n n n E for d f O. E * Therefore, using the ratio of (3.13b) and (3.7b), By/Bn * will exist if Bn # 0, and 2 (1'0 )0 (l-dpp) * (3.14) By/Bn = in Y! n gn xx (1 ' pinpxxpyy)(l - dnpgn) * This expression for By/Bn (3.14) will exist if: 1.) pan # :_l (Needed for ¢_1 to exist. Also, see Note * with condition 1 for existence of BX/BE above.) 2.) a O n # That is, a = u(l) — “(0) # O = u(l) # u(0). This is r) T) T) Tl 1') needed only to guarantee the existence of the specific * formulation for By/Bn being used. A variation of the * expression of By/Bn will be examined for an = 0. 4-". the ‘1‘ RV”. 64 3.) 1 - d O = d 1 no n f ann 7‘ E This along with the second requirement above is needed * to guarantee that Bn # 0. Note since 02 > 0 then p > O, and 02 > 0 then p > 0. There- E XX n YY fore the effective ran es for p and are: 9 XX, pyy pan —1 < p < +1 (with one possible exception for one of the ratios depending upon the value of dE or dn). * * Ex ressions (3.12) for and (3.14) for will be P Bx/BE By/Bn the primary expressions of interest for the work below. However, a close examination of expressions (3.12) and (3.14) shows an identical structure for each expression. Because of this identical structure, * the expression for By/Bn (3.14) can be found from the expression for * Bx/Bg (3.12) by merely interchanging the x's and y's as well as * the 5's and n's in the notation for Bx/B . E ‘k * result derived for Bx/Bg will have a corresponding result for By/Bn Therefore any algebraic which can be simply stated, rather than derived, using this property of interchangeability of x and y (and g and n as well). It is important to note that the values of p and dE which an: Dxx: pyy * * produce a given value of Bx/Bg say R (i.e., Bx/Bg = R for the given p and d ) will not in general also produce a En, pXX' pYY * value of R for By/Bn. That is, in general, for a given situation (i.e., a specific set of values for p , p , p and d ) the En xx W E 65 * * values of Bx/Bg and By/Bn will not be identical. However, by the use of the property of interchangeability of x and y it is possible to identify a different situation (i.e., different values for * and dg) where By/Bn = R. If we let p p En' DXX' YY * and d5 represent the situation where Bx/Bg = R and p pg”! pxxr pyy I ! I En' DXX' pYY * and dé represent the generally different situation where B /Bn = R then Appendix B.l demonstrates that the two situations have the follow- ing relationship: (3.15a) ' = p pEn En 3.15b ' = ( ) pxx pyy 3. 5 ' = ( 1 C) pyy pxx (3.15a) d' = a n E * Therefore, it will be necessary to examine only Bx/B in E detail across the universe of situations (i.e., values of pin' pxx' * pyy and dg)' Corresponding results for By/Bn can be obtained through the use of expressions (3.15a) - (3.15d). The prime in the notation will rarely be used unless the interchanging of x's and y's becomes ambiguous without its use. * Since only values of Bx/B need be examined in detail and E since d = l/dn let d = d = l/dn be used where no ambiguity will E E result. Appendix B.l also demonstrates that only values of d 3_O * 5' * Bx/Bg with d < O is the reflection through the line pEn = 0 of * the expression for Bx/Bg with Idl > 0. That is, for d < 0, there need be considered in the examination of Bx/B The expression for 66 will exist values d" (d" > O) and pg“ which produce the same value * of Bx/Bg as d and pg”. These values d" and pg” have the following relationship to d and OED. (See Appendix 3.1 for details.) (3.16a) d" = -d (since d < 0, d" > O), and 3.16b " = - ( ) pEn 0 En * g 'k be examined since results for other situations for Bx/B and all E * situations for BY/Bn can then be derived using expressions (3.15a) - Thus in the work which follows only Bx/B for d :_0 .will (3.15d) or (3.16a) and (3.16b). Results for special case situations, i.e., d = O (i.e. d = d undefined), d = d = O, = O, n ' E E pEn xx = pyy = 1, pxx = 1 and pyy < 1, and pxx < 1 and pyy = 1, are presented in Appendix B.2. * III. Presentation of the Approach to the Examination of Bx/Bg As indicated in section A of this chapter the interest for this area of the research is to determine for what situations Bx is * an overestimate, an exact estimate or an underestimate of B . To E * pursue this question comparison of the ratio Bx/Bg to l is to be examined. There is an algebraic expression which will aid in this examination. Let (3.17) Q = 02 Enpxx(l - o ) - do (1 - oxxpyy) + (l - oxx) YY En for O < < 1 O < < l, -l < < +1 and an d such that d l. pEn # To see how Q can aid in the search for relationships between * Bx/Bg and one consider: b) 2 l - 1 - d EE._ 5%.- ( pEn)pxx( pEnpyy) - 1 ’"ln *_(1 2 )(1 d) Ba 8: panpxxpyy ”an 1 - l - d - l - l - d e ( p )p ( pE pyy) ( pgnpxxpyy)( pEn) _ 2 _ 3 e pxx pgnpxx panoxxpyy panoxxpyy -l- -d +d3 pEnpxprY pEn pEnpxprY o - - - d + d + 1 - a ognpxx pgnoxxpyy pg" panoxxpyy oxx eo=2p (1- )-d (1- )+(1- ) pEn XX pyy pEn pxxpyy pxx ' Therefore 8 X —.=1=0=Q. Be 8 e (1-p2)p(1-dpp) l - l - d 8g 8g ( pgnpxxpyy)( pg”) . 2 1) 1f 1 - d > 0‘9 d < l and sin e O < l - < 1 0En pEn C pEnpxprY'— then x 2 —<1©1- l-d 8* ( pan)pxx( pgnpyy) E <(1-2 )(l-d) pEnpxxpyy pEn ° Using algebra from a) above with appropriate attention for the inequality yields: m l 1 then pEn pEn m C) 68 ‘03 I» ma»): < 1 a (1 - 92 ) - d gn)pxx(l p Enpyy 2 3 (l - pinpxxpyy)(l - dog“). Using algebra from a) above with appropriate attention for the inequality yields: B _%;< 1 e o > Q . 8g 2 B B (l ‘ pg )0 (1 - do P ) —f>l¢=-—§= 2“ xx Enfl>l l - l - d 85 8g ( pgnpxxpyy)( pin) 1) If 1 - dp > 0 “ dpgn < l and since En O < l - 92 p p < 1 En xx YY" 8x 2 -;-> 1 e (1 - pan)pxx(l - dpgnpyy) B E 2 > 1 - l - d . ( pEnpxxpyy)( pEn) Using algebra from a) above with appropriate attention for the inequality yields: 8 -—§ > l¢= O > Q. 8 E 2) 1f 1 - doEn < 0‘9 doEn > 1 B x 2 Bi.) 1'“ (1 - pEnmxxu - dpEnpyy) E <(1-o2 p )(l-dp ). 0 En xx YY En Using algebra from a) above with appropriate attention for inequality yields: 69 B -§-> l a O < Q. B E Therefore, combining results from a), b) and c) above pro- duces: If dpgn > 1 then * . . * (3.18a) Q < O a Bx/Bg < l i.e., Bx underestimates Bg' * . . * (3.18b) Q - O o Bx/Bg — l i.e., Bx exactly estimates 85' * . . * (3.18c) Q > 0 a Bx/Bg > 1 i.e., Bx overestimates BE . If d < 1 then pEn * . . * (3.19a) Q < 0 a Bx/Bg > 1 i.e., Bx overestimates BE, * * (3.19b) Q = O a Bx/Bg = l i.e., Bx exactly estimates Bg, (3.19c) Q > O a Bx/B; < l i.e., B underestimates 8;. * * = 1, then Bx/B is undefined since B = O. E E Note: It is also possible to consider dpEn as the ratio of If dpEn two slopes. The numerator of the ratio represents the slope of the pooled within categories regression line of g on n. The denominator of the ratio represents the slope of the line joining the midpoints of the joint distributions of g and n between the two categories. For more information about this interpretation see Appendix 8.9. Thus the examination for relationships between Bx/8; and one can be pursued by examining the relationship between Q and zero. The questions now are, for what values of p pyy and d will Q an, pxx' be less than zero, equal to zero, and greater than zero. The approach 70 to answering these questions will be to consider three of the four variables ( , and d) as fixed thus can be considered pg“ pxx' pyy I Q solely as a function of the fourth non—fixed variable for the given combination of the three fixed variables. Although any one of the four variables could be selected as the non-fixed variable, the most interesting and useful information has come from fixing pxx' pyy and d and examining Q (and hence * Bx/Bg as well) as pEn .varies from -1 to +1 for various combinations of pxx' pyy and d. Following this approach Q is clearly a quadratic function in DEN. * IV. The Search for Categories of Distribution of Bx/Bg as a Function of - pEn Consider expression (3.17) for Q as a function of p for En fixed values of p , p and d; XX YY 2 .20 = - - d l - . (3 ) Q 0 (l D ) p€n( pxxp ) ) + (1 ‘ pEn xx W W pxx 2 = + + = - Let Q axpEn bxpEn cx where ax pxx(l pyy), bx = -d(l - p pyy) and cx = (1 - p ). Expression (3.20) clearly illustrates that Q is a quadratic function of p. . As a quadratic function of p n' Q will possess En E two roots call them DQQX) and pEQX) which are defined as: 2 2 -(x) d(l - pxxpyy) -‘Jd (l - pxxpyy) - 4pxx(l - pyy)(1 - pxx) (3.21a) pEn = 2 (1 _ ) pxx pyy d(l- )+\ld2(l- )2-4 (1- )(1- ) (3 21b) +(x) = pxxpxy DXXEYY pXX pyY pxx En 2pxx(l - pyy) 71 '(X) +(X) . . . -(x) '+(x) Both roots and will eXist w1th < ' pEn pEn pEn - En 4p (1 - p )(1 - p ) if Id] 1 xx yy 2 xx . (See Appendix B.4 for details.) (1 - o p ) xx YY -(x) +(x) Since the existence of the roots pan and pEn as real numbers is important, the quantity on the right of the existence expression will be used frequently. As an abbreviation in notation, let _ 4oxx(l - o y)(l - pxx) (3.22) : ~21 . x (1 - p p )2 xx yy Here the square root sign indicates that the quantity involved is a square root and the x indicates that the expression is related to 8/85. .. + . Since DEQX) and pgéx) represent pOSSible values for the correlation between the two true predictors, pEn where - + -l < pEn < +1, the existence of pgéx) or OEQX) in the interval from -1 to +1 is as important as their existence as real numbers. Therefore, from Appendix B.4: pgéx) will exist with "(X) (-1, +1), for o < p < 1, 0En e yy 1 . (3.23a) for 0 < p < -——-—- if 1 < d xx —-2 - p YY (3.23b) for -—-l;-—-< p < 1 if < d 2 - pyy xx -' X‘— + or if —1 < d < — J", and p (X) will —' x En + (X) (-1, +1) for o < p < 1, exist with pEn e yy (3.24a) for 0 < p < -—-————- if d < -l 72 (3.24b) for ___2;__.< p < 1 if Ix §_d < 1 or if d 5_- J;‘. * E En' equality only when pxx = 1 too. (See Appendix 8.2 for Note 1: When pyy = 1, 0 < Bx/B :_l for all p pxx, d Wlth proof.) Note 2: When p = 1, xx . -(x) +(x) if 0 < d < 1 then = 0, = d . -(X) +(x) f d > 1 then = 0 -1 +1 . Note 3- I < l with e ual't if and onl if = -——l-—- ° x - q 1 y y pxx 2 - pyy ° (See Appendix 3.3 for proof.) *- Prior to identifying general categories of Bx/B it will be E worthwhile to examine the relationship of Q to 0 for various com- binations of situations since as noted above in subsection III the relationship of Q to 0 provides some direct information about the * relationship between Bx/BE and 1. Much of the derivation for the results which follow has been developed in Appendix B.4. If ldl > J— then —- x (3.25a) Q < 0 for max(-l, p-(X)) < p < min(+l, p+(x) En En En [By B.4.3a] ) -(X) . -(X) (3.25b) = 0 for = rov1ded that -1 +1 Q pan pEn p pEn e ( . ) [By B.4.3.b] + or for pEn = pgéX) provided that pE£X) (—l,+l), [By B.4.3b] 73 (3.25c) Q > 0 for -l < p < max(-l, p-(X)) [By B.4.3c] En En or for min(+l, p+(X)) < p < +1. (By B.4.3c] En En “ (3.25d) If Idl < Ix then Q > o for all pEn e (-1,+1). [By B.4.4] Now combine results (3.23a-b) or (3.24a-b), (3.25a-d) and (3.18a-c) or (3.19a-c) to derive general categories of distributions Examination of these expressions will pro- of Bx/B; versus pan. duce three general categories of distributions. For exploration for the first general category, let d > 1 -(x) and 0 < p < 1. Therefore p e (-1. +1) by (3.23a) but (3.24a) En En + and (3.24b) indicate that pEQX) ( (-1, +1). Using results (3.25a-d) and considering all possible values of pan, pan 5 (-1, +1) produces: (3.26a) for —l < p < p—(X) then Q > 0 En En (3.26b) for p = p-(X) then Q = 0 En En (3.26c) and for p-(X) < p < 1 then Q < 0. En En (3.27) Note: For d > 1, pg;X) < %-. (See Appendix 8.5 for proof.) Therefore since d > 1 (hence %-< I) combine the results from above and from (3.18a—c) and (3.19a-c) to produce information * about B /B for values of (-1, +1). x E pEn E The following information will be presented for values of which cover the whole interval from minus one to plus one. pEn a) Consider d > 1 and D such that l/d < pEn < +1. En Therefore dpEn > 1. 74 By (3.27) pgéx) < l/d. Hence pgéx) < l/d < pEn < +1. - (x) En ' Q < 0' By (3.26c) for values of p > 9 En * By (3.18a) when dp > 1 and Q < 0, Bx/Bg < 1, En * (3.28a) Therefore, when d > 1 and l/d < p5n < +1 then Bx/Bg < 1. b) Consider d > 1 and DEN such that pEn = l/d. Therefore d = 1. When d = l, B = 0. “an pEn a * (3.28b) Therefore, when d > 1 and p = 1/d, Bx/Bg is not defined. En * In this case IBxl is an overestimate of B6 unless Bx is also zero. c) Consider d > 1 and p such that -1 < pEn < l/d (where En l/d < 1). Therefore dpEn < 1. By (3.27) pgéx) < l/d. Therefore there are three subintervals of values for pan here which must be examined. I) For pgéX) < p < l/d, then Q < 0 by (3.26c). By En * (3.19a) when dp < l and Q < 0, Bx/BE < 1, that is En * [Bxl is an overestimate of IBEI for correlations in . -(x) the interval (pEn , l/d). (3.28c) Therefore, when d > 1 and pg;X) < pan < l/d, then * Bx/BE > 1- II) For Dan = pggx), then Q = 0 by (3.26b). Thus by (3.19b), B /8* = 1, that is when p = p-(X), B = 8*. x E En En X E (3.28d) Therefore, when d > 1 and p = p-(x) (where p-(x) < l/d) En En En then Bx/B; = 1. III) For -1 < pan < pg(x), then Q > O by (3.26a). Thus n by (3.19c) Bx/B; < 1. (3.28c) Therefore, when d > 1 and -l < pEn < pgéx) (Where 75 p"(X) < l/d) then s /s* < 1. En X E * To determine the relationship between Bx/Bg and.zero for the range of values for p , apply results (B.6.4a-d) from Appendix En B.6 for d > 1. Since 0 < pyy < 1, then l/d < l/dpyy. Therefore, the Appendix B.6 results produce: (3.29a) for -l < pan < l/d then Bx/B; > 0 [from (B.6.4a)], (3.29b) for l/d then Bx/B; is undefined [from (B.6.4c)], * (3.29a) for l d < < min(l l d ) then B B < 0 / p . / pyy x/ E En [from (3.6.413) J I (3.29d) for ll 0 * l/dpyy then Bx/Bg [from (B.6.4d)], and * (3.29e) for min(l, l/dp < 1 then Bx/BE > 0 YY) < pEn [from (B.6.4a)]. Combining results (3.28a-e) with corresponding results from (3.29a-e) yields general category one (G.C.I.) of distributions for Bx/Bg' * General category One (G.C.I) of distributions for Bx/B as E a function of pan has the following form as values of 0E0 vary across the interval (-1. +1). For d > 1, any pxx' pyy # l and (3.30a) for -l < p < p-(X), 0 < B /B* < 1 [from (3.29a) En En X E and (3.28c)] (3-30b) for p = p-(x) B /B* = 1 [from (3.28d)] En En X (3.30c) for p‘(X) < p < l, 3 /B* > 1 (from (3.28c)] En E d X E I 76 * (3.30d) for OED = l/d, Bx/Bg is undefined since B = 0 [from (3.28b) or (3.29b)] (3.30e) and for l/d < DEN < l, Bx/B; < 1 [from (3.28a)]. Result (3.30e) can be further specialized as follows: (3.30f) for l/d < p < min(+l, l/dpyy). Bx/B; < 0 [from (3.29e)]. En (3.309) for p = l/dp , Bx/B; = 0 [from (3.29d)], En yy * (3.30h) for min(+l, l/dp ) < p < +1, 0 < 3 /s < 1 yY x E En [from (3.29e) and (3.28a)]. * Note that G.C.I for Bx/Bg could actually be considered as * having two subcategories depending on the behavior of Bx/Bg when < < , l/d pan 1 a) If d > 1 is also sufficiently large enough so that dp > 1 (l/dp < 1) then (3.30f) becomes YY YY (3.30j) for l/d < p < l/dpyy, Bx/B; < 0, En (3-309) becomes (3.30k) for p = l/dp * 1 =0! En yY BX/BE and (3 . 30h) becomes b) However, if d > 1 but dpyy < 1 then l/dpyy > 1 and (3.30f) becomes 77 * (3.30m) for l/d < p < +l, Bx/BE < 0, En and (3.309) and (3.30h) are not applicable. * Since the prime interest in examining Bx/B is in relationship to E one, and since when d > 1 for l/d < DEN < l, Bx/B; < 1, there is only academic interest in differentiating between the two sub- categories of G.C.I identified above. Therefore, G.C. I will be considered as a single category of distributions with regard to the relationship of Bx/B; to one. G.C. I for Bx/B; as a function of 0E0 covers all values of p y f l and d > 1. Therefore other general categories will xx, Dy involve values of d where 0 :_d :_1. For exploration of the second general category, let J—_ < d < l, 0 < p < l and -—-l+——-< p < 1. Therefore x - yy 2 - p xx-— . VY - + 053x) 6 (-1. +1) by (3.23b) and pEQX) e (-1, +1) by (3.24b). Using results (3.25a-c) and considering all possible values of pEfl' pEn e (-1, +1) produces: (3.31a) for -l < pEn < pgéx) then Q > 0 [from (3.25c)], (3.3lb) for p-(x) < p < p+(X) then Q < 0 [from (3.25a)], En En En +(X) (3.310) for DE” < 0E” < 1 then Q > 0 [from (3.25c)], -(X) +(X) 3.3ld) and for = or = then = 0 ( pEn pEn pEn pEn Q [from (3.25b)]. Note, since IX §_d < l and pEn < +1 then dOEn < 1. Therefore by (B.6.4a). 83/8; > 0. 78 Thus combining the results (3.3la-d) with (3.19a-c) to pro- * g for values of pEn e (-1, +1), yields * general category two (G.C. II) of distributions for Bx/B£° duce information about Bx/B When < d < l and 0 < p < l X -' YY (X) .. * (3.32a) for -l < 0E” < pan then 0 < Bx/BE < 1 [from (B.6.4a), (3.31a) and (3.19c)]. (3.32b) for p'(X) < p < n+(X) then 3 /s* > 1 [from (3.31b) En En En x g and (3.19a)]. (3.320) for pgéx) < pgn < +1 then 0 < Bx/B; < 1 [from (B.6.4a). (3.31C) and (3.19c)], _ -(X) _ +(X) * = (3.32d) and for pgn - pan or DEN — pan then Bx/BE 1 [from (3.31d) and (3.19b)]. For exploration of the third general category, let 0 :.d < x _ , . -(x) +(x) 0 < < d d . T h pyy 1 an conSi er any pxx hen neit er pEn nor pEn will exist [see (8.4.2) from Appendix 8.4]. By result (3.25d) > o — + . ' < I I < Q for all DE” 6 ( l, 1) Since d x and x __l [by (8.3.2) from Appendix 8.3] then d < l and dogn < 1 also. Therefore, since dpEn < l and Q > 0 for pEn (-1, +1) applying (B.6.4a) and (3.19c) produces the following result: (3.33a) When 0 §_d < Ix , and 0 < pyy < l, for any pxx and any 0 * 5n 6 (’11 +1): then 0 < BX/Bg < 1. This is general category three (G.C. III) of distributions of Bx/Bg . 79 One other set of distributions also fall into G.C. III. Let r“ < < , < < - d O < < 1. Th x __d l 0 pxx _ 1/2 pyy' an pyy en p-(x) ( 2 (-1, +1) and p+ x) t (-1. +1). En En If Pgéx) > +1 or pgéx) < -1, then Q > 0 for pan 6 ('1! +1)- ‘1 *3 an , f Fifi 1V1 9;?) 1 +4 If pEJX) < -l and pgéx) > +1 then Q < 0 for p e (-1, +1). 5n . . . -(x) I -l b ff t h th t f d > I > 0 t wil e su iCien to s ow a or __ x , pEn ‘_ and thus Q > 0 for pEn e (-1, +1). Let d 3_ Ix therefore pg;X) will exist but may not exist in the interval (-1, +1). Therefore the question is, for what values - (X) of , l ' > 0? on pyy 7‘ IS pEn _ d(l- ) - Id2(l- )2 - 4 (1- )(1- ) p—(x) > O” pxxpfl pxxpyy pXX pyy pXX > 0 En " 2pxx(l - pyy) ~— since 2pxx(l - pyy) > 0 for pyy # 1, 8O pan 1 0o d(l p p ,1) d (1 p p ) 4p (1 pyy) (1 pxx) :0 ed(1-pp)>Id2(1-pp)2-4p (1-p)(1-p) xx yy - xx yy xx yy xx 2 2 2 2 w d (1 - pxxpyy) 3_d (l - p pyy) - 4p (1 - p )(l - pxx) O ' e d l - > for d > and l. Sinc ( pxxpyy) __ J;: pyy f - (X) >0“ 4pxx(l-p )(1-pxx) :0. YY But 0 < p < 1 and 0 < p < l by definition, xx —' YY (3.34) Therefore, pgéx) :_0 for all values of pxx and p (pyysfil) when d__>_ 5. Since pgéx) 3_0 when d 3_ Ix , then Q > O for YY -1, +1 . F r < d < l d < 1. Therefore since can 6 ( ) o I: _ . pEn . dp < l and Q > 0 for p e(-l, +1), applying (B.6.4a) and (3.19c) En En produces the following result: 2'0 (3.33b)When I_ 1, YY YY all values of p and all values of p except p = l. G.C. II xx YY YY includes values of d such that Ix :_d < 1, values of pxx such that 1/2 - p < < l and all values of exce t = 1. YY pxx -' pYY p pYY G.C. III includes values of d such that I;-.:_d < 1, values of pxx such that 0 < pxx g 1/2 - pxx and all values of pyY except pyy = l. G.C. III also includes all values of d such that 81 0 < d < lIx for all values of pxx and all values of pyy # l. * When pyy = l, Bx/Bg was examined in Appendix 8.2. When * pyy = 1, then 0 < Bx/Bg < l for all values of p e (-1, +1) and En xx < l [by (8.2.8) from Appendix 8.2]. When pyy = pxx = l, 0 < was examined in Appendix 8.2. When p = p = 1, then Bx/B yy xx Bx/Bg = l for all values of pan 5 (-1, +1) [by Appendix 8.2, Section D]. *MI-‘D The situation when d = 1 will be shown to represent a slight variation of G.C. III for 0 < p < l/2 - p with p # l and to xx- YY YY represent a middle ground between G.C. I and G.C. II for 1/2 - pyy < pxx §_l with pYY # 1. To examine the situation where -(x) d = 1 first determine the conditions for existence of pan and p+(x) En ° Let d = 1, then p’(X) will exist and p'(X) e (-1, +1), En En for 0 < < l and for l 2 - < < l b 3.23b . F pyy / pyy pxx _, y ( ) or -(x) . 0 < < - - + . . pxx __l/2 pyy' pEn g ( 1, 1) by (3 23a) Referring to +(x) (3.24a) and (3.24b) (-1 +1) for an , ( l). . pEn ( , y pxx pyy pyy # Consider d = 1 0 < < l 2 - and 1. Since . pxx __ / pyy oyy # -(x) d = 1, then d 3_ IX (by (8.3.2) in Appendix 8.3) and pEn +(x . - x + x and DEN ) eXist but pg; ) t (-1, +1) and pg; ) j (-1, +1). The t t d f " r same argume presen e or 1;. §_d < 1 when 0 < pxx §_l/2 p pyy # l in general category three is completely applicable here hence * (3.35a) O < Bx/Bg < l. The variation of G.C. III which results when d = l is not obvious yet. It will be identified below. 82 Consider d = 1, 1/2 - < < l, # 1. Since d = 1, oyy pxx __ oyy -(X) (X) ... and in ”an then d > and both p exist but - J:x 9;;3)£; (-1, +1) by (3.23b) while pgéX) g {-1, +1) by (3.24b). . -(x) _ +(x) _ I-(x) +(x) Since pg” 6 ( 1, +1), pan 5 ( 1, +1) and pin fi'pfin then +(x) pan :_+1. Using results from (3.25a-c) and considering values of pin 6 (-1. +1) yields: _ -(x) (3.36a) for l < pan < pan then Q > o [from (3.25c)]. (3.36b) for DEN = pgéx) then Q = 0 [from (3.25b)]: and (3.36c) for pEQX) < pan < +1 then Q < 0 [from (3.25a)]. When d = 1, then dpan < l for pan 5 (-1, +1). Therefore applying results (B.6.4a) and (3.19a-c) to (3.36a-c) yields -(x) 3.35b) for -1 < < ( p can * En then 0 < BX/BE < 1 [from (B.6.4a) and (3.190)) _ -(x) * = (3.350) for pEn - pan then Bx/BE 1 [from (3.19b)], '(X) * (3.35a) for pgn < pan < 1 then sX/eE > 1 [from (3.19a)]. Some similarities to both G.C. I and G.C. II are obvious. More direct comparisons and contrasts require additional work to be presented below. To continue to add information about the three general cate- gories identified above as well as the situation when d = l. The work which follows will examine the limiting case of Bx/B; as a function of p as in pen is allowed to approach various values of interest. 83 For all three general categories as well as the situation i where d = l, the limiting case of Bx/Bg will be considered for values of p in an arbitrarily small neighborhood of negative one in (-l). Since, by definition, p > -1 the only values of p which En En can be included in the arbitrarily small neighborhood of negative one are values which are greater than negative one. The notation which * will be used here is: the value of Bx/B will be examined as E + * pEn + -l . The notation indicates that the value of Bx/Bg is to be examined for values of p which are greater than negative one in (indicated by the + as a superscript) but which are arbitrarily close to negative one (indicated by the +). * .— The value of Bx/B will also be examined as p + +1 for 5 En each subcategory of G.C. I, for G.C. II and G.C. III combined and for d = l. [p + +1- indicates that the values of p which are to fill in be considered are those values which are less than +1 (indicated by the - as a superscript) but which are arbitrarily close to +1]. * For case G.C. I only, the values of Bx/B will be examined 5 - + as pin + l/d and as DEN + l/d . (Recall: for G.C. I, d > 1, * thus l/d < l, and Bx/Bg is not defined for pEn = l/d (i.e. * dp = 1) since 8 = 0). in 5 The approach to the work on limits will be to determine the * limits of 8x and B separately first and then consider the limit 5 * of Bx/Bg based on the work for the separate limits. Therefore consider * bgpxx(l — dpgnp ) B = .YXe [from (3.11b) x 1 _ p2 5n pxxpyy with d = d5], 84 and * b§(1 - dog ) 8* = 2 n [from (3.6b) 5 l ‘ pgn with d = d5]. _ . n . n For general notation let Bxlim — ll: Bx and Bglim - 11: qu. pin ‘q pin * b p (1 + dp ) + As 0 + -1 . Bx + Bx = i fix iyz_ . En lim oxxoyy * Since d > O, 8 > 0 if b > O - X . 5 11m * B < 0 if b < 0. X 6 lim + 2 + As + -l l - d + l + d > O and l — + 0 . “an ' pEn pEn * * Therefore 85 + +w if bg > O * 'k + -w if b < O . E 5 Therefore for any d Z_O, pxx and pyy (i.e. any of the three general categories as well as d = l), as (3 37) —> -1+ 8 /s* + 0" ' “an ’ x E ° Consider the subcategory of G.C. I with d > l/pyy i.e. * dpyy > 1. (See expressions (3.30j—1) for the behavior of Bx/BE when l d < < d < , / pan 1 an d l/pyy) * _ bgpxx(l - do ) As pEn + +1 , B + BX 1 _ yy . x lim pxxpyy Since do > 1 w l - dp < 0, then YY YY B > O 'f b* < o l X lim g * BX < 0 if b5 > O . 85 As pan + +1-, 1 - dpgn + l - d < 0 (since d > l/pyy > 1) and l - DZU +'O+. Therefore * * 8g + +w if bE < O * 'k E + -m if b5 > O . Therefore, for G.C. I when dpyy > 1 - 'k + (3.38) as pgn + +1 , Bx/Bg + O . Consider the subcategory of G.C. I with l < d < l/pyy i.e. * dpyy < 1. (See expression (3.30m) for the behavior of Bx/BE when 1 d < < l d d < l . / pin an /pyy) b p (l - dp ) As pg + +1 , BX + BX = i _ yy . n lim pxxpyy Since dp < l.¢ l - do > 0, then YY YY B O 'f b* 0 > 1 > xlim g B O 'f b* O < i < . xlim 5 As pEn + +1-, 1 - dpEn + l - d < 0 (since d > 1), and 1 - p2 + 0+. Therefore En 8* * + +00 if b < O E E 'k 'k 0 + -w if b > . E 5 Therefore, for G.C. I when dpyy < l, - t - (3.39) as pEn + +1 , Bx/Bg + 0. 86 Consider either G.C. II or G.C. III. In each category d < l. * b p (l - dp ) As pg ++1,Bx+8x =53}: _YY. n lim pxxpyy Since d < l, dp < l w l - dp > 0, then YY YY 8 f * O > O i b > Xlim E 'k 3 < 0 if b < 0 . Xlim 5 As + +1‘, 1 - d + l - d > 0 since d < l a d can can ( ) n 2 + l - + O . pin Therefore, Therefore, for either G.C. II or G.C. III (i.e., d < l), (3.40) as can + +1”, Bx/e; + 0+. Consider the situation where d = l. As pEn + +1-, Bx + Bx = 1 _ p p yygfo lim xx yy 8 0 'f b* 0 > i > Xlim g Bxlim < 0 1f bg < O. * * b l - b As + - * _ 5( pin) * _ ___ . . , 0 +1 , B — + B — by L Hopltal s 5” 5 (l - 2 ) Slim 2 1 pin Rule. Thomas, Goerge B., Calculus and Analytic Geometry, Addison-Wesley Publishing Co., Reading, 1968, pg. 651. 87 Therefore, when d = l B . 2p (1 - p ) _ * (3.4la) as p ++l, 3/3 + *11‘“= "x W . En x E B (1 — oxxp ) Elim yy Note 20 (l - p ) 1 (3.4113) YY < 1 no < p < —— [See (8.3.4a) and l - oxxp -' xx *‘2 - p yy yy (8.3.4b) from Appendix 8.3 for proof]. 20 (1 - p ) xx Xy > 1 ¢,___l;___< p < 1 [See (8.3.4c) from 1 - pxxp 2 - p xx-— yy yy Appendix 8.3 for proof]. Consider G.C. I, where d > 1 and for pEn = l/d (i.e. * dpgn = l) Bx/Bg is not defined. For the arguments below consider d as some fixed value such that d > 1. As p + l/d‘, i.e. dp + 1’, En En b*p (l - o ) 8X + 8X . = g x: p YY ° 11m 1 XXZYY d d 8 >0 if b*>o Xlim g B < 0 if b* < O. Xlim E Since d > 1, l - A-2--> 0, thus as d +ld' ' d 1' * * o+ 'f * , , + + + pEn / 1 e pan Bg Balim 1 bg * E 0 if b 88 Therefore for G.C. I _ * (3.42) as + l/d , Bx/Bg + +w . on + + AS 0 + l/d i.e. dp + l , En En b* (l > ._ prx pyy Bx + 8X - O 0 11m 1 _ xx yy 2 d 1 - p o + Since d > 1, XX yyg> 0, therefore as p + l/d , d2 in 0 'f b* O B > i > Xlim 5 B O 'f b* O < 1 < . Xlim g B§+Bglim + 0' if b Therefore for G.C. I + t (3.43) as + l/d , Bx/Bg + -m. 0&1 Now combining the results on limits (3.37) through (3.43) with * the results on the relationship of Bx/B as a function of p to E in zero and one for each of the three general categories as well as the situation when d = l [(3.30a-e), (3.30j-1), (3.30m) for G.C. I; (3.32a-d) for G.C. II; (3.33a-b) for G.C. III and (3.35a-d) for d = 1], it is possible to describe more fully the characteristics of each general category and to produce for each general category a generic 89 graph which represents the general shape of all distributions in the category. The information summarized below is presented as Dan ranges from near -1, through 0 and finally to near +1. G.C. I, i.e. d > 1, any p (p # l): XX' pyy yy Subcategory a) d > l/pyy i.e. dp > 1 and l/d < l/dpyy < 1, YY (3.44a) as p + -1+ , B /B* + O+ [by (3.37)]: an x E (3.44b) for -l < p < p-(x) , 0 < 8 /B* < 1 [by (3.30a)]: En En x E “ * (3.44c) for p = O , B /8 = p [by section C En x 5 xx Appendix 8.3], (3.44d) for p = p-(x) , B /B* = l rby (3.30h)] En En x E “ ' (3.44e) for p-(X) < p < l/d, B /B* > 1 [by (3.30e)], En En x E (3.44:) as pan + l/d- , (ax/s; + +oo [by (3.42)], (3.449) for pan = l/d , ex/e; is undefined [by (3.30d)], (3.44h) as pEn + l/d+ , Bx/B; + -m [by (3.43)], * (3.441) for l/d < pEn < l/dpyy' Bx/Bg < O [by (3.30])]r * (3.443) for pEn = l/dpyy , Bx/Bg = O [by (3.30k)], * (3.44k) for l/dpyy < pgn < 1 , o < Bx/Bg < l [by (3.301)). and (3 441) +1” * 0+ b 3 38 . + , . . as can Bx/Bg + E y ( )1 Note for G.C.I: since d > 1, pQQX) :_O [by (3.34)]. * Thus the generic graph of Bx/Bg as a function of pgn for G.C. I subcategory a) has the following general shape: 90 8:: $0 # 3° ,3 I t’ O C) %--ugnpn 393' Figure 3.1a Subcategory b) l < d < yy, i.e. dpyy < l and 1/d < 1 < l/dpyy' Since subcategory b) differs from subcategory a) only when l/d < pEn < l, expressions (3.44a-h) for subcategory a) also apply for subcategory b). Therefore all that is needed to finish specifying subcategory b) is: * (3.44m) for l/d < p < l, B /B < o [by (3.30m)], an x E and - * - 3.44 as + +1 , + o [b (3.39)]. ( n) pgn Bx/Bg Y * Thus the generic graph of BX/BE as a function of pan for G.C. I subcategory b) has the following general shape: (3.45a) (3.45b) (3.45c) (3.45d) (3.45e) (3.45f) (3.459) and (3.45h) G.C. II, i.e. 91 +19 vl------ 851 “1.0 h? (75: Figure 3.lb l (i e pyy # l): as pEn + -1+ for -1 < 0&8 < pEQX) for pan = O for pin E;X) a?“ < < pas“ for pin = pg:X) for p+(X) < p < 1 in in as pEn + +1- \lx i-d < 1' 1/2 - pyY < pxx - * + , Bx/Bg + 0 I BX/Bg = pxx : BX/BE = l * r Bx/BE > 1 * r BX/BE = 1 [by (3.37)], [by (3.32a)], [by section C Appendix 8.2], [by (3.32d)]! [by (3.32b)]. [by (3.32d)], [by (3.32C)]r [by (3.40)]. 92 * Thus the generic graph of Bx/B as a function of p for E in G.C. II has the following general shape: tLo 8“ ' 1.0 :g‘ y;{” *1 ‘0 Figure 3.2 G.C. III, i.e. 0 :_d < J::‘ , any pxx' pyy (pyy # 1) or 1 < d < 1 O < < ————‘——v 0 < < ’. . : Ix __ , pxx __2 _ pyy l (l e pyy # l) pYY (3 46a) as + -1+ 8 /s* + o+ [b (3 37)] ° 0&0 ’ x E y ° ' * (3.46b) for -1 < pEn < +1 , O < Bx/B < 1 [by (3.33a) or E (3.33b)] ( ) /* r 3.46c for p = O , B 8 = o .by section C an x 5 xx Appendix 8.2], and (3 46d) 1‘ B 8* 0+ [b (3 40)] . a + + , + . . 8 DE” x/ 6 y * Thus the generic graph of Bx/BE as a function of pin for G.C. III has the following general shape: I (3.47a) (3.47b) (3.47C) (3.47d) (3.46a) through (3.46c) for G.C. III. tion, occurs as 93 +10 db 931 4.0 14.0 W6; Figure 3.3 When d = l and 0 < pxx §_1/ 2 - pyy with pyy # 1, then: as Dan —> -1+ . (Bx/8; + 0+ for -l < pEn < +1 , 0 < Bx/B; < l for pan = 0 , BX/BE = Dxx _ * 2p (1 " p YY as pin + +1 , [Bx/85+ 1 p 0 xx YY [by (3.37)]. [by (3.35a)]. [by section C Appendix 8.2], [by (3.4la) and (3.4lb)] Note that (3.47a) through (3.47c) above are identical to 050 Thus the generic graph of Bx/B * E The only difference, the varia- approaches one, (3.47d) versus (3.46d). as a function of p for Er) d = l, O < pxx i 1/2 - p with pyy f 1, which is somewhat similar to the generic graph for G.C. III, has the following general shape: (3.48a) (3.48b) (3.480) (3.48d) (3.48e) and (3.48f) 94 up Run-311). $1 l.- 31:35: I“, -i~° +1.0 9 We: Figure 3.4a When d = l and 1/2 - pyy < pxx §_l with pyy # 1, then: as + -1“ B /B* + 0+ [b (3 37)] pin I X E . Y ° r for -l < p < p-(X) 0 < B /8* < 1 [by (3.35b)]: an in ' X E * for p = O , B /B = p [by section C En x 5 xx Appendix 8.2]. for = —(X) B /B* = l [by (3 35c)] pan pan ' x e: ' " for p-(X) < p < l , B /B* > 1 [by (3.35d)], En En x E _ 20 (1 - p ) asp ++1 .sx/3*+l’_°‘ yy>1 in E oxxpyy [by (3.4la) and (3.4lb)] Note that (3.48a) through (3.48d) above are identical to (3.44a) through (3.44d) of G.C. I and to (3.45a) through (3.45d) of G.C. II. above with d = 1, when p Like both G.C. I and G.C. II Bx/B -(x) n €n>p€ * E . But unlike G.C. II Bx/B > 1, for the situation * E 95 never gets infinitely large and never gets negative for -(x) . * En < pEn < l, and unlike G.C. III Bx/BE does not approach zero as 058 approaches +1. * Thus the generic graph of Bx/BE as a function of p for En d = 1, 1/2 — p < pxx :_l with p # 1 has the following general YY shape: Ignatu'hfl ’ 1 .....- t-v-m A Y“ -1.0 $1“) +1.0 Figure 3.4b Examples of graphs in each of the three general categories for fixed values of d, pxx and p will be presented below combined * with the work on general categories of distributions for By/Bn as a function of p 610' V. The Search for Categories of Distributions of B of — pin A question which arises immediately relates to the need for * /B as a Function Y n this section based on the results demonstrated in Appendix 8.1 for the property of interchangeability of x and y. In Appendix 8.1 it is wn that iv it ti '. . ’ 3 sho 9 en any 5 ua on (i e , given value of 058' pxx' pyy 96 * * and d ) where the value of By/B is needed, that value of B /B T) n Y n * is identical to the value of Bx/Bg for another situation with the relationship between the two situations provided by (3.15a—d). Hence * E The important thing to note about that result is that in gen- it is necessary to derive detailed results for Bx/B only. * eral the two situations, the one of interest for By/Bn and the * adjusted one for Bx/Bg' will be different situations. That is the i values p p and dn used to get a value for BY/Bn will En' pYY' xx in general not be identical to the values pén, p§y, pkx and dé (as related to pin' pyy, pxx and dn by (3.15a-d) used to get the same * g. Therefore since one of the interests of this research is to value for Bx/B * * examine the ratios 8 /B and By/Bn for the same situations, i.e., x E gn, pyy' pxx and d = d5 = l/dn, it will be necessary to state categories of distributions and important algebraic results for values of p * * BY/Bn which are analogous to those derived for Bx/Bg' The properties (3.15a-d), i.e., the property of interchange- ability of x and y will simplify the work for By/B; considerably. It will be necessary to consider the major results for Bx/B; and apply the property of interchangeability to arrive at analogous con- clusions for By/B;. In simple terms applying the property of inter- changeability to a result for Bx/BE involves replacing every x with a y, every y with an x, every 5 and an n and every n with a 5. Therefore 058 is replaced by p but since there is no variable 05 orderin in a correlation coefficient 0 = p is re laced b 9 n5 En, pXX P Y p p is replaced by p and d = d is re laced b 97 dn = l/dE Therefore for all applications of the property of interchangeability = l/d (i.e., d is replaced by l/d provided d # 0). consider d > 0. The situation when d = 0 will be examined separately. The guiding interest in this phase of the research is to com- * * are and now 8 to one. That is to see for what situa- p ex/sg 3/ n , * tions is BY an overestimate (By/Bn > 1), an underestimate * i 'k (B /B < l) or an exact estimate (8 /B = l) of B . Y n ' Y n n Combining results (3.18b) or (3.19b) with (8.4.2) produces: * there will exist a value of p e (-1, +1) such that Bx/Bg = l 6n 4p (l-p )(1-p ) if |d| > xx YY xx 2 ‘I_- . _' (1 - 0 p )2 x xx yy * The corresponding result for By/Bn becomes by the property of interchangeability: * there will exist a value of p e (-1, +1) such that By/B = l n £8 40 (1 - p )(1 - p ) (3.49) if Ii) 3_ YY xx 2 yy 2 l( ) for d s 0. (1 - p p ) y yyxx Since it would be generally more useful to consider values of Idl rather than Il/dl (3.49) becomes: there will exist a value of * e (-1, +1 su h th t = 1 pin ) c a By/Bn 2 (l - p p ) (3.50) if la] 5_ 4 (le YY~)(1 _ ) s l for d s 0, pyy oxx pyy y oxx f 1, pyy # l. 98 Note that LY) and IY are used as abbreviations in notation. Here the square root sign indicates that the quantity in- volved is a square root and the y indicates that the expression is * related to work on By/Bn. l(y) is a temporary symbol to be used only for the immediate results of the application of the property of interchangeability to I x (defined by (3.22)). The permanent statement of results will involve IY as will be seen below. o < :3 Since Ix __l for all pxx' pyy (except pxxpyy 1) with e ualit for D = l 2 - 0 . Then q y xx / yy < :- l(y) __l for all pyy' pxx (except pyypxx l) with equality for pyy = 1/2 - pxx' which is equivalent to (3.51) > 1 for all p p (exce t p = 1, p = l) \ly -' xx’ yy p xx yy 20 - l l with equality for pxx = ——X§-———u since -——--= IY (by defini- YY ‘I(Y) tion in (3.50)) and 1 2p - l pyy = E—:-E--e pxx = -—§X-—- (by the second result in yy yy (8.3.3b) from Appendix 3.3). * . —(x) ...(X) * For B B if = or = then = l. x/ E pEn pEn pEn pEn Bx/BE —(x) -(y) +(x) +(y) Let be re 1a d b d be re laced b pEn P ce y pEn an pEn p y pEn when the property of interchangeability is applied. * . -(y) +(y) Therefore for if = or = then By/Bn pEn “an ”an 0E0 By/B; = l, and expression (3.21a) for pg(X) becomes 99 1- o p (1-0 p ) __M - yyzxx __ 4p (l'p )(1-p ) YY XX YY p-(y) ___ d En Zoyy(1 - pxx) for d # 0, pxx # 1. Therefore (1- ) — J(1- )2 - 4d2 (1- )(1- ) ' _(y) - pyypxx pyypxx Dyy pxx pYY (3.21a ) pEn - 2d (1 _ ) Oyy pxx for d # O, pxx # l. +(x) and (3.21b) for p becomes En 2 2 1' + 1- . _ 4d _ _- (3 2113') +(Y) ___ ( pyypxx) V( OXLQXX) DYYCL DXX)(1 p ) ' pi" 2do (l-p ) YY YY xx for d # O, p # 1. xx * Note here that the expressions for By/Bn produced directly * from expressions for Bx/BE using the property of interchangeability will not be assigned new expression numbers. The number assigned to 'k the expression for By/Bn will be the same number as that of the * g. * pressions the expression number for BY/Bn will always be primed original expression for Bx/B To differentiate between the two ex- (i.e., given a ' as a superscript). Consider now the conditions for existence of pg;X) e (-1, +1) and hence for the existence of pgéy) e (-1, +1). Result (3.23a) states: pg;X) exists and pEQX) e (-1, +1) f l d or pyy f an 100 for O < pxx §_3-:l;-- if d > 1. YY -(y), p-(y) En ' En -(y) En This becomes for p exists and p e (-1, +1) for pxx # l and 1 < ———————- if l/d > 1. yy 2 pxx for O < p Rearranging this result so that the interval of reliabilities is an interval of pxx values rather than pyy values and so that the con- dition is expressed in terms of d rather than l/d produces: -(Y) exists and p-(y) e (~l, +1) pEn En 2p - 1 (3.23a') for ‘—€?L———-f_pxx < 1 if 0 < d < l. yy This rearrangement results from O < pyy f §—:—;——-a 2p - 1 -x§-—-—-§_pxx < 1 by (8.3.3b) and (8.3.3a) from Appendix 8.3 and YY * from the fact that d > 1 for Bx/BE also implies d > 0 and l/d > 14a d < l. -(x) -(x) Result (3.23b) states: pan exists and 058 e (-1, +1) for +1 and pyy ¢ for l < < 1 if < d or if -1 < d < - 2 — pyy pxx - x - —- J ° -(y), -(y) En pan exists and p-(Y) e (-1, +1) En This becomes for p for pxx f +1 and 1 I 1 l -——————- < 'f < l d 'f - < -< - . for 2 _ pxx < pyy __l i (y) __ / or i l d __ ( ) 101 - (y) En Rearranging this result as above produces: p exists and pggy’ e (—1. +1) 2p - 1 (3.23b') for 0 O, that -l < l/d :.- hy) for Bx/B implies d < O, and that I(Y) = l/ ly . * E + + . Result (3.24a) for pgéx) states: pg;X) exists and +(x) e (-1 +1) for l and for O < pxx : 1/2 - pyy if d < -l. +(y): +(y) En En 6 (‘1’ +1) . + eXlStS and p (y) En This becomes for p for O < p < 1 2 - 0 if 1 d < '1. Rearranging this result produces: p+(y) exists and p+(y) e (-1, +1) En En 2p - l (3.24a') for ——X%————- i-pxx < 1 if -1 < d < O. YY * This rearrangement results from the fact that l/d < -l for Bx/Bg also implies that d < 0. Result (3.24b) for p2£X) states: pgéx) exists and +(x) (-1, +1 for l and pEn ) pyy # 102 for 1/2 - p < p < 1 if ‘lx 5_d < l or if YY d<" 1» -" JX +( - y) p (y) . . +(y) This becomes for : exists and (-1 +1) pEn En pEn e ' for pxx # l and for -—-l—-—-< p < 1 if < l/d < l or if 2 - o W — (y) - xx 1 d < - o / — \I(y) . . +(y) . +(y) Rearranging this result produces: pEn exists and pEn 5 (—1, +1) 2p - 1 (3.24b') for O 1 then pgéX) = O, OEQX) g (-1, +1). 2 rpggpxx * (1 - pg )(1 - d ) Therefore, when pyy = l, By/Bn = n p for d # O (l-pzp )(1-J1) En xx d , (1-p2)(d-D ox) which becomes 8 /8 = in an x for d # O, l - d - ( pEanX)( pEn) and for d > 1 (i.e., 0 < l/d < 1) then p;(y) = O, —' n +(y) = l d pEn / f . -(y) _ or O < d < l (i.e., l/d > 1) then pEn - 0, +(y) -1, l . pEn ( ( + ) When d = 0, expression (8.2.5) represents the appropriate * expression for By/Bn i.e., 2 8 /B* = (1 _ pEn)pyypxx y n (1 _ p2 p p ) for pEn En yy xx 104 Applying the property of interchangeability to the result * presented in Appendix 8.2 (for Bx/B when a = O, i.e. d is un- E E * defined) yields: when d = O, O 1 and dpyy > 1 for any pxx' pyy (oyy # 1). Subcategory b) of G.C. I consisted of situations where d > 1 but d < 1 for an 1 . Therefore sub- pyy Y pxxr p (pyy 5‘ ) YY category a') for G.C. I (y) consists of situations where 0 < d < l (i.e., 1/d > 1) and 0 < d < pxx (i.e., pxx/d > 1) for any pyy' pxx (oxx # l) and subcategory b') of G.C. I (y) consists of situations where 0 < d < l (i.e., l/d > 1) and oxx < d < l (i.e., pxx/d < l). * G.C. II for Bx/B consists of situations where Ix :_d < l E for l 2 - < < l with 1. Therefore G.C. II con- / pyy pxx - pyy ’1 (y) sists of situations where l < d < .I (i.e., I < l/d < l) for 20 - l _- y (Y) — 0 < < -—3§L———— '. ., - < ' . pxx pyy (i e 1/2 pxx pyy :_1) With pxx # l * G.C. III for Bx/Bg consists of two sets of situations; either 0 :.d < ,Ix, for any pxx' oyy (pyy # 1) or I x :_d < 1 for O < pxx :_l/2 - p with pyy # 1. Therefore G.C. III (y) YY 105 consists of two sets of situations; either d > IY (i.e. 0 < l/d < l(Y)) for any pyy' pxx (pxx # 1) or 1 < d :_ lY 2o - l (i.e. < l/d < l) for -—1§L-—- < p < 1. (y) — pyy - xx Detailed information about the characteristics of each gen- eral category as well as information about the situation where d = 1 will be presented in the following tables (3.1a through 3.4b). In * E will be presented on the left side of the table and the corresponding each table the characteristics of each general category for Bx/B information for By/B; will be presented on the right side of the table. Following each table the generic graph of By/B; as a func— tion of pan will be presented for the general category displayed in the table. Although the generic graphs for each category of By/B; are identical to the generic graphs for corresponding categories of * Bx/Bg they will still be presented. When the joint categories for * * Bx/Bg and By/Bn are constructed and generic graphs are presented * there will be less chance of confusion if each category for Bx/Bg * and BY/Bn is clearly identified along with their generic graphs. ************* Insert Table 3.1a Here ************* Based on results (3.44a') through (3.441') the generic graph * of By/Bn as a function of p for G.C. I (y) subcategory a') has Er) the following general shape: 106 s a c c 2+.Sa M++namalagai B+W€$ m++nan31312 s s c s as m\ a v o a v we v xxa\b wow 1.xee.mc H v Wm\xn v o H v we v eb\H wow Axvv.mi .1 c c s c o n am\>m xxexb u we now 1.mvv.mc o n wmxxm » eb\a n we won Aflev.mc s s» c v exam xxe\u v ewe v b wow A.avv.mc o v Wm\xm ebxa v we v b\a wow lave.mc h) c a c a- + m\ m b + cue an a.sev.me a- + wmxxm e\a + we nn Asqv.m. t + I + c in c c bmsaumbss ma .m\ m b u we now A.mve.mv bmsamobss ma wsxxm oxa u we now Amee.mc f: c s\ a -e + spa nn i.wee.mc 8+ + Ws\xa -e\a + we we awae.mc s c s c a\ a e v be v be new l.maa.mi a A ba\xa e\a v spa v be new leav.mc « val « Axvl c cu ( cw . I u x cm 1 cm . *tmx m E..e u a wow A.bes m. H (tax a excue ) a wow Anew me as c c an m\> m o." ewe won i.uve.mv xxe u Nm\xm o u we sow .oev.mv s c s m\ m v o we v cue v H) wow A.bee.mi a v um\xm v o cue v we v H) wow Anev.m. « Axel « Axvl c +0 + cmxsm a- + ewe no 1.nee.m. o + wm\xm H(.+ we mm Anev.mc .... + a. ... AH x xxev xxa mNNe ham .xxa v p v o .H v p v 0 AH x away Nma .xxa zap .H A swap .H A p “.0 muommuMUQSm mm\»m wow AsiHoo Am anommuMUASm as} son 3 H8 8. .A.m cam no wuommumonsm .Ami H .U.@ paw Axe H .U.U mmfluomoumo Hmumsmu .MH.M manna 107 +tJ) -’---‘q I O O I S gap--..-..- g. 8’31 '*‘° (3.43:! Figure 3.5a ************* Insert Table 3.lb Here ************* * The generic graph of BY/Bn as a function of pEn for G.C. I (y) subcategory b') has the following shape: v1.0 I --’-----“- 1‘-- n.- 6p------ «9 ‘4'. A“ -’---r at P O gx‘ " 1.0 W a. 4,; Figure 3.5b 108 c a s no 1. «3 m ..H+ + we mm 123.2 o + 03$ H+ + sue mm 2.36 . ... u s \H o v «3 a H v 5e v e wow 1.54.2 o v wmxxm H v Se v b\H now 33.8 AMH.m mannav AmH.m mannav A.m muommumonsm ou HmoflucmpH A.£).mqv.mv Am whommumonsm on HMUfiusmGH Antmvv.mv : x has as as c % A.n muommuMOQSm : m\ m “on “we How a. un a sanvbvervbvo a» [N AH x >hdv o .xe asp .H v w on .H A p 3 rommumogm I Wm\xm ..Hom 33 How .A.a pom An mmfluowmumonsm .Hhv H .U.U tsp Axe H .0.0 mofiuommumu Hmuosmu .QH.m manna 109 ************* Insert Table 3.2 Here ************* i The generic graph of By/Bn as a function of pEn for G.C. II (y) has the following shape: : I ‘ I . I : : | 1 l L “at -1.0 fig") ’3“) +1.0 . G's/9‘ Figure 3.6 ************* Insert Table 3.3 Here ************* * The generic graph of By/Bn as a function of pEn for G.C. III (y) has the following shape: 110 c» c c o + .m\ m -H+ + 5a mm H.:m¢.mo +o + Wm\xm -H+ +5 5c mm Hnmv.mo c s cm :5 . u x :5 c5 . «m\ a v o H v a v Aso+a you A.mmv mo H v *m\ m v o H v a von+a you Home mo -: 5-5 . I -5 5-5 . H - «mxwm Hmo+a - a “om A.mmv mo H - .m\xm Axo+a - a you Ammv mo c cw cw cm 5 x cw cm :5 a a a . . a a . H A *m\ mlsv+ v v Hmo- now A mmv no H A .m\ m Hx~+ v v Axe-Q you Aomq mo cm : c c c H u m\ m a mg n 5a you A.cmv.mo H u 5m\xm 5a u me you Homv.mo . A o- . Axo- c a u .mxsm o u cwa you A.omv.mo xxa u wm\xm o u cua mom Aom¢.mo =m\mm v o cwa v cua v H- Mom H.nmq.mo H v5u\xm v o cue v awn v H- How Hnm¢.mv . Aso- « Axe- : a o + m\ m H- + cue mm A.mmv.mo ox.5m\xm H- + cwa mm Ammv.mo i + + t + a xx h a xx a» x --Han- v a v o . w w o.v H H x a .H.w a v a - «\H .H v u.w w c m\>m How Ame HH .o.o i w # .A>V HH we can Axv HH 00 mowuomoumo Hmuwcmo m\xm How Ax. HH .o.o .N.m wanna lll c» c x c +0 + m\ m H+ + mo mm A.cmv.mv +0 + wm\ m H+ + we mm Hpov.mv t I a. I a» c s cm xx 5 x cm a u m\ m o u a you H.oov.mo a u m\ m o u a you Homv.mo « s c» c x c H v ¢m\ m v o H+ v mo v HI Ham H.o©v.mv H v Wm\ m v o H+ v we v HI Hum Aaov.mv c a c w x c o + m\ m HI + mo mm A.mmv.mv o + m\ m HI f we mm Ammv.mv + a. + + a. + ax xx xx xl a a l a oI l -I Hx quXavlllll. _ voVH Hxhq.Lvaqvo.Hvovx_ >% H H - am .HO .HO a HH-x xxav qu .5 Q and .o v %_ AH x quv aha .qu acm . x_ v o v o c» x m\ a you Ase HHH .o.o 5m\ u now Axe HHH .o.o « « .Asv HHH .o.o new Axe HHH .o.o moHuomoumo Hmumcoo .m.m mHnma 112 +1.0 L 3: 3“ ‘ LO #1-0 53/5; Figure 3.7 ******************* Insert Tables 3.4a and 3.4b Here ******************* 20 - 1 When d = l and -X%-—--§_pxx < 1 the generic graph of YY * By/Bn as a function of p has the following shape: in 334‘ '1'0 *1.0 53/91 Figure 3.8a 29 - 1 When d = l and O < pxx < ——x§—-—— the generic graph of * YY By/Bn as a function of pEn has the following shape: MU -n~V .m- 0 HAN-u. pH. 113 H.onv.mv Apbv.mv H.05v.mv o How Acnv.mv A.nwv.mv cm H+v QVHI How Annv.mv A.m>v.mv HIT two mm Hohv.mv a» o I N H WxxavoHuw 5m\xm a. u p .wv.m oHndB f! F FA n...~t 114 mxexxe - H aexxe - H xx Hxxe - Hemmee A e - chxem a c c + ”m\ m IH+ + we mm A.mmv.mv + Wm\xm IH+ 1- we on Ammvfiv c c c c H A m\xm H v cue v ue mom A.mmv.mc H A ue\xe H v ue v ue now Hmmv.mc a Ail « CCI c s c c c c H - e\ e ue - ue ecu l.eee.me H - ue\xe ue - ue ecu Aeee.me « A%VI .1 Axvl xx c x c e u «e\ e o u cue you A.omv.mc xxe u Wexxe o u ue now Aomv.mc H v cm\xm v o cue v cue v H- uou H.bme.mc H v uexxm v o cue v cue v H- com Anmv.me a. A%vl ... Axvl o I ce\>e H- I cue mm H.mmv.mc o I ue\xm H-_I cue mm Hmmq.mc + k. + + i + \CHe xx in l xx xxe I IIJHNI v Q 0 .H u U H K Q .H v Q v TH H mu H - em H c x e\ e ue} .— c H u c .nv.m mHame 115 3’“ '1‘ . 93/9; 1 Figure 3.8b Examples of graphs in each of the general categories of x and y for fixed values of d, pxx and pyy will be presented below in section VII. * E * Prior to combining the general categories of Bx/Bg with the VI. Additional Algebraic Results Involving Both Bx/B * d an BY/Bn general categories for By/B; to form joint general categories some additional algebriac derivations involving both Bx/B; and By/B: are needed. In this section three algebraic derivations are to be presented. The first two derivations will be used in subsequent work but also represent interesting results by themselves. The third derivation provides an algebraic justification for a result which can be noted from the generic graphs for each general category of Bx/B; and By/B;. * Derivation 1: Show that when Bx/Bg * = 1, then B /B = and when Y n pYY * * BY/Bn = 1, then Bx/BE = pxx. 116 * Proof: From either (3.18b) or (3.19b) BX/BE = lflQ = O 2 Q = D where p (l ' 0 En xx Therefore consider By/B - d yy) ”6 * pyy’ * for e /3 with l/d = d yields: y n n 2 (1 - p ) tn.” 2 (1 - pgnpxxpyy YY n By/B and 2 1 - d ( p€n)pyy( C9 ) 8 /8n - ognpxx 2 (1 - pinpxxpyy pyy 2 _ 3 + En p D (d - d yy 0 pEnpxx 2 pfinpxxpyy - 3 Enpxxpyy (d - d Dyy 0 in 2 3 d 05n n ( - Dyy pgnpxx O 2 (Ognpxx(1 - d0 0 ) in p YY yypén a p Q = O YYDEn pyy En - t g = 1, then Q = O and thus 8 * Therefore 8 /B = if p Y n But if B /B x (3.52) Therefore if Bx/%:= 1, then By/B: And by the property of interchangeability of x XX * * 3.53 'f 1, th = ( ) 1 By/Bn en Bx/Bg o (1 - n pxxpyy Enpxx 0xx 0 or / F Y 8n ) + (l - pxx) from (3.17). Modifying expression (3.14) )(d - OED) ) 3 + pinpxxpyy - d 2 + -o o ) yy Dan Dan an xx 0 (l - pxxpyy) + (l-pxx)) = 0 O > 0. since Q o YY p . YY pyy' and y: Note that the converses of (3.52) and (3.53) are in general * = 0, Bx/Bg = pxx not true, since if p in * d an By/Bn = p . For YY 117 * * “an x 0. then By/B; = pyy a tax/Bg = 1 and sx/sg “xx.” BY/B; = 1. Derivation 2: Show that Bx/B; and By/B; cannot both be greater than one for any situation, i.e., there exist no values of d, pEn' pxx and pyy such that Bx/B; > 1 and By/B; > 1. For any given situation at most one of the observed weighting coefficients will be an overestimate of the corresponding latent weighting coefficient. The procedure will first locate for Bx/B; and By/B: separately the general categories and then the regions within the general categories (from Tables 3.1a through 3.4b) where each ratio is greater than one. Then the regions will be compared to see if there are any regions where both are greater than one. If such regions exist then more detailed algebraic work will be performed to examine the situations in each region. * Bx/Bg > 1 (3.54a) 1. From Table 3.1a for G.C. I, when d > 1 for any pxx' p and p-(x) < * yy En pgn < l/d then Bx/Bg > 1. (3.54b) 2. From Table 3.2 for G.C. II, when IX :_d < l, - + 1/2 - p < p :_l and p (x) < p < p (X) then YY xx 60 En En * > 1. Bx/Bg (3.54c) 3. From Table 3.4b when d = 1, 1/2 - p < p < 1 and yy xx —- -(X) < < 1 then B /B* > 1 “an “an ' x a ' > B/ (3.55a) 1. From Table 3.1a for G.C. I (y) when 0 < d < 1, any -(y) En < pin pxx' pyy and p * l. (3.55b) (3.55c) * > By/Bn (3.56) 118 2. From Table 3.2 for G.C. II (y) when l < d 5_ ly, 2" '1 -II +() 0 < p < -—j§L———— and p Y < p < p Y then xx 0 En En En yy * B B > 1. Y/n 2p - l 3. FromTable 3.4bwhen d=1,0 1. 5n En y n 2p - 1 Notel: 1/2-p >—11— forall p ,0

1 for all ‘IY'— with equality for (3.51).) (See Appendix such that pxxpyy # l p = ————;-—3 (See Appendix YY 1 such that pxxp # . (By expression * Therefore the only regions where both Bx/Bg > 1 and 1 occur within the region are: when Kid< l and l/2-pyy 1 1f pEn pEn < pEn [from (3.54b)] and 8 /B* > 1 if p-(y) p < d [from (3.55a)]. Y n in in ’ 119 2p - 1 ..JQL___.. (3.57) and, when 1 < d _<_ I; and o < pxx < p . YY Bx/B* > 1 if p-(X) < p < l/d [from (3.54a)] 5 En En ‘ * . -(y) +(y) and By/Bn > 1 if pEn < pEn < pan [from (3.55b)]. Note however, that this second region can be found from the first region by use of the property of interchangeability. Therefore it is necessary to examine in detail only the situations in the first region, since all results for situations in the first region can be extended to corresponding situations in the second region. T 1 and By/B; > 1 simultaneously. By-the property of interchangeability of x and y this result for the region identified as (3.56) also holds for the region (3.57). Therefore it is not possible to find a set of values of I p , p and d so that Bx/8* En YY XX 5 simultaneously. That is in the two category, two predictor case it * > 1 and B /8 > 1 Y n is not possible for both of the observed weighting coefficients to be overestimates of the latent weighting coefficients for the same com- bination of values of p , p , p and d. yy xx * E dp < 0. It is important to note here that this result represents En a sufficient condition only. in * Derivation 3: Show that both Bx/B < l and BY/Bn < 1 when * Consider Bx/BE from (3.12), i.e., 2 - - d B /B* = (1 pgfl)oxx(l ogno ) X (1 - 2 )(1 - d ) pEnpxprY pin * Note that BX/Bg can be considered as the product of two ratios R 2 1 (1 - p )p (1 - do 0 ) and R where R = in xx and R = ED—YY . Thus 2 1 (1 _ 02 p p ) 2 (l - dog”) * En XXYY * B /8 = R ° R . The examination of B /B for this derivation will x g 1 2 x 5 proceed by examining R1 and R2 separately and then combining the results. Work with R1 first. There are three situations to examine. For what values of , and will R (1) be reater than Dan pxx Dyy 1 g 121 one (R1 > 1 ?), (2) be equal to one (R = l ?) and (3) be less than 1 one (Rl < l ?)? 1 2 (l - p )p R > 1 9’ g“ > 1 1 l - 92 p p in xx YY 2 2 Q (l - ) > 1 - since pin 0 pgnpxxo l - > 0 pinpxxpyy 2 > 1 a _ _ ’oxx panoxx panoxxoyy a 0 > o o - 2 p + (1 - ) Enxx pgnxxpyy pxx O>pzp (1- )+(1- ) Enxx pYY pxx but note p2 > O p > 0 l - p > O l — p > 0. En_ I XX I YY_ r 10(- 2 Therefore 0 (1 - + l - 7 ognoxx oyy) ( oxx) (3.58a) and R 7 1 1 (1 - 02 )p 2.) R < 1? R < 1 a 5” xx < 1 l - 1 - (l _ 2 p )'- “an xxpyy using algebra from 1) above 2 . < C: O < 1 — + _ (3 58b) R1 __ 1 _ pgnpxx( pyy) (1 pxx) true for all values of , , and . pan pyy 0xx 2 N t : R = 1 W 0 = 1 - + 1 - . o e 1 pgnpxx( pyy) ( pxx) . T = ' = ' = (3 58¢) hus R1 1 only If pxx 1 and either pyy 1 or 122 Note: if d = O, 2 (1 - p )p B /B* = 5" xx = R . x g l _ 2 p p 1 OED xx yy Work now with R2. Consider only situations where R2 < 1. * Then Bx/BE = R1 ° R2 < l by (3.58b). Note, for some situations where * * > 1 and for other situations Bx/B < l. The purpose 5 E of this derivation is to produce a sufficient condition for Bx/B R2>l-B£$ * E < and not reproduce the exhaustive study which has been done above in 1 'k * subsection IV for Bx/Bg and in subsection V for By/Bn. The situation to be examined then is for what values of p , En ‘ a 9 pyy and d will R2 be less than one (R2 < l .). l - d0 0 R <:1? R < 1 a 5” YY < 1 2 2 l - dpan 1.) if 1 - dpEn > 0 dog“ < 1 then R < l a l - d < 1 — d 2 panpyy pan ado <0 - d in pinpyy R do (1 - o ) < 0 En yY a do n < O for pyy # 1. 5 Since R < 1 if dp 2 n < 1 and dp < 0. Then E in 3.59 R < 1 'f d < o. ( a) 2 1 “an or 2.) if 1 - dpEn < O ¢°dp€n > 1 123 then R2 < 1 a l - dp > 1 - dp Enpyy in a do n(1 - pyy) > 0 E o dpin > O for pyy # l . Since R < 1 if d > 1 and d > 0. Then 2 pin pin 3.59b R < 1 if d > 1. ( ) 2 “an Therefore, if dp < 0 or dpEn > 1 then R < 1. Since in 2 b 3.58b R < f 11 l f d ' y ( ) l __l or a va ues o pan, pxx an pyy and Since * * Bx/Bg = R1 0 R2 then Bx/BE < 1, * (3.60) If dpEn < 0 or dpin > 1 then Bx/BE < 1. For dpEn to be less than zero, d and pEn must have opposite signs. By the property of interchangeability of x and y (3.60) becomes for d # O: * 3.60' If d < 0 r d > 1 then < 1. ( ) pgn/ o n/ By/Bn ”5 For n/d to be less than zero, d and pEn must have opposite “5 signs, i.e., dp < 0. En Therefore if d and p n have opposite signs (dpfin < 0) E then both BX/B; and By/B; will be less than one. Note that this is a sufficient condition only. As noted above dpEn can also be interpreted as the ratio of the slope of the pooled within categories regression line of E on n over the slope of the between categories line joining the mid- points of the distributions in each category of E and n. This interpretation is presented in Appendix B.9 along with a presentation of results which corroborates the results in (3.60). 124 Using the ratio of slopes interpretation of dp in a in situation where dp < 0 indicates that the slope of the pooled En within categories regression line has the opposite sign as the slope of the between categories line, i.e., the direction of the relation- ship between E and n as expressed by the pooled within categories regression line is the opposite of the direction of the relationship between category means as expressed by the slope of the between categories line. Using this interpretation for do < 0 there may be some En question about whether dpEn can be less than zero in practice. The concern at issue here is very similar to the concern involved in the study of ecological correlation where the interest is on using a correlation between group means (similar to the between categories situation here) to estimate either a total group or pooled within groups correlation (similar to the pooled within categories situation here). The study of ecological correlation produced some results which are applicable here as well. If the groups (in this case two groups [categories]) are in- dependent samples from the same population then the correlation in the population of individuals and the correlation in the population formed by the sampling distribution of the group means for a given sample size are identical. In this case the between groups correla- tion and the pooled within groups correlations are less likely to have different signs than the same sign. For the quantal response situa- tion, if the individuals are arbitrarily assigned to each category of the criterion on the basis of a random sampling from a single 125 population of individuals then the relationship between 5 and n as measured by the slope of the pooled within categories regression line will be more likely to have the same sign as the slope of the between categories line joining the category midpoints than to have a different sign. That is, do is more likely to be greater than En zero rather than less than zero. For most quantal response situations it would seem unlikely and even contrary to the intents of quantal response procedures to arbitrarily define categories of a criterion as multiple random samples from some population. For most quantal response situations assignment of a subject to a category of the criterion is based on distinct and non-overlapping membership criteria, e.g., health status of an experimental animal (e.g., a rat) following an administration of an experimental drug (i.e., alive or dead) or group affiliation (Democrat, Republican, Independent, etc.). For these types of situations it is not reasonable to assume generally that the between groups relationship will have the same sign as the within groups situation. That is, for a given situation there is no a priori basis to assume that dp > 0 with any more in likelihood than doEn < 0. Although in many situations the ratio of slopes (dpin) will be positive there will exist situations where the ratio is negative. A hypothetical example can be constructed to illustrate thatthe ratio of slopes can be negative. Consider two elementary schools. Each school represents a category. The mathematics curriculum of school 1 heavily emphasizes 126 work on the basics of computation through rote memory and repeated drill under the assumption that students must have a sound basis of computational skills prior to tackling more advanced mathematics topics. The mathematics curriculum of school 2 emphasizes training in approaches to problem solving and the conceptualization of mathe- matical problems under the assumption that it is important to be able to identify approachestx>the solution of problems and that specific computational skills can be more efficiently learned when the student is confronted with the need to compute as part of the solution of a problem. The predictor variables in this situation are the mathematics computation subscale and the mathematics application subscale of some standardized test. It is reasonable to expect that there is a similar positive relationship between subscale scores on mathematics computation and application within each school, since factors such as general mathematics ability and motivation are likely to be under- lying factors related to performance on both subscales within each school. Therefore, there is a positive within categories relation- ship between the predictor variables. However, it is also reasonable to expect that students in school 1 will do better on the mathematics computation subscale than students in school 2. While, students from school 2 can be expected to do better on the mathematics application subscale than students in school 1. Therefore, the slope of the line which joins the midpoints of the distributions of the two schools on the computation and applica- tion subscale can be expected to be negative. That is, there is a 127 negative between groups relationship. Therefore, the ratio of the pooled within categories slope to the between categories slope is negative. If this or a similar situation were to be analyzed using a quantal response procedure then dp < 0 and Bx and BY will in * * underestimate BE and 8n respectively. Note if the ratio of slopes based solely on the observed predictor is negative then the ratio of slopes for the latent predictors (dpgn) will also be negative. This follows since the errors of measurement will not affect the value of the slope of the line between the midpoints of the categories but will attenuate the value of the pooled within categories slope based on latent predictors (see Appendix 8.9 for details). Thus the magnitude of the ratio based on observed predictors will be smaller than the magnitude of the ratio based on latent predictors but the signs of the two ratios will be identical. * VII. Joint General Categories of Distributions for Bx/Bg and * By/Bn Together In this section the results from section IV (for categories 'k of var us for fi ed values of d, and ) sec- Bx/Bg 5 pin x oxx pyy , * tion V (for categories of BY/Bn versus OED for fixed values of * d, pxx and pyy) and section VI (Algebriac results for Bx/Bg and * BY/Bn together) are combined to derive joint general categories of * E * For each joint general category, the generic graphs of Bx/B and E * By/Bn will be displayed to provide a visual indication of the generic * distributions for Bx/B and By/Bn together for the same situations. shape of the distributions within the category. In addition actual 128 graphs for specific situations (i.e., values of d, pxx and pyy) within each category will be referenced. For a more detailed in- * * dication of the shape of either Bx/Bg or By/Bn within any joint general category see the information from the tables in section IV or section V which applies. An example of the notation for the joint general categories is: G.C. I (x,y). The x and the y in the parenthesis indicate * that it is a joint general category involving both Bx/Bg and B /B* y n' G.C. I (x,y). When 0 < d < IX (Recall, ‘lx is defined by (3.22)), XX YY XX YY * * O < < l 2 - G.C. III f r nd G.C. I for oxx __ / pyy- o Bx/Bg a (y) By/Bn for any p , p (p , p f l) or when IX :_d < l for apply. Ignoring the two subcategories of G.C. I (y), the generic graphs for this category of situations are: Figure 3.9 In 129 * The exact shape of By/Bn when d < pEn < 1 depends on whether 0 < d < p or p < d < 1. See section V for G.C. I (y) xx xx above for details. Figure 3.1la through 3.lld provide examples for 4 specific situations in G.C. I (x,y). G.C. II (x,y). When Ix :_d < l for 1/2 - pyy < pxx < l, G.C. II t 'k for Bx/Bg' G.C. I (y) for BY/Bn and section VI results apply. Again * ignoring the two subcategories of By/Bn, the generic graphs for this category of situations are: Figure 3.10 * The exact shape of By/Bn when d < p n < 1 depends on whether 5 O < d < pxx or pxx < d < 1. See section V for G.C. I (y) above the details. Figures 3.12a through 3.12c provide examples for 3 specific situations in G.C. II (x,y). 130 1 fi‘.° up ' o = .6 xx * h. ..r O = .8 yy d = .2 ”.5 4- IX = .843 HA if 2_: = .833 W No?- -- Figure 3.11a. G.C. I (x.y). pxx = .9 o =-8 YY d=.2 |=.958 X 2_: = 833 YY 131 +1.8 - *Lb - §\I‘ - 91-1 - “.0 u I .L' : : ‘r ‘- l t t 1“. -i.o -.3 -.I.-. -.4 -.‘L In. M“ +.b ma .,2 - .- ..- WM ...), .. -.3 .- -I.O.I “Llfi Figure 3.llb. G.C. I (x,y). 132 pxx = .6 *\‘s -L = .8 YY ¥\eb 'I- d = .8 K: .843 *L‘ .- l = .833 2- . pyy H1- Jr ' o < d < Ix 1 +L° '- 0 < p < xx 2-p . YY ~h8 and p < d < l . +Jo (*3/91 IA «II- (3‘/(3; +JL'i- _l J l l A I g 1 'a L ' T i ' . o .. n. c o. +.‘ 3‘) -.5 .JL--- -.4 .5. ...,“ d- (3: " ..8 4h: Apt -19 J+ ...i.‘ .- Figure 3.llc. G.C. I (x,y). 133 pxx = .6 *1.$dn p = .8 yy IiJo dr d = 843 = , I. ‘Ix 843 IM 1 1 2‘0 = .833 *1»?- .). YY x = d < 1' ' +1.0!- ; l n g l i L J 1 J'— l T— ! r i I T I ‘ 951 oioo I08 ..5 .4 -‘ fit *‘ ‘ +.h *IB +1.0 .55 2 JP ‘hs ' I -.4 w. J - I. J. - (33/ ' l (I, -.8 WI ( .100 'b 41(- Figure 3.lld. G.C. I (X-Y). p = .95 xx p = .7 YY d = .8 J:= .713 = .769 134 +1.6 -- +1.4 -- #1.‘ u- +1.o -b “4"" 3‘ Figure 3.12a. : I :— H— -.g -.7. In. I. u. Is +1.0 -.5 ...: ii- -.4 j- (33 I ...L .1. 1 #51 l 1 -,a H. i 1 -1-0 -b ‘l -1”: .1. G.C. II (x,y). pxx = .9 p = .8 YY d = .97 = 958 x 3‘1 . . . -.s . Figure 3.12b. G.C. II (x,y). 135 I -3 +- '5. i H. i -flJ:-h 4.1 .1- a» «II- A ‘0."me d I- 4r I T‘Ib *o. +“° «n5 ‘w— 136 pxx = .9 I10 -)- p = .7 yy (1 = 888 I“. 'I- HIH—MH ~—:— “I _ of; ¢u4 0J0 9.. viwo +.‘S -|q ‘- -I‘ .)- 02/. ."8 cup fl‘ '1J) mp ~12 (- Figure 3.12c. G.C. II (x,y). 137 2p - 1 G.C. III (x,y). When 1 < d < I for O < p < -—jDL———- then Y XX 0 * * YY G.C. I for Bx/BE' G.C. II (y) for By/Bn and section VI results apply. Note, this general category is identical to G.C. II (x,y) but with the roles of x and y reversed. Therefore Figure 3.10 with the property of interchangeability of x and y applied provides the generic graphs for this category. Figures 3.12a through 3.12c with the property of interchange- ability of x and y applied provide examples for specific situa- tions in G.C. III (x,y). G.C. IV x, . When < d f , 1 or ( 4y) J:: or any pxx pyy (pxx' pyy # ) r * E and G.C. III (y) for By/B; apply. Note, this general category is identical to G.C. I (x,y) but with the roles of x and y reversed. Therefore Figure 3.9 with the property of interchangeability of x and y applied provides the generic graphs for this category. Figures 3.11a through 3.lld with the property of interchange- ability of x and y applied provide examples for specific situa- tions in G.C. IV (x,y). 2p - 1 When d = l for O < p < -—}QL———- with p # 1 Figure xx 0 YY * yy* 3.4a for Bx/B and Figure 3.8b for By/Bn apply, since E (see proof in Appendix B.7). Therefore the generic yy pyy graphs for these situations are: 138 2:1:(i’gux ’ 1. (1° 3" 9113 1:“ (i‘ 39:) ‘ 1 (1 “ f“ 2153 Figure 3.13a 2p -1 When d=l for —-Yy———

xx 11’ so >o l-I P l-Ixxryy W “3' 139 .. > _ 0y (1 pxx) o x(1 oyy) 2p (1 - pxx) prx(1 - pyy) lYY (3.61a) > a D > p 1 Dxxpyy 1 "' pimp”, YY xx 2p (1 - oxx) 20xx(1 - p ) (3.6lb) 1i” 1 1 _ H a p 1 an - pxxpyy pxxpyy yy l = f -—-— < < ‘ - When d l or 2 _ p p l Wlth p f 1 Figure * E perty of interchangeability of x and y to Figure 3.13a will pro- * 3.4b for Bx/B and Figure 3.8a for By/Bn apply. Applying the pro- vide the generic graphs for these situations. Figures 3.14a and 3.14b provide examples for the first two sets of specific situations noted above when d = 1. Applying the property of interchangeability of x and y to Figure 3.14a pro- vides an example for the third set of situations noted above. J Category, Two Predictor Models (J :_2, p = 2) The preceeding work has considered the case of 2 categories (J = 2) and 2 predictors (p = 2). To complete the examination of the two predictor special case, it is necessary to consider the most general two predictor model, that with J categories (J :_2). It is reasonably straightforward to show that results for the 2 category, 2 predictor model extend with only slight modification to the J category, 2 predictor model. pxx = .6 = .8 pYY d = 1 2p - l -—x%-—-= 75 14 ‘ YY § . 1 = 833 2-0 *iot . YY 2p - l 0 < 9xx< "‘L—p I10“ 140 #- vw Figure 3.14a. 141 p = .76 XX = .8 CW d = 1 2p - l ——X§-—— = .750 YY l .833 ”‘0 'I- .580 \ .116 'L[ A' L ‘L_ L _ 4_____ I '0. IA 0.6 +.B +5.0 Figure 3.14b. 142 Consider any category k (k = 1,2,...,J), any vector of ob- served weighting coefficients from category k, gk-j (j # k, j = 1,2,...,J) and the corresponding vector of latent weighting co- efficients g;°j° Since the model under consideration here is a 2 predictor model, r- a 8 . P “W k' . 3(x) * 8k,3(g) eFk'j(YLH eék.j(n{4 Result (2.21) modified for the two predictor case (p = 2) becomes * (“(k) _ u(3)) _ b (“(k) _ “(j)) ' k°j(€) 2 ’ c - b o O E E'n En _E. where bE'n - pan 0 . n 'J _ (k) (j) k‘j _ (k) _ (3) Let a: a “a and n ’ n n Note: 2 2 35_ 2 2 °€ ' bE°nO€n = “a ' (“an 0n)(pEn°€On) = “5‘1 ' p€n)' Thus (3.62a) becomes k-j . a /o . . k-j n n k°j _ k°j Eg_ a (l - -jfi——"D ) * a: an 0' pén g a: 3/0 En Bk '(g) = 2 2 n = 2 2 g .3 - - 05(1 pin) 05(1 pin) .. ak-j/O (u(k) _ “(3))/O Let dk J _ n n = n n n g ‘ k-j/CI ( (k) _ (j))/c ’ a: a “a “a 5 Therefore 143 kc k. ' * 3: j‘l ‘ 6: Jpgn’ (3.62b) B . = . k 3(5) 02(1 - 2 ) E pEn Result (2.17b) modified for the two predictor case (p = 2) becomes (k ' k ' (u ) - u(3)) - b (u( ) - u(3)) x x x-yy y Ay (3°63a) Bk-‘(x) = 2 3 c — b c X X'Y xY 0x where b = p -—- . X°y xy 0 Y Let ak.3 = u(k) - u(3) and ab:I = u(k) — u(j), where x x x y y Y a:.3 ag .3 and a .3 = a:.3 by a result comparable to (3.8). 0 '0 Note: 1) b = p o /o = (p /p p )(—5——)(——X¥I -———' X'Y XY X y in XX YY /——— 0 D n xx 0 = “5'0 0 . 0 En YY n By (3.10), (3.9a) and (3.9b). o 2 2 2) c - o = o - (0 -§) 0 o c ) = c (1 - p ) x x-y xy x xy 0 xy x y XY 2 3§_. 2 - (l - panoxxo ) pxx YY By (3.9a) and (3.10). Therefore (3.63a) becomes k'J k'j .5 a5 n on pgnpyy Bk-j(x) = 2 c8; 2 3;;(1 " pgnpxxpyy) k-j R.) an /o p a (l - "___—1. 0 xx 5 ak°3/O En yy = E E 2 2 c€(1 - pgnpxxpyy) k-j k°j 1 d (3 63b) 8 = pxx g ( g pgn9XX1' ° k'j(X) 02(1 - 02 o o ) t 5 En XX YY k-j .' a /o where dk J n - n ' 5 akOJ/o E E * The ratio of interest Bk-j(x)/Bk°j(€) becomes k-j k-j k'J' k'j * pa(1-d00)a(1'dp) B ' /8 . = xx <5 5 an YL/ E E En k°j(x) k'J(€) 02(1 _ 02 p p ) 02(1 - p2 ) 5 En xx yy 5 5” 2 k°j 1 - 1 - d (3 64) B /B = ( pgn’pxx‘ 5 Dan) En xx yy 5 En k-j (k) _ (j) where dk°j = an /On _ (“n u” )/OD E ak'j/o (u(k) - u(j))/° E E E E 5 Note the close similarity between expression (3.64) for the J category, 2 predictor model and expression (3.12) for the 2 cate— gory, 2 predictor model. The only difference is in the use of dk.3 in (3.64) and dg in (3.12). But note that dg in (3.12) is based 145 on a comparison between predictor means in the only two categories k-j E predictor means from some two of the J categories. in the model whereas d in (3.64) is based on a comparison between Thus in the J category, 2 predictor model any ratio of the * k°j(x)/Bk-j(g) for j'k = 1:2,...,J and j # k. will have a distribution which corresponds to one of the General Categories for form 8 . k'j de din o the val es f d d h th x pen 9 n u o E , pxx' pyy an pEn’ w ere e d2.) value can be treated as a value of di from (3.12), since dk.3 will take on values -m < dk.3 < +m just as d does. 5 E E A generalized property of interchangeability applied to (3.64) * . . f ',k = 1,2,...,J .th k-J(y)/Bk°j(n) or 3 w1 j # k which is an extension of (3.14). produces the ratio 8 _ 2 _ k'J’ _ (1 pgn)pyy(l an agnoxx> (3.65) B . /B . - ' k'j(Y) k°j(n) (1 - 02 p p )(1 - dk.Jp ) En xx YY n 5” . (u(k) - u(j)>/O where dk.3 = g n g n U n Therefore all results which apply to (3.14) can be easily extended to apply to (3.65) with d:.j in (3.65) taking the place of dn in (3.14). Therefore all results noted earlier for the 2 category, 2 pre- dictor model extend simply to apply to corresponding cases in the J category (J 3 2), 2 predictor model. 146 Section D: Summary The purpose of this chapter was to examine the effects of errors of measurement on the weighting coefficients of a Latent Random Predictor Quantal Response Model, given by (2.19) for the most gen- eral case. The approach to the problem involved selecting an arbitrary vector of weighting coefficients associated with some arbitrary category of the criterion variable and examining the in- dividual weighting coefficients associated with each predictor. From an arbitrary vector of latent weighting coefficients associated with some category of the criterion from the model given by (2.19), an individual latent weighting coefficient associated with some latent predictor Tq was selected, call it 8;. From the corresponding vector of observed weighting coefficients associated with the same category of the criterion, the individual observed weighting coefficient associated with observed predictor Xq was q q selected, call it Bq. Note that X and T are related through the measurement model (2.22) such that Xq = Tq + Eq where Eq is * the error portion of the observed predictor. Then the ratio Bq/Bq * was examined. When Bq/Bq > 1, then the observed weighting coeffi- cient (Bq) is an overestimate of the latent weighting coefficient * * * ( ). When = 1 then is an exact estimate of . When Bq Bq/Bq . Bq Bq * * Bq/B < 1, then B is an underestimate of 8 . q q q The research of this chapter included one and two predictor models only. No general results applicable to all models were dis- covered and the approach used in this chapter proved extremely dif- ficulty for use with models involving more than two predictors. 147 For the one predictor models (p = 1) there is only one com- ponent in each vector of weighting coefficients. Result (2.28) indicated that for all one predictor models the value of the observed weighting coefficient will be an underestimate of the value of the latent weighting coefficient by a factor equal to the reliability of the single predictor variable. This result holds true for every pair of related observed and latent weighting coefficients associated with any category of the criterion. The only exception to this result occurs when the latent weighting coefficient is zero. In that case the observed weighting coefficient was also shown to be zero. For the two predictor models (p = 2) no universally applicable result was found, such as that produced for one predictor models. The approach for the two predictor models involved a change to simplify the notation and make it consistent with the notation used by McSweeney and Schmidt (1974). Under this simplification the observed predictors are noted as x and y with the corresponding latent pre- dictors being 5 and n, where x = g + ex and y = n + ey in an adaptation of the basic measurement model (2.22). The major work for the two predictor model was done for the two category (J = 2) case. All results for this simplest case of the two predictor model were then shown to extend easily to the general case (J 3_2) of the two pre- dictor model. In the 2 category, 2 predictor model the observed weighting coefficients were denoted 8x and 8y while the * 5 Therefore, the ratios which were examined as they relate to one were * corresponding latent weighting coefficients were denoted B and B”. * * Bx/Bg and By/Bn. Each of these ratios were shown to be functions 148 of d, pen, pxx and pyy° d is a ratio of the differences between category means for the two predictors where the differences are in standardized latent units. See expression (3.6b) and the ensuing narrative for the definition and explanation of d = d . is 5 ”an the correlation between the latent predictors. pxx and pyy are the predictor reliabilities for observed predictors x and y respectively, and indicate the presence of errors or measurement when either, or both, p or. p are less than one. xx yy * In Appendix B.l, part B, it was shown that Bx/Bg and * By/Bn need to be examined only for d 3_O. Results for d < 0 can be derived simply for comparable results when d > 0 by the use of expressions (3.16a) and (3.16b). Since there were no universally applicable results discovered, the work in this chapter identified four general categories of situa- * 6 were considered as functions of DE” for fixed values of d (d > 0), tions associated with Bx/B; and 8y/8;, where Bx/B and BY/B; pxx and pyy. Within each of these four joint general categories, defined in Section C under sub-heading VII above, the behavior of the * E situations (i.e., values of d, pxx and pyy) included in the category. ratios Bx/B and By/B; follow the same general pattern for all In addition to the four joint general categories three categories re- lated to the special case when d = l are also identified in Section C, sub-heading VII. Three results do apply across all 4 joint general categories and the three special case categories. First, when there is no correlation between the latent predictors, i.e., p = 0, then each in 149 ratio is equal to the reliability of the predictor. That is, when * i = O = and = . This result a lies to all Dan , Bx/Bg pxx By/Bn Dyy PP * * situations when 8 and En are not equal to zero. 5 Second, when the correlation between the predictors, 0 En' and the ratio of standardized category mean differences, d, have Opposite signs then the observed weighting coefficient will under- estimate the latent weighting coefficient for both predictors, i.e., * * < l and < 1. In this case i.e., d < 0, for fixed Bx/Bg By/Bn : 05" d, p and p the amount of the underestimate increases as the xx YY magnitude of the correlation increases. That is, if d > 0 then as * 1' takes on values nearer to -l the ratios and will Dan Bx/Bg BY/Bn become smaller, approaching zero as pEn approaches one. In Appendix 8.9 the interpretation of do is given as a En ratio of the slope of the pooled within category regression line of E on n over the slope of the line joining the midpoints of each category distribution of E and n. The potential for occurrence of a negative ratio of slopes is discussed under Derivation 3 in Section C, subheading VI above. Although in many situations the ratio of slopes will be positive (therefore do > 0) it is possible for the En ratio of slopes to be negative (i.e., dpEn < 0). Third, Derivation 2 in Section C, sub-heading VI above demonstrates that it is impossible for both observed weighting co- efficients to simultaneously overestimate the latent weighting co- efficients for the same set of values for d, p , p and p . At En xx YY most one observed weighting coefficient will be an overestimate of the latent weighting coefficient in any given situation. In addition 150 Derivation l in Section C, sub—heading VI above indicates that for the two predictor model if the observed weighting coefficient for one pre— dictor is equal to the latent weighting coefficient for that pre- dictor then the observed weighting coefficient for the other predictor is an underestimate of its corresponding latent weighting coefficient by a factor equal to the reliability of this second predictor. That f / * th / * * l is i B = 1 en = or if = then I x BE BY 8n pYY BY/Bn * Bx/Bg = pxx' The conversescnfthese statements are true only when 0. can 7‘ An interesting result which occurs only for joint general categories one and two [G.C. I (x,y) and G.C. II (x,y)] occurs for values of p in an arbitrarily small neighborhood of d (in these in categories d is positive and less than one). For values of pan * * arbitrarily near d the magnitude of By/Bn, i.e., [By/8n] is un- boundedly large. When pEn = d the ratio By/B; is not defined since 8; = 0. A similar situation occurs for Bx/B; in G.C. III (x,y) and G.C. IV (x,y) for values of 95“ near l/d (in these categories d exceeds one hence l/d is less than one). In this case when pEn = l/d the ratio Bx/B; is not defined since 8; = O. The importance of this result occurs in interpreting the effects of errors of measurement using Bx/B; when pEn is near l/d or using By/B; when pg” is near d. Consider some situation from G.C. I (x,y) where d, pxx and pyy are fixed. Here d will have some positive value which is less than one. For values of 05” which * are arbitrarily close to d, 'By/Bgl will be arbitrarily large. How- ever, depending on the specific value of the difference between the 151 category means for predictor y, the magnitude of the latent weight- * ing coefficient will be extremely small, i.e., IBnI will be near zero. In this situation the magnitude of the observed weighting coefficient, * [B may also be quite small even though the ratio [By/8n] may be y" relatively large. For example, for some pEn near d,B; might have a value of .005 while By might be .05. In this case By/B; = 10 which represents a rather large factor. Even though By is an over- estimate of B; by a factor equal to 10, the magnitude of the over- * estimate, BY - Bn = .045, is relatively small and in most interpreta- tions a difference of this size for weighting coefficients of this magnitude is meaningfully insignificant. For values of p near in d, 8; must be near zero. However, there is no necessary reason why By need be near zero also. In fact the difference By - B: may be significantly large for some situations. Therefore, the value of By/B: for values of DE” near d * and the value of Bx/Bg for values of p near l/d need to be En interpreted with great care. A relatively large ratio may mask two rather small weighting coefficients which may have a neglible practical difference in magnitude. Or a large ratio may represent a significant * discrepancy between By and 8”. Note however that when p = l/d then the value of 8x con- En sists totally of effects of errors of measurement since in this case * B = 0 indicating no relationship between the latent predictor and E the probability of classification into category one of the criterion. See Appendix 8.9 for an interpretation of this result in terms of the ratio of within group to between group slopes. The same conclusion * applies for By as an estimate of Bn when p = d. En 152 Cochran (1968) reported results for the effect of errors of measurement on regression coefficients in linear regression models. Although the distributional assumptions included in linear regression models are different than the distributional assumptions included in random predictor quantal response models, the expression for a vector of regression coefficients has a structural similarity to the ex- pression for a vector of quantal response weighting coefficients. Because of this structural similarity between vectors of regression coefficients and vectors of quantal response weighting coefficients it is not surprising that some of the results reported from this research for quantal response weighting coefficients have a similarity to results reported by Cochran (1968) for regression coefficients. For the one predictor case Cochran (1968, p. 652) demonstrated that the observed regression coefficient is an underestimate of the latent regression coefficient by a factor equal to the reliability of the predictor. An identical result for the relationship between the observed quantal response weighting coefficient and the latent weighting coefficient is reported from this research. In the two predictor situation where only one predictor is subject to error Cochran (1968, p. 656) provides an expression for the observed regression coefficient as a function of the latent regression coefficients and other parameters describing the two predictors. An identical expression also exists for the observed quantal response weighting coefficient as a function of the latent weighting 153 coefficients and other parameters describing the two predictors. Although the structures of the expressions for the vectors of regression coefficients and the vectors of quantal response weighting coefficients are similar, the derivations of the coefficients, which are based on the distributional assumptions of each model, are dif- ferent. Thus it is also not surprising to discover that some results reported by Cochran (1968) for regression coefficients do not have exact counterparts among quantal response weighting coefficients. For example, in the two predictor situation where the reliabilities of the two predictors are equal, Cochran (1968, p. 656) reported that the ratio of the observed regression coefficient to the latent regression coefficient for either predictor will be somewhat greater than the reliability of the predictor when the correlation between predictors is positive. This conclusion is not true, in general, for ratios of quantal response coefficients. One quantal response counter-example occurs for pxx = pyy = .8, d = d: = .2 and pan = .3. In this case Bx/B; = .782 and By/B; = .310 and both ratios are less than the common predictor reliability even though the correlation between pre- dictor is positive. Cochran (1968, p. 655ff) also reports that the observed regression coefficient, in a multiple linear regression, associated with some one predictor can be expressed as a linear function of the latent regression coefficient associated with that one predictor and the latent regression coefficients associated with any other predictors that are correlated with that one predictor. For the multiple pre- dictor quantal response model no general result comparable to this result was found. 154 Thus, although there are similarities in the results reported by Cochran (1968) for the effects of errors of measurement on the regression coefficients of a linear regression model and the results reported from this research for quantal response models, the con- clusions for the two models are not identical. For the two predictor models (p = 2), reviewing the generic graphs and the tables of results which define each of the four joint general categories and the three categories of the special case d = 1, clearly indicates that for every situation where at least one predictor is less than perfectly reliable, i.e., either pxx or p is less than one, the observed weighting coefficient represents either an over- estimate or an underestimate of the latent weighting coefficient for at least one of the predictors. For the one predictor models (p = l), the observed weighting coefficient is always an underestimate of the latent weighting coeffi- cient by a factor of the reliability. Therefore, for both one and two predictor models the presence of errors of measurement in the predictor variables does have an effect on the determination of the true relationship between a pre- dictor and the probability of classification in a given category of a criterion. In all cases where errors of measurement are present in the predictors, use of the observed weighting coefficient as an estimate of the latent weighting coefficient will result in an in- correct estimate. This applies for at least one if not both pre- dictors in a two predictor model and for the single predictor in a one predictor model. Determination of whether the discrepancy between the 155 observed weighting coefficient and the latent weighting coefficient is large enough to be of practical significance for situations which typically occur in quantal response applications is beyond the scope of this research. Since the use of observed weighting coefficients as estimates for latent weighting coefficients does not provide exact estimates, the work presented in chapter 4 will give a reformulation of the Observed Random Predictor Quantal Response Model (2.2) in terms of parameters from the Latent Random Predictor Quantal Response Model and parameters describing errors of measurement. The associated maximum likelihood estimation procedures which allow the estimation of the latent weighting coefficients from the observed data will also be presented. CHAPTER 4 Section A: Introduction The work in chapter 3 consisted of a theoretical, analytical comparison of the weighting coefficients from two quantal response models. In the Observed Random Predictor Quantal Response Model (2.2) the vectors of weighting coefficients are defined in terms of the variances of the observed predictors which include the error variances. In the Latent Random Predictor Quantal Response Model (2.19) the vectors of weighting coefficients are defined in terms of the variances of the latent predictors which include no error variance. The relationship between the two models is provided by the classical measurement model (2.22). The relationships of interest between the predictors and the criterion are given by the vectors of latent weighting coefficients from (2.19). However, most variables encountered in practice which are reasonable candidates for use as predictor variables contain some errors of measurement. Thus the model (2.19) based on the avail- ability of predictors with no errors of measurement will not typically be applicable. Hence the estimation of the latent weighting coeffi- cients must come from the model (2.2) for observed predictors. For the work to be presented below the Observed Random Pre- dictor Quantal Response Model (2.2) will be reformulated in terms of 156 157 parameters from the Latent Random Predictor Quantal Response Model (2.19), and parameters describing errors of measurement. Then the maximum likelihood procedures associated with the reformulated model for estimating the latent parameters will be described. The estimates of the parameters of the Latent Random Predictor Quantal Response Model can then be used to produce estimates of the vectors of latent weighting coefficients. 158 Section B: Reformulation of the Observed Random Predictor Quantal Response Model The most general case of the Observed Random Predictor Quantal Response Model for J categories of the criterion (J :_2) and p observed predictors (p 3_1) is given by (2.2) and repeated here for convenience. For some category k (k = 1,2,...,J) J Pr{Y = klx} = pk = 1/[1 + jil exp{-(ak.j + fikoj 5)}3 j#k where P. . . __ _l _1 (k)'-1(k)_ (j)'-1(3) dk.j — ln(pk) if x 2 Ex Ex 2 Ex 3 and Ek-j = 2-1(Efik) ’ E;J)) for j f k, j,k = 1,2,...,J. Applying the classical measurement model (2.22) together with some of the properties of the classical measurement model (2.24a) and (2.24b), to (2.2) produces a reformulation of the Observed Random Predictor Quantal Response Model in terms of parameters from the Latent Random Predictor Quantal Response Model (2.19) and parameters describing errors of measurement. For some category k (k = 1,2,...,J) J (4.1) Pr{Y = klg} = Pk = 1/[1 + 3:1 exp{-(ak.j + $le 95)}] j#k where P. . . = _ _J_ _ 1 (k)' 2 -1 (k) _ (j)' 2 -l (J) ak-j ln(pk) 2£HT (0 + W ) ET ET (0 + V ) HT 3 159 and B . = (4 + Y2)-1(gék) - 243’ k0] ) for j # kl jlk = 1'2'ooo'Jo Applying the expanded measurement model (2.26) together with some of the properties of this model, i.e., (2.27a) and (2.27b) to (2.2) produces another reformulation of the Observed Random Predictor Quantal Response Model in terms of parameters of the Latent Random Predictor Quantal Response Model (2.19), parameters describing errors of measurement and parameters allowing for different scales of measure- ment among the observed predictors. This reformulation incorporates the use of replicate observed measurements for each predictor. For some category k (k = 1,2,...,J) J (4.2) Pr{Y = k|§} = Pk = 1/[1 + j:1 exp{-(ak j + §£.j 5)}3 j#k where p. = _ _1_ l (k) , , 2 -l (k) ak-j ln(pk) - §[(AET ) (AQA + W ) (ART ) (Agéj))'(A¢A' + 42)'1(Ag;j))] and _ , 2 -1 (k) _ (j) gk.3 — (A4A + w ) (ART ART ) for j ¢ k, j,k = 1,2,...,J. The work presented below will describe the maximum likelihood procedures associated with the Observed Random Predictor Quantal Response Model for estimating the latent parameters. The term "latent parameters" as used here and in the work which follows includes the parameters from the Latent Random Predictor Quantal Response Model, 160 the parameters describing the errors of measurement and for (4.2) the parameters which indicate a scale factor for each observed measure- ment. The term "observed parameters" includes the elements of 2 and 2:1) (i = l,...,J) as found in expression (2.2) of the Observed Random Predictor Quantal Response Model without application of any measurement model. That is, observed parameters, from Z and 3:1) (1 = 1,2,...,J), represent population variances, covariances and means of the observed predictors with no consideration of latent predictors or errors of measurement. The initial work will determine the conditions for the existence of estimates of the latent parameters and thus will demonstrate the need for (4.2) instead of (4.1) as the reformulation of the model (section C). Then the estimation procedure associated with reformula- tion (4.2) of the model will be described (section D). 161 Section C: Identifiability of the Models for the Covariance Matrix and Vectors of Category Means of the Reformulated Observed Random Predictor Quantal Response Model Before estimation procedures associated with either (4.1) or (4.2) can be described, it is first necessary to determine the con- ditions under which estimates will exist. Consider some model y = f(9) where y and 8 represent matrices of parameters and the elements of y are known to be estimable. Definition (4.3) The parameters of 6 are said to be identifiable if each parameter in 6 can be uniquely defined as a function of parameters of 7. When the parameters of some model are identifiable then the parameters can be estimated. Thus in order to describe estimation procedures for the parameters of the model for Z and Efil) (i = 1,2,...,J) as . . 2 (i) (i) . given in (4.1), X = 0 + W and Ex = ET , or in (4.2), E = AQA' + W2 and 3:1) = A341), it is necessary to show that the latent parameters are identifiable. That is, it is necessary to show that each latent parameter can be expressed uniquely as a func- tion of observed parameters. (i) u NX If the latent parameters for models for Z and (i = 1,2,...,J), whether given as in (4.1) or (4.2), are to be identifiable, definition (4.3) clearly implies that there must be at least as many observed parameters in 2 or g;i) as there are dis- tinct latent parameters in the expression of the model. 162 Thus the approach to determining the identifiability of any model can begin by checking a simple counting condition. If there are at least as many observed parameters in 2 as there are latent parameters in the model for 2 then it is possible, but not guaranteed, that the model may be identified. However, if there are more dis- tinct latent parameters in the expression of the model than there are observed parameters, the model is not identified and thus unique estimates of the latent parameters do not exist. Before proceeding with the detailed examination of identifi- ability for the full models of (4.1) and (4.2) consider two examples. Example 1. Let 2 be a covariance matrix where C‘ “T o o 2 = X1 XIX2 2X2 0 o 2 X X1 X2 t. _J and let the structural model for 2 be 2 = 4 + W2 where 2X2 ZXZ 2x2 F' '3 " T 02 o 02 0 2 T1 TlT 2 E1 4 = 2 and W = 2 a o 0 o l 2 2 2 T T T E . L J t J There are 3 distinct observed parameters in 2, i.e., 2 2 . o , o and 0 Since a = o . There are 5 distinct l -2 2 2 X X XIX X X1 XIX2 latent parameters in the model for 2, i.e., 02 02 o . 02 1' 2' 1 2 T T T T E1 2 . . . and o 2. Thus, Since there are more distinct latent parameters than B observed parameters, this model is not identified and unique estimates of the latent parameters do not exist. 163 Example 2. Let 2 be a covariance matrix where F' “T o o l 2 Z = X XlX ZXZ o 02 XZX1 X2 e. .4 and let the structural model for X be 2 = 4 + W2 where 2x2 2xz 2x2 .7 r- a r02 0 02 0 l . l 2 E T T T 2 ¢ = 2 and W = 2 2X2 0 1 2 to 1 2x2 0 GE L'T T T _j e. .4 ° . . . . 2 There are 3 distinct observed parameters in Z, i.e., 0 1' X 022 and o 1 2. There are also 3 distinct latent parameters in the X X X , 2 2 . . . model for Z, i.e., o 1, o l 2 and 0E. Thus the preliminary counting T T T requirement is satisfied. Now consider whether the latent parameters can be uniquely defined as functions of the observed parameters. r- - r- '1 o o o + o o X1 XlX2 T1 E1 Tsz =+w z = o 02 = o 502 + 02 x2x1 x2 T1T2 Tl L. L. .4 2 2 2 i.e. c 1 = o 1 + o X T o 2 = 8021 + o X T and 0 = o = O . XIX2 XZXl Tsz Therefore, 164 o = o 2 2 TIT XIX 2 o = 2(0 - o ) Tl X1 x2 2 2 2 2 and CE = o 1 - 2(0 1 - o 2) = 20 2 — o l . X X X X X The definition for identifiability (4.3) is satisfied for each latent parameter in the expression of the model for 2. Since the model for Z is identified, estimates of the latent parameters will exist. The two examples above pose two potential models for the same 2, the second of which is identified while the first is not. A more detailed discussion of identifiability of the models for the general covariance matrix of this research will now be presented. Consider a quantal response model with V observed predictors. Then 2, the covariance matrix of observed predictors, assumed homo- geneous across all categories, is a V x V matrix. Since 2 is symmetric, only the lower triangular portion of 2 (including the diagonal) will contain distinct observed parameters. There will be = V(V + 1) 1 + 2 + 3 +...+ V 2 distinct observed parameters in 2. For some model for 2, let r be the number of distinct latent parameters V(V + 1) in the model. If r > 2 then there are more distinct latent parameters in the model for 2 than there are observed parameters V(V + 1) in Z and the model is not identified. If, however, r 5_ 2 the counting condition is satisfied. That is, there are fewer distinct latent predictors in the model for 2 than there are observed para- meters in 2. Thus if each latent parameter can be expressed 165 uniquely as a function of the observed parameters then the model is identified. The question now arises about whether or not the model for X which results from applying the classical measurement model (2.24b), as in (4.1) above, is identified. Recall, if there are p predictors, then 2 is a p x p symmetric matrix. The model for E (2.24b) is: 2 Z = 0 + W PXP PXP PXP where 4 is the covariance matrix of the latent predictors, and W2 is a diagonal matrix of error variances for the p predictors. From example 1 above it was shown that when p = 2, this model for 2 is not identified. The general model for 2 with p pre- dictors is also not identified for any value of p. There are p(p + 1) + 212___ll. observed parameters in 2. There are 2 2 distinct latent parameters in ¢ and p distinct latent parameters in W2 for a total of r = E£E§i_ll.+ p distinct latent parameters in the model for 2. Thus there are more distinct latent parameters than observed parameters so the counting condition is not satisfied. Hence the model for 2 based on the classical measurement model (2.24b) and given in (4.1) is not identified. Consider now the model for 2 based on the expanded measure- ment model (2.27b) and given in (4.2). This model for Z, 2 = A¢A' + W2, allows for the use of replicate measures, i.e., for multiple observed replications for a single latent predictor. This type of replicate measures is what Lord and Novick (1974) call nominally parallel measures. 166 2 In the model for Z, 2 = A 0 A' + W , there are p VXV VxP pxP pXV VXV latent predictors and V observed replications associated with the p V(V + 1) 2 . + 1 . . . meters in 2. There are 2£2§--l- distinct latent parameters in 4, latent predictors by (2.26). Thus there are observed para- V - p distinct latent parameters in A, and V distinct latent para- meters in W2 for a total of r = V - p + 2123:411-+ V distinct latent predictors in the model for Z. Note 1: There are V - p latent predictors in A since each of the V observed replications is assigned a scale factor but at least one observed replication associated with each of the p latent predictors is assigned a scale factor of l which defines the metric of the true score. Note 2: The counting condition will be satisfied when V(V2+ 1) Z.V _ p + p(pz+ 1) + v. Consider the single predictor (p = 1) models. Here assume that there are V observed replications related to the single latent pre- dictor by (2.26). Thus the model for 2 becomes 2 = A a: A' + W2. VxV VXl lxv va V(V + 1) . There are —-—3-———- observed parameters in 2. There are V - l . . . . . 2 distinct latent parameters in A, l latent parameter in ¢, i.e. OT, and V latent parameters in W2 for a total of r = (V - l) + l + V = 2V distinct latent parameters in the model for Z. The counting condi- V(V + 1) tion will be satisfied if 2 3_2V, that is if V 3_3. Therefore there must be at least 3 observed replications of the single latent parameter to satisfy the counting condition. 167 Assume there are a total of K (where K :_3) observed replications for the single latent predictor, i.e., X' = (X1 X2 ... Xx). The model for X is (4.4) 2: = A 4 gy+ ((2 KXK KX1 1X1 lxK KXK where A' = [l A ... A ] lXK 2 k 4 = 0: 1X1 and W2 = diagEo2 02 ... o ] . l 2 K KXK E E E . . , K(K + 1) For this case V = K 3_3. ExpreSSion (4.4) contains -——E;-——- observed parameters in Z and 2K latent parameters in the model for 2 where E-(--I-<--§:'--l‘-)--_>_.2K. since K :_3. Expression (4.4) produces 51531111. simultaneous equations of the form 2 2 for i,j = 1,2,...,K. Solving each of these expressions for latent parameters as functions of the observed parameters produces: (See Appendix C.l for details) OXKxi A. = for i = 2,3,...,K-1 i 0 K 1 X X o XKX2 )‘K=o 2 l 168 2 l K 02 = X X X X1 T 0K2 X X o o o _ o _ szl XKX1 1‘ 1 o E X XKXZ o . o i i l 021 = 021 - X:x X X for i = 2,3,...,K—l E X xle and o 0 K 0 =0 _ XKX2 XXl K K 0 ° 2 E X X Xl Thus a one predictor model is identified if there are at least three observed replications for the single latent predictor. Thus there will exist estimates of the latent parameters. In this model there are K(K + l)/2 observed parameters in Z and r = 2K distinct latent parameters in the model for 2. When K = 3, there are 6 observed parameters and 6 latent parameters and since the model is identified it is said to be "just identified". When K = 4, there are 10 observed parameters and 8 latent parameters and the model is said to be "over-identified". When K > 4 the model will be over—identified. When K = 2, there are 3 observed para- meters and 4 latent parameters and the model is not identified. The model with K = 2 is also said to be "under-identified". Now consider the general model with p predictors (p > 1). It will be shown that in order for the general model with p pre- dictors to be identified eagh_predictor must have at least two observed replications. 169 Suppose there are p predictors. Appendix C.2 demonstrates that there must be at least p + 2 observed replications, i.e., V 3_p + 2, in order for the counting condition to be satisfied. Therefore some one predictor must have at least three observed replica- tions or at least two predictors must have two observed replications. Consider a model with p predictors. Let some predictor i (i = 1,2,...,p) have exactly one observed measurement, i.e., K. = 1. Let each of the other p - l predictors have Kj observed .1 replications where Kj 3_l for j = 1,2,...,p with j # i, such that P V = Z K.m 3_p + 2. “F1 1 1 I I i I I If 5' = [x1 ... XK .... 'Xl :... :Xi ... Xi ] represents W 1 i i l l p the observed replications of the p predictors then the model for Z is (4.5) X = A 0 A' + W2 . VxV pr pxp pr VxV For this model there are V(V + l)/2 observed parameters in 2. There are r = (V — p) + 21231—11-+ V = 2V + BiE§:_ll.‘ distinct latent para- meters in the model. Since V :_p + 2 then V(V + l)/2 3_2V + 21232—11 and the counting condition for identifiability is satisfied. Expression (4.5) produces V(V + l)/2 simultaneous equations of the form Zij = f(A, 0, W2) for i,j = 1,2,...,V. For the pre- dictor with only one observed replication, X1 l' the equation for 170 2 . The parameters Uzi and o i occur together and only in the equa- E T 1 2 2 2 tion for o i (4.6). Therefore a solution for o i apart from o 1 E x1 T 1 as a function of observed parameters will not exist. Thus the definition of identifiability is not satisfied for the model for X as given by (4.5) if even one predictor has only one observed measurement. A more detailed algebraic exploration of this situation is contained in Appendix C.3. Consider now a model with p predictors where each predictor has at least two observed replications, i.e., Ki 3.2 for i = 1,2,...,p. For p > 1, V = 'gl Ki Z_p + 2 and thus the counting condition for 1: identifiability will be satisfied. The model for 2 here has the same appearance as (4.5) only the internal structure differs. (4.7) X=A III A'+‘¥2. VXV VXp po pXV VXV Expression (4.7) produces V(V + l)/2 simultaneous equations of the form Zij = f(A, 4, W2) for i,j = 1,2,...,V. There are + .. r = V - p + p( 2 1) + V = 2V + 2123__ll. latent parameters in the model for 2. Each of these latent parameters can be expressed as a function of the observed parameters of 2. See Appendix C.3 for the details. The results are presented below. The V - p parameters of A are: 0' 2 1 x141. A.=—-—l for j=2,...,K j 0 2 1 l xlx1 (K1 - 1 number of parameters) 171 A. = for i = 2,...,p, j = 2'°°°'Ki P P ( 2 (K - l) = 2 K - (p-l) number of parameters) where K - l + Z Ki - (p-l) II n raw 7< I '0 ll < I 'U The p( + 1) parameters of 4 are: 0 = o . for i 1,2,...,p with i # j - 1,2,...,p U. I (21232.11. number of distinct off-diagonal elements of 0) 1 x2 (1 diagonal element of 4) 0 0' i i i 1 O = x2xl xlxl f . _ 2 i O ' 0r 1- ,ooo’p X 1 X1 (p - 1 number of diagonal elements of 4) where 2125:-ll-+ 1 + p - l = 2123:-—l-. 2 The V parameters of W are: 2 o = o - o (1 element of $2) X 172 N N ...: = _ ____l . ' = o o 1 o for j 2,...,K1 P‘F‘ X P'H- for i = 2,...,p P P -1)+z(1<.-1)=p+zx-p= . l . 1:2 1: where l + (p - l) + (K Thus if eagh_of the p predictors has at least two observed replications then each latent parameter in the model for 2 (4.7) can be expressed as a function of observed parameters of 2. Thus the model (4.7) for X is identified, therefore estimates of the latent parameters will exist. Consider now the model for the mean vector of observed replications for some category i (i = 1,2,...,J), i.e., 3:1). Applying the expanded measurement model (2.26) produces the following (1). model for EX . 173 E;l) = A £41) for some i = 1,2,...,J. X VXl V p le (4.8) . i There are V observed parameters in E; ). There are V - p latent . . i parameters in A and p latent parameters in E; ) for a total of r = V - p + p = V distinct latent parameters in the model. Thus the counting condition for identifiability is satisfied. Because of the special nature of A (2.26), it is clear that A'A will be a diagonal matrix of full rank i.e. F" K j 1 1.2 (4.9) A' A = 1+ 201) 0 . . . 0 pXV VXp i=2 K 2 2‘2 0 1 + Z (1.) . . . 0 . i i=2 . . . . KP L o o ... 1+ 2 QEPJ . i=2 K. i jl Note: A' A will be less than full rank if and only if 2 (A.)=-l, pXV VXp i=1 j for some predictor T (j = 1,2,...,p). This is impossible. Therefore A'A will be of full rank p and thus will possess an inverse, (A'A)-l. Thus it is possible to express 341) as a function of pél) (i = 1,2,...,J) and A, i.e., (4.10) 341) = (A'A)-1A'E;i) for i = 1,2,...,J. Since the latent parameters of A can be expressed as a function of observed parameters in 2 based on the work for 2 above, result (4.10) indicates that the latent parameters in the 174 mean vector for any category can be expressed as functions of ob- served parameters from that same category involving parameters from the covariance matrix and the mean vector for the observed replications. Thus the definition for identifiability is satisfied for E41) (i = 1,2,...,J). (i) Therefore since the models for Z X and B (i = 1,2,...,J) as used in reformulation (4.2) are both identified, it will then be possible to produce estimates for all the latent parameters in the reformulation (4.2). And using appropriate estimates it will be possible to construct an estimate for the latent weighting coeffi— cients. This will be discussed in greater detail in Section D below as part of the description of the maximum likelihood estimation pro- cedures associated with the reformulated model (4.2). Before beginning the discussion on estimation procedures one additional topic relative to identifiability needs to be discussed briefly. Consider some non-identified model which expresses the co- variance matrix 2 in terms of latent predictors, e.g., the model of example 1 above where 2 = 4 + W2 with r- 02 o W r‘02 o T (.02 X1. XlX2 T1 '1‘sz E E = and o 02 o 02 l 2 L XZX X J L TlT2 T2 J L O It was shown above that this model for X is not identified. A question which arises relative to non-identified models such as those in example 1 is whether it is possible to modify or extend the model in some fashion so that the modified or extended model is identified. 175 Two general approaches to the modification of non-identified models to produce an identified model are possible. For convenience, these two approaches will be presented in reference to models for the covariance matrix 2 as considered in this research. The first approach attacks the problem of the non-identified model for Z by attempting to increase the number of observed para- meters in 2 without producing an equivalent increase in the number of latent parameters in the model. This is done by the use of multiple observed replications for each latent predictor. The model for 2 based on the expanded measurement model (2.27b) is an example of the use of this approach. The model for 2 based on the classical measurement model (2.24b) was not identified. By the appropriate inclusion of replicate measures an identified model (2.27b) for X was produced. As noted above, it is not sufficient to indiscriminately include enough replicate measures to satisfy only the counting condition for identifiability. The pattern of replicate measures to be included in order to achieve identifiability of the model is crucial. Since this approach was discussed in detail above for models for X it will not be pursued further here. The second approach attacks the problem of the non-identified model for Z by attempting to reduce the number of distinct latent parameters in the model for 2. This is done by introducing con- straints on the latent parameters. The process for introducing con- straints on the latent parameters is to require that one or more of the latent parameters be given as unique functions of other latent 176 parameters, thus reducing the number of distinct parameters in the model. Typically the constraints involve requiring two or more latent parameters to have the same value. Example 2 above is an example of the use of constraints on the latent parameters to achieve identifiability. The model for example 2 can be produced from the model for example 1 by introducing the following constraints on some of the latent parameters of the model for example 1: 2 2 2 2 2 let 0 2 = 1/2 a l and CE = o 1 = o 2 . T T E E i.e., r r' ‘o 7 2 o 2 o 1 2 GE 0 T TT 2 4 = 2 and W = 2 . o 1 2 to l 0 CE LTT TJ L. -J By introducing these constraints on the latent parameters a non- identified model is modified into a model which is identified. A word of caution is necessary here. The constraints to be imposed on the latent parameters of a model for 2 should be reason- able in terms of the situation to be analyzed. To introduce con- straints which have no support in the situation merely to produce an algebraically identified model will provoke problems in the interpretation of results. Since the number of possible combinations and types of con- straints can be myriad even in a relatively simple model for 2, further discussion for this approach will center on a few specific 177 forms of constraints which may be reasonable in some situations under analysis. Consider the single predictor situation (p = 1). With only one observed measurement of the single predictor, the only type of constraint which will produce an identified model is if the error variance can be considered to be a known function of the true score variance. This is rather unlikely for most situations and will not be pursued further. Consider the multiple predictor situation (p > 1). Sometimes an identified model can be produced from a non-identified model by introducing constraints among the parameters of W2, i.e., among the parameters describing the errors of measurement. The simplest of this type of constraint assumes that some error variance is equal to some other error variance. An example of a non-identified model for X where this simplest type of constraint among the error variances produces an identified model is a model for 2 similar to that given by (4.5). Recall, for this model there is one predictor i (i = 1,2,...,p) which has exactly one observed measurement (Ki = 1). If each of the other predictors has at least two observed replications (Kj :_2 for j = 1,2,...,0 with j # i) then identifiability can be achieved by . . . 2 2 , imp051ng a constraint of the form 0 i = o m for some m # 1, E1 E1 m = 1,2,...,p and some 1 = 1,2,...,Km. This constraint requires that the error variance associated with the single predictor i (i = 1,2,...,p) is equal to the error variance for some replication 1 (l = 1,2,...,Km) of some other predictor variable m (m = 1,2,...,p with m # l). 178 It is possible to express all latent parameters of this model, 2 except Uzi and Uzi (where a i = 021 + 02.) as functions of ob- T E1 X1 T El served parameters using techniques similar to those used to show that the model (4.7) for 2 was identifiable. This is possible with- out the use of the constraint as long as Kj :_2 for j = 1,2,...,p and j # i. Under the imposition of the constraint the expression for 02m as a function of observed parameters will also provide the E 1 expression for Uzi as a function of observed parameters of 2 since E l 2 2 2 2 i = o m = f(A, 4, W2). Thus 0 i = 021 - o i where 021 can be E1 E2 T X1 El E1 expressed as a function of observed parameters of 2. Therefore with this one simple constraint imposed upon the error variances a non- identified model has been modified into an identified model. A more extreme extension of the imposition of constraints on parameters of W2 occurs when all error variances are constrained to be equal across all V observed replications of the p predictors. This constraint can be expressed as W2 = 0:1 where a: is the common value of all the error variances and I is the identity matrix of rank V. An example of a non-identified model which can be modified into an identified model through the use of the constraint 42 = oéI will be presented. Consider a model with p predictors (p > 1). Let some one of the predictors have two observed replications, i.e., Ki = 2 for some i = 1,2,...,p, while each of the other predictors has precisely one observed measurement, i.e., K. = l for j = 1,2,...,p with 179 P 3 f i. In this case V = 2 K = p + 1. As given without any con- m=l straints, the model is clearly not identified since it does not satisfy the counting condition that V 3.9 + 2 (see Appendix C.2 for details). To reduce the number of distinct latent parameters in the model for 2 let W2 = 0:1 where I is the identity matrix of rank 1 V. The two observed replications for predictor i are noted as X1 1 and X2. The single observed measurement for the other j pre- dictors is noted as Xi ,(j # 1). Thus the model for Z is: 2 (4.11) X = A 4 A' + W VXV VXp po pXV VXV where (‘2 '1 Viv = °x1 1 2 t . Oxle OX2 symme ric 1 1 1 —'; ------ g ------------- g2 ------------------------ 1- i 1 i 2 ° ' i Xlxl Xlxl X1 0 . 0 . o o C . . U i l i 2 i i i L xle X2xl xle X2 .. ............................................... .)— ° 2 UXP l Oxp 2 . . . O p i O p i O C O O p 1x1 1x1 xlxl Xlx2 x1 e. .J 180 (.1 0 . . . 0 0 0 . . 0‘1 A = 0 l . . . 0 0 0 . . . 0 VXp 0 0 . . . l 0 0 . . 0 0 0 . . . A1 0 0 . . . 0 I-- __ __ 2 .................. 1 0 0 . . 0 l 0 . . . 0 0 0 . . . 0 0 l . 0 0 0 . 0 0 0 . . l L. _. r. 2 ‘fi 0 1 T ¢ = o 02 symmetric pxp T2T1 T2 OTlTl oTiT2 . . . 0T1 . . . . 2 o l O 2 . . o i . . o b TPT 'rp T TPT Tp r- -\ 02 E 2 W2 = OE symmetric VXV - -L---.._--__-L-_-_-_§ ................ .. 0 0 . . . GE 1 ..................................... _ 2 0 0 . . . 0 . OE L. .4 181 In this model for Q, (4.11), there are V(V + l)/2 = (p + l)(p + 2)/2 = p(p + l)/2 + p + 1 observed parameters in .2 since V = p + 1. There are p(p + l)/2 latent parameters in ¢, one latent parameter in A (i.e., l2) and one latent parameter in W2 (i.e., 0:). Thus there is a total of r = p(p + 1)/2 + 2 latent parameters in the model for 2. Since p > 1, then p(p + 1)/2 + p + l > p(p + l)/2 + 2 and the counting condition for identifi- ability is satisfied since there are more observed parameters in 2 than there are distinct latent parameters in the model for 2. Expression (4.11) produces p(p + l)/2 + p + 1 simultaneous equations of the form Zij = f(A, ¢, W2). Each of the p(p + l)/2 + 2 distinct latent parameters in the model for 2 can be expressed as functions of observed parameters. See Appendix C.4 for additional de- tails. The results are presented below. The one latent parameter of A is: for some Specified i (i = 1,2,...,p). The p(p + l)/2 latent parameters of ¢ are: a k j = O k j for k f j with k,j = 1,2,...,p T T xlxl (p(p — l)/2 number of off-diagonal elements of ¢). 0' O 2 xgx: xix: o i = a for some specified i (i = 1,2,...,p) T x1X1 2 1 (one diagonal element of ¢) for j = 1,2,...,p with j # i (p - 1 number of diagonal elements of ¢). The single latent parameter of W2 is: o . . o . xixl xlxl 2 2 2 l l l . . . . CE = o i - o for some spec1f1ed l (1 = 1,2,...,p). X1 Xixl 2 1 Where r = l + p(p - l)/2 + l + p - l + l = p(p + l)/2 + 2 number of latent parameters in the model (4.11) for 2. Thus the model (4.11) for Z is identified, since each latent parameter in the model for 2 can be expressed as a function of observed parameters in Z. In the work above the only constraints which were considered involved the parameters of W2, that is, the error variances. These are not the only constraints which are possible for use. It is possible to impose constraints on elements of A or ¢ as well as on elements of W2. It is even possible to impose constraints which in- volve elements of any of the three latent parameter matrices in the model for 2 simultaneously, e.g., A: = 025 = 024 . The major ques- E T 2 tion to be answered though, concerns not what constraints are possible but what constraints are reasonable for the given situation. This criterion of reasonableness should be the first priority in any con- sideration of constraints for a proposed model. The brief work above does not even begin to exhaust the possibilities for the use of constraints to modify models to achieve identifiability. The few examples given were merely to illustrate some of the potential of this approach. 183 Summary for Section C This section has included an examination of the identifiability of models for 2 and 3:1) (i = 1,2,...,J). The model for 2 based on the classical measurement model (2.24b) as included in (4.1) was shown to be not identified. Thus unique estimates for the latent parameters of the model will not exist. However, by the inclusion of multiple observed replications for each predictor (with at least two observed replications for each predictor) the model for 2 based on the expanded measurement model (2.27b) and the model for Efii) (i = 1,2,...,J) based on (2.27a) were shown to be identified. Two approaches to the modification of non-identified models in an attempt to produce identified models were presented. One approach involved the inclusion of replicate observed measurements for the predictors. The other approach involved imposing constraints on the latent parameters of the model. In many situations the most appropriate procedure to modify a non-identified model to produce an identified model will involve a combination of both approaches. That is, include observed replicate measurements and impose constraints on latent parameters of the model. Any model for 2 in terms of latent parameters which is to be used in an estimation procedure should first be examined carefully to ensure that the model is identified. This examination for identifiability should be conducted whether or not observed replica- tions of the predictors are included or whether or not constraints are imposed on the latent parameters. 184 For the remainder of this research, unless otherwise in- dicated, the assumption will be made that all models which involve a structure for 2 have been checked and found to be identified. 185 Section D: Maximum Likelihood Estimation Procedures Associated with the Reformulated Observed Random Predictor Quantal Response Model For this section maximum likelihood estimation procedures associated with the Observed Random Predictor Quantal Response Model (2.2) will be described. In this model the vectors of category means, (i) x (i = 1,2,...,J), and the covariance matrix, 2, have structures given by (2.27a) and (2.27b) based on the application of the expanded measurement model (2.26). Expression (4.2) results from (2.2) when the structures of the parameter matrices are displayed. The models of interest here will be assumed to be identified and thus estimates of the latent parameters in (4.2) will exist. The structure imposed on the parameter matrices by the applica- tion of the expanded measurement model (2.26) is not apparent in the expression of the model given by (2.2). Thus the model (2.2) has the same appearance as the general case model examined by McSweeney and Schmidt (1974). Therefore the derivation of the likelihood function and the logarithm of the likelihood function produced by McSweeney and Schmidt (1974) is appropriate for presentation here. Recall first that in (2.2) g is the V x 1 vector of observed replications for the p predictors which has the structure 5 = A2 + g, from (2.26). For each category of the criterion g is normally distributed with V x 1 mean vector Eéi), where Egi) = géi) (i = 1,2,...,J) from (2.27a), and V x V covariance matrix X which is assumed homogeneous across all categories, where Z = A¢A' + W2 from (2.27b). 186 In order to apply the maximum likelihood estimation procedures associated with reformulation (4.2) of model (2.2) it is necessary to have a random sample of subjects from each category of the criterion with nj subjects from category j (j = 1,2,...,J) of the criterion. Thus there is a total of n subjects from all categories, i.e., J -—<') n = Z n.. Let § 3 i=1 + for the observed replications in category j and Sj represent the represent the V x 1 vector of sample means V x V matrix of sums of squares and cross-product deviations about the respective means for the observed replications in categoty j (j = 1,2,...,J). Based on the presentation in McSweeney and Schmidt (1974, p. 13) the effective part of the logarithm of the likelihood function can be written as: J-l J-l J-l (4.12) lnL' = X n.1np. + (n — Z n )ln(l - E p.) j=1 J J j=1 J j=l J J -51n|z[ -l 2 tr(X-lS) 2 2 . F1 1 J —«j) (j) -1-(j) (j) - E-jil “j(§ - Ex ) z (x - Bx ) . ( ) The maximum likelihood estimators for pj and Ex] are then indicated: :14? for j = 1,2,...,J (4.13) and Qéj) = £13) for j = 1,2,...,J. ~ The procedures presented by McSweeney and Schmidt (1974) for the estimation of 2 will be of little help in determining estimates 187 of the latent parameter matrices (A, ¢ and W2) in the structure for 2. Thus consider the effective part of the logarithm of the likelihood (4.12) for estimating components of Z, i.e., J (4.14:1) lnL" = - 52‘- 1n|z| - i 2 tr(Z-IST) - J J=l 1 J -(j) (j) -1 —(j> (3') -3321an -gx )'Z ()5 '5x ). Consider the last term in (4.14a). Let C = - %-.gl nj(:*j) - géj))'Z-l(g%j) - g;j)). McSweeney and Schmidt (1974) have shown that Q;j) = 3(j) will maximize ln L" and L, the likelihood function. Thus Efij) = 2(j) is the maximum likelihood estimate for Hfij)' Therefore there is no need to continue to include C in the expression for In L", since its contribution to maximizing .(j) = gTj) ln L" occurs for EX , i.e., C = 0. + Note: 8. = anj where Sj is the sample covariance matrix 3 of the V observed replications in category j. J J - + - Note also: 2 tr(£ 1S.) = 2 tr(Z 1n.S.) j=1 J j=l J J J -1 = tr{ 2 (2 n.S )} i=1 3 -1 J = tr{ 2 ( Z n.S.)} j=l J J -l = tr 2 n S { p} = n tr{X-ls } P J where n = Z n. j=1 J 188 and S = 2———————- i.e., Sp is the pooled sample covariance matrix of the observed replications. Therefore (4.14a) can be rewritten as: (4.14b) ln L" = - E-1n|2| - 2-tr{2'ls }. 2 2 p Or when the structure for Z is indicated (4.14b) can be reformulated as: 2 1 (4.14c) ln L" = - g-ln|A¢A' + w - g-tr{(A¢A' + w2)' sp}. The problem now is to find values of A, O and W2 which will maximize In L". Let F = -ln L", thus maximizing In L" is equivalent to minimizing F where F can be written as: (4.15) F = g-ln|A¢A' + wzl + g-tr{(A¢A' + w2)'1 sp}. The values of the elements of A, ¢ and W2 which minimize F and thus maximize 1n L", for the given pooled sample covariance matrix Sp, will be the maximum likelihood estimates of the latent parameter elements of A, ¢ and W2. The problem of minimizing an expression F such as (4.15), which is a function of a covariance matrix X with a given structure, is a common problem encountered in the set of procedures termed Analysis of Covariance Structures (ANCOVST). Wiley, Schmidt, and Bramble (1973) indicate that "Covariance structure analysis is a term used to describe 5 recently developed series of procedures and 189 models which are used for the structural analysis of covariance matrices" (p. 317). Both Jbreskog (1970) and Wiley, Schmidt and Bramble (1973) indicate that the minimization of F as a function of the elements of A, ¢ and W2 in the structure for E can be carried out by an application of the numerical method of Fletcher and Powell (1963). The application of this numerical method requires expressions for the derivatives of F with respect to the elements of each of the latent parameter matrices, A, ¢ and W2. These derivatives are pre- sented by JBreskog (1970) for a more general model of the structure of 2 than that employed in this research. The results presented by Jareskog (1970) for the derivatives of F have been verified by derivations contained in Appendix C.S and are: (4.16a) ii = 0 if Aij = constant 13 2(2-l[£ - S JZ-1A¢).. if A.. = parameter P 1] 13 for i = 1,2,...,V j = llzro-orp 8F 2(A'z'ltz - s Jz‘lA) for i g j (4.16b) a¢ = p ii 13 (A'z'ltz - s Jz'lA).. for i = j P l] for i,j = 1,2,...,p (4.16c) 3F = 2(z'ltz - s JZ-lW).. for i = 1,2,...,v BWii p 11 where W2 = W - T and Wii is the ith diagonal element of W. 190 A numerical approximation procedure is typically needed to produce values of the estimates of the latent parameters in X when ANCOVST procedures are being employed. When a structure is hypo- thesized for 2 such as (2.27b) the standard maximum likelihood estimation procedures will typically not be applicable, since the set of simultaneous equations gained by setting equal to zero the derivatives of F with respect to the elements of the parameter matrices in the structure for X will not, in general, be explicitly solvable. Since the structure being hypothesized for Z for this area of this research, that is, Z = A¢A' + Y2 (2.27b) is completely con- sistent with a special case of the general model presented by J6reskog (1970) and with model (8) presented by Wiley, Schmidt and Bramble (1973), the estimation procedures described in either re- ference (which differ only in minor details) will apply for the model for Z for this research. Thus numerical values for the estimates of each latent para- meter can be produced. That is, the maximum likelihood estimates A, 8 and @2 will exist. As noted above the values of these estimates will be the values which minimize F. The original interest of this chapter was to develop estimates for the latent weighting coefficients, g;.j (j # k, j,k = 1,2,...,J) of the Latent Random Predictor Quantal Response Model (2.19) using estimates of latent parameters from the reformulated Observed Random Predictor Quantal Response Model (4.2). Recall that by a result derived in Appendix A.2 only a base set of J - l vectors of 191 weighting coefficients associated with some arbitrarily selected category need be derived. All other vectors of weighting coeffi- cients can then be produced from linear combinations of vectors of weighting coefficients in the base set. Since any category can be selected to provide the reference for the base set of vectors, select the first category for convenience, that is, the category associated with Y = 1. Therefore, the J - l vectors of weighting coeffi- cients in the base set will have the form: * _ -l (l) _ (j) . _ (4.17) fil'fi - ¢ (ET ET ) for j — 2,3,...,J. * In order to estimate the elements of §1.j, estimates of ¢ (hence ¢-l) and £41) for i = 1,2,...,J are needed. The ANCOVST estimation procedures, applied to ¢, described by Joreskog (1970) or Wiley, Schmidt and Bramble (1973) will produce an estimate for ¢, call it 5. In order to estimate the vectors of latent predictor (1) means, ET for i = 1,2,...,J, recall that (4.10) provides a formula- (i) tion for 341) as a function of A and EX , i.e., 341) = (A'A)-1A'E;J) (i = l,...,J). An estimate of A, call it A, will be available from the ANCOVST estimation procedures applied to E. An estimate of 3&1) (i = l,...,J) was derived by McSweeney and .(i) *1i) Schmidt (1974), that is, EX = X (4.13) where 35(1) is the sample mean of the observed replications in category i (i = l,...,J). (i) .(i) Therefore an estimate of ET , call it HT , can be written as: .(i) _ , -1 .(i) (4.18) ET - (A A) AEX or (i = 1,2,...,J). | > E? > le 192 Thus the estimates of the vectors of weighting coefficients for the base set will have the following formulation: 8‘1 (A'A)’l 1’ A — - (A'A)-1A g(j) 15* K _( él.j z or a‘lmn‘lmg‘“ - g‘j’) (j = 2......» A* (4.19) g1.) where the estimates 5 and A will be produced from the application of ANCOVST numerical approximation procedures to the structure for Z. 193 Section E: Summary The purpose of this chapter was to describe models with their associated estimation procedures which would produce estimates of the latent weighting coefficients from the Latent Random Predictor Quantal Response Model (2.19). Since the variables which are available for use as predictors typically contain errors of measurement, direct application of the Latent Random Predictor Quantal Response Model is not appropriate. In section B two major reformulations of the Observed Random Predictor Quantal Response Model were provided. The reformulation (4.1) is based on the application of the classical measurement model (2.22) while the reformulation (4.2) is based on the expanded measure- ment model (2.26) which allows for multiple observed replications of the predictors. To determine whether or not estimates will exist, the identifiability of various models for Z and 3:1) (1 = 1,2,...,J), as contained in the two formulations, was examined in section C. Since the model for 2 contained in reformulation (4.1) was not identified, no unique estimates of the latent parameters in the model for X can be found. However, the models for Z and Egi) (i = 1,2,...,J) contained in reformulation (4.2) were shown to be identified under several combinations of inclusion of replicate measures and imposition of constraints. Thus the estimation procedures pre- sented in section D were those associated with reformulation (4.2) of the Observed Random Predictor Quantal Response Model. 194 In section D, the estimation procedures described by McSweeney and Schmidt (1974) were shown to provide estimates of the uncondi- tional probability of occurrence of each category, i.e., p. 3 (j = 1,2,...,J), and the vectors of means for the observed replica- tions, i.e., fléj) (j = 1,2,...,J). In order to provide estimates for the elements of the latent parameter matrices, A, ¢ and W2, in the structure for Z, the ANCOVST procedures described by J6reskog (1970) and Wiley, Schmidt and Bramble (1973) are needed. The approach involved in these procedures was outlined in section D. Since pro- duction of the desired estimates of the vectors of weighting coeffi- cients requires the use of components estimated through the applica- tion of ANCOVST procedures and since ANCOVST procedures typically re- quire the use of numerical iteration in the calculation of the maximum likelihood estimates, the use of a computer program is a necessity if values of the estimates are to be produced. Chapter 5 will briefly describe a computer program applying ANCOVST procedures described in chapter 4, which can provide the estimates of a base set of vectors of latent weighting coefficients in the form (4.19). The base set of latent weighting coefficients will be the set associated with category Y = 1. Also, included in chapter 5 will be an illustration of the application of the program. CHAPTER 5 Section A: Introduction The conclusion from chapter 3 indicated that when errors of measurement are present in the predictors, the observed weighting coefficient will not provide an exact estimate of the latent weighting coefficient for a great variety of situations. The relationship of interest for the quantal response analysis being considered in this research is given by the latent weighting coefficient, but the only observable data typically available for use as predictors contains errors of measurement. Therefore chapter 4 described a reformulated model (4.2) and its associated maximum likelihood estimation pro- cedures which would allow for the estimation of the latent weighting coefficients based on observed data. However these maximum likeli- hood estimation procedures for elements of the model for 2 belong to a set of procedures (ANCOVST = Analysis of Covariance Structures) which typically require the use of a computer program to provide the numerical iteration procedures needed to produce the estimates of the elements of the model. The purpose of this chapter is to describe, briefly, a computer program which can provide estimates of the elements of the structural model for X and using these estimates then provide estimates for the latent weighting coefficients. 195 196 In addition to the program description two examples will be presented to illustrate various estimates of the latent weighting co- efficients, in particular, the maximum likelihood estimates produced by the computer program. 197 Section B: The Computer Program (TQUANER) The major task in producing estimates of the latent weighting coefficients from an identified model is to produce estimates of the covariance matrix of the latent predictors, ¢, and of the scaling parameters, A. The production of estimates for elements of O and A involves the structural analysis of the model for 2, that is ANCOVST. As there are several computer programs already available which allow for a structural analysis of a covariance matrix there was little utility in developing a totally new program. Thus the approach here was to produce a program (TQUANER) for estimating latent weighting coefficients by extensively modifying, specializing and ex- tending one of the existing programs. The program modified to produce TQUANER is ACOVSM: A General Computer Program for Analysis of Covariance Structures Including Generalized Manova by Karl G. Jbreskog, Marielle van Thillo and Gunnar T. Gruvaeus from the Education Testing Service, Princeton, New Jersey. The version of ACOVSM which was modified was the CDC 6500 conversion by Judy Pfaff dated January 1975. Both this version of ACOVSM and TQUANER are Fortran IV programs suitable for use on the CDC 6500 computer at Michigan State University. ACOVSM provides estimation procedures associated with a gen- eral model for 2 described by Jbreskog (1970). It is a more general program with a more complex model for the covariance matrix, 2, than is needed for the quantal response estimation task. And, of course, ACOVSM does not contain the appropriate algebraic manipulations needed 198 to produce the estimated latent weighting coefficients from the appropriate estimated parameters from the structure for 2. Thus the task of producing TQUANER from ACOVSM was two-fold: first, to extensively modify ACOVSM to reduce the complexity of the model being considered by eliminating or bypassing unneeded components and to delete entirely the considerable portion of the program dealing with the generalized MANOVA; and second, to add the programming needed for the input of the sample category means for each observed replica- tion, for the estimation of the latent predictor means and for the production of the estimated weighting coefficients from the appropriate estimates of elements of the models for Z and Eéi) (i = 1,2,...,J). These adjustments not only specialize the program for quantal response analysis but also realize an economy in both operational cost and amount of space occupied by the program in the computer. Since TQUANER is based on ACOVSM, many of the input and output characteristics of ACOVSM were carried over to TQUANER. The only changes made were to facilitate special requirements related to quantal response analysis. Therefore a user who is familiar with ACOVSM should have little difficulty using TQUANER. The information that TQUANER will accept as input consists of descriptions of the component matrices (A, ¢ and W2) of the model for Z, the vectors of sample means for each observed replication in each category of the criterion and the sample covariance matrix. TQUANER provides four options for the input of the sample covariance matrix. The sample covariance matrix for each category can be entered separately or a pooled sample covariance matrix can be entered once. 199 For either of these choices the individual matrix can be entered in rectangular form by rows or in packed form (i.e., as a vector con- sisting only of elements, taken by rows, in the lower triangular portion, including the diagonal). TQUANER will produce and print out the estimates of the para- meters in the model for 2, the estimated vector of latent predictor means for each category and the base set of vectors of weighting coefficients associated with the category identified as the first category by the user. Various options, most of them also common to ACOVSM, allow the user to request additional printed or punched output. Among these options are included the technical output which describes the behavior of the iterative procedure in the covariance structures analysis, the matrix of residuals for E and an option which allows punch card output of the final solution for the estimates of the elements of the parameter matrices in the model for Z, i.e., A, 5, and @. 200 Section C: Two Examples The purpose of this section is to illustrate the use of the computer program, TQUANER, in the productionxxfestimates of the para- meters in the model for Z and of the latent weighting coefficients. Estimates of the latent weighting coefficients from two other sources will also be identified and derived. It is important to note at the outset the limitations of the interpretations which can be drawn from this presentation. Only two specific and similar situations were selected. For each of these two situations two random samples were generated for each category. One sample included fifty (50) subjects per category and the second sample included three hundred (300) subjects per category. Thus the range of situations and samples is far too narrow to generalize the results pre- sented below beyond the situations involved in the examples. These examples are provided solely as illustrations of the use of TQUANER and two other procedures for producing estimates for the latent para- meters in the model for 2 and for the latent weighting coefficients. It is well beyond the scope of this research to provide a definitive study of the properties of these estimates across even a representative sample of situations. The results from these examples may, however, suggest directions for further study. Before presenting the two examples a general description of the procedure used to develop each of the examples will be presented. Each example is a special case of the simplest multiple predictor quantal response model, i.e., the criterion has two categories and there are 201 two latent predictors each with two observed replications for a total of four observed variables. The latent predictors are denoted as T1 and T2 where xi and x: are the observed replications associated . l 2 2 . . 2 . . Wlth T and, X1 and X2 are assoCiated with T . This 18 a special case of model (4.7) from chapter 4 and thus is identified. The situation for each example was selected first. This * * involved selecting values for b 1 = b6 (defined by (3.6b)), d = dg T (3.6b), the value of correlation between latent predictors, pg“, and the reliability coefficients for each predictor. The reliability coefficients are noted as pxx and pyy where the selected values are used solely to identify the ratio between the true variance and observed variance for the first observed replication associated with . . 1 2 . . each latent predictor, i.e., X and X . The ratios of true variance 1 l to observed variance for the second observed replications, x: and x: are selected to be nearly the same, but not identical, to the ratios for the corresponding first observed replication. It is also necessary to select values for the latent predictor variances, the vector of latent predictor means for one of the categories and the scaling factors (elements of A). All other parameter values can then be calculated to provide values of the population latent parameters (1) for each element in the models for 2 and R (i = 1,2) as well x as population values for the parameters of Z and 3:1) (i = 1,2). Using the population values of the parameters of Z and (i) x (i = 1,2) two random samples of size 50 and 300 were generated for each category from a multivariate normal distribution with mean (1) vector Ex (i = 1,2) and covariance matrix 2. A data generation 202 program, GENDATA, developed and checked by Verda Scheifley for use with her doctoral research was used to generate the samples. The sample vectors of category means gfi) (i = 1,2) and covariance matrices Si (i = 1,2) were then entered into TQUANER as data. TQUANER produced estimates for the elements of A, ¢ and y (where W - W = W2) as well as estimates for the latent weighting coefficients. The population values of the elements of A, ¢ and w2 will be displayed along with the values of two estimates for each element. The values of the maximum likelihood estimates from TQUANER for each sample will be displayed as well as the values of the heuristic estimates. The values of the heuristic estimates are derived directly from the expressions for the latent parameters as a function of observed parameters produced to show that the model (4.7) was identified. To produce the estimate of the latent parameter the observed para- meters in the function will be estimated by their sampled counter- parts. . . 2 . For instance, the expreSSion for o as a function of Tl observed parameters is: 0' 0' l l 2 l 2 X2x1 xlxl o l = o . T X2X1 l 2 . . . 2 .2 . Therefore, the heuristic estimate of o l' o 1' is: T T 6 6 S S l l 2 l l l 2 l X X X a = 2x1 1 1 _ szl xlxl l “ - T oxle szx1 1 2 l 2 203 where S . is some element of the sample covariance matrix Sp )9": i which corresponds to the element 0 of the population covariance xix1m matrix 2: . 1 k These heuristic estimates will, in general, not be the values which maximize the likelihood function (or minimize F (4.15)), that is, the heuristic estimates will typically not be equivalent to the maximum likelihood estimates. If the model is just identified, i.e., there are an equal number of latent parameters in the model to be estimated as there are observed parameters, then the maximum likeli- hood estimates will be equivalent to the heuristic estimates. But when the model is over-identified, i.e., more latent parameters than observed parameters, the maximum likelihood estimates will not equal the heuristic estimates. In this case, the maximum likelihood estimates will generally be more accurate since they incorporate all the observed data simultaneously where the heuristic estimates do not. The advantage of the heuristic estimates is their relative ease of computation. The basic algebraic manipulation to express each latent parameter as a function of observed parameters should be done as part of the determination of identifiability and thus should be available for use in producing the heuristic estimates. Therefore computation of heuristic estimates can be done without the use of a computer program. A question which needs to be pursued is how well the heuristic estimates approximate the parameters they estimate over the total range of situations which typically occur in quantal response analysis. In fact, the more important question may be to determine whether there 204 are recognizable situations where the heuristic estimates will perform nearly as well as the maximum likelihood estimates and thus because Of their relative ease of calculation be a reasonable alternative to the maximum likelihood procedure. Determining the responses to these questions involves work beyond the scope of this research and thus these questions will not be pursued further here. For the weighting coefficients, the population latent weight- ing coefficients for each predictor will be displayed along with the population observed weighting coefficient based on the first observed replication of each predictor only. From each sample the values of three estimates of each latent weighting coefficient will also be displayed. The estimates of the latent weighting coefficients are produced (1) from the maximum likelihood estimates from the computer program, TQUANER, (2) from the heuristic estimates of the latent parameters described above and (3) from the estimated observed weighting coefficients based on the first observed replication of each predictor as derived by McSweeney and Schmidt (1974). As argued above the estimates of the latent weighting coefficients based on the maximum likelihood estimates of the latent parameters can be expected to be the most accurate estimates over the population of all possible samples. Since the estimates of the latent weighting coefficients based on the heuristic estimates of the latent parameters do involve attempts to include errors of measurement it is reasonable to expect that these estimates would be generally more accurate than estimates using the observed weighting 205 coefficients to estimate the latent weighting coefficients especially in situations where the observed predictor variance can be expected to contain a relatively large proportion of error variance (i.e., when one or more predictors have a relatively low reliability). For each of the two examples the situation (i.e., values of * 2 2 (l) (l) l 2 . b , d, p , p , p , o , o , u , u , A and A ) will be 2 T1 xx yy an Tl T T1 T2 2 2 stated. The population parameter values for A, ¢, W2, 2, (1) and p(l) (i = 1,2) will also be given. The values of the vectors of sample means, 2‘1) (i = 1,2), for each category and the pooled sample covariance matrix, Sp, will be given for each sample, as generated. And finally, the various population parameter values and estimates of the latent parameters and latent weighting coefficients will be tabled. Exam 1e 1 For this example the situation is: b = 1, d = 2, = 8, = .7, = +.3 Tl pxx pYY pin 2 2 ' ' 1 2 o 1 = 16, o 2 9, u(:) — 82, (g) = 54, )2 = .7 and )2 = 1.4. T T T T Using these values the remaining population parameter values can be calculated. Thus the values of the population latent parameters are: F‘ -\ A = l 0 4x2 16 3.6 0.7 0 ¢ = 2X2 3.6 9 , 0 l ‘_o 1.44 , 2 w = diag{4.0 2.16 3.857 5.36}, 4X4 206 (1) - 82 and (2) 3 97 ET ‘ 54 RT 78 ° 2x1 2x1 (1) Under the models Bx = from (2.27a) and (2.27b), the values of the population observed para- meters are: From each category of this population one random sample of size 50 and one random sample of size 300 were generated with the following pooled covariance matrix and vectors of observed means: N (i) u A1~T (i = 2 1,2) and Z = A¢A' + W (1) _ u 50/category r' ‘~ 19.15 symmetric = 9.715 8.495 3.435 1.69 10.44 6.105 3.355 9.77 20.905 L. _J r- '5 20.0 symmetric 11.2 10.0 3.6 2.52 12.857 5.04 3.528 12.60 23.0 C. _J r- - 82.01 r.97.o 57.4 and :2) = 67.9 54.0 78.0 75.6 109.2 L- _J \- ._J 207 r- '1 r" '5 ' 82.0 97.33 3"“ = 57.45 and gm = 68.07 53.71 77.60 L_74.51_J x_108.85_J . N = BOO/category P a 20.635 symmetric S = 11.485 10.225 P 4.47 3.29 12.97 L__5.305 4.06 12.325 22.98_J r- -1 r- a 81.64 97.58 g”) — 57.56 and gm = 68.10 54.16 77.97 75.68 108.89 . \— _J \— _J Table 5.1 presents the population parameter values and for each sample, the heuristic and the maximum likelihood (from TQUANER) estimates. ************ Insert Table 5.1 here ************ Table 5.2 presents, for each latent predictor, the population latent weighting coefficient value and the population observed weight- ing coefficient value based solely on the first observed replication of each predictor. These results for the observed weighting coefficient are those that would be derived if the classical measurement model ZCNB mm.v0 mn.mm 00.0m v0.00 00.5 v0.N oo.d 9N.m MH.0H mv.v an.mH NN.H mp. 05Hn> muuaaumm voosadoxwa aseaxm: No.00 00.mm hh.mm mv.00 mm.m mm.N hh.n n0.m mn.0H 6v.v 00.ma 0H.H vn. os~n> mumsfiumm oaumwusom icon I 2. ~ mamaom oo.vo n0.n0n 0v.vv Nm.hm hm.m 00.m mN.n 9H.H vv.m av.n H0.0H 00.a vm. ODHO> ”mafia—Huang coonnauxaa afifiaxmx 0H.mo NM.m0d oh.¢¢ 05.00 mm.m v0.v Nh.n 00.0 0m.m vv.n mh.mH mh.d 0v. osam> ouafiwumm 04064850: iom u 2. H mamsmm .H oamaaxu no“ A~.H u we lawn 05 N i .ouon on now an: aw ow .o>wummo: no: ouuaduuu O 0.05 N 0.50 H 0.vm N 0.Nw H on.m vv 5mm.n mm 0H.N NN o.v Ha 0.0 NN 0.m Hm 0.0a Ha v.H «v s. an usao> umwuunnsw umumauuum ususoum cowuoasmom .0 .< no auuumaduam acoumq may no mouuaduum wouoauuum .H.m wanna. 209 (2.22) were applied to relate each latent predictor to the first observed replication associated with that latent predictor. Under that interpretation the situation for this example falls into joint general category four (G.C. IV (x,y)) from chapter 3. The information from the generic graphs and tables in chapter 3 for G.C. IV (x,y) suggest that the observed weighting coefficient for the observed predictor associated with T1 (observed predictor x in chapter 3 notation) may be a reasonably good estimator of the latent weighting coefficient for T1 for moderately sized positive values of the correlation between predictors, p < l/d but 0 is En' En En not "too close" to l/d here). The same information suggests that (since 0 the observed weighting coefficient for the observed predictor associated with T2 (observed predictor y in chapter 3 notation) will be an underestimate of the latent weighting coefficient for T2. Table 5.2 also contains, for each sample, the three estimates of the latent weighting coefficient described above. To help judge the accuracy of each estimate for the given sample the value in the parenthesis below each estimate is the ratio of the estimate over the population latent weighting coefficient value. As in chapter 3 ratios greater than one represent overestimates, ratios less than one represent underestimates and ratios equal to one represent exact estimates. fi*********** Insert Table 5.2 here ************ 211) a u> co m cod 8 anon o a u u H .u a I nwmonucouwm :w Owumu 05Hm> pounEwumm u osau> ucmumH newuaasmom msHu> 00>uomno COMHMHstm I mfimosucoumm a“ caucus .uouowcwum acouma some by“: coUMwUOmmm newuuuwammu om>uomno uuuwu on» so aawHOm panama .ewm.v hN¢.~ Ammh.. mmN. ms~m> 060804000o0 ucmumq voosaaaxaa asaaxmz .mmm.0 Avno.v .H~v.H. Ammv.av Ammm.. Ammo.v mom.~ mom.a mnm.m Hmm.n ~m~.~ , mvh.a mam.~ me Anom.0 Amo.av Anew.. Aomm.v oxmaa.av namBH.H. 0mm. mov. mmm. mHN. mav. one. Hum. Ha osaw> osam> wsHm> msam> osao> mosam> os~n> uouoacwum uncaowumoou acowoaumoou ucmwuwumooo ucmeowuuooo acowowuuoou powwowuuoou ucwfiowuwwou acouuq ucmumq cm>ummno 060065 60 mangaumm 0:006; «0 60500600 unauaaams 6280:6803 ooumsflumm omuoswuum poonwflwxaq mumswumm nouusuuum oo>uomoo unsung owumfiusmm Esaaxdz ofiumwuswm cofiumHsmom cowuodsmom loom a z. m uwmsum iom u 20 H mamsum .H mamsmxm you mucwwofiuwooo mafiunwwmz mo moumfiwumm 0:6 mmsHm> umuwfimumm .~.m wanna 211 For this example and the given sample of 50 subjects per category the estimated observed weighting coefficient based solely on the first observed replication provides the most accurate estimate of both population latent weighting coefficients, with considerably less accuracy shown by both the heuristic and maximum likelihood estimates of the latent weighting coefficients. For the sample of 300 per category, the estimated observed coefficients only slightly overestimate the corresponding latent coefficient for T1 but provide a rather substantial underestimate of the corresponding latent coefficient for T2. For each latent coefficient, the heuristic estimate is nearly identical to the maximum likelihood estimate. For T1 they both provide slightly less accurate estimates than the estimated observed coefficient. However for T2 they both provide considerably more accurate estimates than the estimated observed coefficient. Considering both latent co- efficients the maximum likelihood and heuristic estimates seem to perform equally well and better than the estimated observed coefficient. One reason for the relatively strong performance of the estimated observed coefficient as an estimate of the latent coeffi- cient here is due to the relationship between the population observed coefficient and the population latent weighting coefficient. Derivation 1 from chapter 3, section C, subsection VI, indicates that when the population observed coefficient is equal to the population latent coefficient for one predictor the population observed coeffi- cient for the other predictor will underestimate the population latent coefficient for that predictor by a factor equal to the reliability 212 of the observed predictor. In this case, the population observed coefficient is nearly identical to the population latent coefficient for T1 and the estimated observed coefficient is also nearly identical to the population latent coefficient. Since the relie ability of the second observed predictor Xi (associated with T2) is not particularly low (pyy = .7) the estimated observed coefficient would not be expected to be too inaccurate in this case. Example 2 involves a similar application of Derivation l, but since the pre- dictor reliabilities are much lower than in this example, the estimated observed coefficient proves to be a poor estimator of the latent coefficient for predictor T2. Example 2 Example 2 involves a situation somewhat similar to the situation examined in example 1 but with lower predictor reliabilities. Thus since a greater proportion of observed variance consists of error variance for this situation than for the situation in example 1 it does not seem likely that the estimates of observed weighting coefficients should provide as accurate estimates of both latent weighting coefficients as in example 1. For this example the situation is: = = = . = . - +. le 4r d 2: pXX 4! pYY SI pgn 4 O 1 = 100, O 2 - 64, “(1) = 80, U(:) = 60, A: = .8 and A: = 1.3. T T T T Using these values the remaining population parameter values can be calculated. Thus the values of the population latent parameters are: 213 r- '5 1 o 100 32 A = 0.8 0 0 = 4x2 32 64 , o 1 0.0 1.3J , w2 = diag{150.0 101.0 64.0 111.84}, 4x4 3(1) = 80 and 342) = 480 T 60 700 . 2Xl ~ 2Xl (1) Under the models Ex (1) 2 = A RT (i = 1,2) and X = AOA' + W (from (2.27a) and (2.27b)), the values of the population observed parameters are: 250.0 symmetric 2 = 80.0 165.0 32.0 25.6 128.0 46.8 37.4 83.2 220’0_, , \— r- H r- a 80 480 (i) _ (2) _ Ex _ 64 and Ex _ 384 60 700 l 78 910 . L. ..J 0. .4 From each category of this population one random sample of size 50 and one random sample of size 300 were generated with the following pooled covariance matrix and vectors of observed means: 214 N = 50/category r- '3 267.455 symmetric S = 92.495 149.47 P 49.61 50.25 153.425 59.805 57.98 96.945 210.075 ; _J r' - r- ‘~ 81.43 479.72 gm = 64.50 and g9) = 382.48 60.42 700.63 78.13 908.95 . K— ..J L. _J N = 300/category r- H 257.595 symmetric S = 77.275 153.64 P 31.27 36.785 133.995 48.32 51.65 91.455 231.585 , g 4 r- - r- 1 79.05 480.32 gm ___. 63.08 and gm = 385.18 59.40 700.36 77.99 911.64 L. _J L. .4 Table 5.3 presents the population parameter values and, for each sample, the heuristic and the maximum likelihood (from TQUANER) estimates. ************ Insert Table 5.3 here ************ .215 00.0mm N0.NNO NN.0Q> 00.Nnh 0.00m N mv.oov m0,amm vN.mmv 00.nwc 0.00v H ANWX no.0m HA.nm MN.V0 00.n0 0.00 N mh.ow 0m.vo b$.Mh vv.mh 0.00 H AHWI mm.00H Hm.om v0.h0 Ha.mm vm.aHH vv vN.0h Hm.vh 0N.0h H0.Mh 0.¢o mm 0h.hm 0h.No 0N.0m oh.mm 0.HOH NN m0.00a Hm.ama 0m.m#a vH.obH 0.0md HA N) mh.mo ma.mm 0N.mm Nv.00 0.vo NN mm.Nn FN.am Nm.om Ho.mv o.Nn AN mm.mw mm.m0 hm.vm Nm.Hm 0.00H HA 0 vv.H mm.H 5H.H HN.H m.H Nv HA.H 0H.H mm. H0.H 0. AN < osau> ouufiwunm msan> wsao> oumeumm os~m> os~u> umauomosm mmxm 63:38.3 0065000 68:39.3 0068003 000 080.03 0.8808 03086.03 fisswxd: owumwuso: essaxuz oaumwuso: noduoasmom Son .. E N 385.0 6... u z. 0 0360.0 m can a .o .< uo nuouusouum unsung Gnu mo moumswumm .m.m ounce B . 0 068% no . u a N H m u AN a .0 .0. N 216 Table 5.4 presents information for example 2 which is compar- able to the information presented in Table 5.2 for example 1. ************ Insert Table 5.4 here ************ It is interesting to note that the population observed weight- ing coefficient for the single observed replication associated with latent predictor T1 is nearly identical to the population latent weighting coefficient for T1 and that the estimated observed weight- ing coefficient is the most accurate estimate of the latent weighting coefficient for T1 across both samples. But also note that the population weighting coefficient for the single observed replication associated with latent predictor T2 is about half the size of the latent weighting coefficient for T2. The estimated observed weighting coefficients are also less than half of the latent weighting coeffi- cients across both samples. This illustrates result (3.52) from chapter 3, that if Bx/B; = 1 'then BY/8; = pyy' Here BX/B; : 1 thus sy/s; : pyy = .5. For the sample of size 50/category, both the heuristic and maximum likelihood estimates associated with T1 have the wrong sign. The heuristic estimate for T2 is the most accurate but only slightly more accurate than the maximum likelihood estimate. For the sample of size BOO/category, the maximum likelihood estimate for T1 is slightly more accurate than the heuristic estimate but for T2 the maximum likelihood estimate is slightly less accurate than the heuristic estimate. 217 . wsau> ucmumH nodumasmom os~m> omumsuumo I mammnucouom a“ Oaumu osam> ucouma nodumasmom ms~m> 00>meno cOAumasmom U I uwmwnucmuam Ga 036.» n .uouowooua ucuuma some sum: vuquUOmno :OMuuofiamou oo>homno umuau on» so aaoaou commas Ahmm.. maa.m .Hm0.0 H00. os~s> unaduwuuooo ucwund voosuaoxaa asauxnz .omm.v Amhv.v .0H0.0 0mm.m mvm.v hNh.m ”mom.. Ammo.av Amam.u. 0mm. m00.H m50.| wsHu> 05Hm> msHm> ucuwowumwou ucowowuuoou ucwwoflwmmou ucmumq oo>ummno woman; no ouMEwumm pouafiwunm pounfidumm voozaamqu owumuusum 5:5«xoz loom u z. N ommeam Ammm.0 .NHv.0 Hom.m nNm.m va.m Na Ammm.nv uxmmn.0 Nvm.u Hon. mmm. as wsam> osam> 03H6> wouUAUwum ucowowumwou ucowofiuuoou unmaoawuooo ucoaowuuwoo acmuoq 060000 00 60>0omno 0:00:6003 ouneaumm couoswuum ucouuq owumausom :ONuMHsQom sawuuasmom low a z. a mamamm .N onmem uow mucoNOMMHmou mcwunmwoz no noumswumm 0:6 mosam> houmsuumm .v.m canon. 218 The result which is reinforced by this example is that the use of estimated observed weighting coefficients as estimates of the latent weighting coefficients in a two predictor model can provide pre- cise estimates for at most one latent weighting coefficient. The other latent weighting coefficient will be underestimated by the estimated observed weighting coefficient by a factor approximately equal to the reliability of the observed predictor, which in this example is rather low (p.yy = .5). In the first example, where the reliability of the second predictor was higher, (pyy = .7), the observed weighting coefficient provided a slightly more accurate estimate across both samples. 219 Section D: Summary This chapter has presented a brief description of a computer program, TQUANER, which can provide estimates of the latent parameters in identified models for Z and E;i) (i = 1,2,...,J) and estimates of the vectors of latent weighting coefficients using the procedures described in chapter 4. Two similar examples were presented in section C of this chapter to illustrate the use of the computer program, TQUANER, to produce estimates from actual data examples. The two examples are insufficient to allow generalization of results and conclusions across a broad category of situations to which quantal response analysis may be applied. Further research is needed to determine and describe the dis- tribution of the maximum likelihood estimates of the latent weighting coefficients for samples of various sizes. Part of this research could include an examination of the relative accuracy and utility of the heuristic estimates since they are relatively easy to calculate, not requiring the use of a computer. Another part of this examination might also focus on the relative accuracy and utility of the estimated observed weighting coefficient, for the most reliable observed replication of each predictor, as an estimate of the latent weighting coefficient. This examination might be restricted to those situations where each of the predictors has a reasonably high reliability since it is clear that problems can exist when even one predictor has a low reliability. But this restriction may not be a major drawback since 220 many situations where quantal response analysis is to be employed may use predictors with reasonably high reliabilities. CHAPTER 6 Section A: Summary and Conclusions In chapter 1, a general discussion of research on various quantitative and qualitative data analysis models indicated that the presence of errors of measurement in the variables under analysis can cause problems in interpretation of data analysis results. The purpose of the research reported here was then identified as expand- ing this previous research to include investigation of the effects of errors of measurement on a quantal response analysis technique. The particular quantal response technique to be examined involves a qualitative criterion with two or more categories and one or more quantitative predictor variables, where the predictor variables are assumed to be random variables possessing a normal distribution (a multivariate normal distribution for two or more predictors). For this particular quantal response technique, the interest is focused on the weighting coefficients associated with each predictor. The weighting coefficient associated with each predictor provides a measure of the relationship between that predictor and the proba- bility of classification into one category of the criterion versus classification into some other category of the criterion. Classifica- tion into categories of the criterion is assumed to be without error. However, the predictors are assumed to be measured with error, where 221 222 the presence of errors of measurement in any predictor is indicated by a reliability coefficient less than one for that predictor. The weighting coefficients of interest for this approach to quantal response analysis are the weighting coefficients derived from the use of the latent (error-free) predictors and given by the Latent Random Predictor Quantal Response Model (2.19) of chapter 2. However, the variables which are available for use as predictors typically contain errors of measurement. Therefore direct application of°a model (2.19) based solely on error-free predictors is not generally appropriate. The Observed Random Predictor Quantal Response Model (2.2) based on predictors which contain errors of measurement, called observed predictors, is also presented in chapter 2. It is this model which is generally appropriate for use. To relate the model (2.2) based on observed predictors (i.e., with errors of measurement) to the model (2.19) based on latent predictors (i.e., with no errors of measurement), two measurement models, (2.2) and (2.26), are presented in chapter 2. Properties of each of the measurement models which are useful in relating the two quantal response models are also pre— sented. In chapter 3, the classical measurement model (2.2) is used to relate the two quantal response models (2.2) and (2.19). The classical measurement model associates one observed predictor with each latent predictor. Thus the weighting coefficient associated with each latent predictor called the latent weighting coefficient for a given category of the criterion has related to it a single weighting coefficient associated with the corresponding observed 223 predictor (called the observed weighting coefficient). The relation- ship between corresponding observed and latent weighting coefficients was examined in chapter 3. Since no generally applicable results of the effects of errors of measurement were found for the most general case of the quantal response model, i.e., for a polychotomous criterion and multiple pre- dictors, the research in chapter 3 focused on one and two predictor models with a polychotomous criterion. For one predictor model, the observed weighting coefficient was shown (3.3) to be an underestimate of the latent weighting co- efficient by a factor equal to the reliability of the single pre- dictor. Thus, the use of a single predictor with low reliability in a quantal response analysis can lead to misinterpretations if the observed weighting coefficient (from (2.2)) is used as an estimate of the latent weighting coefficient (from (2.19)). For two predictor models, no universally applicable result as found for one predictor models was discovered. Since no universally applicable results was found for two predictor models, the approach used in chapter 3 involved the search for general categories of situations where the relationship between the observed and latent weighting coefficients would have a similar pattern for all situa- tions within a general category. As used in chapter 3 a "situation" is completely defined when values sufficient to specify the relation- ship between the observed and latent weighting coefficient are given. For the approach used in chapter 3 then a situation is completely defined for values of the relationship between standardized category 224 mean differences on each of the predictors, and for values of the reliabilities of each predictor. For given situations, values of the relationship between the observed and latent weighting coefficients for each predictor are considered as functions of the correlation be- tween predictors. In chapter 3 four general categories of situations were des- cribed. In addition to the four general categories three special case sets of situations were also described. Although there are some similarities among results for the various categories only a few re- sults are found which apply across all four general categories and the three special case situations. First, when there is no correlation between the two latent predictors then each of the observed weighting coefficients is an underestimate of the corresponding latent weighting coefficient by a factor equal to the reliability of the given observed predictor. This result applies in all cases where the latent weighting coeffi- cient has a non-zero value. This result is comparable to the result for one predictor models, which is not surprising since if there is no relationship between the two latent predictors the value of one latent predictor cannot be expected to influence the value of the latent weighting coefficient of the other latent predictor. The lack of correlation between latent predictors then results in the value of the latent weighting coefficient for each predictor being pro- duced as if that predictor were the only predictor in the model, i.e., almost as if each predictor was included in a one predictor model. See Appendix B.2. section C, for details. agar 225 Second, when the ratio of the slope of the pooled within categories regression line of one predictor on the second predictor over the slope of the between categories line joining the midpoints of the joint distributions of predictors within each category is negative, then both of the observed weighting coefficients are under- estimates of the latent weighting coefficients. See Appendix 8.9 for further details on this interpretation and Derivation 3 under subheading VI of section C in chapter 3 for the proof and an illustrative example. Third, it is not possible for both of the observed weighting coefficients to be overestimates of the corresponding latent weight— ing coefficients for the same situation. That is, at most one ob- served weighting coefficient, in a two predictor model, can over- estimate the corresponding latent weighting coefficient. In addition, Derivation 1 under subheading VI of section C in chapter 3 proves that if the observed weighting coefficient is equal in value to the corresponding latent weighting coefficient for some one predictor then for the other predictor the observed weighting coefficient will be an underestimate of the latent weighting coefficient by a factor equal to the reliability of that observed predictor. The only nearly universal conclusion for the two—predictor model is that for only a few special case situations will the value of either observed predictor precisely equal the value of the corresponding latent predictor. Since the observed weighting coefficient typically does not precisely estimate the corresponding latent weighting coefficient 226 (at least for one and two predictors models), there is some utility in describing procedures for estimating the latent weighting coeffi- cients from observed data. Chapter 4 presents two reformulations of the Observed Random Predictor Quantal Response Model (2.2) in terms of latent parameters, i.e., parameters from the Latent Random Predictor Quantal Response Model (2.19), parameters describing errors of measurement, parameters indicating a relative scale of measure for the observed predictors, and the vectors of latent predictor means for each category of the criterion. The first reformulation (4.1) is based on the applica- tion of the classical measurement model (2.22) while the second re- formulation (4.2) is based on the expanded measurement model (2.26) which includes the use of multiple observed replications associated with each latent predictor. Much of the work in chapter 4 (section C) defines and examines a sufficient condition for the existence of estimates of the latent parameters, primarily from the model for the population covariance matrix from (2.2). The sufficient condition for existence of estimates of the latent parameters is that the elements of a model be identifiable (see expression (4.3) for the definition). If a model is not identified, two approaches were discussed for the modification of the model to produce an identified model, (1) use of replicate observed measurements for each predictor or (2) imposition of con- straints upon the latent parameters of the model. By a careful use of one or both approaches a non-identified model can usually be modified into an identified model. 227 The model for the covariance matrix based on the classical measurement model (2.22) was shown to be non-identified. However, under a variety of conditions involving either imposition of con- straints or the use of replicate observations, or both, the models for the covariance matrix and the vectors of observed predictor means for each category, based on the expanded measurement model (2.26), were shown to be identified. Thus for reformulation (4.2) unique estimates of the latent parameters exist. Maximum likelihood procedures associated with reformulation (4.2) were described in section D of chapter 4. These procedures in- volved the use of covariance structures analysis (ANCOVST). As with most situations where ANCOVST procedures are employed the estimates of the latent parameters in the model for the covariance matrix can- not be expressed as specific functions of the observed data. There- fore the derivation of the values of the maximum likelihood estimates must be accomplished by application of a numerical iteration process. Since ANCOVST procedures typically require the use of a computer pro- gram to perform the necessary numerical iterations, chapter 5 des- cribes a computer program (TQUANER) which was programmed to provide the maximum likelihood estimates for the latent parameters and the latent weighting coefficients. The computer program (TQUANER) is a modification of ACOVSM: A General Computer Program for Analysis of Covariance Structures Including Generalized Manova by Karl G. Joreskog, Marielle van Thillo and Gunnar Gruvaeus. In addition to describing the computer program (TQUANER), chapter 5 provides two simulated data examples to illustrate the use 228 of the program. However, no general conclusions about the distribu- tion or accuracy of the maximum likelihood estimates of the latent weighting coefficients can be drawn from the limited application of TQUANER to the two simulated data examples. This research has shown that errors of measurement in the random predictor variables of a quantal response analySis technique can cause problems in using the observed weighting coefficients as estimates of the latent weighting coefficients which represent the relationships of interest between the error-free predictors and the criterion. A quantal response model based on observed data was pre- sented with its associated estimation procedures with provide for the estimation of the latent weighting coefficients from observed data. And a computer program (TQUANER) which can produce estimates of the latent weighting coefficients was described and its use was illustrated on two simulated data examples. 229 Section B: Recommendations for Further Study I’ One obvious possibility for further study is to attempt to extend the results of chapter 3 to models with more than two pre- dictors. Such an extension is easy to identify but will be difficult to do. Approaches to this extension which were tried with little success involved a general matrix manipulation approach and an algebraic derivation of each individual weighting coefficient in the base set of weighting coefficients. No generally useful results were discovered from the matrix approach. The algebriac derivation approach proved an extremely tedious way to extend the results since there is no guarantee that the patterns of results for three pre- dictor models will extend to four or more predictor models, although at the present, this seems to be the most promising approach to ex- tending the results beyond two predictor models. A second possibility for further study also based on chapter 3 results, is to restrict the examination of the effects of errors of measurement to situations which are likely to occur in practical applications of quantal response analysis. The question of interest here is whether, within the set of typically occurring situations for the application of quantal response analysis, the effects of errors of measurement are sufficiently severe to prevent the use of the ob- served weighting coefficient as an estimate of the latent weighting coefficient. A third possibility for further study is to examine the value of the heuristic estimates of the latent parameters described in chapter 5. The question here is whether the heuristic estimates are 230 reasonable competitors of the maximum likelihood estimates especially considering that the maximum likelihood estimates require the use of a computer program on a reasonably sophisticated computer system while the heuristic estimates can typically be produced using no more than a simple calculation. A fourth possibility to further study is to examine the properties of the sampling distribution of the maximum likelihood estimates of the latent parameters and the latent weighting coeffi- cients. This will probably entail a Monte Carlo simulation study. Four possibilities for further study have been given above. No priority is implied by the given ordering. In fact, some combination of the second and fourth possibilities may be the most fruitful approach for further study. APPENDICES APPENDIX A.1 IDENTIFICATION AND JUSTIFICATION OF THE EXISTENCE OF A BASE SET OF WEIGHTING COEFFICIENTS IN THE MULTIPLE OBSERVED PREDICTOR, POLYCHOTOMOUS CRITERION MODEL Consider any category of the criterion k (k = 1,2,...,J) where l =P = = pk rob{Y klg} J 1+ - + ' x .2 exp{ (Gk.j Ek-j )} 3=1 j#k P. , . , . = _ _J_ _ l_ (k) -1 (k) _ (j) -l (3) where ak-j ln(pk) 2 [EX 2 Ex RX 2 EX 1 _ -1 (k) (j) and ékoj - 2 (Ex EX ). Associated with category k are J-l vectors of weighting . . -l (k) (j) . . . t = - . . coeffiCien s ék-j 2 (Ex EX ) w1th j # k, i e g'k°l' ék-2'°"'~k-(k-l)' gk-(k+l)""'§k°J' Consider any category k' where k' f k, and consider any vector of weighting coefficients associated with category k', 8k, 1 (2 f k', £,k' = 1,2,...,J). Therefore gk'°£ = 2-1(Efik') - R;£)). 231 232 -1. (k') (2) If 1 f kr §k|.2 = 2 (Ex - 0x ) = 2-1(Rék') _ E)(k) + Egk) _ E5(2)) _ X-1(E;k') _ ng)) + z-l(R;k) _ E;£)) _ ‘2-1(B;k) _ ng')) + z-1(E;k) + Egm) = - + . ékok' £k°2¢ Therefore fik'-£ = where fik-l and gk°k' are 8k.g - fik°k' vectors of weighting coefficients associated with category k. _ _ _ -l (k') _ (k) If 1 — k, £k'-£ - §k'-k - 2 (Ex Ex ) -l (k) (k') Therefore gk'-k = -gk-k' where gk-k' is a vector of weight- ing coefficients associated with category k. The above proof shows that given the J-1 vectors of ob- served weighting coefficients for some one category of the criterion k (k = 1,2,...,J), then all vectors of weighting coefficients associated with each of the remaining J-l categories of the criterion can be expressed as combinations of the vectors of weighting coefficients associated with category k. APPENDIX A.2 IDENTIFICATION AND JUSTIFICATION OF THE EXISTENCE OF A BASE SET OF WEIGHTING COEFFICIENTS IN THE MULTIPLE LATENT PREDICTOR, POLYCHOTOMOUS CRITERION MODEL. Consider any category k (k = 1,2,...,J) of the criterion, where P* P b{Y le} l k- r0 - ~ - J 'k *l 1 + .2 exp{-(<1k.j + Ek-j 2)} J=1 j#k * p' I . ’ . = _ _j_ _ l_ (k) -l (k) _ (j) -l (3) where ak-j ln(pk) 2 [RT 4 RT RT 0 ET ] * _ -1 (k) _ (j) and Ek-j — 0 (HT ET ). Associated with category k are J-l vectors of weighting . . * _ -l (k) (j) . . . coeffICients fik-j - 0 (ET - ET ) With 3 # k, i.e. * * * t * 8k.1’ ék-2"°°'ék-(k-l)' 8k.(k+1)""’§k.J' Consider any category k' where k' # k, and consider any * vector of weighting coefficients associated with category k', Ek' g (l # k', 11k. = 1,2,...,J). Therefore fik'ol = 0-1(Bék') - Eé£))° * ... I If I # k, Ek' z = T 1(Eék ) - R;£)) _ -1 (k') (k) (k) (2) 7 ¢ (ET ' ET + ET ' ET ) _ ¢-1(H;k ) _ Bék)) + ¢-1(Rék) _ R1(1)) _ s - g = -6 1(Eék) - Eék )) + 4 1(Bék) - E; )) * * ' -~k-k' + ék'k ' 234 'k * * * * Therefore §k°£ = fik-l - Ek-k' where ékoz and Ek-k' are vectors of weighting coefficients associated with category k. 2 * 'k If ” k' Ekn). gk'-k -1. (k') (k) = (LIT ~T ). 'k * * Therefore 8 = -B where B is a vector of ~k'-k ~k-k' ~k-k' weighting coefficients associated with category k. The above proof shows that given the J-1 vectors of latent weighting coefficients associated with some one category of the criterion k (k = 1,2,...,J), then all vectors of weighting coeffi- cients associated with each of the remaining J-l categories of the criterion can be expressed as combinations of the vectors of weight- ing coefficients associated with category k. APPENDIX B . l A. Development of the Property of Interchangeability of x and y * Consider expression (3.12) for Bx/Bg and expression (3.14) * for By/Bn. A close examination of these expressions shows an identical structure for each expression. To demonstrate the rela- tionship between these expressions, consider some given situation, * i.e. values of and d such that 0. For this pin' pxx, pyy 5 BE 2 (1 - 02 )o (1 - d p p ) * situation 8 /B = g” xx 54;” yy takes on some value X (1 - pzp )(1 - do ) Enp xxp YY En call it R. Consider now some second situation p' , p' , p' and En xx YY d' where: E B.l.l ' = ( ) pin pan ' = (8.1.2) pxx pyy (8.1.3) pyY = pxx (8.1.4) d' = d . n E The two situations are not, in general, identical. * Consider now the value of By/Bn for the second situation. (1 2)p' (1 - d' o' 0' ) BY/B; = 0&3 yy n in xx [from (3.14)] 1 _ I I I _ I I ( pgnpxxpyy)(l dnpfin) 2 (1 - D )0 (1 - d O p ) = M3” xx 5 5” YY = R [from (B.1.1)-(8.1.4)J. 1 - _ ‘ ”En yypxx)(1 dapan ’ 235 236 * But Bx/Bg = R for p p , p and d , i.e. the first situa- En’ xx yy 5 tion. Therefore, for an iven situation , and d Y 9 050 Dxx: Dyy E t which produces a given value for Bx/Bg there exists a second, typically different, situation related to the first situation by * expressions (B.l.l)-(B.l.4) which produces the same value for By/Bn * E. A careful consideration of expressions (B.l.l)-(B.l.4) in- as the first situation produces for Bx/B dicates that the right side of each expression can be produced by replacing each x with a y, each y with an x, each 5 with an n and each n with a g in the left side of each expression. In fact if this same procedure of interchanging the x's and y's as well as the 5's and n's were done on expression * * (3.12) for Bx/B , the result would be expression (3.14) for By/Bn. 5 * Thus it is necessary to examine only Bx/Bg' All results * t for Bx/Bg can be translated into results for By/Bn by the use of this property of interchangeability of x and y discussed above. * E B. Justification of the Need to Examine BX/B Only for Values of d > 0. * . Let d be some 5 E < . . . 0) Therefore the Situation 050' pxx' pyy Consider expression (3.12) for Bx/B negative number (dg and dE (dg < 0) produces some value for 2 , (1 - pgnmxx B /B = x E (1 - 02 p 9 En xx yy (1 ) ’ dapanpyy )(1 - a call it R. Spin) 237 Consider now some second situation 0" , p" , p" , and d" En xx yy 5 where: B.l.5 " = - ( ) pin pan (3'1'6) 0xx = 0xx (B.l.7) " = pYY pYY 8.1.8 d" = - d i.e. d" > 0 since d < 0 . ( ) E g ( g , g ) t The value of Bx/BE for this second situation is: (1 _ pII2)pII (l - d" p" D" ) B /8* = in xx 5 En xx x E 2 (1 _ II II II 1 _ a" II pin pxxpyy)( E 060) 2 (1 _ p€n)pxx(l - dgpénpyy) . "2 2 2 = Since 0 = (-o ) = p (l - 02 o p )(1 - d o ) En an En En xx yy 5 En a d d" " = -d - = d n Epfin ( )( pan) can = R. 8* f f But = R or , , and d d < 0 . Thus or an Bx/ g can oxx pyy g ( g ) y situation where dg < 0 there exists a comparable situation related by (B.l.5)-(B.l.8) with dE < 0 which produces the same value for * Bx/Bg' Note that expressions (B.l.5) and (B.l.8) represent the only differences between the two situations. These changes can be ex- * pressed as: the expression for Bx/B with d < 0 is the re- 5 E flection through the line 0&0 = 0 of the expression for Bx/B * E with Id > o. g l 238 C. Summary * Therefore, part A indicates that only Bx/B need be ex- 5 need be examined only for values * E * t f d = d > 0. V 1 es of or val es of 'th d < 0 o E __ a u BY/Bn u Bx/BE wi can be simply stated from results for Bx/B with d :_0 by the use amined and part B indicates Bx/B * E of expressions (B.l.l)-(B.l.4) or (B.l.5)-(B.l.8). Examination of Bx/B A) d = 0 (i.e. d = d = n E C) pgn = 0, D) pxx = pyy = pXX < 1' pYY = l- A.) d = 0 (i.e. d = d n 5 a /0 Recall d = E E an/On (1) (0) = - = ,tht as “a “a O a d = d = an/C” E ag/og not appropriate. 8; (3.6b) which were used since their derivations involve a division of In fact the expressions for Bx APPENDIX 8.2 * for special case situations 5 l/dn is undefined), B) d = dg = 0, l, and E) pxx = l, p < l or YY l/dn is undefined) . Therefore dn (1) __ (0) . pg g . But if ag 0 requires that is - = 0 then * is undefined and expression (3.12) for Bx/BE is (3.llb) and to produce (3.12) are not appropriate a Thus it is 5' * necessary to derive new expressions for Bx and 85 from (3.11a) and (3.6a), respectively, for a5 = 0. Consider (3.11a) B = 1 35.- 32352. x 1 _ 2 02 0x0 ° ny x Y With ai = ax = 0, B = l -3235! x 1 _ 02 chy ° XY 240 Using expressions (3.8) - (3.10) and expressing 8x in terms of latent parameters: a o o 0 (8.2.1) B = 1 - g 5” xx XX 2 0 0 (l - ognoxxpyy) E n Consider (3.6a) 8*: 1 _§-__fl_..§fl. g 1 - 02 02 Gian En > 6 With a = 0 a 0 (3.2.2) 8* = __J;__ - _n_§_rl . g 1 - 02 0500 En * Consider now Bx/Bg formed from (8.2.1) and (B.2.2): (8.2.3) BX/B* = (1 - pg”)pxxpyy . (l - panpxxpyy) Expression (8.2.3) for Bx/B; when ag = 0 will exist if: 1. 050 f :_l (Needed for 0-1 to exist.) Note: 0 VS——E_-, therefore lpxyl §_Ip€nl. xY: pin XX Hence, if # :_1 then pxy # i'l and 2‘1 will ”an exist. 2. a # 0 n * If a = 0 as well as a = 0, then = = 0 and n , 5 8x Bg * by (3.7a) and (3.13a), By =8n = 0. That is, if there is no mean difference between categories on either of the predictor variables then there will be no information gained by the use of the predictor variables. In this 241 case the unconditional probability of classification (assuming no knowledge of the predictors) will be equal to the conditional probability of classification (assuming knowledge of the predictors). pin 5 0 * If pan = 0 and a5 = 0, then Bx = 85 = 0. That is, the observed predictor x contributes no weight to the probability of classification. Any value to the use of the conditional probability of classification over the unconditional probability (i.e. in the use of the pre- dictors) must come solely from predictor y. * g as given by (B.2.3) has the Note: When a5 = O, Bx/B following property: 'k 0 < Bx/BE :_1 With equality only if pxx = pyy = l. * Proof: When ag = 0, expression (B.2.3) for Bx/Bg is: 2 l - * _ ( pan)oxxpyy B /8 - . x g l _ 02 0 En xxpyy 2 4 (1 — p )o o * 1.) Is Bx/Bg > 1? BX/(sg > 1 a 25” "x H > 1 1 - pénpxxpyy 2 0 - > 1 — pxxpyy pfinpxxpyy pgnpxxpyy a > 1 which is im ossible. Oxxpyy P * Therefore 8 /8 f l. x E (1 - 02 )o o * * En xx = = Q = 2.) Is Bx/8€ l? Bx/Bg l .12, l. l - 02 o p in xx yy 242 Using algebra from part 1) above provides: * Bx/BE = l a pxxpyy = 1 a pxx = pyy = l ' 2 (1 -p )o p * * 3.) Is Bx/Bg < 1? Bx/Bg < 1 c: 25” "x H < 1. 1 - pinpxxpyy Using algebra from part 1) above provides: 88* 1 1 1 1 < a < a < or < . x/ g pxxpyy pxx pyy * Therefore Bx/BE :_l for any pxx' pyy when a = 0 with E equality only for pxx = pyy = 1. When aE = 0, both the numerator and denominator of expression (B.2.3) for Bx/BZ are positive. Hence Bx/Bg > 0 for any pxx' pyy when ag = 0. Thus 0 < Bx/B; §_l with equality only if pxx = pyy = l. * To consider BY/Bn where dn = 0, the property of inter- changeability of x and y can be applied to expression (3.12) for * Bx/Bg . Using expressions (3.15a) - (3.15d) and interchanging the x's and y's and the 5's and n's produces an expression for * with d = O: By/Bn n 2 (1 - p )p * (3.2.4) 8/8 = 5” YY . y n (l - 02 o p ) En xx YY This same result can be obtained directly from expression (3.14) with dn = 0. Therefore the conditions for existence of expression (3.14) for By/B; also apply for expression (8.2.4). Since expression (8.2.4) for By/B; represents a special case of expression (3.14) no special considerations are needed for 243 * By/Bn when dn = O (i.e. a = 0). However, when dn = 0 (i.e. E a = 0) where d = d is undefined, then expression (B.2.3) must 5 E be used for Bx/B; instead of (3.12). B. d = d = 0 ’ a a /0 Recall d = d = n n . Therefore d = 0 requires that E ag/og E - p(l) - p(o) = 0 that is, p(l) = u(0). Therefore, the general n n n n n * expression (3.12) for Bx/BE is applicable, that is: 2 , (1 - p )p 'k B /B = in xx hen d = d = O. x g l - 92 o o g in xx yy * To consider BY/Bn when d = d = O, the property of inter- changeability of x and y cannot be applied to expression (3.12) * * for B /B to derive a comparable expression for By/B . When x E a /o n d = d = 0, then a = 0 thus d = -———§- is undefined. The E n n an/on derivation of expression (3.14) for By/B;' which also results from applying the property of interchangeability of x and y to expression (3.12) for Bx/Bg' requires an # 0. However, the property of interchangeability of x and y can be applied to expression (B.2.3), since this expression for * Bx/Bg depends on a5 = O (i.e. an = 0 when the property of interchangeability is applied). Applying the property of inter- * changebability to expression (B.2.3) for Bx/Bg when a5 = 0 pro- * duces the following expression for By/Bn when an = O: * (3.2.5) By/sn = 5” YY xx . 244 Existence conditions comparable to those for expression * (B.2.3) for Bx/Bg' with the property of interchangeability applied, apply for (8.2.5). Thus when d = d = O (i.e. an = 0), expression (3.12) is E * appropriate for Bx/B . But expression (8.2.5) must be used instead 5 * of (3.14) for By/Bn since dn is undefined. C.) pEn = 0 * When pEn = 0, then expression (3.12) for Bx/Bg becomes (1-2) (l-d ) * _ “an pxx pinpyx _ Bx/BE _ (1 - 2 )(1 - d ) - pxx. pinpxxpyy pan This result is identical to the result from the one pre- dictor case. That is, in the two category, two predictor case if there is no correlation between the latent predictors (hence no correlation between the observed predictors either since pxy= pan/S;;_;;) then the observed weighting coefficient will be attenuated by a factor equal to the reliability of the predictor, * i.e. Bx = pxng. Using the property of interchangeability a similar result * and conclusion emerges for By/Bn- * * 8y/Bn - pyy or By - pyan ° These results are not surprising, since if there is no correlation between the predictors then the value and characteristics of one predictor cannot be expected to influence the conditional weighting coefficient of the other predictor. 24S D°) pxx = pyy = 1 For any p e (-1, +1) and pxx = p = 1, that is both in YY predictors are perfectly reliable, then (3.12) becomes * Bx/Bg = 1 that is Bx H m m * 0 And (3.14) becomes /8* 1 h B * = t at is = . BY n Y 8 Again this result is not surprising. Logic suggests that if there are no errors of measurement in either predictor (i.e. 2 2 2 2 = = l or o = o o = 0 or x = = then the observed predictor model (3.4) and the latent predictor model (3.5) are, in fact, identical and thus the observed weighting coefficient for each predictor will be equal to the corresponding latent weight- ing coefficient. E.) pxx = l, pYY < l or pxx < l, pyy = 1 Section D) above examined the case where both predictors are perfectly reliable (i.e. have no errors of measurement). This section examines the case when one predictor is perfectly reliable and the other predictor is fallible (i.e. has a reliability coefficient of less than one) which means that errors of measurement are present in only one of the two predictors. If p = 1, but p < l and p xx yy En # O the expression * (3.12) for Bx/Bg does not simplify. That is, (3.12) becomes 246 2 1 - 1 - d B /B* = ( p€n)( pgnoyx) x g _ 2 _ ° (1 panoxxpyy)(1 dog") This expression has no simple interpretation relative to one even though it represents the ratio of conditional weighting coefficients associated with an error-free predictor. This indicates that, based on the work so far, the observed weighting coefficient is neither a consistent underestimate or overestimate of the latent weighting co- efficient. More work on this special case is included in Chapter 3 of this research. * If however, pyy = l and pxx < 1 the expression for Bx/Bg simplifies to become: * (1 - pgnmxx B /B = for dp # l . x 5 (1 - 2 p ) 5“ pin xx * In this case Bx/B can be shown to have a ratio between E zero and one for all situations where dpan # 1 indicating that the observed weighting coefficient will be an underestimate of the latent weighting coefficient for the fallible predictor. Proof: Let pxx < l, pyy = l, and dpgn # l. * > ? 1. Is Bx/BE l 2 (1 - p )o t s /B > 1 a 5” xx > 1 x g l _ 2 pEnpxx 2 2 e (1 - pan)°xx > 1 - panoxx Q - - pxx pgnpxx > 1 pgnpxx a pxx > 1 which is impossible. Therefore Bx/B; 7 1. 247 2. Is Bx/B; = l? 2 (1 - p )o B /s* = 1 a 5" xx = 1 . x g 1 _ 2 pfinpxx Using algebra from part 1 above produces: * = C3 = Bx/Bg 1 pxx l. 3. Is Bx/B; < l? 2 (l - o )p B /s* < 1 a 5" xx < 1 . x g l _ 02 p in xx Using algebra from part 1 above produces; * Bx/Bg < l e pxx < 1- (8.2.6) Therefore, when pyy = l, pxx < l, and dpgn # 1, then * Bx/Bg < l. * Consider now the relationship of BX/BE to zero. 2 = < = When pyy 1, pxx l, and dpin # 1, Bx/Bg 1 2 - O 0 25h xx ' - < + < < < < Since 1 < pEn l, and O pxx 1, then 0 pan 1 and O < p2 p < 1. Therefore both the numerator and denominator En xx * of Bx/Bg are positive. - , + d EU 6 ( l 1), such that pEn # 1, “k = > ' O < pxx < 1 and pyy l, Bx/Bg 0 [Note. even for (8.2.7) Hence, for any p * pXX ‘ 1' BX/Bg = Dxx > 0.] Combining results (8.2.6) and (8.2.7) produces: 248 (8.2.8) For pyy = 1, O < pxx < l and p n e (-1, +1) such that E * dpfn # 1, then 0 < BX/BE < l. Comparable results for By/B; could be stated using the property of interchangeability for x and y. Therefore in a two category two predictor model with one error-free (pii = l) and one error-ful (pjj < l) predictor the results above and the property of interchangeability of x and y indicate that the ratio of conditional weighting coefficients associated with the error-free predictor (pii = 1) has no simple interpretation while the ratio of conditional weighting coefficients associated with the error-ful predictor (pjj < 1) will be less than one indicating that the observed weighting coefficient will always be an underestimate of the latent weighting coefficient for the error- ful predictor. APPENDIX B.3 ALGEBRAIC EXAMINATIONS OF RELATIONSHIPS BETWEEN EXPRESSIONS NEEDED FOR WORK IN APPENDIX B.4 Relationships among three expressions will be of interest for the algebra to be produced in Appendix 8.4. The three expressions are: 1 - 2 1 - J__= yy)( pxx) . f = pxx( pyy) . 1 x I I 0 2 1 - (1 - o ) x pxxpyy a) Consider ‘r; R l i.e. find the values of p and pyy such XX that: 1) J;'= 40(1-o)(1-p) J;'= l u xx Ayy 2 xx = 1 (1 - pxxoyy) 4oxx(l - pyy)(l - pxx) Q 2 = 1 (1 - ) oxxoyy a4 (1- )(1- )-(1- )2 pxx pyy pxx pxxpyy 2 - - + a 4pxx 4pxxpyy 4pxx 4pyypxx 2 2 _ l - 20xxpyy pxx yy 2 2 2 2 = + - + - + a O pxxpyy 4pXX 4pyypxx prxpyy 4pXX 1 0 u 2 2 Pxx(Pyy - 4ryy + 4) + 29&x(f&y - 2) + l 249 250 2 2 a 0 - - 2 + 2 - + oxx(pyy ) pxx(pyy 2) 1 2 e 0 - (oxx(p - 2) + 1) 9 = pxxmyy - 2) + 1 a (2 - pYY)pXX = a 2pxx - pYprx = 1 ” 9 = 1 Q 29 - 1 - 0 xx 2 - p xx _ xx yy YY 2pxx - 1 c: =0 pxx YY I 1 szx - 1 (8.3.1) Therefore x = 1 e pxx = §—:—S__.e p =.________ yy YY pxx 2) .l < 1 x 4p (1 - D y)(1 - pxx) l < 1 u xx y 2 < 1 X (1 - pxxpyy) 4p (1 - p )(1 - p ) “ xx YY 2 xx < 1 since 0 :_ IX (1 - p p ) xx YY e 40 (1 - p )(1 - p ) < (1 - o o )2 xx yy xx xx yy oxxoyy # l . Using algebra from a) 1) above with appropriate attention to the inequality here produces: < 1 9 O < [p (O - 2) + 132 which is true for all J): xx yy 1 values of pxx’ pyy except for pxx - 2 _ p 2p _ 1 YY (or p .=._321___9. YY 0 xx 251 3) £31 4p (1 - D )(1 - pxx) > 1 a xx yy > 1 ‘Jx 2 (1 - o p ) XX YY Using algebra from parts a) l) and a) 2) above produces: 2 . O I > 1 a 0 > [p (p 2) l] impOSSible Therefore 4oxx(1 - pyy)(1 - pxx) (8.3.2) 0 < 2 < 1 for all values of p , p '_ (l — o p ) —. xx yy xx yy 1 2‘)xx - l with equality to one (1) when p = -——-——- (or p = -——-——-) xx 2 - p yy 0 yy xx and with equality to zero (0) when either pxx = l or pyy = l but oxxpyy # l. b) Consider I; R fx i.e. find pxx and pyy such that 1) ‘J;'= fx 4p (1 - oyy)(1 - oxx) 20 (l - p ) = a xx = xx yy \lx x 2 <1-pp) (1 pxxoyy) xx yy 4 (1- )(1- ) 42(1- ) a pxx pyy 0xx = 0xx 0yy 2 2 (1 - pxxpyy) (1 - pxxpyy) since I; :_O and fx :_0 2 2 e 4oxx(1 - oyy)(l - pxx) — 4pxx(1 - pyy) «no—42(1- )2-4 (1- )(1- ) oxx pyy pxx oyy oxx e O = 4oxx(l - oyy)[oxx(1 - pyy) - (1 - pxx)J e 0 = 40 (1 - p )[o - p - l + oxxJ xx yy xx xxpyy 252 e 0 = 4oxx(l - Dyy)(20xx - pxxoyy - 1) O = 4 - - — . Q oxx(1 pyy)(pxx(2 pyy) 1) (8.3.3a) Therefore ‘I = if 1 - p = O a p = l X X YY YY 2p - 1 or if pxx(2 - p ) - l = 0 e pxx = 3—:l—--e p =-—J%§—-—-. YY pyy' YY xx 2) J;'< fx f 4 1 - 1 - 2 1 - < f “ 0xx( oyy)( pxx) < pxx( 92y) x x (l _ )2 l - pxxp pxxpyy yy 4 (1- )(1- ) 42(1- )2 a pxx 0yy 2 pxx < pxx pyyz (1 - oxxoyy) (l - oxxoyy) since ~IZZO and f :0 2 e 4oxx(1 - oyy)(1 - pxx) < 4pxx(l - oyy) Using algebra from b) 1) above with appropriate attention to the inequality produces < f a < - - _ ‘IX X 0 4pxx(l oyy)(pxx(2 oyy) 1) Therefore (3.3.313) J_< f if p 7‘ 1 and x x yy pxx(2-p)-l>0apxx>—-——¢p < YY 253 3) l > f X x 4p x(1 -p l - [-'> f a x yy)( pxx) > xx yy X X 2 .. ° ( ) 1 p pyy l - pxxpyy Using algebra from b) 1) and b) 2) above produces: (Ix > fx ‘1' 0 > 4Dxx(l - pyy) (pxx(2 - pyy) - 1). Therefore (8.3.3c) ,lx > fx @ pyy # l and 2 - l oxx(2 - p ) - 1 < 0 e o x < §-—l--e p > pxx . YY X pyy YY pxx c) Consider f R l, i.e. find values of p and p such that: x xx yy l) f = l x 20 (1 - p ) fx=lc9 x}: 21:1 pxxpyy 2 -- = - ” pxx(l pyy) pxxpyy @ 2C)xx - pxxpyy = - xxpyy a 2pxx - pxxpyy - = e pxx(2 - pyy) - l = 0. Therefore 20 -1 1 xx (B.3.4a) fX-lQpXX-Z-p @pyy_ p 254 x 1 - pxxpyy e 20xx(1 - pyy) < 1 - o p for 9 xx yy xxpyy # 1' Using algebra from c) l) with appropriate attention to the inequality produces: fx < l w pxx(2 - pyy) - l < 0. Therefore 20 - 1 (8.3.4b) fx < 1 a pxx < §—:l———-e p > ——§§———— . pYY YY pxx 3) f > 1 X x l - pxxpyy Using algebra from c) l) and c) 2) above produces: fx > 1 a pxx(2 - pyy) - l > 0. Therefore 1 prx - l . .4 f -——————- -———————- (8 3 c) x > 1 a pxx > e p < 2 - Dyy YY pxx Combining results from parts a), b) and c) above produces: for (8.3.5) 255 1 —————-—— < < l < 0 < pxx j-2 - p ' O -fx - x -1 YY with the rightmost two equalities occurring only if =_.._l__ 0xx 2 ' [Hence -1 i - < -f i 0.] [Hence -f < -1 < - < 0.] x J x - APPENDIX B . 4 2 Enpxx(l - pyy) - dog (1 - pxx) as a function of p Examination of Q = p n(l - p p ) + xx YY . Identification and determination in of existence conditions of the roots of Q as a function of p €n° Expression (3.17) defines Q as: = 1 - - d l - + 1 - . Q panoxx( pyy) p€n( pxxoyy) ( pxx) Expression (3.20) for Q as a function of pan is: Q - a 2 + b + he e - (l ) xpgn xpgn Cx w r ax - pxx pyy bx = -d(1 - pxxpyy) and cX = (l - pxx). This clearly indicates that Q is a quadratic function of pin' 3 32 Note: —2—-= 2a p + b , Q = 2a . Span x an x 302 x in 2 Since a = p (l - p ) > O, é-2-= 2a > O. x xx yy 302 x in Therefore, Q as a function of pén is concave upward and will possess a minimum value at the point where %§——-= 0 i.e. -bx d(1 - pxxp Y) 5” where pE = §;-= 2 (1 _ Y ) , provided that n x 0xx pyy d(1 - p p ) -1 < xx yy < +1. Zf;x(1 - pyy) 256 257 However, since the concern here is only with the relationship of Q * E (3.18a) - (3.18c) or (3.19a) - (3.19c)) as to zero (thus the relationship of Bx/B to one by expressions DEN varies over its domain, the existence and precise location of a minimum value for Q is of little importance. Since Q is a quadratic function of pEn for fixed values of pxx' pyy and d, Q will possess two roots identified in Chapter -(x) +(x) . . 3 as pEn and pan defined by expreSSions (3.21a) and (3.21b) respectively: d(1- )-\Jd2(1- )2- 4 (1- )(1- ) p-(x) = pxxpyy pxxpyy pxx pyy 0xx in 2pxx(l - pyy) d(1- )+\[d2(l- )2-4 (1- )(1- ) 0+(X) _ pxxpyy pxxpyy pxx pyy pxx En 20xx(l - oyy) - + Using expression (3.20) for Q, p53X) and EQX) can be expressed as: -(X) -bx - bx - 4axcx (8.4.1a) p5n = zax where ax = pxx(l - pyy) - + - =— ... +(x) bx bx 4axcx bx d(1 pxxpyy) (B.4.lb) p = in 2a x c = (l - o ) x xx ° Therefore examining the relationship of Q to 0 will begin -(x) in such that Q = 0, using expression (3.20) for Q. +(x) in by determining the conditions for existence of p and p = + -I- = . Q 0 e a bxp cx O 2 ngn in 258 p-(x) +(x) En in not necessarily in the interval (-1, +1), if b: - 4axcx 3_O. Therefore and p will exist as real numbers but 2 2 2 bx - 4axcx :_0 e d (l - p O ) - 4pxx(1 - p > 0 xx yy - yy)(1 - pxx) 2 > 4pxx(1 - pyy)(1 - pxx) a f - > d (1 - p p )2 or 1 pxxpyy 0 XX YY pxxpyy < 1 o p f 1. XX YY 4p (1 - o )(1 - p ) (3.4.2) b2 - 4a c > o e |d| > X“ W "x s x x x - - (1 _ p p )2 x xx YY [definition by (3.22)] -(x) = p+(x) En En ' To translate this into information about the relationship be- When b2 - 4a c = 0 then p x x x tween Q and 0, ignore temporarily the requirement that values of correlations, pan, need to be in the interval (-1, +1). Thus if Id] :_.J;- (that is, if b: - 4a cx :_0) then there exists p‘(x) x in +(x) -(x) +(x) and pEn where pgn §_p€n . . _ 2 . Since Q — axpgn + bxpEn + cx is concave upward then for -(X) +(x) p€n < pen < can . Q < 0 -(x) f°r pan < pEn Q > O +(X) ”an > “an -(x) and for p = p an in Q = O . +(x) pin pin 259 Now considering the requirement that p e (-1, +1), if in b2 - 4a c > 0, then p-(x) and p+(x) will exist and x x x —- En En -(x) . +(x) 8.4.3a < O for max -1 < < min +1 ( ) Q ( ’pgn ) pin ( .ogn ) (8.4.3b) Q = O for p = p-(X) provided that -l < p-(x) < +1 En En En +(x) . +(x) r = roVided that -l < +1 0 pEn pin p can (8.4.3c) Q > 0 for -l < p < max(-l,p-(x)) En En . +(x) or min +1 < < +1. - + If, however, b2 - 4a c < 0 then neither p (x) nor p (x) x x x an in exist as real numbers, that is, there does not exist any pan such that Q = 0. But since Q is concave upward for all pEn then 2 - .. C: (8.4.4) Q > O for all pin 6 ( 1, +1} when bx 4axcx < 0 Id] < ‘J;: Now consider the conditions under which pgéx) e (-1, +1) or that pzéx) e (-l,+l). The approach to this problem will be to first examine pgéx) when |d| :_ ‘J;Z The first question to be determined -(x) En e (-1, +1) is: for what values of d, p and p will p xx yy -(x) [i.e. pEn e (-1, +1)?]. The second question will then be: for what values of d and will +(x) e (-1 +1) [i e I pXX pYY Dan I . . +(x) e —1 +1 ? . pEn ( , ) 3 Both pgéx) and pgéx) will exist with pgéx) §_p2£X) if and only if |d| > 4DXX(1 - OYX)(1 - pxx) _. 2 ' (l - pxxpyy) 260 4pxx(1 - pyy)(1 - pxx) Let E and J2: (1 - p o )2 xx yy f = 29xx(l - pyx) x 1 - pxxpyy The task here is to find values of d, pxx and pyy such that l) DE:X) e (-1, +1) and 2) pZ;X) e (-1, +1). Consider only situations where pgéx) and 02:0 exist, that is, when |d| _>_ ‘1'; - x 1) pg: ) e (-1, +1)? -(x) En ' a) Consider -l < p 2 2 < d(1-oxxpyy) - Jd (l-pxxp ) -4pxx(l-p )(1-pxx) 2 1 - oxx( pyy) -1 < pgrfix) C9 -1 e -2pxx(1 - pyy) < d(1 - oxxpyy) - / since 2pxx(l - pyy)> O for pyy # l. e o < J < d(1 - p _. xxpyy) + 2pxx(l - p ). YY Therefore (1) O < d(1 - o ) + 20 (l - p ) xxpyy xx yy -20xx(1 - p ) e d > 1 yy = -fx pxxpyy _ + _ and (2) / xx *yy XX 3' l a d > I or d < - l —- (1 _ p p )2 x —- x - x xx yy ~20 (1 - o ) and d 3_ 1 Exp YY = -fx xxpyy and d > -1 . Using the results (8.3.5) from Appendix 8.3 produces: -(x) pin -(x) for p f 1 exists and -1 < pan yy 1 xx 3’2 - p YY if (p <1 if d :_ d“;' d.: J';' or if -1 < d < - J:x 262 1) (cont'd.) -(x) b) ConSider pEn < +1 d(1- )-\Jd2(1- )2—4 (1- )(1— ) p-(X) < +1 a pxxpyy oxxpyy pxx pyy pxx < +1 En 20xx(l - pyy) e d(1 - oxxpyy) - / < 20xx(l - oyy) e d(1 - pxxpyy) - Zoxx(l ~ Dyy) (I) p-(x) < +1 if d(l - p p ) - 20 (1 - p ) < o in XX YY XX YY [since V 3_OJ 2p (1 - ) a d < XX pYX = f 1- oxxoyy -(x) . + - .. _. or (II) p n < 1 if 0 §_d(l pxxpyy) 2pxx(l pyy) < J 1 O < d 1- - 2 1- ( ) __ ( oxxoyy) oxx( pyy) 2oxxfl - p y) a d Z-(l - ) = fx pxprY and (2) d(1 - pxxpyy) - 2pxx(l - pyy) ( J 4342(1-0 p )2+4o (1-p ) -4do (1-0 )(1-0 p xx yy xx xx yy xx yy < d2(1- )2 - 4 (1- )(1- ) pxxpyy pxx pyy 0xx 2 e 0 < -4pxx(1-pyy)(l-pxx) - 4pxx(l-pyy) + 4d (1- )(l- ) pxx p pYY pYY p O < 40xx(l_p )[pxx-l-pxx(l-pyy) + d(l-pxxpyy)J 263 e o < 4pxx(1-pyy)[—(l-pxxoyy) + d(1-pxxoyy)] a O < 4pxx(l-pyy)(l-Dxxpyy)(d - 1) a O < d - 1 for p # 1 YY w d > 1 for l. oyy # Therefore p-(x) will exist and p-(X) < +1 En if |d|.: J2: and either En d < f X or d > 1 and d Z-fx' Using the results (8.3.5) from Appendix 8.3 -(X) -(x) (8.4.5b) exists and < +1 for 1 Dan Dan ' pyy # l . for 0 < p < -———-——- if d > +1 xx -2 - p YY or if d < - - ‘Jx for -——l———-< < 1 'f d < 2 - p pxx —- l - J){ YY or if d :_+ r;' Combining results (8.4.5a) and (8.4.5b) produces: -(x) exists and p-(X) e (-1, +1), for p ¢ 1, 8.4. ( 5C) pan En yy for O < p < -—-l¥——- if d > +1 XX -2 - p YY 264 2) 0;;X) (-1, +1)? a) Consider -l < DZéX). d(1- )+-Jd?(l- )2-4 (1- )(1- ) _1 < p+(x) a _1 < oxxpyy oxxoyy oxx 93y oxx En 20xx(l - pyy) ) + J a — _ _ 20xx(l oyy) < d(1 Dxxpyy Q— _ — — 2pxx(l pyy) d(1 pxxpyy) < / +(x) (I) -l < pin if -2pxx(1-pyy) - d(1-p ) < O xxpyy [Since V > 0] -2p (1 - p ) g d > xx yy = _f l - pxxpyy +(x) . 7-""“ or (II) -1 < 0&0 if 0 5_-2pxx(l-pyy)-d(l-pxxpyy) <, X (l) O j_-20xx(l - pyy) - d(1 - pxxoyy) -20 (l - p ) Q d i. 1 XX Y1, = -f pxxpyy X and (2) -2pxx(l-pyy) - d(1-p p )< V XX yy 2 2 2 2 - + - + - _ e 4oxx(l oyy) d (1 pxxoyy) 4doxx(l oyy)(l oxxoyy) 2 2 < d (l-pxxpyy) - 4pxx(l-pyy)(1-oxx) 2 2 “ 4oxx(1-oyy) + 4doxx(l-pyy)(1-o xxpyy) + 4oxx(l-pyy)(l-oxx) < 0 e 4oxx(l-pyy)[pxx(1-pyy) + d(1-oxxpyy) + l-oxx] < O 265 e 4pxx(1-pyy)[ d(1-oxxoyy) + (l-oxxpyy)] < 0 e 4DXX(1-oyy)(l-pxxpyy)(d + 1) < O h d + 1 < O for p # 1 YY (___—__1 e d < -1' for p # l. ‘—--- YY Therefore p+(x) will exist and -l < p+(x) if p # l and En En YY if Idli \l—x- and either d > -fx or d < -l and d_: -fx. Using results (8.35) from Appendix 8.3 produces: +(x) . +(x) (8.4.6a) eXists and -l < for l pEn pin pyy # for O < < -——l——— if d > 0XX -2 - p - 4:: YY or if d < -l for --lF--< < 1 if d > 2 - p pxx — — \Jx YY or if d §_- IX + b) Consider p (x) < +1 En d(l- ) +\ld2(l )2-4 (1- )(l- ) p+(x) < +1 Q pxxpyy pxxpyy pxx pyy pxx < +1 En 20xx(l - pyy) .. + - e d(1 pxxoyy) V < Zoxx(l oyy) e V < Zoxx(1 - oyy) - d(1 - oxxoyy). 266 2) b) (cont'd.) (1) 0 < 20xx(1 - o ) - d(1 - pxxp ) YY YY 2p (1 - p ) a d < xx YY = f l - pxxpyy x and (2) / < prx(1 - pyy) - d(1 - o ) XXp yy < 2 — _ - f" oxx(1 pyy) d(1 pxxoyy) 2 2 2 2 e d (l-pxxpyy) -4oxx(l-pyy)(l-pxx) < 4pxx(l-pyy) 2 2 + - - - - d (l oxxo ) 4dpxx(1 pyy)(l p ) YY XXOYY 2 2 e 0 < 4oxx(l-pyy)(l-pxx) + 4pxx(l-pyy) - 4doxx(l-pyy)(l-pxxpyy) e 0 < 4p (l-pyy)[1-oxx + p (1-0 )-d(l-p )1 xx xx yy xxpyy e O < 4pxx(l-pyy)[l-pxxp - d(1-pxxoyy)] YY e 0 < 4oxx(1 - pyy)(l - p p )(1 - d) XX YY e O < 1 - d for pyy # l e d < l for pyy # 1. Therefore p+(X) will exist and p+(x) < +1 for p in in YY if |d| 1 J1: and if d < f x and if d < 1. Using the results (8.3.5) from Appendix 8.3 produces: + + (B.4.6b) p (x) exists and p (K) < +1 for p # 1 En En for 0 < p < -——l;——- if d < - l xx—2-pyy — X or if d < - . —- \Jx Combining results (8.4.6a) and (8.4.6b) produces: +(x) . +(x) (8.4.6c) eXists and G (414 +1) for l pEn pEn pYY # l . for O < p < if d < -1 xx—Z-p YY Summary 40 (l - D )(1 - p ) _ If Idl < IX E xx YX1.2 xx then neither OE;X) (l - pxxpyy) +(x) . nor pEn eXist but Q > O for all pEn 5 (-1, +1). [8y (8.4.4).] If ldl > then —' J x p-(x) will exist with p-(x) e (-1, +1), for p g 1 in in YY l . for O < p < -——————- if d > +1 XX‘Z-p YY l for < p < 1 if d > J_- 2 - 0 xx —- —. x YY or if -1 < d < - J x [By (B.4.SC)]. and 268 +(X) . . . Hit) 11 t th -1 +1 for 1 pin wi eXlS W1 pg“ 6 ( , ), pyy 7e 1 . for O < p < ------ if d < --1 xx — 2 - p YY l . for ___

1. d(l- ) -\Jd2(l- )2 - 4 (1- )(1- ) _(x) = oxxpyy Lumpyy on on, pxx «En 2pxx(1 - pyy) l. pEQX) 3_l/d when d > 1? That is, do there exist values of p , p and d > 1 such that p-(x) > l/d? XX YY En '— Let d > 1 and f l. pYY d'l- ) -\[d2(1- )2 - 4 (l- )(l- ) -(x) ) pxxpyy pxxpyy pxx pyy pxx pa 11“” 2 (1- ) 31“ n pxx pyy 9 42(1 - p o ) - d/ > 24 (1 - p ) XX YY -' XX YY since 2 l - > O for l oxx( oyy) oyy # and d > O by definition. 2 e d (1 - pxxpyy) - prx(l - pyy) 3_d/ p‘(X) > l/d for p y 1 if En - yy I.) d2(1 - p ) - 2p (1 - p ) xxpyy xx yy |v O 2pxx(1 - p ) e d2 > 1 yy = f pxxpyy x and II ) d2(l - ) - 2 (1 - ) > d/ ' pxxpyy pxx pyy '— 4 2 2 2 2 e d 1- ) + 4 1 - ) - 4d 1- ) 1- ) ( pxxpyy oxx( pyy pxx( oyy ( pxxoyy 4 2 2 < - - - - —-d (1 pxxpyy) 4d Dxx(l pyyHl pxx) 269 270 1. II.) continued )>0 2 2 2 2 e 4oxx(1-pyy) -4d pxx(l-pyy)(1-pxxoyy)+4d pxxu-pyyHl-pxx _ 2 2 . e 4oxx(1-oyy)[pxx(1-pyy)-d (1-oxxpyy) + d (1-pxx)J.: o 2 2 2 2 e 4oxx(l-pyy)[pxx(1-pyy) - d + d pxxpyy + d - d oxx] :_0 2 )[pxx(1-p ) - d p (l-p )] > o e 4 (1—0 pxx yy xx yy '- YY 2 2 2 a 4pxx(1-pyy) [l-d J 3_O 2 2 2 e l-d > 0 since 4 (l- > O for l _, pxx pyy) pyy # a 1 > d2. -(x) 2 But d > 1 therefore d £_1. Therefore pgn Z_l/d. -(x) En Since (8.4.5c) indicates that for d > 1, p -(x) in from 1) p 2) Thus p < l/d for d > 1 for any px , p (p # l). X YY YY -(x) 6n 5 (-1, +1) for any pxx, p (pyy # l), the algebra results YY -(X) -(X) in I l/d guarantee that p an 1 when d > 1. (pyy # ) will exist and p < d l/ for any pxx, pyy APPENDIX B.6 * * Examination of the arithmetic sign of Bx/Bg and By/Bn. * Consider expression (3.12) for Bx/Bg * sx/sg - where 2 (l - (l - d ) pgnmxx pgnpyy 2 (l - )(1 - d ) pinpxxpyy pEn 2 O < l - < 1, since -1 < < +1 0 < l - 2 p p < 1 since -1 < < +1 ”an xx yy —' ' pan ' + O < pxx :_ 1 0 < < +1. pYY —' * Thus the arithmetic sign of Bx/BE depends solely on the arithmetic sign (B.6. That (8.6. (8.6. (8.6. When of l) is, 2a) 2b) 2c) 1 - d pgnoyy . 1 - dpfin - do 0 5" YY > o e 3 /B* > o l - dogn X E - do 0 g” YY~= o e s /8* = o l- dpén x g 1 - do 9 gm yy * 1 _ dpgn < O a Bx/Bg < 0. Consider the denominator of expression (8.6.1) i.e. 1 - dp 1 - do En > O (dpin < 1), consider the numerator of (8.6.1). 271 Sn 272 If 1 - dp p > 0, (8.6.1) is positive and by (8.6.2a), En yy * Bx/BE > o. If 1 - d = 0, 8.6.1 is zero and b 8.6.2b ognpyy ( ) y ( ). ID "x ID m I C If 1 - do 0, (8.6.1) is negative and by (8.6.2c), anpyy m ’K m m A .0 when l - dpan < O (dpgn > 1), consider the numerator of (8.6.1). If 1 - dp p > 0, (8.6.1) is negative and by (8.6.2c), in YY * B£$€( m If 1 - d = 0, (8.6.1) is zero and b (8.6.2b), pgnflyy Y * Bx/Bg = 0. If 1 - dpgnpyy < 0, (8.6.1) is positive and by (8.6.2a), * > Bx/Bg O. * When 1 - d = 0 (d = 1), neither (8.6.1) nor / are de- pfin pin BX BE fined. Note 1: (B63a) 1 dpp >0 dp <1 a. " C) — En yY in p YY (8 6 3b) 1 - dp p - 0.; do _ _1_. En yy En oyy (B63C) l-dpp —l— ' ' En YY in YY where 3.1 since 0 < p < 1. Y— YY Y Note 2: For an interpretation of expressions (B.6.3a) through (8.6.3c) in terms of ratios of slopes of lines see Appendix 8.9. Note 3: When the numerator of (8.6.1) is zero, then Bx is zero. That is, 1 - dp = 0:: 8 = O. Enpyy x 273 Combining results from above produces: * 1 8.6.4 > 0 'f d < l d > ( a) Bx/Bg 1 can or pgn p YY * l (B.6.4b) B /B < 0 if 1 < dp < x 5 En p YY * 8.6.4c) is undefined if d = l ( Bx/Bg can ( 6 4d) / * O f d 1 f O l 8. . = i = —-- or < < . Bx BE , pin oyy pYY * * In order to compare the distributions of Bx/Bg and By/Bn for common situations, it will be useful to have the properties of * By/Bn expressed in the same parameters as the properties of Bx/B§° * Thus instead of using dn in the expression of properties of By/Bn, l/d (where d = d5) will be used. Note that d = dg in expressions * for Bx/Bg will be replaced by dn when the property of inter- changeability is applied. But since dn = %—-= éy d in expressions 6 * for Bx/Bg will be replaced by l/d in comparable expressions for * By/Bn‘ Using the property of interchangeability expressions (8.6.4a) - * * (8.6.4d) for Bx/Bg become expressions for By/Bn as follows: From (8.6.4a) * p D 8 /B > 0 if -§D-< 1 or -§fl-> 1 . Y n d d o Therefore * (B.6.5a) By/Bn > O for d > 0 if 0 From (8.6. Therefore 4b) for * BY/Bn < 0 [Note: d E_ d , here.] pxx d < 0 if > d or < ”an “in 0 [Note: d 3_ , here.) pxx pg“ if 1 < d < l/pxx . 274 Sn * (B.6.5b) By/Bn < O for d > 0 if d < pEn < d/pxx From (B.6.4c) 'Therefore (8.6.50) IProm (8.6. Therefore (B - 6.5d) 4d) 8* BY/ n * B /B Y n BY/Bn * BY/Bn for d < 0 if d/p < xx is undefined if p in is undefined if p if if p in in /d in = l/pxx for d/p XX /d p < d. in II P < < O pXX for O < p < 1. xx APPENDIX B . 7 COMPARE 1 2 - TO 2 - 1 / pyy oyy /oY Y 1 2p - 1 1.) _ __XX____ ? 2 - o p YY YY 1 2p - l ————=—XY-—ep =(2p -l)(2-p ) Note: 2-p >0 2- Dyy pyy YY YY YY YY and p > O. YY 2 8 D = 49 - 2 - 2 + YY pYY pYY 2 e 2p - 4p + 2 = 0 YY fi 2(p2 - 2p + l) = O YY YY 2 8 2(0 - 1) = 0 YY e = l. pYY Therefore: 2p - l 1 (8.7.1) 2 _ p = y: when p = 1 . YY YY yy 1 2p - l 2.) )L? 2 - o p YY YY 1 2p - l —--—>-—XY———<=p >(2p -1)(2-p ) since 2-p >0 2 - oyy oyy yy yy yy yy and p > O. YY Using algebra results from 1) above produces 1 2022 ' 1 2 _) p C:2(pyy-l) )Océp <1. 2 _ pyy yy YY 275 2p - 1 . (8.7.2) Therefore ——1—-—)—XZ—— for p < l. 2 - p D YY YY YY 2p - 1 3.) 2 1 < p 9 pyy yy 2p - 1 1 yy < cap <(2p -l)(2-p ). 2 _ pyy pyy YY YY YY Using algebra results from 1) above produces: 2p - l l <. oyy 2-0 o a 2(py - 1) < 0 which is never true for YY YY Y an values of . Y pyy (8.7.3) Therefore -—1———-< -—)QL-- is never true for any p . 2 - pyy pyy yy Therefore (B.7-4) ‘——1—-—-> -)QL-- for all p , O < p 2 - o -' p yy yy e ualit onl if q Y Y pyy 1 with [A ll H O APPENDIX B . 8 Comparison of pgéx) to d d(1-o p )+\Jd2(1-o p )2-40 (l-p )(1-p ) +(x) = xxyy xxyy xx yy xx ”an szxu - ow) ' (x) For the purposes of this appendix p2” will not be re- stricted to the range (-1, +1) for the initial algebraic work. Let d :_ Ix, pyy # l. +(x)_ 1) pin — d? +(x) =d 0511 2 2 d(erxxpyy) +\ld (1-pxx3yy) -4pxx(1-pyy)(1-pxx) = 2pxx(l-pyy) ed(l-po)+v’ =2do (l-p) XX YY XX YY for pyy # l e/ =2dp (1-p)-d(1-pp) XX YY XX YY +(x) = d pin if a) 2dpxx(l - pyy) - d(1 - pxxpyy) :_O 1='d[2‘pm{(l-p)--(1-pxxpyy)]>0 YY -’ W dEZpr - prxpyy - 1 + pxxpyy] _ O 8 deoxx - oxxpyy - 1] > 0 278 l) a) (cont'd.) l . a 2_-—p__ : pxx Since d _>_ 0. YY 2 2 d b d 1 - - - - an ) ( pxxp ) 4pxx(1 pyy)(1 pxx) YY 2 2 2 — 4d oxx(l - oyy) 2 2 + d 1 - ( oxxpyy) - 4d29 (1 - o )(1 o o ) XX YY XX YY 2 2 2 2 e 0 = 4d - - - - oxx(1 pyy) 4d pxx(1 pyy)(l pxxpyy + 4oxx(l - pyy)(1 - oxx) r 2 e O - 4pxx(1 - pyy)Ld pxx(1 - Dyy) 2 - d (l - pxxpyy) + (l - pxx)] 2 2 o = 4 - - a p (l pyy)[d pxx d p xx xxpyy 2 2 _ - d + d p o + (1 - o )J xx yy xx 2 e 0 — 4pxx(l - pyy)[-d (l - pxx) + (l - oxx)J _ 2 e 0 - 4oxx(l - oyy)(1 — pxx)(l - d ) a pXX = l or ldl = 1. Therefore p+(x) = d (for d > J-—) in - x (B.8.la) if d = 1 and p > ———1 YY (8.8.ab) or if p xx ) 279 2) pzéx) > d? +(x) > d pEn 2 2 d(1 oxxpyy) d (l pxxp y) 4o 2 (1 - ) pxx oyy “ xx(l-pyyHl-pxx) Using algebra from 1) above produces: +(x) > d e J > 2d 1 - - d 1 - . pin oxx( oyy) ( oxxoyy) +(x) . > d if a 2d 1 - - d 1 - < 0. pin ) pxx( oyy) ( oxxpyy) Using algebra from 1) a) above produces: e d[oxx(2 - oyy) — 1] < O a p <——— for d>0 or if b) J > 2dpxx(l - D ) YY - d(1 - oxxoyy) 3 o 2 - - - > 1) dpxx(l pyy) d(1 oxxoyy) _ o l a p > for d > 0 xx -2 - p ‘- .YY and II) V > 2dp (l - p ) xx yy Using algebra from 1) b) above produces: (B.8.2a) (B.8.2b) 280 2 e 0 > 4pxx(l - oyy)(l - pxx)(1 - d ) e O > 1 - d2 e Idl > 1 Q d > 1 since d :_O by definition. +(x) Therefore > d for d >‘l 1 if 0 < p < 1 xx 2 - p YY . 1 or if -——————-< p < l and d > 1. 2 - p —- xx:— YY +(x) 3 < d? ) pin Using algebra from 1) and 2) above produces: + p(x) O can ) oxx( pyy) ( oxxoyy) __ e dfpxx(2 - pyy) - l] > O l e p > -——————- for d > 0 xx -2 - p YY and b) J < 2dp (1 - p ) - d(1 - p p ). xx yy xx yy Using algebra from 1) b) above produces: 2 e 0 < 4oxx(1 - 0W) (1 - oXXHl - d) 2 9 O < l - d for pxx # l e O < d < l] for pxx # 1. +(X) Therefore < d for , l, d > 051) ( on CW 7‘ _ Ix ) (8.8. (8.8. (8.8. (8.8. (8.8. (8.8. 281 . 1 < 3) if 2 _ p _pxx and O < YY Summary (for d :_ I x' pyy # l) l +(x) < 4a) When 0 pxx < 2 _ , then pgn YY When 1 < p < 1 2 - p - xx YY 4b) for o < d < 1, then pESX) +(x) 4c) for d = l , then p En 4d) for d >1 , then p+(X) En 4e) When p = 1 then +(x) xx ' En . . +(x) Restricting pan to the range (-1, A V +1): [from [from [from [from [from (8.8.2a)] (B.8.3)] (B.8.la)] (B.8.2b)] (8.8.lb)J. i.e. applying results (B.4.6c) from Appendix 8.4, produces adjustments in results (8.8. (8.8. (8.8. (8.8. 4a) - (8.8.4d) as follows: When d > O and l __ oyy # 1 +(x) - + 5a) for O < pxx < 2 _ p y then 0&8 ¢ ( l, l) y r' O < d < J:: then 1 5b) for 2 _ p i-pxx < 1 and4 IX §_d < 1 then YY d.: 1 then K. F +(x) 0 < < d I x pin ¢ ( +(x) SC) for p = l and < d < 1 then p d xx J)<- En +(x) d > 1 th _. en can ¢ ( +(x) pan ¢(-ll+1)l +(x) pfin +(x) ogn ¢( 1,+1) -1, +1) -1, +1). APPENDIX B . 9 An Interpretation of dp as a Ratio of Two Slopes En Recall (from 3.6b) (l) (0) d = d = an/On - (un un )/on E a /0 (l) (0) ’ 5 6 (Ha ha )/°g Therefore a /o a o o / (3,9,1, do =__1__n..p = ”.0 4:151:31, 6n ag/og En ag in o aE/an Consider the regression of E on n within each category i e E = b*(i) + b* n + e i = O 1 . . 5'0 €°n r I * 05 (3.9.2) where b€°n = pan ‘3- is the regression coefficient, T) assumed identical in each category, *(i) . . and bE'O = the constant in the regre551on for category i (i = 0,1). Consider also the midpoint of the bivariate distribution of values of g and n within each category i.e. (uél)' p(l)), i = 0,1. 5 For this two category case the line between the midpoints can be portrayed as: en (» 9‘1 'l‘g) ( b‘ ‘0‘ ) J": J“ 282 283 where the slope of the line between the two midpoints can be de- noted as m and is defined: 5 “£1) ' “£0) a: (8.9.3) mg = “(1) - (O) = a; . n n Therefore using (B.9.2) and (8.9.3) in (3.9.1) produces * = DEnOE/on = b€°n 5” aa/an ma (B.9.4) dp * Thus dp n has been expressed as the ratio of two slopes. b€_n is the slope of the pooled within categories regression line *(i) * . . . E = b€°0 + bg.nn + e (1 = 0,1) and mg is the slope of the line between the midpoints of the distributions of n and 5 within the categories. * Note also expression (2.20) for 85 becomes C* 2 o o * _ ll (1)* (O)* _ 5 En Bg 'TST (bE-O bE-O ) where ® — 2 o O in n 2 2 2 ¢ = o l - l l 05 n( pan) * 2 and C11 — on . 'k * * (3.9.5) B = 1 (hm -b(0) ). g 02(1 - 02 ) g-o 5'0 En Similarly, expression (2.15) for px becomes 1 (l) (l) (i) . . = - = + + (B 9 6) BX 2 2 (bx.0 bx 0) where x bx-O bx-yy 0 (1 - D ) x xy is the regression of x on y for category i (i = 0,1). 284 2 2 2 0 = a Note 1. 0g pxxox' g < o ___pr =I l>l I=l-2<1-02 pan p on pxy pin xY° xxpyy 2 2 2 . . (B.9.7a) Therefore o€(l - pin) < o (l - pxy) i.e. the denominator is less than the denominator 01*XN of expression (8.9.5) for B . 2 of expre551on (B.9.6) for Bx for all values of ca, pgn, pxx and pyy' Note 2: b = b X'Y pW €°n * (B.9.7b) Therefore, lb I < lbg'nl i.e. the magnitude of the slope x-y of the regression line for the observed predictors will be less than the magnitude of the slope of the regression line for the latent predictors. Note 3: a a (B.9.7c) mX = ;§-= ;§-= mg i.e. the slope of the line joining the midpointsyof tge distributions of the observed predictors between the two categories is equal to the slope of the line joining the midpoints of the distributions of the latent pre- dictors between the two categories. Now examine Bx and 85 for various combinations of situations of dpgn = b;.n/mg using a pictorial approach. Let mg > O. Comparable results for mg < O can be found * * easily by considering -b . in place of b€°n. En 285 * b€°n ma < O =idp < O. * 1 Let b < O = ) S-n in Here the within groups slope is negative while the between groups slope is positive. (o\‘ (d. .b .o 36 (l) (0) (l) (O)* (B.9.8a) Here 0 < bx-O bx-O < bg-O bE-O , and thus * * Bx > O and B > 0. Thus the numerator of B is greater 5 E * than the numerator of Bx and the denominator of B is g less than the denominator of BX. * Hence combining (B.9.8a) and (B.9.7a) produces 85 > B > 0: x * 8 * bgon (8.9. b) O < Bx/Bg < 1 when m — dpgn < O. 5 ... * b€°n 2) Let b5.” - 0 = pin — o =» m - o = dpan - o 286 ‘0‘ (d , _ 9;“)i‘) __‘3-q‘° Since pEn - 0 * ‘ 0 both b = O and mu hm. bus 5‘» ’0 t-g'9 5°11 bx... ‘e. . "O ‘I. b = 0. X°y 5? ca 9‘ WW) (1)* (O)* _ (1) (O) (B.9.9a) Here bE-O bg-o — bx-O bx-O > 0. * * Since the numerators of B6 and an are equal and positive * the ratio Bx/Bg formed from (3.9.5) and (B.9.6) when pEn = 0 produces (B.9.9b) <1)* (0)1"< (1) (0) (B.9.10a) Here 0 < bg-O - b£°0 x00 - bx-O' and thus * 85 > O and 8x > 0. Thus both the numerator and denominator * of Bg will be less than their counterparts in Bx' 287 * (B.9.10b) Hence, the relationship of BX/BE to one is not clear * when 0 < b < m (i.e. O < d < 1 . 5'71 5 “an ’ * * b5. 4) Let b = m =’ = 1 a do - 1 ° 5 mg in bt~°.bt-o ’0 t- c“ b"? 4?} ma 5““ f3 5 3' 3' (l)* (O)* _ * _ . (B.9.lla) Here b€°0 - b€°0 — O = Bg - 0, while Bx > O. * I I * o (B.9.llb) Thus Bx/BE is not defined for bi'h - mg (i.e. * d = 1 since = O. can ) BE * * ha.” 5) Let b > m = > 1 =»dp > 1. €°n E mg in * First map out the situation for the latent parameters b€.n and m . Then examine three subcases for b . E x-y 288 (l)* (O)* * 8.9.12a) Here b - b < O =' < O. ( 5-0 a-0 BF. * Subcase a b > m as well as b > m -—-—‘————- X’Y E €°n E * * b = b . Thus b > m = b > m x-y pyy €°n x'y E pyy €°n E * b O =9-5—fl-> (i.e. dp > l/p ). Here the slope of the m p En YY a yy. line x = bgfé + bx ya + e for the observed predictors . _ (i)* * - _ and the slope of the line E - bg-o + b€°nn + e (i - 0,1) for the latent predictors are both larger than the slope of the between categories line (m ). E U3 (a) bmo' ~o“° u I! b;:°(‘;‘° (l)* (O)* (l) _ (O) * O (B.9.12b) Here bg-O - bg-O < bx-O bx-O < O and BE < , (1)* (O)* (l) (0) . Bx < 0. But lbg O - b5.O I > lbx-O - bx-Ol > 0. That is * I the numerator of 85 has a greater magnitude than the numerator of Bx' 289 * Since the denominator of B has a smaller magnitude E than the denominator of 8x, (B.9.7a) together with (B.9.12b) * * produce 8 < Bx < O (and [B a gl > lsxl). Therefore * * (B.9.12c) O < BX/Bg < l for b > b > m (i.e. do > l/py ). €°n X°y 5 5n Y Subcase b b = m ___-___ xoy b* h b b* b = . T us = m = = m X°y pyy €°n X°y E pyy E'n a * 1 bE°n 1 a dpgn = 0 = m = 5— ° W E W That is, even though the slope of the regression line for the latent predictors exceeds the between categories slope, the slope of the regression line for the observed pre- dictors equals the between groups slope. (B 9 12d) Here b(l)* - b(0)* < 0 thus 8* < O and ° ° 5'0 €°0 ' E b(l) - b(0) = 0, thus 8 = O. X'O X'O x 290 * (B.9.12e Hence = O for b = m i.e. d = l . ) ex/Bg x,y g ( “an /pyy) * Subcase c b < m while b > m -————————' X'Y E €°n E * * * o b =p b . Thusb mE =' i n > 1, bx-y < m ='1 < —§—fl'< pl 5 E YY i.e. l < d < l . ( pan /oyy) (B.9.12f) (3.9.129) That is, the slope of the regression line for the ob- served predictors is less than the slope of the between categories line, while the slope of the regression line for the latent predictors is greater than the between categories slope. Here b(l)* - b(0)* < O < b(l) - b(0) g-o g-o x-O x-o ' thus * 85 < O and Bx > O. * * Hence < O for O < b < m < b BX/BE X'Y E €°n (i.e. 1 < dp < l/Dy an Y)' APPENDIX C.1 The model with P = l predictor and V = K 3.3 observed replications is X = A ¢ A' + W2 , KxK KX1 IXl 1XK KXK that is, r- H r- fi ’- 2 , 2 o Symmetric l 0e 1 l 2 2 0x x 0x A2 2 Ce 2 = l ... + l 2 OT AZ AR 2 . . 2 0x x 0x x 0x A L—kl k2 k-J gka C. or q F02 x1 Symmetric 2 O x 0x x21 2 2 0x x 0x x 0x = 3 l 3 2 3 ° 2 0x x 0x x 0x x 0x k l k 2 k 3 ° ° ° k L... ..4 291 292 ,- A 02 + 02 T el 2 2 2 2 A 2 0T A2 0T + 0e2 2 2 2 2 2 A 3 0T A2 A3 OT A3 0 + Ge 0 3 . A 02 A A 02 2 2 2 k T 2 k T A3AkoT ... AkoT + 0e C 1&1 This model for 2 results in K(K + l)/2 equations which re- late the parameters in Z to the 2K parameters in the model for 2 2 2 2 , Z, e.g. o = o + o , o = A o , etc. These equations can be x1 T el x2x1 2 T found by equating the corresponding elements of the two matrices in the equation just above. O 2 x x3 A3AkoT Therefore = -———3—-= A3. In a similar fashion it is xkxl >‘kOT possible to see that o 2 o 2 xkxi AiAkOT _ . xkxz A2)‘k°'r = -—-———-- A. for i = 2,...,k-l and = -—-——-= A . o x A 02 l o x A 02 k xk 1 k T x2 1 2 T Thus expressions for the k-l parameters of A, in terms of para- meters from Z, exist. Consider: This is one of several possible expressions for the single parameter in ¢. Finally consider, 293 0x x O x 2 2 1 xk 1 2 2 2 2 o - o = OT + o - o = 0e ; l xkx2 l l o o 2 2 2 kai xixl 2 2 AiAkOTAiOT 0 - = A o + o - i 0x x l T i A 02 k l k T 2 2 2 2 =A + _ iGT ° . A1% 1 2 . = 0 for i = 2,. .,k-l, ei and 0x x 0x x A A 02 - A 02 2 k 2 k 1 2 2 2 2 k T k T o = A o + o - xk 0x x k T 8k A 0 2 1 2 T 2 2 2 2 = + — AkoT o AkoT k 2 = 0e . k . 2 . Thus expreSSions for the k parameters of ? , in terms of parameters from Z, exist. Since it is possible to express all 2k parameters of the model for Z in terms of parameters in Z, the model (4.4) is identified. APPENDIX C.2 The necessary condition for identifiability will be satisfied when V(V + l) + p(p + 1) > - + 2 _V p 2 v, where V is the total number of observed replications and p is the number of predictors in the model. 2 + V(V 1) > V _ p + p(p + l) + V = 2V _ p + P 2+ 2 2 —' 2 3 = 2V + 23-- E- 2 2 V2 + V :_4V + p2 - p 0 2 2 V - 3V :_p - p . Since each predictor must have at least one observed repre- sentative, then V = p + A where A is the number of observed replications beyond the original representatives (A 3_O). 2 2 V - 3V 3_p - p 8 2 2 (p+A) -3(p+A):P 'P 2 2 8 2 p + 2pA + A - 3p - 3A :_p - p 2 8 2pA + A - 3p - 3A :_-p 8 A2 - 3A :_2p - 2pA = 2p(1 - A). 294 295 For A < l, l - A > O. . A(A-3)> °° 2(1-A)-p’ =A(A - 3) < : ___—__— But A l =1A 0 2(1 _ A) __3_ 2 MIL») But - :_p is impossible since by definition therefore A # 0. For A = l, A(A - 3) -2 and 2p(1 - A) = O. A(A - 3) _>_ 2p(1 - A) n -2 3_O. Impossible. Therefore A # l. . A(A- 3) " 2(l-A) —p' A(A 3) -2 For A = 2, 571—:—XT-= :§-= 1 :_p. This is possible. For A = 2, 2:? : A; = O :_p. This is possible. For A > 3, A - 3 > O and l - A < 0, therefore %%%—E—%%-< O < p. This is possible. Thereforetfluecounting condition for identifiability is satisfied when A :_2 R V :_p + 2, since V = p + A. That is, the counting condition for identifiability is satisfied if there are at least two additional observed replications beyond the original set of p observed measurements. APPENDIX C.3 EXAMINATION OF IDENTIFIABILITY FOR TWO MODELS FOR 2: A) MODEL (4.5) FROM CHAPTER 4, AND B) MODEL (4.7) FROM CHAPTER 4. Although each model, (4.5) and (4.7), has (a similar external appearance their differences will become apparent upon close examina- tion. Consider the model with p latent predictors (p > 1) where each predictor, Ti, has Ki (i = 1,2,...,p) observed replications, and where V = .gl Ki with V being the total number of observed 1: replications (including all predictors). The model for Z is: Z = A Q A' + W2 VXV VXp po pXV VXV where 296 297 r- ‘j A = 1 0 . . . O pr l O . . . A2 0 A: O . . . O 1 0 l . . . O 2 0 A2 . . . O 2 0 AK .. o) 2 0 O . . l P O O . . A2 0 0 AP , L. .1 r- - 2 . ¢ = O l Symmetric PXP T 2 02102 T T T o c 02 1 2 ' TpT TPT Tp t. J and I I l 2 2 2 I 2 2 2 l :2 2 2 W = DIAG o lo 1 ... o 1 t o 2 o 2 . o 2 I .. oEp Opp ... 03p x A I . VV E1132 EK‘ 1:31 E32 EK: , 1 2 l : 2 Thus 2 expressed in terms of parameters of the model becomes: 0“” 298 + “on“ Noog" ”A . . . 0“” D. N D. v< + Not. so. No o O- 0.): Non!- Q‘N '< O ----¢—-----—§—-b--—_—-——-d —)_-——d————-- N a: N! [-i N m g D N >0 + ..4 E" “‘ m “‘ P‘ “L. ”xx ”be “9 oh ON NA 01- ON N“ N O Nx on. ’< Nx N r< 0- " Nx QN 9.x v .< v< "< . ' I I I | O o ‘ t o I . I . I I O NN. N O + "B N NNE-o N [-c N F at" N” O a 0L 0 x N 0.- (MN 0 09 A N” '< NN N o 4 0- 4 N8 NN QN v »< u -< ¢< "< I I NNa-n N O N E" NH N E-c 0“ N N [-a N D“ N E‘ .' D II Ell! 000 o N O N o D: N E-a NN Nx o.N CL" A a: O —< j -) + HM Ill D + a-o "‘ u-O .4 E" H! o :4 DE-0 0 a ab 9 x N N ... Qt— o x” E‘ v-t ~12 H "‘ H O H“ ... K O ax on. ’< Hg .4 .< ... ...4 A O. 4 H2 NN Nx Ax (IN 0.): v 4 »< .< e< "< ' HN o o I o o o ' 0 ND” .0 o u c a o ' HN * ... H . F‘E‘ x H [- Ne. 'E-a 8;, N E4 H N t-o o-l 0% O N in {-0 OE-o o 0 N D HN O "N 4‘ '00 H” E. HN no. .I H” 000 ’< #6! O N o 4 0. —< H HN NN Nx HN (LN 9‘ V H“ v< ,< —< .< H H H [a c-c 5" MN Nae NAB H NE. N9 H [-4 0‘. o o a as o t— at: 0 H N N a!“ o G. n. HN 000 A“ E-ONN N“ DIN 000 K '( D K O .< K o-c N ‘3' H“ M NH NH N); H akN 0*“ x 0.0 ”a x ... x on a 00. 299 V(V + 1) 2 + There are V - p parameters in A, p( 2 1) parameters in ¢ and In this model 2 contains observed parameters. . 2 - V parameters in W for a total of R = 2V + (pz 1) parameters in the model for 2. PART A: For model (4.5) it is assumed that Ki = l for some predictor i = l,...,p and Kj 3.1 for j # i, j = l,...,J such that P 2 Km :_p + 2. m=1 Note: If p < ll 2 and K1 = 1, then the counting condition for identifiability will not be satisfied unless K2 3_3. If K2 = 2 there will be V = 3 total observed replications which gives 6 observed parameters in 2. But there will be 1 para- meter in A, 3 parameters in ¢ and 3 parameters in W2 to be estimated. Since there are 7 latent parameters in the model for X with only 6 observed parameters in Z, the model is not identified. P Since V = 2 K :_p + 2, the counting condition for iden- m=1 tifiability is satisfied. To show that the definition of iden— tifiability is not satisfied, it is sufficient to show that there exists one latent parameter of the model which cannot be expressed as a function of the observed parameters in Z. By examination of the expression for Z in terms of the 2 2 T1 and 0 i will each occur in only one location and they will occur E 1 2 2 2 together, i.e., o i = o i + o .. Since there is only one equation Xl T El 2 2 relating the two unknown latent parameters 0 i and o i to observed T El latent parameters of the model it can be seen that if Ki = 1, O 300 parameters in 2 there will exist no unique solution for either of the latent parameters separately and therefore the model (4.5) with Ki = l for some predictor (i = 1,2,...,p) is not identified. Models which have several predictors with only one observed measurement will have the same problem identified above with each ex- pression for the predictor with a single observed measurement and thus will not be identified. PART B : For model (4.2) it it assumed that Ki :_2 for all predictors (i = 1,2,...,p). Since p > 1, the counting condition for iden- tifiability is satisfied. To show that the definition for iden- tifiability is satisfied, it is necessary to show that each latent parameter can be expressed as a function of observed parameters in 2. By observation of expression (C.3.1) for Z in terms of latent parameters of the model (4.7), the following result is easily obtained. (C.3.2) o . . = o for i,j = 1,2,...,p with i # j. 1 J i j T T xlxl p(p - 1) Off_ These expressions (C.3.2) solve for the 2 diagonal parameters of ¢. 0 2 1 l XlX. (c.3.3) A. = -———1- for j = 2,...,K j C 2 l l xlxl O . i XIX: i = 2,...,p (C.3.4) A. = 3e1——- for . 3 xixl j = 2,...,Ki 1 1 301 These expressions (C.3.3) and (c.3.4) solve for the V - p P P parameters in A (since K - 1 + X (K. - l) = 2 (K. - l) = l . i . 1 i=2 i=1 v-p)o 0 2 xix: 1 (C.3.5) 0 l = 1 where A2 is given by (C.3.3). T A 2 O i i 2 x2xl (C.3.6) 0 . = . for i = 2,...,p i i T A2 where A: is given by (c.3.4). These expressions (C.3.5) and (C.3.6) solve for the p dia- gonal elements of ¢. for i II 0 I Q 2 (C.3.7) o l 1,2,...,p 2 . . . where o i is given by either (C.3.5) or (C.3.6). T 2 2 ' (C.3.8) o l = o i - (A5202i for i = l,...,p E. x. 3 T 3 3 j = 2,...,Ki where A; is given either by (C.3.3) or (c.3.4) and 2 . . . 0 i is given either by (C.3.5) or (C.3.6). T These expressions (C.3.7) and (C.3.8) solve for the V para- 2 P meters of W (since p + Z (Ki - l) = p + V - p = V). i=1 Thus all R = 2123:_11.+ v - p + p + v = 2v + 21333—31- latent parameters in the model for 2 can be expressed as functions of the observed parameters in 2. Therefore, when there are at least two observed replications for each predictor when p > 1 the model (4.7) for Z is identified. 302 CONCLUSION For models with more than one predictor the model for 2, given by Z = A¢A' + W2, will be identified ifenuionly if there are at least two observed replications associated with each latent pre- dictor. APPENDIX C.4 Consider the model for 2 with p predictors where Ki = 2 for some i = 1,2,...,p and Kj = l for each j = 1,2,...,p where j # i. Here V = p + l. The model for Z is Z = A T A' + W2 VXV VXp po pXV VXV As given the model is not identified. To reduce the number . . 2 2 . . of parameters to be estimated the constraint W = OEI is intro- duced, where I is the identity matrix of rank V. Notice that this constraint will not be reasonable for all situations. Let g be the V X 1 vector of observed replications for I 1 2 i-l the p predictors. Therefore, g' = Xle ... 1 Ex X I i i l 2 Thus 2 Z = A¢A' + OeI becomes 303 P + H 1:221 304 r- ‘s x1 2 1 0x1 1 x2 2 SYMMETRIC 1 “2221 022 1 1 1 i 2 2 = X1 °XiX1 Oxixz ' ' Oxi = l 1 1 l 1 i 2 X2 ° i 1 0 i 2 ° O i i O i x2xl x2xl x2xl x2 Xp 0' C O O 0'2 1 p l 2 ° ° ' i p i ' ° ' p X1X1 XIX1 Xixi x1X2 X1 L. .J P l 2 X o + o 1 T1 E X2 o 02 + 02 2 2 l T T1 T Xi 0 O 02 + 02 1 . . l TiT T1T2 Ti E X2 A o l l A o i 2 . A202l (A2)202i + 02 T T T T T T XI 0 1 o 2 . . o i A20 1 . . . 02 + A: ‘_ TpT TpT TpT TpT Tp First check about the counting condition for identifiability. (p+l)(p+2) = p(p+l) There are 2 2 + p + 1 observed parameters in Z. in A (all other elements are either zero + . 2 2 or one), El%_ll. parameters in ¢ and one parameter in W , OE, . i There is one parameter, A2, + for a total of El§_ll.+ 2 parameters in the model for X. Since p > 1, the counting condition for identifiability is satisfied. 305 Now check the definition foridentifiability. (C.4.l) O k j = o k j for k # j with k,j = 1,2,...,p. T T Xle This expression (C.4.l) solves for the p(2-l) off- diagonal parameters of ¢. 0 X hard X unw- i (C.4.2) A2 — o X for some given i (i = 1,2,...,p) . X P‘H' H H This expression (C.4.2) solves for the single parameter in A. 0’ i X X 1 w P- (C.4.3) A l i for some given i (i = 1,2,...,p) 2 I-BN >2 where A: is given by (C.4.2). This expression (C.4.3) solves for one of the diagonal para- meters of ¢. 2 (C.4.4) o = o . - o i for some given i (i = 1,2,...,p) 2 . . where 0 i is given by (C.4.3). T This expression (C.4.4) solves for the single parameter in (C.4.5) ozj = O . - CE for j # i with j = 1,2,...,p where a: is given by (C.4.4). This expression (C.4.5) solves for the remaining p-l dia- gonal elements of @. 306 Therefore all p(2+l) + 2 parameters in the model for 2 can be expressed as functions of the observed parameters in Z. 2 Thus the model (4.11) with the constraint W2 = OBI is identified. APPENDIX C.5 Derivatives of F =£ml£| + tr{Z—lsp}, where Z = A®A' + W2 . The expressions for vector and matrix derivatives employed in this appendix are taken from Chapter 2, Section 5 of Multi- variate Statistical Methods in Behavioral Research by R. Darrell Bock (1975), and are referenced by the chapter, section and statement number used by Bock. A.) Derivatives of F with respect to elements of A(V x p) Recall: A is a V x p matrix of scale factors, where P V = 2 Ki with Ki being the number of observed replications i=1 for predictor i (i = 1,2,...,p) i.e. r' -\ A = l O 0 . . . 0 pr Al 0 0 0 2 . A: 0 0 0 ..--l .................. 0 l 0 . . 0 2 0 AK 0 0 0 O 0 . O P 0 O 0 . A2 P O 0 0 AK P L. .J 307 308 The work on finding the derivative of F with respect to A will first assume that A is a general V x p matrix with elements Aij’ i = 1,2,...,V and j = l,...,p. The derivative desired for the A of this research will then be a special case of the general deriva- tive with the necessary adjustments in notation. -l 3_F_=30/n2 + 3 tr{2 59} 8A 3A 3A ° Consider a) é—l%%§L-= §§?L§L . 13' For some element of A, Aij, agnhfl __ -1' 82' 3A,. - tr 2 22.. (Bock 2.5 32) ij 13 2-1 32 8A.. 13 tr since 2 (hence 2-1) is symmetric. 32 _ 3(131' + 22) 3(131') 3(12) _ 3(131') 31.. ’ 31.. = 31.. 31.. “ 31. i 1] ij i 13 3 j 3¢A' 3A 31.. + 31.. 13 13 A ¢A' (Bock 2.5-3) 31' 33 31 A ¢ 31.. + 31.. + 31.. 13 13 13 A®l!. + 1..¢A' where 1.. is the V X p matrix which i] ij 13 has 1 as the ijth element and zero as all other elements. '. + 3. ' ' ' . A¢lij (A®llj) recall ¢ is symmetric 309 + ' QJELEL‘- tr{Z-1[A¢1!. 1] '° 8A. (131!.)']} . ij 13 = tr{Z-1A¢13.} + tr{z'l(1¢1!.)'} ij 13 -1 = tr{Z-1A®l!.} + tr{(A¢1!.Z )'} l] 1] l = tr{Z- A¢lij} + tr{A¢1ijz-l} since tr{A} = tr{A'} = tr{z’11¢1!.} + tr{z‘11¢1t.} 13 13 _ -l l — 2 tr{£ A¢l..} 13 = 2(2'113).. J. 3 ' -31@L§L - 22'113. .0 8A — Now consider 3 tdz -15 } 3 tr {(TlsP-TTIM} P _ _ _ b) 8A — 3A (Bock 2.5 22) where the bar over 2.1 indicates that 2-1 is "to be regarded as constant for the purposes of differentiation." (Bock, 1975, p. 69) 1S :51 = C where C is considered constant with P respect to the differentiation. Let E“ -1 3 z . t“ Sfi=_aumm=_ienii °° 31 31 31 2 2 _ _ 3 tr{(A¢A' + w )c _ _ 3 tr{A¢A'C} _ 3 tr{w c} ‘ 31 ‘ 31 31 II o _ _ 3 tr{A¢A'C} = _ 3 tr{1$1'c} _ 3 tr{A¢A'C} 31 31 31 Let A$.= D1 and A'C = D2 with both D1 and D2 con- stant with respect to the differentiation. 310 . 3 tr{D11'c} 3 tr{A¢D2} 3A 3A 3 tr{CD1A'} 3 tr{A¢D2} 3A 3A -1 I I 3 tr{z SP} 3 tr{ADlC } 3 tr{A¢D2} 3A 3A 3A -(DiC')' - (¢D2)' (Bock 2.5-15) = _ - I CD1 D2¢ -l 1 1S 2-1A¢ P -z s 2‘ p 13 - 2' 3 tr{z'ls } . £L_= _ - .. 3A 22 l s 2'113 p Combining results from part a) and part b): -l 3 tr{Z S } §§_= fiflpjzj + P 3A 3A 8A = 22'113 - 22'15p2'113 = 22-l(I - s 2'1)A¢ P . §§__ -1 -1 .. 3A — 22 (Z Sp)£ A¢ This result for %%- is precisely appropriate only if all elements of A are latent parameters. Since many of the parameters in A of the model for Z of this research are fixed values, A slight adjustment is needed on the above expression for g%- to make it suitable to the A of this research. The adjustment is described in Chapter 4. 311 B.) Derivatives of F with respect to elements of ¢(p X p) Typically T, as used in this research, will contain no fixed values and thus a general expression for EE- will be appropriate. 8¢ If, however, in some applications T does contain one or more fixed . . . . 3F values then modifications, suggested in Chapter 4, are needed for 23. '1 22.- EBWIZI + 3 tr{2 SP} 3¢ 3T 3T 2 Zn Consider a) L121: a__J_Zl 3T 8¢.. 1] for some element of ¢, ¢ij aflnl I _ -1 32 22.. - tr{2 3¢,.} (Bock 2.5 32) 13 13 U 2 U 2 32 = 8(A¢A + T ) = 3(A¢A ) + 3(W ) 3¢.. 3¢.. 3¢.. 3¢.. 13 1] l] 1] u 0 BA' 3(A¢) 8¢ 3A = -——-+--————- ' = -———-+ ————- ' A¢ 33.. 33.. A A3¢.. 33.. ¢ A 1] ij ij ij 1 n O .1 32 = Al?%A' where l?% is a p x p matrix where a¢ij ij 13 elements (ij) and (ji) are equal to one and all other elements are equal to zero. = 1:; since T is symmetric.) .2 31hi§i-= tr{2'111?¥1'} = tr{1'z'11 1?? } 3¢.. ij 1] l] r- . BBnIZI =< (A'Z-lA)ij for i = j, i,j = 1,2,...,p 2(1'2'11) for i s j, i,j .. 1,2,...,p \ 13 312 .2 39g$§i-= 2A'z’1A - DIAG{A'2'1A} 3 tr{z'ls } <9 33 Now consider b) Before proceeding with this section it is necessary to establish a result which will be needed and is not given by Bock. If X is an r X r symmetric matrix of variables and C is an r X r matrix of constants then 3 tr{XC} = ' + - 3x c C DIAG{C} and if C is also symmetric E—EEiEEl-= 2c - DIAG{C}. 3X Proof: r th By definition tr{Xc} = 2 [XC] where [XC] is the k k=l kk kk diagonal element of the matrix product XC. r [XCJkk = 1:1 xkficfik by definition of matrix multiplication r r I. tr{sc} = z z c R k=1 i=1 xk gk For i # j, i,j = 1,2,...,r r r 3 3 tr{xC} 3 = (x..C.. + x..C..) -—————-—-==-———- 2 Z X C 3X.. l] )1 31 1] I C a I 3X1] Xi3 k=l £=1 k£ 1k 1] N t - 3 (x c ) - 0 1 k - ' d 2 - ' o e. axij kg 1k — un ess - i an - 3 or k = j and R = i 3(X..C..) 3(x,.C..) 3xij axij ' 313 But since X is symmetric x.. = X... l] 31 . 3 tr{XC} _ . . . . _ .. axi. - Cji + Cij for 1 # j, 1,] — 1,2,...,r J and, for i = j, i,j = 1,2,...,r 3 tr{XC} = 3 E E X C = 3 (Xiicii) _ 3 (Xiicii .. x.. 2 2 .. ’ .. 3X1] 3 l] k=l £=l k k x1] Xii (X C ) Note: 5§2- kfi Rk = 0 unless k = i and 2 = i. ii . 3 tr{XC} . _ . . . _ .. 3X.. - Cii for l — j, 1,] — 1,2,...,r. 1] . 3 tr{xc} 3 tr{xc} . .. ————————— = -———————— = + - . 3x axij c c DIAG{C} If C is symmetric, then C' = C and E—EELEEl-= 2c - DIAG{C}. 3x 3 tr{z‘lsp} 3 tr{(E“1spE‘l)z} Now 8¢ = - 8¢ ‘ (Bock 2.5-32) Let C = Ehlspfbl as in Section A) part b.) above. 3 tr{2-ls } - p _ _ 3tr{CXl _ _ 3 tr{ZC} 33 ‘ 33 ‘ 33 = _ 3 tr{(A3A' + wz)c} 8¢ 3 tr{A3A'c} 3 tr{W2C} 33 ' 33 I‘ o _ 3 tr{3A'CA} 33 ‘ ) 314 Since ¢ is a symmetric matrix of parameters and A'CA is a symmetric matrix of constants (with respect to the differentia- tion) the result proved above applies. 8 tr{Z-lS } .2 33 .p = -[2(A'cA) - DIAG{A'CA}] = -2A'z'lspz-1A + DIAG{A'Z-lSpX-1A} . Therefore combining results from part a) and part b) 3_F = ”MEL + 3tr{Z-lsp}' 33 33 33 = 2A'z-1A - DIAG{A'X-1A} - 2A'2'lspz-lA + DIAG{A'Z-lSpZ-1A} = 2(A'z‘lA - A'z'lspz'lA) - DIAG{A'2'1A - A'z’lspz’lA} = 2(A'z'l(1 - spz'l)A) - DIAG{A'Z—1(I - spz-1)A} 2A'Z-l(Z - sp)z'lA - DIAG{A'2'1(£ - sp)z'lA}. C) Derivatives of F with respect to elements of V(V X V). Since W2, hence V, will be a diagonal matrix for virtually . . . . . 3 all applications of the model of this research, the derivations, 5%-, will be found with respect to the diagonal elements alone. -1 3 tr{2 S } L=39712 + p for i=l,2,...,V. 33.. 33.. 3W.. 1]. ll 11 2 Note the use of ?.. rather than W.. here. 11 ii Consider a) 2J21§l-= tr{2-l 32 } 3W.. 8W,, ii ii 32 _ 3(A3A' + 32) _ 3(A3A') _ 3(32) 3W.. 3W.. 8W.. 8W.. ii ii ii ii a O 315 Note: 32 = 3 and since 32 is diagonal 2 2 (3 ) .. = E O3)..] ii ii 2 32 _ 3(3 ) _ " 33.. ’ 33.. ’ 2Wiilii 11 ii . 3%IZI __ -l _ -l _ -1 .. 33.. - tr{£ 2311111} — ZWiitr{Z l } — ZWii ii ii -1 -l . . = 22..W.. where 2.. and 3.. are the ith diagonal ii ii 11 ii elements of 2-1 and 3 respectively. And since 3 is diagonal 3J2J§i—= 2[z'13].. i = 1,2,...,v ii 33.. ii where [2-1Y111 is the ith diagonal elements of -1 of the matric product 2 ”3. Consider now b) 3 tr{z'lsp} 3tr{(EhlsEE'l)z} 33.. = - 33.. (Bock 2.5-22 with respect ii ii to a single element of 3) Again let C = Eblspffil. 3 tr{Z-lS } 2 p _ _ 3 tr{CZ} = _ 3 tr{zc} _ _ 3tr{(A3A' + 3 )c} 33.. 33.. 33.. " 33.. ii ii ii ii _ 3 tr{A3A'c} _ 3 tr{32c} - 33.. 33.. ii ii n O 316 _ §_E£i!!Efi._ 3 tr{33c} 3 tr{33E} . 33,_ " 33.. 3111 (Bock 2.5 20 with respect to a __ ___ single element of 3) 3 tr{3c3} 3 tr{33c} 33.. 33.. ii By a result gained in the process of the proof in section B) i.e. M: C." x.. 11 ii 3 tr{3C33 _ 33.. ‘ (CW)ii ii 3 tr{33C} d -————————-= . an 33.. (3C)il ii 3 tr{Z-ls } .2 p = -(c3) . - (3C) 33.. i1 ii ii 0 -1 -l I O I 0 But Since C = E SpZ is symmetric and 3 is diagonal, C3 is symmetric and C3 = 3C. -1 . atr{2 S } P 33.. ii _ _ -1 -l - 2(C3)ii - 2(2 5px 3) ii Therefore combining part a) and b) -1 3 tr{Z S } 33‘ =mlfl. P for i=1,2,...,v 33.. 33.. 33.. ll 11 11 = 2(2' 3).. - 2(z'ls z"l3>.. ii p 11 = 2(2-13 - z-ls 2-13).. p ii 0 3F -1 -l n .. =2 - .I = '2,...' O 33. 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