C! a); at: . a , {.mdmflx‘hl ”Mummy? fidhmw . «Mafia‘w . ar‘mfiw; ; ma. 4mm? .a .5.“ 5.5% J.» .5 . .... I. » Pt 2.... . . _ a? u ”mm rip" us». 4. . than... MTWVVNWQI: M A... .1 Mm»- 3.1.7}.th . . ,. X . #fljmfiwrmnfl.,. .. m i . . gfififig. TEES!“ 1 w IJBRARY 20° '5 Michigan State University This is to certify that the thesis entitled PARAMETER ESTIMATION AND INTERPRETATION IN SPATIAL AUTOREGRESSION MODELS presented by JIQIANG XU has been accepted towards fulfillment of the requirements for the PhD. degree in Counseling, Educational Psychology and Special Education Cassandra Book, EDU, Associate Dean Major Professor’s Signature 1 998 Date MSU is an Affirmative Action/Equal Opportunity Employer PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 K:IPrq/Acc&Pres/ClRC/DateDue indd PARAMETER ESTIMATION AND INTERPRETATION IN SPATIAL AUTOREGRESSION MODELS BY J iQiang Xu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Counseling, Educational Psychology And Special Education 1998 ABSTRACT This dissertation explores Spatial Autoregression Models (SAM). It gives the definition of the autocorrelation coefficient p in SAM, supplies the technique for the computational precision for parameter estimation in SAM, and makes the SAM model applicable in practice. Based on SAM models, the autocorrelation coefficient p turns out to be the correlation coefficient between a matrix W and a vector Y, a new measurement in statistics for the social sciences; the classical Factor Analysis Method is also generalized to the non- variance - covariance matrices. Copyright by JIQIANG XU 1998 To the memory of my late mother and father ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to his dissertation committee, Drs. B. J. Becker, K. A. Frank, D. Gilliland and S. Raudenbush for their valuable advice, guidance and help in the preparation of the manuscript. The author is specifically indebted to his advisor Dr. K. A. Frank for the suggestion of the dissertation topic which leads to the following outcomes. The author is also thankful to the Measurement and Quantitative Methods (MQM) program, to the Department of Counseling, Educational Psychology and Special Education (CEPSE) for their encouragement and financial support. TABLE OF CONTENTS LIST OF TABLES ....................................................................... LIST OF FIGURES ..................................................................... LIST OF SYMBOLS AND ABBREVIATIONS .................................. INTRODUCTION ...................................................................... CHAPTER 1 A GENERAL REVIEW IN THE LITERATURE OF SPATIAL AUTOREGRESSION MODELS (SAM) .......................... 1.1. The importance of SAM model in education ......................... 1.1. The MLE approach to SAM models .................................... 1.2. Methods other than MLE ................................................. CHAPTER 2 EXISTING PROBLEMS AND NEW PROBLEMS WHEN APPLYING SPATIAL AUTOREGRESSION MODELS TO THE SOCIAL SCIENCES .. 2.1. The theoretical limitation, the old problems .......................... 2.2. Technical constrains, the new problems when applying the model to the social network analysis .......................................... CHAPTER 3 A NEW TECHNIQUE FOR ESTIMATION OF PARAMETERS p AND 02 .. 3.1. The “Separation” of the log-likelihood function ...................... 3.2. The “Far End” method of the initial value selection .................. CHAPTER 4 A TRANSFORMATION FROM THE W, Y SYSTEM TO THE A, Z SYSTEM ................................................................ 4.1. A phenomenon: Why does the estimate p approach an extreme value easily? ............................................................... 4.2. The technical reason ...................................................... 4.3. The theoretical reason .................................................... vi viii ix X WMNN 16 16 19 21 21 40 44 44 46 47 CHAPTER 5 THE DEFINITION OF PARAMETER p AND THE W(a) FAMILY OF THE WEIGHT MATRICES ........................................................... 54 5.1. The definition of the parameter p ....................................... 54 5.2. The geometrical reasoning of the definition of the parameter p 55 5.3. A comparison of SAM with linear models ............................. 56 5.4. The W(a) family ........................................................... 57 5.5. The dynamic system in a social setting of the trio: estimate p, W and Y ................................................................. 63 CHAPTER 6 DATA ANALYSIS ..................................................................................... . 66 6.0. The source of data ........................................................ 66 6.1. Part I. Data simulation ................................................... 67 6.2. Part II. A practical example .............................................. 75 6.3. About the formulas for the standard deviation of p .................. 84 CHAPTER 7 DISCUSSION ............................................................................................ 86 7.1. The comparison of the new technique with the OLS technique 86 7.2. The comparison of the new technique with factor analysis: the extended Factor Analysis Method ................................. 88 7.3. The next stage work ...................................................... 92 7.4. Conclusion ................................................................. 93 APPENDICES Appendix3A: Example 1 .......................................................... 96 Appendix3B: The calculation of the conjugate eigenvalues 97 Appendix6A: The raw data matrix W(24x24) ................................. 98 Appendix6B: The comparison of the results from different initial value selections ........................................................... 99 Appendix6C: A SAS program (1) ................................................ 100 Appendix6D: A Flow Chart ...................................................... 104 Appendix6E A SAS program (2) ................................................. 105 Appendix6F: A SAS output from program (1) ................................ 109 Appendix6G: A SAS output from program (2) ................................. 125 Appendix6H: The eigenvalues of W(0,1,2,3,4) ............................... 141 Appendix6I: The comparison of moral levels ................................ 142 Appendix6J: The racexsexxmoral table ....................................... 144 Appendix6K: The eigenvalues of W(0,1) ...................................... 146 REFERENCES .......................................................................... 148 vii LIST OF TABLES Table3A ............................................................................ TableSA ........................................................................... Table6A ........................................................................... Table6B ........................................................................... viii 23 58 78 82 LIST OF FIGURES Figure3A ........................................................................ Figure3B ........................................................................ F i gure3C ........................................................................ Figure3B ........................................................................ Figure3B ........................................................................ Figure3F ........................................................................ Figure3G ........................................................................ Figure3H ........................................................................ Figure3I ......................................................................... Figure3J ......................................................................... Figure3K ........................................................................ FigureSA ........................................................................ Figure6A .......................................................................... Figure6B ........................................................................ F i gure6C ........................................................................ 26 27 28 29 3O 31 32 33 34 35 41 62 72 73 74 LIST OF SYMBOLS AND ABBREVIATIONS MLE: Maximum Likelihood Estimation. NR method: Newton - Raphson iteration method. OLS method: Ordinary Least Square method. SAM: Spatial Autoregression Model. INTRODUCTION The spatial autoregression model (SAM) represents the relationship between the vector Y representing attributes of subjects, and the association matrix W that represents the relationships among all subjects. In the model, the two main estimands, the 2, are uniquely autocorrelation coefficient p and variance of the random error term 0 determined by the matrix W and the vector Y. But the commonly used techniques for the estimation of p produce many values outside of a sensible range for p, and even the interpretation of the parameter p itself is not yet well understood. All these facts restrict the use of SAM. This dissertation explores the interpretation and estimation of p as well as 02, and emphasizes the importance of the extent of the consistency between W and Y which is captured by p. This estimation of p is especially useful in the social sciences where the data could be high in dimension (say around 10 or higher), and the data are possibly highly correlated. Key words: MLE method, Newton-Raphson (NR) iteration, autocorrelation coefficient p. Chapter 1 A GENERAL REVIEW IN THE LITERATURE OF SPATIAL AUTOREGRESSION MODELS (SAM) Spatial Autoregression Models (SAM) are important in social sciences including in education. In this chapter, I first briefly introduce the parameters in SAM, and approaches researchers have applied to parameter estimation. I cite part of Ord’s (1975) technical work on a maximum likelihood estimation (MLE) method, to which my new development in SAM is tightly related. 1.1. The importance of SAM model in education. Researchers are interested in how students gain their academic achievement (an outcome vector Y). The variable Y could be caused by their personal variables (X1) such as age, gender and race. Y could also be caused by their family variables (X2) such as Social Economic Status (SES), income level and his / her mother’s education. Again, Y could be caused by their school variables (X3) such as private / public school, number of teachers in this school, etc. To answer the question how students gain their academic achievement, the multivariate regression is commonly used. When doing so, we assume that there is no mutual connection between individual students, or we are assuming that students’ academic activities are mutually independent. For example, in Shavelson (1996), the first assumption for conducting the multivariate regression analysis (MRA) is “Independence: The scores for any particular subject are independent of the scores of all other subjects” (page 536). It is the same for conducting the analysis of variance (ANOVA) (page 378). However, such kind of assumptions might be not appropriate. For example, Cressie (1993) said in his book focusing on the Spatial Models: “The notion that data close together, in time or space, are likely to be correlated (i.e., cannot be modeled as statistically independent) is a natural and social phenomena” (page 3). Also, Duke (1993) said in his article about Network Effects Models: “Researchers in the sociology of education have long recognized the importance of peer influences in shaping the academic achievements, aspirations, and educational attainments”, and “. .. have also been very aware of the limitations of the methodologies they have employed in investigating peer influences” (page 465). Here, “Spatial Models” and “Network Effects Models” are exchangeable in the literature depending on the different topics. When we start the educational quantitative research work based on the data set of Y, {X1, X2, X3} and {W1, W2, W3} like this, we usually conduct the multivariate regression treating Y as outcome and {X1, X2, X3} as predictors. Now new questions arise: what can we do with those {W1, W2, W3}? Is there any kind of relation between Y and {W1, W2, W3}? Or between {X1, X2, X3} and {W1, W2, W3}? We think that there should be some kind of relation, strong or weak, between Y and {W1, W2, W3}, also, between {X1, X2, X3} and {W1, W2, W3}. Treating W, Y and X as individual variables, we hope to find some kind of relation between W and Y (or X). The new questions make sense. Subjects are more or less mutually connected and influenced within a network (a matrix W). The connections could be geographical (W1) such as the distances between the seats of pairs of students (so that W. is symmetrical). It could be personal (W2) such as the level of friendship (thus W2 is not necessarily symmetrical). The connection could also be social (W3) such as the sports team membership of the same interest / club (then W3 probably would be symmetrical). Here is a practical example for teachers. In Frank (1995, 1996), the data collected from a group of 24 high school teachers included an association (weight) matrix W and some vectors Y. The author identified cohesive subgroups among those teachers based on W, the level of the frequency of their professional discussions. In these two articles, however, the relations between the weight matrix W and each of those vectors Y, such as the teachers’ gender, race, year of teaching, and their moral agencies, were not considered due to the lack of available techniques. So that, we are not sure whether or not the weight matrix W, or the pattern of those teachers’ professional communication, is more or less associated with some personal vector(s) in that school. Is the pattern of their professional communication mainly associated with individual’s orientation to teaching? Or mainly because of their race and gender? Practically, we are concerned that the subjects’ variable (a vector Y) might be more related to one of the W matrices than to others. That is, the student achievement might be more related to the pattern of their fiiendships than to the geographical distances between their homes. Or, we are concerned that the pattern of subjects’ fiiendship (a matrix W) might be more likely related to one of the X vectors than to others. That is, the pattern of subjects’ fiiendship might be more associated with their gender than with their ages. Theoretically, the first concern above regards a comparison of relations between one Y with different Ws; the second concern above means to compare the relations between one W with Y and different Xs. Both of those concerns are essentially focusing on the same topic: the relation between a matrix W and a vector Y which can be addressed through Spatial Autoregression Models (SAM). Substantial developments have been made by Mead (1967, 1971), Ord (1975), Doreian (1981, 1982, 1989), Cressie (1993), Duke (1993), Marsden and F riedkin (1994), Leenders (1995) and others to the literature of Spatial Autoregression models (SAM). However, the remaining difficulties, both theoretical and technical, restrict the practical application of SAM in many instances. This dissertation makes the SAM model applicable in practice. We will be able to measure the general relation between a matrix W and a vector Y, to compare the differences of relation between one W with different Y and Xs. Also, we may compare the differences of relation between one Y with different Ws. We’ll find the essence of the relation between W and Y in SAM. 1.2. The MLE approach to SAM models. The SAM model was initially applied to studies in geographical and agricultural economics such as Whittle (1954), Mead (1967), and Ord (1975). Working with the spatial autoregression model, researchers explored the relationship between a vector Y representing attributes of subjects, and a weight matrix W, or the association matrix in different contexts, describing the mutual relationship among those subjects. In SAM, the subjects can range widely including plants, counties or persons; the relationship between subjects is often given in terms of geographical distance. For example, in Ripley (1981), possible subjects mentioned were trees, towns, birds’ nests, imperfections of metals, galaxies and earthquakes. When the weight matrix W represents the mutual relationship between individuals in an organization, and Y represents an attribute of those individuals in the organization, we are applying the SAM model in a sociological and psychological sense. Then we may find the important application of SAM models in education. Once the subjects are determined, the researcher needs to specify the strategy for choosing a measurement for the relationship, either geographical or interpersonal. In spatial autoregression models, the connections among subjects extend in all directions, so that even geographical distance might not be unique depending on the definition of “all” directions. For example, Anselin (1988) suggested geographical ways such as “short path” or “neighboring” for the definition (page 18). Mead (1967) introduced ways of making geometrical connections on a plane in different covering sizes (page 193). Both the above approaches are objective. When trying to apply the SAM model in social sciences, we might be dealing with the interpersonal relations in many different ways, either objective or subjective. For example, when a subject is making a decision, this decision could be influenced by the frequency of phone calls made to others in the social network, then this connection is objective. A subject’s decision could also be influenced by his level of intention towards those he’d like to engage, then this connection is subjective. In general, we will deal with many different types of connections, which are represented by the weight matrix W. SAM initially deals with effects through spatial autocorrelation. In Cliff and Ord (1973), the authors illustrated that concept: “If the presence of some quality in a county of a country makes its presence in neighboring counties more or less likely, we say that the phenomenon exhibits spatial autocorrelation” (page 1), although no direct definition of “autocorrelation” was given. Anselin (1988) acknowledged that spatial autocorrelation, or spatial dependence in his words, “is best known and acknowledged most often, particularly following the pathbreaking work of Cliff and Ord (1973)”. The author also said: “it is generally taken to mean the lack of independence which is often present among observations in cross-sectional data sets” (page 8). Recently Leenders (1995) described spatial autocorrelation “either a variable or of an error is the situation where the observations of variables or the values of the error terms for different actors are not independent over time, through space, or across a network”. All of the above give us an understanding of spatial autoregression models although a clear definition of autocorrelation is unavailable in the extant literature. We also notice that in recent decades, a lot of excellent research work has been done with the autocorrelation of error terms, but relatively less has been done with the autocorrelation of variables in spatial autoregression models. 1.2.1. The parameters p and 0'2 in the SAM model. In Ord (1975), the weight matrix W (nxn) in SAM is assumed with entries w}, 20 ( i at j) and w,-,- = 0 for any i,j=l,2, ..., n. Ord’s assumption is carried on in this dissertation from chapter 1 through chapter 4, and is developed to a more general case where wij 20 is not required and w,-,- = a at 0 is considered in chapter 5. With W = {Wij} , an (nxn) set of non-negative weights which represent the degree of association between the jth subject and the ith subject, and with Y = {y,}, an (nxl) set of observed outcomes, the first order spatial autoregression model is y, =a+p2wijyj+8i (1.1) j=l where 8 ~ N(O, 021) with parameters or, p and 0'2. Equation (1.1) can be reformulated in matrix notation by taking or = 0 suggested by Ord (1975) (page121), as Y = pWY + s (1.2) where W is the (nxn) matrix of weights and Y, s are (nx 1) vectors. We notice that W has entries Wy' 20 ( i ¢j) and w,-,- = 0 for any i,i=1,2, ..., n. This is the simplest first order spatial autoregression model in which the parameters p and 0'2 need to be estimated. We’ll mostly focus on the relationship between W and Y, and the estimate of p as a function of W and Y. For the SAM model, there are different methods of estimating the parameters p and 62. One of the commonly accepted methods is the maximum likelihood estimation (MLE) method. In MLE for the parameters p, the Newton-Raphson iteration is conducted repeatedly until convergence or divergence of parameters p is obtained by some rules. In each step for the SAM model, the researcher has to obtain an estimate of parameters p which is carried into a matrix for computation, then have the matrix inverted, and obtain a new estimate for the next step. Without a computer or when the dimension of the weight matrix W is high, say around 10 or higher, it is difficult to obtain a maximum likelihood estimate of parameters p and 62. So until the theoretical development of Ord (1975), this method was rarely considered practical due to computational difficulties. 1.2.2. The “once and for all” technique for the Newton-Raphson (NR) iteration in calculation. In Ord (1975), writing A = A(p) = I - pW which is in full rank, the log-likelihood function for p and 02, given Y is 1 )Y'A'AY+ln|A|. (1.3) or We may write the term Y'A’AY as ll AYllz, and specifically write the term |A| = ldetAl where “detA” is the determinant of A. Thus 1n |A| is always defined. But in order to be consistent with Ord’s article, I keep using 1n |A|. A brief note is given later in this chapter. Then the ML estimators are obtained as 62 =n-'Y'A'AY=(1)[(Y—5WY)(Y-pWY)], (1.4) n and the p estimate is the maximizer of [(5,6‘ 2) =const - (able; 2 ) — $1144]. (1.5) Equivalently, it is to minimize f(f>)= —;21—lnlA|+ln(62) = G(fa)+ H(fi), (1.6) .. 2 A .. where G(p) = —;lnlAI and H(p) = ln(oz). I separate the function f (p) into two parts G(p) and H(p) for the reasons given in chapter 3 (§3.1). This separation is a key step, which helps us to draw conclusions from the equation (1.6) by applying the Cauchy mean value theorem. The possible minimizer of (1.6) would be the solution to the equation f '(p) = O (1 -7) or those points on which f ’(p) does not exist. In the process of iterating to minimize f (P), lAl , the polynomial in p has to be evaluated afi'esh at each iteration. As Ord (1975) said, when n is large, or A is irregular, this process becomes computationally intensive. Thus Ord created a new computational procedure as follows. Since |A| = 11K] — pit ,) where 2.3 e {2.3}, the eigenvalue set of W, then i=1 ln|A| = lnfi(1 — phi): :lnfl— pki). M i=1 In the Newton-Raphson iteration process to find the minirnizer of f (p) in (1.6), writing YL = WY, the iteration is taken as p... = p. -f’(p.)/f"(p.) where f'(p)= (321:7/(1- p7.,.)+ 2(p(Y,)' Y, — Y'YL) /s2 and 10 n f "(p)= (9200’ /(1 — pt.)2 +2(Y,)' Y, /s2 —4(p(Y,)' Y, — mg)2 /s4 where i=1 s2 a 32 (p): Y'Y— 2pY'Y, + p2 (Y, )' Y, If pr converges when r --> 00, it converges to the ML estimate of the parameter p. The advantage of Ord’s technique of writing a determinant |A| into an algebraic product is obvious. Because we need to repeatedly calculate the value of |A| which changes in each step, we need to deal with a lot of matrix computation. But now with Ord’s technique, whole matrix computations become simple algebra because the eigenvalues of W need to be evaluated only once. That is why this technique is called the “once and for all” technique. 1.2.3. A note in case the weight matrix W is asymmetrical. We know that a 2 = n"Y'A'AY = —1-[Y'(I — 5W)'(1 — my] n — 1[Y'Y-pY’WY-13(WY)'Y+132(WY)'(WY3]- _ Z When W’ = W, this form will be the same as that in Ord’s Appendix A for the following computation. The function (1.6) is exactly equivalent to “5): (—%):ln(l— pr,)+1n(Y'Y— sz'Y, +§2(Y,)'Y,)—1nn. (1.3) i=1 where the author uses YL to represent WY. 1] But when W' at W, a possible case in social sciences, the above formula (1.8) and its derivatives need to be slightly adjusted. The new version of f (p) after adjustment should be A 2 n A A A2 ' f(p) = (_ £2an - pr,)+ 1n(Y'Y— pY'(W+ W')Y+ p (WY) WY) — Inn. (1.8)’ The expression of the minimizer of f (p) in the Newton-Raphson iteration process is the same as above, namely 9... = p. -f'(p.)/f"(p.)- But others are now as follows: f'(p)=(%);K,/(l-pki)+(2p(WY)'WY—Y'(W+W')Y)/SZ and 2 n ' I 2 f~(p)= (;)Z(7t,)2/(1— pk,)2 + 2(WY) WY s2 - (2p(WY) WY— Y'(W+ W')Y] /s4 i=1 where s2 a 52 (p): Y'Y— pY’(W+ W')Y+ p2(WY) WY. Here I use WY instead of Y1, to clarify the difference between W and W'. In the following chapters, I am not going to emphasize the difference between symmetrical and asymmetrical Ws repeatedly. But in chapter 6, the data analysis involves an (8x8) asymmetrical weight matrix, and the formulas for computations are the new versions. 1.2.4. Problems when |A| is non-positive, and how to avoid the problems. 12 We know that I A | = | 1 — pW | is an nth polynomial of p, and may have values either positive, or negative, or possibly zero when p ranges on (-oo, oo). In some formulas of this chapter, e.g. (1.3), (1.5), the logarithm of |A| is taken, this logarithm may make no sense if the value of |A| is zero or negative. Also, in chapter 4, |A| appears in a log-likelihood function which may also make no sense if the value |A| is zero or negative. That is why I mentioned early in this chapter to write |A| as |detA| to avoid such a technical problem. In the following, I treat |A| as |detA| without further notification. 1.3. Methods other than MLE. In Ord (1975), the MLE method is compared with the ordinary least square (OLS) method or generalized least square (GLS) method. Ord showed (page121-122) that the OLS estimator is inconsistent for a general weight matrix W, and is consistent only when W is triangular, a limited and non-interesting case. 1.3.1. Anselin’s Bayesian method. In Anselin (1988), the author introduced some other methods for comparison with MLE method. The author concluded (page 81): “the maximum likelihood approach to estimation and hypothesis testing in spatial process models is by far the better known methodological framework”. For the Bayesian method, the combination of prior information about the distribution of the parameters, namely the autoregressive coefficient p and the error variance 02 in the model, are needed. When we start to work on an SAM model, it might be difficult to get such information in advance. Without any prior information, however, this method is not 1 l3 appropriate (not a Bayesian one). As Anselin said (page 89-90), “following the standard approach in econometrics, diffuse prior densities for these parameters are expressed as: P(o)o< 1/0',0<0'<+oo. P(p) o< constant, -1 < p < +1.” There might be flaws in these priors. The second prior density means that p is uniformly distributed on the interval (-1, l) which is the boundary for the model after the weight matrix W has been standardized as some authors suggested. However, the relationship between W and Y is not considered which can strongly affect the estimate p as I will show in the following chapters. Also, as we know, the absolute values of the maximum and minimum eigenvalues of the weight matrix W are not equal in most cases, so that (l/kmgn, l/itmax), the range of p, is not symmetrical to the origin in most cases. This second prior density seems not to be appropriate. Anyway, Anselin said (page 81) that in spatial models, the implementation of alternative approaches (other than MLE method) including the Bayesian method “has been rather limited”. 1.3.2. Cressie’s asymptotic property approach. Cressie (1993) gives details of spatial modeling and parameter estimating regarding the lattice models in the chapters 6 and 7 of his book. The author said (page 458), for models in chapter 6 of his book where the first order spatial autoregression model is included, “estimation of model parameters is not always so straightforward. Of course, finite- sample properties, such as sufficiency, completeness, ancillarity, unbiasedness, minimum mean-square error are still desirable; however, they are even more elusive than for the i.i.d. paradigm.” As a conclusion, the author said that “methods of estimation are usually 14 assessed via their asymptotic properties” while those finite-sample properties are not guaranteed. 15 Chapter 2 EXISTING PROBLEMS AND NEW PROBLEMS WHEN APPLYING SPATIAL AUTOREGRESSION MODELS TO THE SOCIAL SCIENCES This chapter is mainly a list of problems: theoretical or technical; old or new in the literature of Spatial Autoregression Models (SAM). When dealing with the agricultural and geographical data decades ago, the SAM model arosemainly as a theoretical work with limited practical meaning due to the computational difficulty and the lack of understanding of the model itself. When trying to apply spatial autoregression models in social network analysis in recent years, some new problems occurred in the model application. I list these old and new problems in the following, and am going to solve them in the following chapters. 2.1. The theoretical limitation, the old problems. (a). The definition of the parameter p has not been clear. 0 As cited in chapter 1, Cliff and Ord (1973) said that “If the presence of some quality in a county of a country makes its presence in neighboring counties more or less likely, we say that the phenomenon exhibits spatial autocorrelation”. But no 16 definition of autorcorrelation or illustration of the autocorrelation coefficient p were given. Mead (1967) wrote (page 191) that “7. is defined to be the competition coefficient” in his interplant competition model where the coefficient p was written as 7.. In another paper (1971), the same author said (page 18) that “7. is a competition coefficient which, if the ‘correct’ form of f} is found, would be expected to remain constant at a particular time in a particular environment”. There, f} is the element of the weight matrix (or the association matrix). We noticed that in Mead (1967), the author cited a discussion about the parameter p from another early paper, Kira et a1. (1953). When dealing with a simplified setting, Kira et al. “found that most values of the (autocorrelation) coefficient were positive, which they interpreted as showing co- operative rather than competitive situations”. This interpretation began to be considered as the meaning of the parameter p in a limited sense. Ord (1975) briefly says (page 120) that together with 0 as the estimate of the standard deviation for the random error, “0 and p are parameters”. In Doreian (1982), the author says (page 240) that “p is the spatial effects parameter”. In his other paper (1989), the author says (page 285) that “p is the network autoregressive parameter”. In Anselin (1988), on page 33, the first order spatial autoregression model is introduced with no statement dealing with the parameter p; on page 35, the author says that “p is the coefficient of the spatially lagged dependent variable (of WY)”, giving no more details; on page 58, the author says that “p is a spatial autoregression coefficient”. 17 0 When tracing back the history of p, Leenders (1995) says (page 55) that “p is a scalar”. The author also mentions (page 167) that p “should be considered a descriptive parameter rather than a governing parameter” with no further explaining on being “descriptive”. 0 Duke (1993) gives a statement about the parameter p by saying “p is the parameter representing the strength of the context effect”. Also, no further statement was given in that paper. From the above, we may find that these statements about parameter p are mainly in a sense of naming, rather than giving a definition. A new definition will be given in chapter 5 (§5.1). (b). The necessity of the boundary for the p estimation is not clear. The boundary of the estimate p is a key issue concerning the autoregression model in the literature. The estimated value of parameter p from a computer direct search frequently is large, and researchers have tried to find an appropriate boundary to restrict the divergence. The commonly accepted range (1/lmin, l/itmax) or similar ones can be found in Ord (1975), Doreian and Hunmon (1976), Anselin (1982). Recently, Leenders (1995) suggested that parameter p could truly take large values with no further reasoning. I will address this issue in chapter 4 (§4.3). (c). The range and interpretation of the estimate of p have not been discussed. The estimate p is the minimizer of the likelihood fimction f(p), so is a solution of the equation f '(p) = 0. We know that f '(p) = 0 is a non-algebraic equation with multiple 18 roots, and an analytic expression of the estimate of p is difficult to obtain. We also know that the interpretation of the estimate of p was least talked about so far in the literature. The meaning of p being high or low, positive or negative, in which sense, and in what kind of scale, etc. has not been clearly stated. The range and interpretation of the estimate of p will be discussed in chapter 4 (§4.1). 2.2. Technical constraints, the new problems when applying the model to the social network analysis. Because the matrix W could be asymmetrical, or the value wij of W rrright be negative, we are facing some new problems in the field. (d). The matrix W might be asymmetrical, so that there might be (and most likely will be) complex eigenvalues. On the other hand, when dealing with the boundary problem, some researchers such as Ord (1975), Doreian (1981), Duke (1993) and Leenders (1995) suggested the “row normalization” for matrix W. Once W is row normalized by setting w”. . .*= WU n 2 W.)- j=I for any i,j =1, 2, ..., n so that Zwy' =1 for any i= 1, 2, ..., n, the boundary j=l of the estimate p would be simply | p | s 1. However, the normalized matrix W* would become asymmetrical in most cases and cause the complex eigenvalue problem, although the original matrix W could be symmetrical. This issue is addressed in chapter 3 (§3.1.2). (e). So far in the literature, the weight matrix is treated as w),- = 0 for i=1,2, ..., n. In social sciences, however, the case w),- = a at 0 could make sense but has not yet been considered. We need to explore the essence of the case w,-,- = a at 0, and the relationship 19 6‘ 99 between the estimates p for cases w),- = 0 and w), = a at 0. If we treat this a as a constant which may take zero, positive or even negative values, we are actually talking about W(a), a family of weight matrices in a more general sense. I will address this issue in chapter 5. 20 Chapter 3 A NEW TECHNIQUE FOR ESTIMATION OF PARAMETERS p AND 0'2 TECHNICAL IMPROVEMENTS In this chapter, I explore the construction of SAM model. Based on the understanding of the construction of SAM, I obtain two results regarding the parameter estimation of SAM. One is that in the interval (I/Xmin, l/kmax), the value of Y'WY and estimate p will take the same sign, which helps to understand the construction of solution space of the estimate p (§3.1). Another is the so-called “far end” method of initial value selection for the estimation of p, which helps to control the estimate p not “flying out” of the required boundary (§3.2). 3.1. The “Separation” of the log-likelihood function. The key work here is to use a “separation” technique which helps to draw conclusions from the log-likelihood equation in the MLE method. That is, I separate the function f ’(p) into two parts: g(p) and h(p). In function (1.6), we had the function f(p) = G(p) + H(p). Taking the derivative of function (1.6) gives f '(p) = G'(p) + H'(p) = g(p) + 11(9) where g(p)= (Eli-(fig; and n i=1 21 p(Y,) Y, — Y'Y, . Y'Y-ZpY'Y, + p2(Y,) Y, h(P)= 2- with YL= WY. Clearly, the value of g(p) is determined by p and W's eigenvalues {7.3}, and the value of h(p) is determined by p, and both W and Y. Now we have the derivative of the log-likelihood function as we had in chapter 1, P(YL) YL —Y’YL Y'Y-2pY’Y, +p2(Y,) Y, f'(p)= [SET—l— pit-F2) i=1 I discuss the solution of equation f '(p) = 0 which will be the solution to our parameter estimation. 3.1.1. Assuming W has all real eigenvalues. Case 1. Assuming Y'YL > 0. Applying the Cauchy mean value theorem shows that both Y'YL and estimate p will have positive values in the boundary (I/Kmin, 1/Xmax). [Table3A about here] I briefly explain Table3A as follows. The second column from left is checking the value of function f '(p) when p = 0. Since g(0) = (2m): 0.3 = 0 and h(0) = -2(Y'YL)/ YL ’YL < 0, we have f '(0) = g(O) + h(O) < 0. The fourth and fifth columns from left are checking the value of function f '(p) when p = p' = min(pv, l/Amax) where pV = (Y'YL)/ YL 'YL, the OLS estimate. When pV s l/itmax, we have 22 Table3A The location of p, the solution of the equation f '(p) = 0. Assuming Y'YL > 0 so that pV = Y'YL / (YL)’YL > 0. There must be p' such that 0 < P. < min(pv, UM) p' = min(pv. UL...) with pv g 1/7.,,,,. p' = min(p". UM...) with pV> 10.4.... g(p) =(2/n)): 7.- 41-974) g(O) = (21m): 7.- = o g(p') = g(p") > 0 g(p.) = g(l/lmu) = +°° h(p)=2(P(YL)'YL - Y’YL) / (Y 'Y- 2pY'YL +PZ(YL)'YL) h(0)= -2Y'Yl/Y'Y < 0 MW) = Mo“) = 0 h(p') = h(l/AmM) < 0 f '(p) = sin) + h(p) f'(0) = 8(0) + 11(0) < 0 f '(p) = 0 f'(P') = f'(pv) > 0 fr(po) = f’(I/}Lmax) = +430 23 g(p') = g(pv) > 0. and h(p') = h(pv) = 0 so that f '(p°) = g(pv) + h(pV) > 0- When p‘7 > 10.4mm, we have g(p') = g(l/itmax) = +00, and h(p') = h(l/kmax) < 0 so that f '(p') = g(l/kmax) + h(l/kmax) = +00. At last, we have f '(p') > 0 no matter pV 31/)...”1x or p" >1/7.,,,,,.. Since f '(p) is continuous on (0, p') with f ’(O) < 0 and f '(p°-0) > 0, we must have at least one point, namely p‘, within (0, p') with f '(p‘) = 0 by Cauchy mean value theorem. That is, both Y'YL and estimate p will have positive values in the interval (l/kmin, l/kmax). Clearly to say, if Y'YL > 0 given W and Y, then the estimate p > 0. This conclusion is expressed in the third column from left in Table3A. Case 2. Assuming Y'YL < O. In the similar way, we see that Y'YL and the estimate of p will both be negative in the boundary (l/Amin, l/lmax). Case 3. Assuming Y’YL = O. In case Y'YL = 0, Y'YL and the estimate of p will be both zeroes. Now, we see that the sign of the estimate of p is uniquely determined by the sign of Y'YL which is determined by the relationship between W and Y since Y'YL = Y'WY. Thus, we make the conclusion that the value of Y'YL and estimate of p, the mininrizer of the equation f '(p) = 0 will have the same sign in the boundary (l/kmin, l/lmax). This gives three parts of the solution space of the estimate p taking zero, positive and negative values respectively. 24 An example is given in appendix (Appendix3A) where the dimensional number is 3. I choose Y1 = (1 1 l) ' and Y2 =(10 -1) ' so that we will get Yl'YlL = Y1'W Y1=16 > 0 and Y2’Y2L = Y2'W Y2 = - 6 < 0 respectively. In the first case, the estimate of p will be positive, and in the second, the estimate of p will be negative. The separated parts of its . 2 . .. log-likelihood firnction G(p) = —;lnlAI and H(p) = ln(o 2 ) , and their corresponding derivatives are g(p) and h(p). In Figure3A-Figure3J, I show the likelihood functions and their corresponding derivatives for two different Y vectors in order to demonstrate how the positive / negative estimates of p will be located. [Figure3A — Figure3J about here] I briefly explain Figure3A — F igure3J as follows. Figure3A is the first part of the log- likelihood function, namely G(p) = -(2/n)ln(I - pW). In the figure, the y axis represents G(p). Figure3B is the derivative of the first part of the log-likelihood firnction G(p), namely g(p) = G’(p). Both functions G(p) and g(p) are independent of Y. The second part of the log-likelihood function is related with Y. The shape of the figure will vary when the given Y varies. Figure3C is the second part of the likelihood function H(p) = ln(0'2) when Y = (1 1 1)’, and Figure3D is the derivative of the second part of the likelihood function, namely h(p) = H’(p) when Y = (1 1 1)’. Now sumrrring up part one and part two of the log-likelihood fimction, we get Figure3G, the figure of the likelihood firnction, namely f (p) when Y = (1 1 l)’, and F igure3H, the figure of the derivative of the likelihood function, namely f ’(p) when 25 G(p) 2 G03) = -;ln(1 - PW)- The frrst part of the log-likelihood function, 11 = 3. Figure3A 26 g(p) 100- 50' g(p) = G'(p) =(-%ln(1- 9W7),- ....50 . —100* The derivative of the first part of the log-likelihood function, 11 = 3. Figure3B 27 Hl(p) = ln(62 (VI). The second part of the log-likelihood function where y] = (1 1 1)’, n = 3. Figure3C 28 111(1)) 20- 10- h1(P)=H. (p)=(1n(62 y,))- The derivative of the second part of the log-likelihood function Where y] = (1 1 1)’, n = 3. Figure3D 29 H203) = 111(62 y, )- The second part of the log-likelihood function where y2 = (1 0 -1)’, n = 3. Figure3B 30 h2(1)) h2(P)= H2 (P)=(1n(62 y,))- The derivative of the second part of the log-likelihood function Where y2 =(10 -1)’, n = 3. Figure3F 31 f1(p) 2 f1(p) = G(p)+ H1(p)= -;;ln(I-pW)+ln(cr2 2). The log-likelihood function where y1 = (1 l 1)’, n = 3. Figure3G 32 MP) 60* 40- 20* 5.152 D -20 . -60 . . 2 ' 7 f: (P) = g(P)+h.(P) = (-;1n(1- PW)) +(1n(6’ y,))'- The derivative of the log-likelihood function where y] = (1 1 l)’, n = 3. Figure3H 33 f2(p) -1- 2 f2(P) = G(p)+Hz(p) = -;1n(I-pW)+ln(0 (3.1.1) Y'YL = 0 (3.1.2) Y'YL < 0 (3.1.3) respectively. It’s easy to see that Y'YL = Y’WY = Y'PAP'Y = (P'Y)'A(P'Y). Let P'Y = Z, so that Y = PZ because P'P=I. The transformation between Y and Z is orthogonal and Y'YL is L, 0 0 transformed into Z'AZ. Corresponding to A = 0 L2 0 , I write Z = (Z. Z2 Z3)’ 0 0 L3 where 23 is the subset of vector Z and get Ll 0 O Zl 3 Y'YL = Z'AZ =(zl Z2 2, o L, o 22 = 22,. L2, 0 o L, z, "' 38 Recalling that L2 is a nil block, L1 and L3 are both diagonal with all positive and all negative eigenvalues respectively, I transform the equations (3.1.1), (3.1.2) and (3.1.3) into: S,(7.,2)=Z7., -z,2 +Zr,-z,’>o (3.2.1) ta], ta], S,(7.,Z)=Z7., -z,.2 +22, ~z,.2=0 (3.2.2) is]. re], S,(7.,Z)=Z7., 2,2 +27, -z,.’<0 (3.2.3) rel. is], where 23 (ieIl , 13) are the elements of 2’s subset Z], Z3 respectively. Clearly, now we get the solution space of the estimate p when p equals zero from (3.2.2) which is actually an n-dirnensional conicoid. The “inside” and “outside” of the conicoid would be the solution spaces from inequalities (3.2.1) and (3.2.3) respectively. Thus, we get the construction of the solution space of the estimate p from a weight matrix W given any observation vector Y. By the orthogonal transformation, we transform W (and Y) from W and Y space into A (and Z) in A and Z space where each axis Z,- (weighted by the corresponding eigenvalue 7.)) represents the ith factor. We notice that A is diagonal. To make the space visible, we draw a conicoid in a three dimensional space (see Figure3K). This figure is based on the data from the example 1 (see Appendix3A) where W is a 3x3 matrix on X or Z1 (7.1 = 5.50963), Y or Z2 (7.2 = -4.55287) and Z or Z3 (7.3 = -.956762) coordinate system within a range of (0 3 XS 2,-2 g Y s 2). 39 [Figure3K about here] Figure3K gives the construction of the estimate of p in the Z space based on the weight matrix W given in example one. When a given Y is transformed into Z space, and if the corresponding Z is exactly on the surface of the conicoid, we get estimate of p = 0. If the transformed Z is “inside” or “outside” the cone, we will obtain estimates of p > 0 and < 0 respectively. When the dimensional number n is larger than 3, we are unable to draw a real figure, but the construction will be the same as an n-dimensional conicoid. 3.2. The “Far End” method of the initial value selection. 3.2.1. Ord (1975) suggested using Y'YL / Y'Y as an initial value for the Newton - Raphson iteration. In §3. l , we obtained the graphical impression that the estimate of p would be pretty close to Y'YL / YL'YL, not Y’YL / Y'Y. It is possible that the latter can be larger than 1 / kmax, and out of boundary (I/Lmin, l/Amax) before any iteration, producing an immediate problem. It could also be the case that the latter is negatively larger than l/Amin, and out of the boundary as well. Both cases will be verified in chapter 6 when I run a data simulation. On the other hand, it is also possible that although we selected an initial value within the sensible range (llkmin, 10.4mm), we may not get to a “good” place within the boundary, and the estimate p in the next steps may go out of the boundary and take huge values easily. It is “flying out” or “falling out” of the boundary, a difficulty bothering researchers for long. In the following, I talk about the initial value selection problem in detail, and suggest a “far end” method. 40 5.509632,2 —455287Z,2—.956762Z32 = 0. Figure3K 41 3.2.2. A new method of the initial value selection based on the construction of W and the relationship between W and Y. We know that for f '(p) = g(p) + h(p), the derivative of the likelihood function, g(p) is determined by W only, and h(p) is determined by both W and Y. Looking at the equation for h(p) = O, or 90’.) Y. - w. Y'Y—2pY’Y, + p2(Y,) Y, h(p)=2- =0, we see that the numerator of the left side is linear in the parameter p while the denominator is a quadratic form of p, so that the equation has one and only one intersection point p‘7 = Y'YL / (YL)'YL at which h(p) = 0. We may look at the graphs Figure3D and Figure3F from the example 1 given in the appendix to get an intuitive impression. In the graphs, we see plv > 0 and p2V< 0. But in both cases, we see that when p > pV, we have h(p) > O, and when p < pv, h(p) < 0. There are two “peaks” symmetrical to the intersection point. Simple calculation shows that the peaks have width {(Y'Y)((YL)'YL) - (Y'YL)2} 1’2/ ((YL)'YL), and height ((YQ'YL) / {(Y'Y)((YL)'YL) - (Y'YL)2} ”2. The product of the width and height is a constant one. In case the width is small, the peak would be close to the intersection point pV = Y'YL / (YL)'YL and is ‘steep’, just as h|(p) shown in Figure3D. Otherwise, when the width is not small, the peak appears relatively far from the intersection point pV and is 42 ‘flat’, just as h2(p) shown in F igure3F . Notice that the scales in the figures are different due to the limitation of the computer figuring function. Assuming Y’YL > 0 so that pV > O, we know that the solution of the estimate p to the equation f ’(p) = O is located between 0 and pV. If the width is small, then the peaks would be close to pV, and are relatively ‘steep’. We see that there would be a twist around the peak area. Once an estimate pr fiom an iteration falls into this peak area, and it happens that the twist is “sharp”, the tangent taken for the next iteration by the Newton tangent method may cut the p axis far away, or “fly out” easily. Our best choice (based on intuition) is to take the initial value around .90 of the value min(pv, l/lmax), or even .95 of the value. That is, to take it between 0 and p' = min(pv, l/lmax) and very close to the latter. The selection is the same for the case Y'YL < 0. We may call this selection as “far end” method. At last, the estimate of parameter p is obtained, and the estimate of parameter (52 follows via formula (1.4) from chapter 1. The verification of the difference between Ord’s and my new initial value settings is in chapter 6. 43 Chapter 4 A TRANSFORMATION FROM THE W, Y SYSTEM TO THE A, Z SYSTEM A DEVELOPMENT OF THE SAM MODEL In this chapter, I will address problems regarding the parameter p, both the technical aspect regarding the estimation of p, and the theoretical aspect regarding the function of p in the model. There is a new understanding about the boundary regularity (l/Amth/kmax) for the parameter p. 4.1. A phenomenon: Why does the estimate p approach an extreme value easily? One big issue regarding the SAM model is the interpretation of parameter p, either the real effect of p in the model, or the estimate of p. In the literature, researchers have mainly focused on the fact that Newton-Raphson maximum likelihood estimates may take extreme values. Efforts were made to restrict this estimate p within specified boundaries. Different boundaries were suggested by Ord (1975), Doreian and Hunmon (1976), Anselin (1982), and Leenders (1995). Recently, some began to consider the situation that the real parameter p may take any real values while carrying the idea of finding the boundary for estimate p. Commenting on the boundary problem, Leenders (1995) first 44 said that “... the ‘appropriate‘ regularity condition is that p estimate may attain any value, except 1/7.,-, i=1,2, ..., g for it) real” (pageIOO, Appendix to chapter 3). Then after citing different versions of restrictions in the literature such as “-1/|).max | S. p S I/Amax” from Ord (1975), also from Doreian and Hunmon (1976), and “1/7.mi,, < ‘3 < 1” from Anselin (1982), Leenders emphasized that the case when the p estimate falls out of the boundary “is not even a rare case”, and said that “Substantively, however, it may be difficult or even impossible to interpret values of 6 that falls outside of the unity interval” as a conclusion (page71 ). We know that theoretically, there are n solutions of .3 to the equation f ’(p) = 0 as we see in Figure3H and Figure3J intuitively. By using the “separation” and “far end” methods I introduced in last chapter, we may find any of these solutions without technical difficulties. But we have a problem of making choices. That is, which one(s) of {5 should we choose to be the solution to the model (1.2)? We know that one of the n solutions is located in the range (1/Xmin,1/3.max). We also know that pV = Y'YL / (YL)'YL is the global minirnizer of the random error term of the model (1.2), and pV can be located in or out of the range ( 1/7.miml/7.max) based on both W and Y. We may address the issue of boundary now. The phenomenon of {5 “flying out of the boundary” comes from two different situations. Situation 1. If pV is located in the range (l/kminJ/kmax), then we may start the iteration of ML estimation with an initial value which is ideally located in the range (“Mind/Kmart). The random error term is to be minimized in a global sense. But even 45 doing so, we often get a large value of (3 that is far out of the range as a result. If this is what we may call “flying out”, this is actually a computational error as I said in §3.2.2., and is easy to avoid by applying the “far end” technique. Situation 2. If p’7 is not located in the range (l/lmiml/itmax) but another one, say (l/kbl/km) where one of 10.3 or l/lm could be either -oo or +00, then we have a problem of making choices. First choice is, we may prefer to choose the [3 which is located in the same range as pV is. By doing so, we have the random error term minimized in a global sense, but the p is out of the boundary (l/lmiml/kmax) and could be very large. Noticing that the similar computational error of “flying out” mentioned in situation 1 may also happen here, we still need to avoid this computational error. Second choice is, we may prefer to choose the ,3 which is located in the boundary (I/AmmJ/Amax) so that the (3 could not be large, but the random error term is minimized only in a local sense because pV is not in the boundary as we know. In social network analysis, we prefer to make the second choice. The reason is given later in this chapter. 4.2. The technical reason. Here is the technical reason as to why the estimate of p approaches an extreme value: the relationship between W and Y may cause the estimate of p to fly out in computation. It’s just a technical error. In the above, we have mentioned that the estimate of p may “fly out” of the range (1/7tmin, llkmax) caused by the twist from the relationship between W and Y. What is more, when W's dimension is relatively large, and the matrix W is not very sparse, it is 46 usual that the eigenvalues 7.4m and 2min might be relatively large in the absolute value sense, then the range (l/Xmin, l/Xmax) would be pretty small, and the “flying out” would occur rather easily. To avoid this technical error, we need only to select an initial value by picking from “far end” as we have said in §3.2.2. 4.3. The theoretical reason. Here is the theoretical reason as to why the estimate of p takes an extreme value: a problem of making choices. The original autoregression model is Y = pWY + e, where the error term a is assumed to have an identically independent (i.i.d.) normal distribution N(0, 0'21). Now, an orthogonal transformation from W, Y system to A, Z system will help to find a rule in the new A, Z system. That is, the standard deviation of each ith factor Z is adjusted by both the autocorrelation p, and the corresponding ith eigenvalue 7.3. Let’s explore it. As I have said in chapter 3 (§3. 1 .3), mathematically, any square matrix W can be transformed into a Jordan canonical standard form which might not always be diagonal with an orthogonal transformation matrix P. But if the order of the eigenvalues of W is fixed, the orthogonal matrix P is unique when W is non-singular. With an orthogonal transformation matrix P for W = P’AP, and writing PY = Z, the SAM model is transformed from the W, Y system to the A, Z system as Z= pAZ+Pa. 47 It’s well known that the new error term Pa remains i.i.d. normal, and we may rewrite it as a for the convenience without confusion. We know that the likelihood function for the model in W, Y system is |A| ——l——l"'A AY We 2 ’ (4.1) whereA=|l~pW|. Ltp. oz) = When it is transformed into the A, Z system, the likelihood function is now l___1 —pAl— :32u— pA)(l— pA)Z (Jere—i 140.62) (4.2) 1 As I have said in §1.2.4, we may simply rewrite |A| as «Al2 )A to avoid the negative value problem in both (4.1) and (4.2), and the likelihood function (4.2) becomes fi((1— p7.,)2)’4 Zz,(1— p7.,)zz,. L(p, 02) = i=l exp _ I=I (fine—2) 202 = " 1 exp _ 2" . (4.3) H 1/2n(%1— p7. )) 2 20%] — pm) 2 Now we see that for 2, ~ N(O, 0'2 /(1-p7.i)2), i=1,2,..., n, this likelihood function is a product of densities of all {23 (the ith factor)} which are independent but not identical because their variances are different from each other. I discuss the variance 632 = (52/(l-p}.i)2 below. 48 (l) p cannot be a reciprocal of any eigenvalue of W, (namely l/7i.i) for any index i, or the function (4.3) will make no sense. (2) When p approaches (1/7.i) for any index i, the corresponding variance 0'32 approaches infinity, the function (4.3) is converging to zero in a limit sense. This process can be seen in different ways. In mathematics, it is simply a process of limitation. In physics, this is the case when the factor z, is becoming a white noise, while all other factors are dysfunctioned. In social sciences, this might be the case in which only the factor 2 = 62/(I-pAi)2 approaching 23 is becoming a dominating figure with its variance at infinity while all other factors are muted with their variances 032 = 0'2/(1-p7.i)2 remaining finite. (3) When p = 0, all 2, have the same variance 02, so that all 23 are i.i.d. normally distributed. (4) Now p at 0. First for the case p > 0, assuming W has a positive eigenvalue set {A1, 7.2, , 7.1,} in a descending order. (i) For all negative eigenvalues 7.1, we have 1 < 1"ij , so that all the corresponding variances 01-2 are shrunk. (ii) When 0 < p < 1/7tl so that 0 < p < l/7ti for i = 1,2, ..., k, we have 1 > 1-p7ti> 0 for all i. In such a case, the variance 032 is enlarged for all i = 1,2, ..., k. (iii) When 1/7.1 < p < 1/7.2, then for i = 2,3, ..., k, we have 1 > 1-p7.i> 0 and the variance 632 is enlarged for i = 2,3, ..., k only. The variance 0,2 behaves as follows. (a) If l/7.1 < p < 2/7.1, then 0 > 1-p7.1 > -1, the variance 012 = (52/(1-p7.i)2 is enlarged. 49 (b) If 2/7.l 1-p7.., the variance 612 = 62/(l-p7ti)2 is shrunk. From (a) and (b), we see that for 012, when p is increasing away from 1/7.., the variance 0'12 was larger than 0'2 but decreasing to 0'2, and is then decreasing and shrunk less than 0'2. (iv) The discussion for the following 7.2, , 7.k would be the same as for 7., in part (ii) and (iii). (v) For the case p < 0. We will get the similar series of results as in (i-iv). Let’s summarize the above (1) - (4) and talk about the change of the variance 0,2 in the following. Case 1. When p = 0, for all weights 7.3, positive or negative, their variances 632 = (32/(1-p7ti)2 remain the same as 02. Neither enlargement, nor shrinkage would occur. This is a “fair and natur ” status. Case 2. When 0 < p < l/7.1 where 7.1 is the 74m, for all negative weights, their variances are shrunk. The larger the eigenvalue is (in an absolute value sense), the greater the shrinkage will be. When 0 < p < 1/7.1, for all positive weights, their variances are enlarged. The larger the eigenvalue is, the more the enlargement will be. This case shows a kind of “bias”. In this case, the factors with negative weights contribute less than it should, and the factors with positive weights contribute more than it Should. 50 We notice that in classical factor analysis, variances are proportional to the corresponding eigenvalues where all eigenvalues are non-negative. But in SAM models, variances are linked to both positive and negative weights, especially to those whose absolute values are large. Case 3. When 1/7.. < p < 1/7.2 where 7.. is the 74m, for all negative weights, their variances are shrunk (more than they were in case 2). The larger the eigenvalue is (in an absolute value sense), the more the shrinkage will be. When l/7t. < p < l/7.2, for all positive weights, their variances are adjusted differently. For all positive weights 7.2, 7.3, ..., 7..,, their variances are enlarged (more than they were in case 2). The larger the eigenvalue is, the more the enlargement will be. When 1/7.. < p < 1/7.2, for the positive weight 7.., its variance is frrst being enlarged, then shrunk, and shrunk more and more when p is becoming larger and larger. This case shows a kind of “bias and manipulation”. In this case, the factors with negative weights contribute much less than they did in case 2. Those factors located right of p with positive weights will contribute much more than they did in case 2, while for Z., the factor located lefi of p with positive weight corresponding to 7.., its variance is first being enlarged, then being shrunk. That is, Z. will contribute first more then less and less. Case 4. When p is becoming larger and larger, similar to the case 3, the variances of all negative weights are shrunk more and more. 51 The counts of factors with positive weights located right of p will be less and less, the variances of these remaining positive weights are enlarged much more. Their contributions are artificially increasing. The counts of factors with positive weights located left of p will be greater and greater, the variances of these positive weights are first enlarged but then shrunk much more. Their contributions are artificially first increasing then decreasing. This case also shows a kind of “bias and manipulation” in a more serious situation. It is biased more and more against the original weights. Case 5,6,7. These are cases for 0 < p. They are similar to the case 2,3,4 for p > O, and I don’t write in details. Seemingly, all the above cases are essentially a problem of choices. We may take any value of parameter p in the model. Taking any non-zero p means a kind of bias. When this p is becoming larger and larger (in an absolute value sense), the weights of this system are much more seriously adjusted, and less original. We may choose an extreme value for p for a special reason, but it is mostly less meaningful. One example might be the case of a congress meeting where each congressman is represented by a factor. Each one is with his eigenvalue either positive or negative depending on the case. Senior ones carry large eigenvalues and large variances, and influence the system more than those junior ones who carry small eigenvalues and influence the system less. In case parameter p goes out, say positively, of the boundary, then some special situation happens. Assuming the parameter p is now between the 52 reciprocals of the largest and the second largest eigenvalues, then not only the variances of all negative eigenvalues are being shrunk, the variance of the largest eigenvalue is also being shrunk. The variance of the second largest eigenvalue, and the following ones, are being enlarged at the same time. Back to the congress meeting, such a situation means that not only the influences of all congressmen from negative side are shrunk, but the influence of the first leader from positive side (represented by the first factor) is also shrunk, while the influence of the second leader from positive side (represented by the second factor) is enlarged, and so are those following positive ones. It might be because the first leader from positive side is constrained, or his role is ignored, either one is causing the system unstable. When the parameter p moves rightwards further and further, this situation is becoming more and more serious. At last, it could be such a case that the variances of all factors with negative eigenvalues and almost all factors with positive eigenvalues are shrunk, while the variances of the remaining one or two factors with very small positive eigenvalues are highly enlarged. In the congress, it means that only one or two very junior persons are making decision while all others, senior and junior ones are constrained. We should believe that this kind of decision could not be very much stable, and the system itself is not stable at all. 53 Chapter 5 THE DEFINITION OF PARAMETER p AND THE W(a) FAMILY OF THE WEIGHT MATRICES In this chapter, I first give parameter p a general definition as an indicator of consistency between W and Y in the model. Second, I develop a W(a) family that is a general format of the weight matrix W with a constant “a” as the principal diagonal elements. The assumptions of wy- 20 ( i at j) and w.,- = 0 carried on in chapter one through chapter four are not required now. Specifically, we now consider W’s principal diagonal elements “a” as a constant, taking zero or non-zero values. Based on the understanding of the W(a) family in SAM models, I try to extend the concept of the classical Factor Analysis method to a more general level. 5.1. The definition of the parameter p. In Xu (1996), an early paper working on spatial autoregression models, I described “The autocorrelation parameter p in the model indicates the extent of the members’ communication-cooperation within this group and is important to estimate.” Now I give the parameter p a definition below. 54 In spatial autoregression models (SAM), the parameter p is an indicator of the extent of consistency between a weight matrix W and an observation vector Y. With 7..,... and 7..,“... as the maximum and minimum eigenvalues of W, if W and Y are more consistent, the estimate of p goes positively higher, and intends to reach 1/7.max, the maximum of the range of estimate of p. If W and Y are less consistent, the estimate of p goes negatively higher, and intends to reach 1/7.min, the minimtun of the range of estimate of p. The word of “consistency” is in an n-dimensional Euclidean distance sense based on SAM. Now we may consider the parameter p as a correlation coefficient between a general matrix W which is not necessarily a variance — covariance matrix, and a vector Y. It seems to be a new measurement in statistics for the social sciences. 5.2. The geometrical reasoning of the definition of the parameter p. The transformed Z space as in chapter 4 is spanned by factors Z., Z2, ..., Zn with the weights of the corresponding eigenvalues 7.., 7.2, , 7... respectively. Notice that the Z space has its combined direction 2* which is essentially the combination of all these axes weighted by their directional weights {7..}, the corresponding eigenvalues. When an observation Y° is transformed into Z°, the estimate p is actually a projection of 2° on 2*, the combined direction of {2.} in the Z space. If the value of this projection is positive and high, we see that Z° is consistent with Z*, the combined direction of Z space, and we say that Y° is positively consistent to the matrix W. If the projection is negative and high, we say that Y° is negatively consistent with the matrix W. That is why 55 we may consider the parameter p as a correlation coefficient between a vector Y and a weight matrix W, a new concept in statistics. Z space is a multidimensional space. It could happen that a vector Y is exactly transformed onto one axis Z. in Z space. But in most cases, the corresponding Z from Y is usually “between” those axis 2.. So that mathematically, I would prefer to call this new technique as a “non-eigenvector analysis”. 5.3. A comparison of SAM with linear models. In a sense, the function and behavior of estimate of p seems to be similar to the estimate r of the correlation coefficient between two vectors X and Y. This estimate p is substantively a kind of correlation coefficient, but it is between a vector Y and a matrix W. In TableSA, the second column refers to a correlation coefficient analysis between two standardized variables X and Y. The model is Y. = rX. + 3., 8. ~ N(O, 0'2), i=1,2,..., n with cov(e., a.) = 0 when i at j. The parameters r and 02, the variance of the random error term need to be estimated. The third column in TableSA refers to a spatial autoregression model with a weight matrix W and an observation vector Y. The model is Y = pWY + a, e ~ N(O, 621). The parameters p and 0'2, the variance of random error term need to be estimated. All values a, c, B, D in TableSA are constants. 56 [TableSA about here] I briefly compare the differences between these two models as below. Some more comparisons can be seen in chapter 7 (§7.1). - The solution of SAM model is not an algebraic one, so that an analytic formula for the solution of .3 is not available. 0 The range of r in linear regression model is [-1, 1] when X and Y are standardized. But in SAM model, the range of .3 is (1/7.m.n, 1/7.max) which can not be standardized to [-1, 1]. o The properties are easy to verify. 5.4. The W(a) family. In order to consider the estimate of p from weight matrix W in which all elements of the principal diagonal are a non-zero constant “a”, we need to develop the W(a) family, a more general concept of the weight matrix W for the following reasons. As we know, w.,- is usually treated as zero in the literature. This property guarantees that the summation of W’s eigenvalues equals to zero, or 27.,- = 0 which helps us to obtain the conclusion in chapter 4 that “the values of estimate of p and Y'WY have the same sign within the boundary”, and to obtain the construction of the distribution of the estimate of p in space. At the same time, the importance of the condition of w.,~ = O, and the case where w,-,- i 0 might not be emphasized enough in the literature especially in social sciences. For instance, Leenders (1995) says (page 54) that “An entry w.) of W denotes 57 TableSA estimate r estimate p model Y. = rX. + a. Y = pWY + a 8. ~ iid normal 8 ~ N(0, 0'21) cov(X., a.) = 0 cov(Y, a) at 0 definition correlation coefficient correlation coefficient between a vector Y and a between a vector Y and a vector X matrix W analytic formula available not available range ['1’ +1] (10min, 1/7vrnax) unit and scale no scale no scale sign “+” means positively “+” means positively consistent and vice versa consistent and vice versa PIOPerty r(aX+(3,b)'+1)) =r(X,Y). P(aYer) = (1/b)p(Y,W) IIISI- I/thn + co, we have from (5.1) that p(a) —) 0 if p(0) #- 0. Also, when a —> O, we have p(a) —-> p(O) for any p(0). When the value of “a” varies, we may obtain information fiom these subjects as they relate with others in the network setting. When a = 0, these egos (y.) are simple information processors. They don’t have ideas from themselves, but make decisions totally based on other subjects’ attitudes. We see that the case “a = 0” means these subjects are totally objective. When a > 0, these subjects in the model become more than simple information processors. The model effect is not purely objective. These egos (y,-) make decisions not 64 only based on others’ information but also based on their own. When “a” is positively increasing, the weights of these subjects’ own information are increasing. That is, the positively increasing “a” indicates that these subjects are more and more self-centering. The extreme case when “a = + co” means these subjects are totally subjective. It’s easy to understand that in matrix W, a —) + co actually means that all wy-( for any i at j) are approaching 0, and we find that these egos (y.) don’t want to accept information from any others for their own consideration. They are purely subjective. Practically, subjects in a social network are mostly between two extremes. Neither “a = 0” nor “a = + 00” are practical and ideal. We prefer to have “a” at an appropriate level between 0 and + 00. We see that when a is close to 0, these egos are more likely cooperative and less self-centered; when a is becoming larger and larger, these egos are more likely self-centered and less cooperative. 65 Chapter 6 DATA ANALYSIS In this chapter, I conduct a data analysis. There is a data simulation in part I and an example in part II. I use my new Newton-Raphson technique in both parts. The result is the realization of the definition I gave in chapter 5 to the parameter p in the SAM models. 6.0. The source of data. In chapter 4 of F rank (1996), a weighted data set collected in January 1993 was introduced. The initial motivation for using this data set was to identify cohesive subgroups from professional discussions among teachers from a high school named “Our Hamilton High” located in the Chicago area. In this set, a group of 24 teachers of the school were surveyed. In Frank (1996) (page106), the collected data include these teachers’ gender, race, years of teaching in the school, and their level of moral agency which was one of four measures “based on each teacher’s extent of agreement (1 = strongly disagree to 4 = strongly agree) with items referring to the teacher’s sentiments and orientation towards teaching”. Another variable p, the level of the frequency of their professional discussions, was collected such that “each teacher listed the five (or fewer) teachers with whom he or she had most often discussed professional 66 matters during the 1992-1993 school year. The teachers were asked to weight the frequency of discussions (1 = less than once a month, 2 = two to three times a month, 3 = once or twice a week, and 4 = almost daily)” (page 99) (see Appendix6A). Based on the levels of the frequency of their discussions, an association (weight) matrix W (24x24) was constructed in the following way. If the ith teacher indicated engaging in professional discussion with the jth teacher at level p, the frequency of discussions (p might take values either 1, 2, 3 or 4), then the entry w.) = p. If the ith teacher didn’t talk with the jth teacher, then Wy’ = 0. Noticing that the conversations were not mutually initiated, we observe that the weight matrix W is asymmetrical. Now I will use the information mentioned above to conduct my data analysis in both part I and part 11. Other available information collected from that school included the teachers’ subject field (the courses they were teaching) and their office numbers, but I do not use them here. 6.1. Part I. Data simulation. 6.1.1. The goal of the data simulation. The goal of the data simulation is to check the level of goodness and efficiency of my new technique and to evaluate the effects of using different initial values in the estimation. Ord (1975) suggested using the initial value (Y’YL / Y’Y), and I suggest using (Y’YL / YL’YL) as I have said in chapter 3 (§3.2.1).. 6.1.2. Simulating data in a spatial autoregression model (SAM). In an initial SAM model 67 Y = pWY + a (6.1) where 8 ~ N( 0, 021), the weight matrix W is known, Y is observed. The researcher estimates the parameter p by optimizing the likelihood function from model (6.1). In this part, I first create a multivariate random error term which is i.i.d. normal. With a given weight matrix W, a pre-chosen value of the parameter po (1 denote the value as po here to tell the difference from the original p) within the boundary of (1/7.min, 1/7.max), I calculate Y using the formula Y=a-pow)"e (6.17' which is from model (6.1). With the calculated Y as observed, I start the process of estimating the parameter p by optimizing the likelihood function from model (6.1) to get the estimate p (which would be different from the pre-chosen po). For the same pre-chosen p0, I repeat the above steps a number of times by using different random errors I created, so that the calculated Y would be different. Then I obtain a series of parameter p estimates. By getting the distribution of the obtained series of estimate of p and comparing with the pre-chosen pe, I may find the extent of efficiency of the maximum likelihood technique for the parameter estimation. I would also repeat the above steps a number of times using other different pre-chosen p0 values to find the extent of efficiency of my new technique when the pre-chosen p0 is ranging within the preferred boundary of (l/7.m..., 1/7.max) from left side to right side. Again, I would use the same series of multivariate random error terms I created before, and the same pre-chosen p0 values to repeat all the previous procedure. But this time, I 68 use Ord’s (1975) initial value (Y’Y. / Y’Y) instead of mine. Thus, I would have two series of estimates p for each pre-chosen 60. By comparing the two different series, we may have a better understanding of my new technique. 6.1.3. The plan of the work. In an early version, I used a subgroup A(4x4) of the W matrix from Frank (1996) (see Appendix6A) and found important differences. It was found that using Ord’s initial value setting, we got about 23% of estimates p within the required range (1/7.m.n, 1/7.max) and the percentage of violations to the requirement is 77%, a fairly large percentage (see Appendix6B). While using my initial value setting with the “far end” technique, we got all 100% of estimates of p being within the required range (1/7.mm, l/7.max). Seemingly, the new initial value selection method gives much better results. Since this A(4x4) is rather small, now I use a combination of subgroup A and subgroup B from Frank (1996) for my example. Teacher ID 17 and ID 8 in subgroup B were removed for simplicity, and I rewrite this matrix as A(8x8). A is asymmetrical and contains a pair of conjugate eigenvalues. Its maximum eigenvalue is 6.87348 and the minimum eigenvalue is —3.4321 1. The boundary from the reciprocal of the maximum and minimum eigenvalues then is (-.291366, .145487) = (BL, BR) where B. and BR represent left bound and right bound of the required boundary. I use nine initial values as the pre-chosen p0 in the following way. They are listed from left to right as {po,-} = {.9OBL, 7OBL, .50B., 308., 0, 308.3,, 503.3, 708.2, .90BR} (6.2) That is, (p0,) = {-.26223,-.20396,-.14568,-.O874,0, .04365, .07274, .10184, .13094} (6.2)’ 69 We see that the left four are negative, and right four are positive. The middle one is zero. With a seed =3837, mean value = 0 and variance = l, I use SAS programming to generate a series of random vectors (8x 1) as the random error term a,- for i = 1 to n. For simplicity, I chose the number n =30, not a large one. But even with n at this level, we may find that the obtained estimates p are fairly normally distributed around the pre- chosen p0 values respectively. By using each of these nine pre-chosen initial values from (6.2), I calculate Y,- for i = l to 30 based on (6.1)’. With A known, the Y,- calculated as observed, I obtain nine series of estimates p,- (i = 1 to 30) for each of these nine initially pre-chosen values. The distributions of these nine series of estimates of {pi} (i = 1 to 30) are easy to obtain. Actually, all the above steps of the plan was conducted in my SAS program 1 (see Appendix6C). The flow chart of the program is shown in Appendix6D. I also prepared another SAS program 2 (see Appendix6E). Program 1 and 2 are substantially the same except one minor difference. In program 1, the obtained estimate of the parameter p is restricted within the preferred boundary, namely (1/7.min, 1/7.max). If the estimate is out of the boundary, I treat it as missing so that the number of obtained estimates of the parameter p is usually less than 30 (see Appendix6F). But in program 2, this restriction is taken off, so that the total 30 estimates will be printed out which can be located anywhere on the real line, within or out of the boundary (see Appendix6G). 6.1.4. Discussion of the results. 70 In general, the new technique gives most of the estimate p located within the boundary. Generally but not necessarily, when the pre-chosen p0 is close to the left bound, some of the estimates are out of the left side of the boundary; when the pre-chosen p0 is close to the right bound, some of the estimates are out of the right side of the boundary. 6.1.5. Comparisons between different strategies of the initial value selections. As I planned above, I re-conduct the same program again using Ord’s suggested initial values. Now I use (Y’Y., / Y’Y) instead of (Y’YL / YL’YL) as the initial value. I found that the difference in results is obvious. A large portion of estimates of p flew out of the boundary when I used Ord’s method of the initial value selection. Figure6A helps to understand the difference between these two strategies of initial value selection. Remember I got a total of 270 8-dimensional y vectors in the above simulation. 1 used both Ord’s and my strategies of initial value selection to get initial values for iteration. Now, each of the 270 y vectors corresponds to a dot in Figure6A with the coordinate (x, y) = (Xu’s initial value, Ord’s initial value). Noticing that the scales for the horizontal and vertical axes in F igure6A are not equal, I also give individual histograms to compare the two strategies of the initial value selection in detail (Figure6B and Figure6C). [Figure6A, 6B and 6C about here] Figure6B is obtained from the initial values using my method. We may find that when using my method, the initial values distributed pretty well within the boundary. That is, 71 0RD 021$. geese a- 3 65° 0:11:10 0 SD 53000 D O D o O o on Ch 4« ° 0° $0990 a o o at! D 0 no 0 O 2. 63%|; no 325330 Cl cnfp POD 00 00$ co 0 I? O 0-: a QIchoP o 0080 we"? 8 a 8 CEO 0 o no '2‘ E a: :1 com 390 a 0 ° o o 09 0° on: :9 Q: 0 0 0Q: 0,000 can .4. D 8 O . 0%: g 0 O @O 3& «refer .5- 3? -s l 1 .03 '02 '01 '00 .1 XU For each dot in the graph, the X coordinate is Xu’s initial value, ranging (-.245376, .178482), the Y coordinate is Ord’s initial value, ranging (-6.0202, 7.363602). The required boundary is (-.1692563, .145487) which is between two arrows on each coordinate. Figure6A 72 Std. Dav I .13 Moon I -.001 N I 210.00 -.050 .000 -.225 -.175 -.125 -.075 -.025 .025 .075 .125 .115 Distribution of Xu’s initial values. The required boundary is (-.1692563, .145487) which is between two arrows. About 15.93% of Xu’s initial values are out of boundary. Figure6B 73 Std. Dov I 4.23 Hm I .70 [ill N I 270.00 foa’eefoafeefoa’ °o °°o'°°o"’°o"°a"°a" °o fee °o 0RD Distribution of Ord’s initial values. The required boundary is (-.1692563, .145487) which is between two arrows. About 97.78% of Ord’s initial values are out of boundary. Figure6C 74 about 84% of the estimates based on my initial values are located within the boundary, about 5% are out from right side and 10% are out from left side. We know from chapter 3 that the Cauchy mean value theorem plays an important role in locating the estimate p. With an initial value within the boundary, the estimate would be also located within the same boundary, an ideal result. Figure6C is obtained from the initial values using Ord’s method. Noticing the scales of Figure6B and F igure6C are different, we may find that when using Ord’s method, the initial values are not distributed ideally. That is, about only 2% of that initial values are located within the boundary, 50% are out from right side and about 47% are out from left side. Once the initial value is out of the boundary in the beginning, by the Cauchy mean value theorem, the final estimate is most likely to be located in the same interval, thus a poor result occurs. We may change the seed value when creating the random error term, and the result might be a little different, but the situation of “large portion being out of the boundary” will remain the same when using Ord’s method of initial value selection. 6.2. Part II. A practical example. 6.2.1. The goal of the practical work. Here I apply the new technique to work on a practical data set. The theoretical base is, the parameter p in a spatial autoregression model is an indicator of the level of consistency between W, the weight matrix and Y, the observed real outcome. We want to compare the different levels of consistency between W, and four observations (Y). 75 6.2.2. Steps of the practical work. 6.2.2.1. To obtain the eigenvalues of W. I use the whole weight matrix W (24x24) from Frank (1996) (see Appendix6A). This W matrix is asymmetrical, and I used F ortran programming to get {7.}, the eigenvalue set of W. (see Appendix6H). 6.2.2.2. In this school, “sex”, “race”, “year of teaching” and “moral agency” are four variables obtained from these teachers. Writing them in a vector format Y (24x1), I use each of these Ys in SAM model to estimate p, the correlation coefficient between W and each Y. In such a way, I may find an answer to the question: among those variables “sex”, “race”, “year of teaching” and “moral agency”, to which one(s) is the pattern matrix W most highly consistent? Coding sex, I have male = 1 and female = 2. Coding race, I choose two different versions W1NW2 and NB1B2. In W1NW2, I code white =1, non-white = 2. In NB 182, I code non-black = 1, black = 2. The reason why I do so is because of the existence of the two “others”. They might be coded as non-white, or as non-black. We may even code them with the middle value 1.5 which might be a little strange but make some sense. There are only two “others” in this school. I found that their level of moral agency are relatively low, and different coding technique may affect the analysis result especially when dealing with moral agency. A comparison shows that black teachers had higher moral agency than whites (and others) 76 with a mean difference around -.25 which is not significant with p value around .078 (see Appendix6D. Note the value of moral agency for teacher ID 24 who is a white male is missing. In such a case, I may redo the analysis at a dimension number 23 (one unit less than 24) level, and may still obtain general relationship between W and Y. But it would not be consistent with analyses based on other variables (sex, race and year of teaching). To be consistent, I assign teacher ID 24 a value of moral agency .21391 from teacher ID 11 who was similar to ID 24 in terms of race, gender, and year of teaching. Also, I assign teacher ID 24 another value 0.19042 that is the average value of moral agency over all white males. As a contrast, I assign him a value of moral agency —.01690 from teacher ID 18 who is a black male, teaching the same course of physical education for similar years as a contrast. 6.2.2.3. The real work of estimation using model (6.1). With W given and Yme, Ysex (with two coding methods), wa and Ymom. (with moral24 = .21391, as well as moral24 = .19042 and moral24 = -.Ol690), I estimate those ps. Using Ord’s formula (1975, page 124), I also calculate the standard deviation of those ps. Ord’s formula might be problematic, and a brief discussion is in §6.4. [Table6A about here] The result is shown in Table6A. We find that both psex and prace are a little larger than .06, pyea, is slightly below .06. These three values of pm, pm, and pyealr are pretty close. 77 W’s entries are 0, 1, 2, 3, 4. Table6A NB 1B2 W1NW2 Year of Moral Moral Moral Sex teaching Non-black White=l (ID 24= (ID 24= (ID 24= =. Nomwhite .21391) .19042) -.01690) Black=2 =2 Estimate .06051 .06147 .06340 .05910 .04653 .04644 .04503 p ' (.01909) (.01784) (.01498) (.02076) (.02932) (.02936) (.02990) % of Estimate 85.08% 90.24% 89.13% 83.09% 65.42% 65.29% 63.31% p to the (UAW) I Figures in parentheses are standard deviations. 7mm, = 14.059360 and l/Amax = .0711270. 7min = -7.701774 and 1/7tm.n = -.1298402. 78 All of these psex, prace and pyear are seemingly larger than pm... which is at around .046 level, no matter ID 24 takes which value for his moral agency. All estimates are within two standard deviations of one another, even if using the smallest standard deviation of .015. We notice that all pm, pm and pm.r are selection effects, effects of people choosing to have professional discussion to others like themselves. That is, people may decide the level they choose subjects to talk with just because of that person’s gender, or race, or because of that person’s seniority. The pmm. is the only effect which could be considered as influence. 6.2.2.4 Discussion of the table. I make the following observations: 0 W is positively associated with race with pm at .06 level (89%). It might mean that when these teachers choose subjects to initiate a professional discussion, the race is a main element to consider. They prefer to choose subjects of same race to have professional discussion. 0 W is also positively associated with sex with psex = .0605 (85%). The levels of the associations of W with sex and with race are close although sex is slightly lower. It might mean that when these teachers choose subjects to initiate a discussion, sex is also a main element to consider. They prefer to choose subjects of same sex to have professional discussion. It is not clear whether or not sex and race are associated among those teachers. A correlation coefficient matrix of all variables including sex, year, two different versions of race, three different versions of moral agency shows that there is no strong evidence to show that sex and race are highly correlated with r 79 = .299 and .293 for two different versions of races (W1NW2 and NB1B2) (see Appendix6D. 0 With pyear = .059 (83%), W is also positively associated with years of teaching but at a slightly lower level compared with race and sex. This is a relative comparison. It does not confirm that year of teaching is not important compared with race or with sex. In next chapter (§6.2.5), I would explore firrther. 0 With pmora. at .046 level (65%), Moral agency is less consistent with W than three other Ys in the analyses. There is not much difference among the three different values I assigned to the ID 24. To summarize, we get a range of 60% ~ 90% of the ratio (estimate of p) / (1/7mx) in the p estimation. 6.2.3. Compare W(0,1,2,3,4) with W(0,l). Dealing with the weight matrix W expressing connections between the regions in spatial regression models, Ripley (1981) suggested (page 98) that “This can be just a binary matrix giving 1 if the two regions have a common boundary, 0 otherwise, or it could depend on the length of the common boundary, the distances between the regions or transport costs between them”. Comparing the binary measure with the length of the common boundary, we see that the author is actually talking about the coding or re- coding of the weight matrix W. Ripley also said that “Bartels (1979) suggests that simple binary weights have proved as adequate as more complex schemes”. Let’s try to explore this idea. 80 From now on, we consider another weight matrix W* which is related to the weight matrix W I used in the above. In §6.2.1, the weight matrix has cell ranging from 0 to 4. Now I dichotomize the value as 0:0, 1-—)4:1. That is, I re-code W into W“ as w*,-J- = 0 if w.) = 0, and w*,~,- = 1 either w.) = 1,2,3 or 4. We know that the level of wij indicates the frequency of the teachers’ discussions. When W is re-coded into W* in such a way, the different level of frequency is reduced to a simple “yes or no” level, a binary one. I repeat the same work for W* as I did for W in §6.2.2.3. The eigenvalues of W* are different from those of W (see Appendix 6K). Using W*, I re-estimate the autoregression coefficient p by using the same Y values: Yracc , Yscx , cha, and Ymond, and obtained the results shown in a table (Table6B). Interesting enough, we find that for the W”, the ratios of pm, pm, (two methods of coding) and pm... to (l/Xmax) are similar to the corresponding values obtained from W. We notice that W and W“ have different eigenvalue sets. The 7.max and 7...... from W is 14.059360 and —7.701774; while the 7.max and 7,...“ from W* is 4.051115 and —2.083131. It is appropriate to use the ratio of estimate p to the bound 1/7.max or l/7tm.n to compare the results as we introduced in §5.4 and applied in §6.2.2.4. The results are shown in Table6B. The ratios of pm... to (10......) are relatively lower in W* than in W: they were at around 65% levels from W, but now they are at around 51% levels from W*. The ratio of pm, to (1/7.max) and that of psex to (10%,...) are remaining at the same levels in W* as they were in W. [Table6B about here] 81 W’s entries are 0, 1 only. Table6B NB 1 BZ W1NW2 Year of Moral Moral Moral Sex teaching Non-black White=l (ID 24: (ID 24: (ID 24: =. Nomwhite .21391) .19042) -.Ol690) Black=2 =2 Estimate .2097 .2198 .2274 .2033 .1284 .1281 .1232 p ' (.05746) (.04475) (.03344) (.06418) (.1010) (.101 1) (.1022) % of Estimate 84.97% 89.05% 92.10% 82.36% 52.04% 51.89% 49.90% p to the (l/Amax) I Figures in parentheses are standard deviations. 7mm. = 4.051115 and I/Amax = .2468456. Am... = -2.083131 and l/Am... = -.4800466. 82 How can we interpret this difference between W* and W regarding pmom? We know that in W*, the different levels (0, 1, 2, 3, 4) of fiequency of professional discussion are simplified to two levels (0, 1). When the percentage of pm... from W and Y is higher than that from W* and Y, it demonstrates that the teachers’ moral agency level is associated with how frequently they initiate a professional discussion. The relatively higher pmom. from W and Y compared to that from W* and Y indicates that the fi'equency of professional discussion is positively associated with teachers’ moral agency level. When different levels of frequency are replaced by a simple “yes / no” choice, professional discussion is not that positively and highly associated with teachers’ moral agency. This pleasant conclusion seems to be acceptable in a common sense. As we have said in chapter 5 (§5.2), the parameter p plays a role as the correlationship coefficient between a vector Y and a matrix W, then different strategies of coding / re- coding either Y or W may lead to changes to the value of parameter p. Specifically to the matrices W and W*, one is having categorical entries from O to 4 and another is re-coded in a binary way, we have found a difference between these two matrices. There might be other differences if we re-code the matrix W in still other ways. But here we are not going to do further. 6.2.4. The cause — effect relationship between W and Y. In Holland and Leinhardt (1981), the relationship described is Y as a function of W. But now we may emphasize that there might not be a direct cause-effect relationship between W and Y. We just say that W and Y are associated. That is, it is possible that Y is the cause of W. It is also possible that W becomes the cause of Y. For example, when 83 W is the association or pattern matrix among a group of subjects, and Y is their gender, then usually W should not be considered as a cause of Y since by no means the subjects’ behavior can be the reason that they have their own gender. However, in a social behavior sense, it rrright be better to assign some subject(s) a “social gender” which is different fiom his / her biological one. 6.3. About the formulas for the standard deviation of p. In TableSA and TableSB, I calculate the standard deviation of p using Ord’s formula. There are different versions of formulas regarding the standard error term of parameters in SAM in the literature. I list some of them below. Doreian, P. (1982) (page249) with 4 parameters. Duke, J. B. (1993) (page 472) with 3 parameters. Both of the above formulas can be simplified to a formula with 2 parameters as below. Ord, K. (1975) (page 124) with 2 parameters. Doreian, P. (1981) (page 367) with 2 parameters. The details of Ord’s formula can be seen in Doreian, P. (1981). Actually, all those formulas are essential the same. I notice the following facts as comments. 1. Those formulas are asymptotic; 2. The importance of the relation between W and Y was not emphasized in the formulas; 3. The fact that the boundary of p is not symmetrical to the origin when dealing with W(O) was not considered in the formulas; 84 4. Those formulas are functions of p and p is multiple-valued. It is not clear how to pick out the value(s) of p fiom a group of many for the functions. Because of the reasons above, it might be problematic to apply Ord’s formula to calculate the standard deviations of p in my work. So are the comparisons with the standard deviation of p. Since there is no other choices, Ord’s formula is used, and further consideration is necessarily expected. 85 Chapter 7 DISCUSSION In previous chapters, we have obtained a new understanding of the SAM model itself, and a new technique for parameter estimation in SAM model. Here I am going to compare the new understanding with some other multivariate analysis methods in statistics (§7.2 and §7.3.1 - §7.3.5) and plans for the further consideration. 7.1. The comparison of the new technique with the OLS technique. In the following models, Y is the outcome, X is the predictor, W is the weight matrix, r is the correlation coefficient, p is the autocorrelation coefficient, and s is the i.i.d. normal random error. I want to make comparison of the new Newton-Raphson (NR) technique of parameter p estimation in the SAM model with the ordinary least square (OLS) technique of the correlation coefficient parameter r estimation. (a). Y = rX + a This is a general linear model with constant setting to 0 for convenience. (b). Y = pWY + a This is a general autoregression model with Y appears in both sides. (c). Y = p(WY) + s 86 This is a general autoregression model but treated as a linear model by computing WY and using it as X, so that actually, Y appears at left side only. Comments. Model (a) is a typical regression model with no intercept. Model (b). If Y is one of the eigenvector of W, then the error term a happens to be zero. But practically, it’s unlikely especially when the dimensional number of W is high. Model (c) is an abuse of the spatial autoregression model Y = pWY + s by treating (WY) as an observation X and solving for the normal equation for p and o2 estimates. Calling this estimates of p and 02 as a “normal” solution, this “normal” solution is different from the solution we obtained in §3.2. We have to notice the fact that the “normal” (1'2 estimate is always smaller than that from §3.2. Can we say that this smaller estimate of the error term is better? Then why bother working on the spatial autoregression model? The answer is simple: we must keep Y equal in both sides of the equity of the spatial autoregression model, whereas the “normal” solution violated this requirement although models (b) and (c) look like the same. Let’s talk a little more below. In model (a) above, we are trying to get the solution of I; = rX where r is obtained by minimizing the error term a. In model (b), we are trying to get the solution of I} = p f' where p is obtained by Optimizing the likelihood function from model (b). We notice that in model (b), I; appears in both sides of the equity. Now in model (c), when we abuse the model by writing WY = X, we are actually trying to get the solution of 17: pX, or 87 I’ = p(WY) by minimizing the error term a. We notice that in the left side of this expression of I; = p(WY), we have I; , the predicted Y whereas in the right side of this expression, we have Y, the original observation. Seemingly, I’ is not necessarily to be the same as Y is, so that the requirement of the model (b) in which the same Y must appear in both sides of the equity is violated. 7.2. The comparison of the new technique with factor analysis: the extended Factor Analysis Method. Substantively, the new NR technique is pretty close to the classical factor analysis. I am going to talk a little more about the comparisons of the classical factor analysis with my new technique in SAM models. The history of factor analysis technique can be traced back to the beginning of the century with the early work starting at one or two factor levels. In Harman (1960), the author said that “a principal objective of factor analysis is to attain a parsimonious description of observed data”. The author also said that “while the goal of complete description cannot be reached theoretically, it may be approached practically in a limited field of investigation where a relatively small number of variables is considered exhaustive” (page 5). In factor analysis, we are going to transform the original data set into a new factor space, and use less factors to express the original data set via the so- called “factor extraction” step. We know that the main goal is to reduce the dimension number, but when doing so, the original data set will lose some information, while applying my new Newton-Raphson (NR) technique, we will lose less information. 88 Actually, in factor analysis, the variance-covariance matrix D is a specification of W(a) as I said in chapter 5. Here D is semi-positively defrrrite so that all eigenvalues of D are non-negative with a total summation equal n, the dimension number. Some eigenvalues are big or much bigger than 1 whereas some are small, close to O or even equal 0 if the rank of matrix D is less than n, the dimension number. The step of “factor extraction” means that those factors whose corresponding eigenvalues are at the “bottom level”, namely those which are close to be 0 or equal 0, should not be considered for the factor list. Only those factors whose corresponding eigenvalues are at the “top level”, namely those which are the biggest ones, should be considered for the communality. We know that those “top level” factors are important and should be considered because of their large variation. But we can not say that “bottom level” ones are not important, and should not be considered in the data analysis. When we are dealing with the power balancing as we do in the SAM model, these negative factors (especially those factors corresponding to the negatively large eigenvalues) must be considered. That is, in SAM models, both those factors corresponding to the positively large and negatively large eigenvalues play the same important role. They are equally important. In a broad sense, the technique of classical factor analysis can be applied for many types of matrices especially for those with principal diagonal elements equal based on the SAM model. This means that the matrices we are working with in factor analysis can be not only variance-covariance matrices, but also more general ones. We surely assume all elements on the principal diagonal equal, because a meaningful result can be obtained based on this assumption. In the SAM model 89 Y = pW(a)Y + 8 where 3 ~ N(O, 0'21), if “a” equals zero, this is exactly what we were talking about in previous chapters. If “a” is not zero, then we may standardize the model as Y = (ap)(W/a)Y + 8. So that the new weight matrix will have all elements on the principal diagonal one while the parameter p does not substantially change. Now, if all entries of the standardized weight matrix W/a are less than or equal to one in an absolute value meaning, this W/a is really a variance-covariance matrix. All its eigenvalues will be positive with some zeros possibly. But if some of the entries are bigger than one in an absolute value meaning, or in other words in case the value of “a” is decreasing to zero, then W will have some negative eigenvalues, and will have more if “a” continues decreasing. When “a” is negatively large enough, we will have all eigenvalues negative. The above statement can be verified easily applying the W(a) family properties 1-3 from §5.4. We make arrangement for the whole process: from “a” equals to positive infinity, to positively large values, to positive 1, then less than 1 but still positive, to zero, and then negatively increasing until “a” reaches negative infinity. In whole this process, the corresponding eigenvalue sets are stable, with only a constant “a” adjustment. At the same time, the important fact is, their corresponding orthogonal matrix P and factors remain the same when W is non-singular. Now we see, in a classical factor analysis work, we pick out the top level factors with large variation, and delete the bottom ones with the least variation. This is because “a” equals one. When “a” takes value zero, we know that both top level factors and bottom 90 level factors must be considered because both of them carry large variation now, although the proportion of the variations are not equal as the positive and negative eigenvalues are not symmetrical in an absolute value meaning. It is now those factors whose eigenvalues are at the middle level, or close to zero, that should be removed because they carry little variation only. This process will continue to work when “a” moves in either direction. Say, when “a” is becoming negatively large, then those bottom eigenvalues and factors are becoming leading as they carry the major part of the variation. Those top ones should be out of consideration as the proportion of the variation they carry is now small. All we have found here is, the whole set of factors is in a dynamic process when the value “a” is varying from + co to -w. Here I want to emphasize that the bottom ones need to be considered because they may have potential importance while the top ones always do. In general, top ones and bottom ones are more likely important comparing with those in the middle places. A question might be asked: does it make sense when we are talking about the value of “a” being + 00 or - 0°? The answer would be “yes”. We see that in a network, if those subjects are purely objective, we have a = 0 and obtain the results before. When “a” is positively increasing, those subjects are becoming more self-confirmation centered, more subjective, and less influenced by others. When a = + 00, this network becomes a small world of purely self-confirmation centered subjects. When “a” is negative, and negatively increasing, those subjects are becoming more self-negation centered, more likely influenced by others. When a = - co, this network becomes a small world of purely self- negation centered subjects. Both of the above positive / negative extreme examples might be found in some special societies. 91 7.3. The next stage work. 7.3.1. We may apply the Newton-Raphson technique for the autocorrelation coefficient parameter p for the multilinear regression. We know that Doreian (1981, 1982, 1989), Duke (1993), Leenders (1995) were working on the model: Y = pWY + XB + a where a is the random error term. This is a linear model combined with the autoregressive information. With the newly developed Newton-Raphson technique for the autocorrelation coefficient parameter p, we may apply this new technique in practice without technical difficulties. 7.3.2. The original SAM model is time-independent, but it can be viewed as a result from a long period of negotiation and balancing. That is, we may understand that the original model is the result from the time—dependent models such as Yr+t = pWY. + a in which the observed Y is time-dependent but the information of interaction W is time- independent. When t —> 00, we get the SAM model. More generally, the model can be Yr+1 = pW.Y. + a in which the observed Y and the information of interaction W are both time-dependent. If W. converges to a matrix W, we get back Y = p WY + a if Y also converges. 7.3.3. Cressie (1993) introduced the stationary processes in a plane prescribed by Whittle (1954). With a set of translation operators defined on a plane, and a translation function 92 of operators based on the connection matrix W, Cressie shows that the SAM model is exactly a spatial analogue of an autoregressive time-series model (page406). It follows that we may directly apply the newly developed technique working on the translated model without difficulties. Similar approaches were discussed by Bartels (1979) and others. 7.3.4. We see that the above time-series models are more advanced, and have appeared in a wide range of application, namely autoregressive moving average (ARMA) models. It is worth to explore the possibility to apply my new technique of the parameter p estimation into the time-dependent model studies. 7.4. Conclusion. There is some limitation to the application of the new technique of the parameter p estimation in SAM models. So far, we can only work on those matrices W(a) with the diagonal elements identical. When dealing with W(O), the summation of W(O)’s all eigenvalues {7..} is zero so that we have 7..,... < 0 and 7.“... > 0. However, we have no technique to have the maximum and minimum eigenvalues equal in an absolute value sense. Consequently, the corresponding boundary (7mm, 7..,“) is not symmetrical to the origin, and we can not standardize the boundary to be [-1 , 1] as we do for the correlation coefficient r between two vectors. This would cause the interpretation less intuitive, and a little computationally complicated. We now have the newly developed technique to estimate the parameters p and o2 in SAM models. We also have the definition and interpretation of parameter p estimate. 93 SAM models represent the relationship between a vector Y and a weight matrix W. When there is association among a group of subjects, we always have chances to explore the relationship between the pattern of the mutual motion and observations obtained from the group of subjects. 94 APPENDICES 95 Appendix3A Example 1 0 l 3 7., 550963 1 1 W=1 0 4,7.= 7.2 = —455287 ,Y,=1,Y2= 0 3 4 0 7.3 -0.956762 1 -1 2 " 7.. g, 4) . ( ) n r=l (1- pki) _ (2) 550963 + —4.455287 + -0.956762 3 l-550963p l+4.455287p 1+0.956762p ' Y. IYiL —Y.Y.L h,(p)=2- p L) , ,(i = 1,2) so that XZ-2PYY+92I1( L) Ii. h(p)_2. P(YIL)YiL—Y1'Y1L _ 2(90p—16) l - I — _ 2 Kx—szrmpzmrm ‘3 32"”) , , 2 12 2 2' hh—zpnn.+p Y::)13, ” 9+ 7“ Solve for f,'(p) = g(p)+h,(p) = o (i = 1,2), we get .3, z.152 and A p2 z—JZ. 96 Appendix3B The calculation of the conjugate eigenvalues We need only check f ’(p) and f "(p) with one pair of conjugate complex eigenvalues here. Assume that M2 = u i iv with v at 0, then for any real p, i A, _ u+iv + u—iv _2[u(1—pu)—pv2] .=1 l-pk _1-pu-ipv 1-pu+ipv_(l-pu)2+(r>V)2 22: 1,2 _ (u+iv)2 + (u—iv)2 .-=.(1-p7~..)2 (l-pu-iPV)2 (1-pu+ipv)2 = 2{2 -2]—4puv2(1 — pu» _ [(1- pu)2 +0»)le Both terms involving the eigenvalues remain real, and so do f '(p) and f "(p) as well, since their remaining components are real. 97 Appendix6A The raw data matrix W(24x24) N Group And Actor ID 24 IAAAAIBBBBBBICCCCCCCCIDDDDDDI | l | I | I |22 21 I 111 11 I1*1121I Group IDI7129I326078|55043694I8*1312| --------- +—--—+--——--+-------—+--—-—-+ 1 A 7|A213I ...... I ........ I...1..| 1 A 1I4A3.I ...... I.4 ...... I ...... I 1 A 2|33A.| ...... l ........ I ...... I 1 A 9I433A| ...... I ........ I ...... I --------- +—-—-+---—--+——-—---—+-----—+ 2 B 23I 2 IB443 I ........ I ...... I 2 B 22I 1 I4B I 4 I 2 | 2 B 6| l4.B I ........ | ...... I 2 B 20I I33.B. I ........ I 1 l 2 B 17| 3 l3 .3B.I ........ I 3 2 I 2 B 8| I .lBI ........ I ...... I --------- +-——-+—-—-—-+--—--—--+------+ 3 C 5|....I ...... IC...3.33I.3....| 3 C 15I.4..|..4...I.C.4..4.I4 ..... I 3 C 10l....I ...... I33C.4.3.I..4...I 3 C 14I.4..I.4....l444C....I ...... | 3 C 3|3...I.4....|4.44C...I ...... I 3 C 16l.1..| ..... 4|3.2.3C..I ...... I 3 C 19I....| ...... I444..4C4| ...... I 3 C 4|....I ...... l3..3.44CI ...... I --------- +-—--+------+-----——-+-—————+ 4 D 18I 1 I ...... |.1 ...... ID 1...I 4 D **l l ...... l4.3 ..... l3D4...I 4 D 11I I ...... I4 .4...4l44D...| 4 D 13I. 3 I 1 I ........ I. .D3.I 4 D 21| I 3 | ........ |.343D.I 4 D 12I 1 | 1 l ........ |.3..3DI The value in each cell represents the extent to which teacher in the row indicated engaging in professional discussions with the teacher in the column (a value of 4 is almost daily and a value of 1 is less than once a month). 98 Appendix6E The comparison of the results from different initial value selections Initial value of p % of the estimate % of improvement of p being within from last step (l/Min, ln‘vmax) Ord’s p° = v'vL / Y’Y 23 p°1 = 0.0 59 36 p°2 = v'vL / (YL) 'vL 89 30 p°3 = .75min or 96 7 .75max * p°4 = .90min or 100 4 .90max * * min = min(pv, 1mm) if Y’YL > o and max = max(pv, l/Kmin) if Y'YL < 0. 99 Appendix6C A SAS program (1) *All estimates will be within the boundary, or treated as missing. options nocenter; proc iml; n=8; nr=30; nk=9; w={ \ OOOOQWDO OOHNWUJON OOOOUJOWH 000000000 wa00000 w00b0000 000-50000 v‘ ‘ ‘ H (I) H (8); *print w; *print 18; zr={6.87348 6.80376 0 —.895568 -1.72068 -l.72068 -3.43211 -5.9082}‘; *print 2r; zc={O O O 0 .687622 —.687622 0 O}‘; *print zc; BR=.145487; *print BR; BL=-.l692563; *print BL; rrO={-.15233 -.11848 -.084628 -.050777 0 .043646 .072743 .10184 .130938}; print rrO; r0=j(l.1,0); ee=j(n,nr.0); e0=j(n,1,0); seed=3837; do i=1 to n; do j=1 to nr; ee(|i,j|)=0+1*normal(seed); *if abs(ee(|i,jl))>1 then eeIli,jl)=.; end; end; *print ee; 1amr=j(n,1,0); lamc=j(n,1,0); 100 y=j(n.nr,0); *y=0; a=j(n,nk,0); b=j (n,nk, 0); c=j(n,nk,0); a=l-zr*rr0; *print a; b=zc*rrO; *print b; c=a#a+b#b; *print c; dcc=j(1,nr,0); ecc=j(l,nr,0); gcc=j(l,nr,0); yyc=j(n,1,0); Rho=j(nr,nk,0); ah=j(n,1r0); bh=j(n,1,0); h1=j(n,1,0); hO=O; j0=0; fh=0; fj=0; XX=O; af=j(n,1,0); bf=j(n,l,0); fl=j(nrllo); aaf=j(n,1,0); bbf=j(n,1,0); ffl=j(n,1,0); XX=O; start GetRho; m=0; eps=1.0E-O6; if abs(gc)eps & abs(gc)>eps then do; af=(l-XO#zr)#zr-XO#zc#zc; bf=(1-XO#zr)##2+(XO#2C)##2; f1=af/bf; sZ=dc-2*XO#ec+XO#XO#gc; ff=2*f1[+]/n+2*(XO#gc-ec)/52; if abs(ff)eps then do; aaf=((l—XO#zr)#zr-XO#zc#zc)##2+zc##2; bbf=((l-XO#zr)##2+(XO#zc)##2)##2; ffl=aaf/bbf; lOl sZ=dc-2*XO#ec+XO#XO#gc; sff=2*ff1[+]/n+2*gc/sZ—4*(XO*gc-ec)/52/52; if abs(sff)eps & m<60 then X1=XO-ff/sff; XO=X1; goto repeat; end; end; finish GetRho; ***********************~k*~k**~k***~k***~k********************************* ak=j(n,1:0I; bk=j(n,1,0); Ck=j (n, 1'0); II=j (1'1, n! 0); WI=j(n,n,O); IW=j (1'1, n! 0); do k=l to nk; r0=rr0[k]; ak=a[,k]; bk=b[.k]; Ck=c[lk]; IW=inv(I8-W*r0); *print IW; do j=1 to nr; e0=ee[,j]; *print e0; dc=j(1,l,0); eC=j(1,1,0); gc=j(l.l,0); yyc=IW*eO; *print yyc; dccljl=yy0‘*yyc; ecoljl=yy0‘*(W*YYC>; gcc[j]=yyc‘*(w‘*w*yyC); dc=dcc[j]; *print dc; ec=ecc[j]; *print ec; gc=gcc[j]; *print gc; XO=.9995*ec/gc; run GetRho; RhOIIj,k|)=XX; 102 if Rho(|j,k|)(BL then Rho(|j,k|)=.; if Rho(|j,k|)>BR then Rho(|j,k|)=.; end; end; print Rho; varnames={rhol rhoZ rho3 rho4 rhoS rho6 rho7 rhoB rho9}; create norms from Rho (|colname=varnames|); append from Rho; close norms; quit; proc means data=norms; var _all_; output out=normdata mean=mrhol mrhoZ mrh03 mrho4 mrhoS mrh06 mrho7 mrho8 mrho9 var =vrhol vrhoZ vrh03 vrho4 vrhoS vrho6 vrho7 vrh08 vrho9; proc print data=normdata; var _all_; title'Empirical Distribution of Rho estimation'; proc univariate plot data=norms; var _all_; run; 103 Appendix6D - A Flow Chart @ Get W, Y Get eigenvalues {All} I Get functions f' (p) , f"(p) I Get EPS, M Get Initial value p0 p0 = p0 _ fl (p0)/fr( (p0) M=M+l NO I f' (p) Il then ee(|i,j|)=.; end; end; *print ee; 1amr=j(n,1,0); 1amc=j(n,1,0); 105 y=j(n.nr.0); *y=0; a=j(n,nk,0); b=j (n! nkl 0); C=j (n; nkr 0); a=1—zr*rr0; *print a; b=zc*rr0; *print b; c=a#a+b#b; *print c; dcc=j(l,nr,0); ecc=j(l,nr,0); gCC=j(1,nr,O); yyc=j(n,1,0); Rho=j(nr,nk,0); ah=j (nl 1(0); bh=j(n,1,0); hl=j(n.1,0); hO=O; j0=0; fh=0; fj=O; XX=O; af=j (nr 1! 0); bf=j(n,1,0); fl=j (n! 110); aaf=j(n,1,0); bbf=j(n,1,0); ffl=j(n,1,0I; XX=O; start GetRho; m=0; eps=1.0E-O6; if abs(gc)eps & abs(gc)>eps then do; af=(1-XO#zr)#zr-XO#zc#zc; bf=(l-XO#zr)##2+(XO#zc)##2; f1=af/bf; sZ=dc-2*X0#ec+XO#XO#gc; ff=2*fl[+]/n+2*(XO#gc-ec)/52; if abs(ff)eps then do; aaf=((l-X0#zr)#zr-XO#zc#zc)##2+zc##2; bbf=((l-XO#zr)##2+(XO#zc)##2)##2; ffl=aaf/bbf; 106 sZ=dc—2*XO#ec+XO#XO#gc; sff=2*ff1[+]/n+2*gc/52-4*(X0*gc—ec)/32/32; if abs(sff)eps & m<60 then Xl=XO-ff/sff; XO=X1; goto repeat; end; end; finish GetRho; ********************************~k~k***********************************~k ak=j(n,l.0); bk=j(nlllo); Ck=j(nlllo); II=j(nInIO); WI=j (nrnl 0); do k=l to nk; r0=rr0[k]; ak=alrk]; bk=b[.k]; Ck=C[,k]; IW=invII8—W*r0); *print IW; do j=1 to nr; e0=ee[,j]; *print e0; dc=j(1,erI; ec=j(1,l,0); gc=j(l.1,0); yyc=IW*eO; *print yyc; dCCIj1=yy0‘*yyc; ecc[j]=yyC‘*(W*YYC): gcc[j1=YYC‘*(W‘*w*yy0); dc=dcc[j]; *print dc; ec=ecc[j]; *print ec; gc=qcc[j]; *print gc; XO=.9995*ec/gc; run GetRho; RhOIIj,k|)=XX; 107 if Rho(|j,k|)BR then Rho(|j,k|)=.; end; end; print Rho; varnames={rhol rh02 rh03 rho4 rhoS rho6 rho7 rhoB rho9}; create norms from Rho (|colname=varnamesl); append from Rho; close norms; quit; proc means data=norms; var _all_; output out=normdata mean=mrhol mrh02 mrh03 mrho4 mrhoS mrh06 mrho7 mrhoB mrho9 var =vrhol vrh02 vrh03 vrho4 vrhoS vrho6 vrho7 vrh08 vrho9; proc print data=normdata; var _all_; title'Empirical Distribution of Rho estimation'; proc univariate plot data=norms; var _all_; run; 108 Appendix6F A SAS output from program (1) RRO -0.15233 -0.11848 -0.084628 -0.050777 0 0.043646 0.072743 0.10184 0.130938 RHO -0.093043 -0.035922 -0.004561 0.0225311 0.0578763 0.1328653 -0.152541 -0.112992 -0.070801 -0.030338 0.0239434 0.0649262 0.0899643 0.1133124 0.1342584 -0.14385 . . -0.088461 -0.064811 -0.039001 -0.012986 0.0306876 0.1094544 . . . -0.067272 -0.025934 0.0351362 0.1132646 -0.157897 -0.138627 -0.118766 -0.093317 -0.039522 0.0184154 0.0587202 0.0975185 0.1322099 . . . -0.044463 -0.004264 0.0516837 0.1204356 -0.144655 -0.103632 -0.076379 -0.057642 -0.031859 -0.001434 0.0268882 0.0627848 0.1035768 -0.029778 0.0225825 . 0.1330539 -0.150079 -O.102038 -0.053859 -0.014943 0.0284294 0.0593437 0.0797859 . 0.1254203 -0.151816 . -0.101576 -0.077795 -0.050537 -0.02298 0.0207714 0.1023134 -0.076481 -0.006553 0.01267 0.0262203 0.0476586 0.0704985 0.0888963 0.1101619 0.1335608 . -0.088569 -0.05954 -0.007212 0.0415269 0.0738519 0.104953 0.1336305 . -0.11541 -0.083249 -0.024229 0.0301886 0.0653107 0.0981717 -0.130321 -0.10168 —0.069988 -0.017111 0.0317415 0.0646765 0.0969821 -0.153238 -0.118188 -0.077435 -0.034343 0.0264485 0.070925 0.0961682 0.1181363 0.1371757 109 0.0858966 0.0592215 -0.127586 0.0996438 -0.145711 0.1087552 -0.067926 0.1094199 -0.154478 0.1075293 0.0158846 0.0531585 -0.055392 0.0749359 -0.157611 0.0534711 -0.084654 0.095576 0.0148812 -0.152127 0.103535 -0.161596 0.0485434 -0.140211 0.1181952 Variable -0.091266 0.1099969 0.0954299 -0.05403 0.1173267 —0.089183 0.1248541 -0.00001 0.1237517 -0.113364 0.1242923 -0.090035 -0.094694 -0.069018 0.1310436 0.0406473 0.0919801 -0.039437 0.1013587 -0.136778 0.0920938 —0.029044 0.115246 0.0523493 -0.108562 0.1217698 -0.145931 0.0912364 -0.067346 0.1285706 -0.062899 0.1375327 0.0940635 -0.049254 —0.016049 0.1342313 -0.012775 0.0168752 0.1347104 -0.034598 0.0095697 0.1382911 0.0251898 0.0448484 0.1374021 -0.01334 -0.05806 -0.036156 -0.096187 0.1294309 -0.024876 -0.007934 0.1309394 -0.11481 -0.088274 0.1296356 -0.007296 0.0137483 -0.060382 -0.014673 0.136639 -0.127747 -0.104224 0.1303182 -0.006049 0.0374953 0.1371136 Std Dev 0.026863 0. -0.02249 0. 0.0534261 0. 0.0586682 0 0.0720933 0. 0.0479039 0. -0.013602 0. -0.041528 0 0.0215263 0. -0.037288 0. 0.0466771 0. -0.053363 -0 0.0431066 0. -0.053415 0. 0.0805432 0 Minimum .1250335 .0805083 .0575235 .0349216 .0055903 .0343597 .0616695 .0407659 .0493466 .0447200 .0473422 .0462534 .0479813 .0424770 110 .1615960 .1459314 .1277471 .1042238 .0777951 .0672717 .0259340 0621555 0246738 0815975 .090647 0947013 0870594 0024012 .0142268 0516421 0155642 0758931 .016234 0821144 0053668 .105402 RHOB 27 0.0915260 0.0324845 0.0207714 RHO9 26 0.1274066 0.0123797 0.0940635 Variable Maximum RHOl -0.0297784 RHOZ 0.0225825 RHOB 0.0251898 RHO4 0.0448484 RHOS 0.0805432 RHO6 0.1054020 RHO7 0.1181952 RHOB 0.1285706 RHO9 0.1382911 OBS _TYPE_ _FREQ_ MRHOl MRHOZ MRHO3 MRHO4 1 0 30 -0.12503 -0.080508 -0.057524 -0.034922 OBS MRHOS MRHO6 MRHO7 MRHOB MRHO9 1 .0055903 0.034360 0.061669 0.091526 0.12741 OBS VRHOl VRHOZ VRHO3 VRHO4 VRHOS 1 .0016619 .0024351 .0019999 .0022413 .0021394 OBS VRHO6 VRHO7 VRHOB VRHO9 1 .0023022 .0018043 .0010552 .00015326 Univariate Procedure Variable=RHOl Moments N 20 Sum Wgts 20 Mean -0.12503 Sum -2.50067 Std Dev 0.040766 Variance 0.001662 Skewness 1.149984 Kurtosis -0.04093 USS 0.344243 CSS 0.031575 CV -32.604 Std Mean 0.009116 T:Mean=0 -13.7165 Pr>|T| 0.0001 Num “= 0 20 Num > 0 0 M(Sign) -10 Pr>=|M| 0.0001 Sgn Rank -105 Pr>=lS| 0.0001 Quantiles(Def=5) 100% Max -0.02978 99% —0.02978 75% Q3 -0.08885 95% -0.04259 11] 50% Med -0.14518 90% 25% Q1 -0.15289 10% 0% Min -0.1616 5% 1% Range 0.131818 Q3-Q1 0.064041 Mode -0.l616 Extremes Lowest Highest -0.1616( 29) -0.08465( -0.1579( 5) -0 07648( -0.15761( 25) -0 06793( -O.15448( 21) —0 05539( -0.15324( 15) -0 02978( Missing Value . Count 10 % Count/Nobs 33.33 Stem Leaf -2 0 -4 5 -6 68 -8 35 -10 -12 8 -14 884332206540 -16 2 ----+---—+---—+-—--+ -0.06166 -0.15775 -0.15975 -0.1616 Obs 26) ll) 20) 24) 8) # Boxplot 1 I l | 2 | 2 + ----- + I I l I + I 12 * ----- * l I Multiply Stem.Leaf by 10**-2 Univariate Procedure Variable=RHOl —0.03+ * +++++ | * +++++ I * *++++ I **++++ | +++++ I +++++ * I * ~k *+**+*~k** *** —0.17+ +++++ +—---+—-—-+----+----+----+—---+----+-—--+-——-+——-—+ -l 0 +1 +2 Univariate Procedure Normal Probability Plot 112 Variable=RHO2 Moments N 21 Sum Wgts 21 Mean -0.08051 Sum -1.69068 Std Dev 0.049347 Variance 0.002435 Skewness 0.638473 Kurtosis —0.6549l USS 0.184815 CSS 0.048702 CV -61.2938 Std Mean 0.010768 T:Mean=0 -7.47641 Pr>ITI 0.0001 Num “= 0 21 Num > 0 1 MISign) -9.5 Pr>=IMI 0.0001 Sgn Rank —112.5 Pr>=IS| 0.0001 Quantiles(Def=5) 100% Max 0.022582 99% 0.022582 75% Q3 -0.03944 95% -0.00001 50% Med -0.09127 90% -0.00655 25% Q1 -0.11336 10% -0.l3678 0% Min -0.14593 5% —0.13863 1% -0.14593 Range 0.168514 Q3-Q1 0.073928 Mode —0.14593 Extremes Lowest Obs Highest Obs -0.14593( 29) -0.03592( 1) -O.13863( 5) -0.02904( 26) -0.13678( 25) -0.00655( 11) -0.13032( 14) -0.00001( 20) —0.118l9( 15) 0.022582( 8) Missing Value . Count 9 % Count/Nobs 30.00 Stem Leaf # Boxplot 2 3 1 I 0 | -0 70 2 I -2 969 3 + ----- + —4 4 l | l -6 7 1 I I -8 109 3 *—-+--* —10 833942 6 + ————— + —12 970 3 I -14 6 1 I ----+—-—-+——--+----+ 113 Multiply Stem.Leaf by 10**—2 Univariate Procedure Variab1e=RH02 Normal Probability Plot 0.03+ * ++++ | ++++ | *+*++ -0.03+ * *+*++ | *+++ I +++* -0.09+ +++*** I *+*** *ir I * *+*+ -0.15+ * ++++ +-—--+—-—-+———-+--—-+--—-+-———+----+-——-+----+-———+ -2 -1 0 +1 +2 Univariate Procedure Variab1e=RHO3 Moments N 23 Sum Wgts 23 Mean —0.05752 Sum -l.32304 Std Dev 0.04472 Variance 0.002 Skewness 0.177019 Kurtosis -0.99104 USS 0.120103 CSS 0.043997 CV -77.7421 Std Mean 0.009325 T:Mean=0 -6.1689 Pr>|T| 0.0001 Num “= 0 23 Num > 0 2 M(Sign) -9.5 Pr>=IM| 0.0001 Sgn Rank -127 Pr>=|SI 0.0001 Quantiles(Def=5) 100% Max 0.02519 99% 0.02519 75% Q3 -0.01277 95% 0.01267 50% Med -0.06038 90% -0.00456 25% Q1 —0.09469 10% -0.11541 0% Min -0.12775 5% -0.ll877 1% -0.12775 Range 0.152937 Q3-Q1 0.08192 Mode -0.12775 Extremes Lowest Obs Highest Obs 114 -0. -0. -0. -0. 12775( .11877( 11541( 11481( 10168( 29) 5) 13) 25) 14) Missing Value Count % Count/Nobs Stem -0 -2 55 -4 -6 -8 59 10 9552 2 5 0 3 12 8 Leaf 23. -0.0073( -0.00605( -0.00456( 0.01267( 0.02519( 26) 30) 1) ll) 20) # Boxplot 1 | 1 | 4 + ----- + 2 I I 3 I + | 5 i: _____ * 2 + ----- + 4 l l I Multiply Stem.Leaf by 10**—2 Univariate Procedure Variab1e=RHO3 Normal 0.03+ | | I -0.05+ | + I ++** | * +~k+~k * -O.13+ *+++++ +---—+——--+----+---—+ -2 -1 Univariate Procedure Variable=RHO4 Moments N 26 Sum Wgts Mean -0.03492 Sum Std Dev 0.047342 Variance Skewness 0.046978 Kurtosis USS 0.08774 CSS Probability Plot 26 -0.90796 0.002241 -1.32498 0.056032 115 CV -135.567 Std Mean 0.009285 T:Mean=0 -3.76126 Pr>|T| 0.0009 Num “= 0 26 Num > 0 7 M(Sign) -6 Pr>=IMI 0.0290 Sgn Rank -113.5 Pr>=|SI 0.0021 Quantiles(Def=5) 100% Max 0.044848 99% 0.044848 75% Q3 0.00957 95% 0.037495 50% Med -0.03234 90% 0.02622 25% Q1 -0.08325 10% -0.09619 0% Min —0.10422 5% -0.10158 . 1% —0.10422 Range 0.149072 Q3-Ql 0.092819 Mode —0.10422 Extremes Lowest Obs Highest Obs -0.10422( 29) 0.016875( 18) -0.10158( 10) 0.022531( 1) -0.09619( 23) 0.02622( 11) -0.09332( 5) 0.037495( 30) -0.08846( 3) 0.044848( 20) Missing Value . Count 4 % Count/Nobs 13.33 Stem Leaf # Boxplot 4 5 l I 2 367 3 l 0 047 3 + ----- + -0 65538 5 l | -2 640 3 *--+--* -4 8 1 | | -6 090 3 I I -8 63883 5 + ----- + -10 42 2 | —-—-+-—-—+——-—+--——+ Multiply Stem.Leaf by 10**-2 Univariate Procedure Variab1e=RHO4 Normal Probability Plot 0.05+ ++++* I *+*+* 116 I *~k*++ I ***~k+*+ —0.03+ **+++ I ++** I +++* * I *+*+*** —0 11+ * ++*+ +---—+----+—---+—---+----+----+----+---—+---—+-—-—+ -2 —1 0 +1 +2 Univariate Procedure Variab1e=RHO5 Moments N 27 Sum Wgts 27 Mean 0.00559 Sum 0.150939 Std Dev 0.046253 Variance 0.002139 Skewness -0.12845 Kurtosis -1.29253 USS 0.056468 CSS 0.055624 CV 827.3815 Std Mean 0.008901 T:Mean=0 0.628024 Pr>|TI 0.5355 Num A: 0 27 Num > 0 14 M(Sign) 0.5 Pr>=|MI 1.0000 Sgn Rank 32 Pr>=|SI 0.4524 Quantiles(Def=5) 100% Max 0.080543 99% 0.080543 75% Q3 0.047659 95% 0.072093 50% Med 0.021526 90% 0.058668 25% Q1 -0.03729 10% -0.05342 0% Min -0.0778 5% -0.06481 1% -0.0778 Range 0.158338 Q3-Q1 0.084947 Mode -0.0778 Extremes Lowest Obs Highest Obs -0.0778( 10) 0.053426( 18) -0.06481( 3) 0.057876( 1) -0.05342( 29) 0.058668( 19) -0.05336( 27) 0.072093( 20) -0.04153( 23) 0.080543( 30) Missing Value . Count 3 % Count/Nobs 10.00 117 Stem Leaf # 8 1 1 6 2 1 I 4 3788389 7 2 24678 5 -0 747 3 -2 7242 4 —4 3320 4 -6 85 2 ----+-—--+—---+--——+ Multiply Stem.Leaf by 10**-2 Univariate Procedure Variable=RH05 Normal Probability Plot 0.09+ +++* | ++++* I ****+** 'k I *****+++ 0.01+ +++++ I ++ir-k-k I +***** I +++** -0.07+ * ++*+ +--—-+-——-+--—-+---—+----+--——+----+-—--+—---+---—+ —2 —l 0 +1 +2 Univariate Procedure Variab1e=RHO6 Moments N 28 Sum Wgts 28 Mean 0.03436 Sum 0.962071 Std Dev 0.047981 Variance 0.002302 Skewness -0.52076 Kurtosis -0.64462 USS 0.095216 CSS 0.06216 CV 139.6444 Std Mean 0.009068 T:Mean=0 3.78927 Pr>IT| 0.0008 Num “= 0 28 Num > 0 22 M(Sign) 8 Pr>=IM| 0.0037 Sgn Rank 139 Pr>=|SI 0.0006 Quantiles(Def=5) 100% Max 0.105402 99% 0.105402 75% Q3 0.073409 95% 0.094701 118 50% Med 0.036634 90% 25% Q1 0.003884 10% 0% Min -0.06727 5% 1% Range 0.172674 Q3-Q1 0.069525 Mode -0.06727 Extremes Lowest Obs Highest -0.06727( 4) 0.082114( -0.05054( 10) 0.087059( -0.04446( 6) 0.090647( -0.039( 3) 0.094701( -0.01623( 27) 0.105402( Missing Value . Count 2 % Count/Nobs 6.67 Stem Leaf 10 5 8 22715 6 25016 4 229 2 502 0 25468 -0 61 -2 9 -4 14 -6 7 ----+-——-+—--—+----+ 0.090647 -0.04446 -0.05054 -0.06727 Obs 28) 21) 19) 20) 30) Boxplot Multiply Stem.Leaf by 10**-2 Univariate Procedure Variable=RHO6 O.1l+ * I I *~k***++ 0.05+ ***+++ I ***+ I ****~k -0.01+ +*+* | ++++* I +++* * -0.07+ +++* +---—+----+—---+----+----+---—+----+—-—-+———-+----+ -2 -1 0 +2 Normal Probability Plot 119 I—‘NI—‘wamefiI—‘at: z;- I l + I I :I- Univariate Procedure Variable=RHO7 Moments N 28 Sum Wgts 28 Mean 0.061669 Sum 1.726746 Std Dev 0.042477 Variance 0.001804 Skewness -0.7597l Kurtosis -0.44793 USS 0.155204 CSS 0.048716 CV 68.87855 Std Mean 0.008027 T:Mean=0 7.682367 Pr>|TI 0.0001 Num 0: 0 28 Num > 0 24 M(Sign) 10 Pr>=IMI 0.0002 Sgn Rank 189 Pr>=|SI 0.0001 Quantiles(Def=5) 100% Max 0.118195 99% 0.118195 75% Q3 0.095872 95% 0.10942 50% Med 0.069581 90% 0.108755 25% Q1 0.037716 10% -0.01299 0% Min -0.02593 5% -0.02298 1% -0.02593 Range 0.144129 Q3-Ql 0.058156 Mode —0.02593 Extremes Lowest Obs Highest Obs -0.02593( 4) 0.103535( 28) —0.02298( 10) 0.107529( 21) -0.01299( 3) 0.108755( 19) -0.00426( 6) 0.10942( 20) 0.014881( 27) 0.118195( 30) Missing Value . Count 2 % Count/Nobs 6.67 Stem Leaf # Boxplot 10 048998 6 I 8 069066 6 + ----- + 6 5545 4 *--+-—* 4 93399 5 | | 2 7 1 + ————— + 0 56 2 I -0 34 2 I -2 63 2 I 120 ----+----+---—+—-—-+ Multiply Stem.Leaf by 10**-2 Univariate Procedure Variable=RHO7 Normal Probability Plot 0.1l+ **+* * * I ******+ I ****++ I *****+ I ++*+ I ++++* * I +++++ * * —0.03+ +++* * +----+--—-+—---+----+----+--——+----+—---+----+----+ —2 -l 0 +1 +2 Univariate Procedure Variable=RH08 Moments N 27 Sum Wgts 27 Mean 0.091526 Sum 2.471203 Std Dev 0.032485 Variance 0.001055 Skewness -0.94129 Kurtosis —0.37333 USS 0.253616 CSS 0.027436 CV 35.4921 Std Mean 0.006252 T:Mean=0 14.64031 Pr>|T| 0.0001 Num “= 0 27 Num > 0 27 M(Sign) 13.5 Pr>=IMI 0.0001 Sgn Rank 189 Pr>=|SI 0.0001 Quantiles(Def=5) 100% Max 0.128571 99% 0.128571 75% Q3 0.117327 95% 0.124854 50% Med 0.098172 90% 0.124292 25% Q1 0.062785 10% 0.035136 0% Min 0.020771 5% 0.030688 1% 0.020771 Range 0.107799 Q3-Q1 0.054542 Mode 0.020771 Extremes Lowest Obs Highest Obs 121 00000 .020771( .030688( .035136( .040647( .051684( 10) 3) 4) 22) 6) Missing Value Count % Count/Nobs Stem 12 11 NWAUIO'NQCD Multiply Stem.Leaf by 10**-2 Leaf 24459 003578 15 1225788 10. Univariate Procedure Variable=RH08 0.125+ +** * * * I ***** I *~k*++ I ***~k*~k*++ | +++ 0.075+ +++ I ++* I ++* * l +++ * I ++* * 0.025+ ++*+ +—---+----+--—-+----+ -------- +-—--+----+----+—---+ -2 -1 +1 +2 Univariate Procedure Variable=RHO9 Moments N 26 Sum Wgts 26 0.12177( 0.123752( 0.124292( 0.124854( 0.128571( 00 \INCDU‘I=§= I—‘Nl—‘NI—I 28) 20) 21) 19) 30) Boxplot Normal Probability Plot 122 Mean 0.127407 Sum 3.312571 Std Dev 0.01238 Variance 0.000153 Skewness -1.53732 Kurtosis 1.311749 USS 0.425875 CSS 0.003831 CV 9.716681 Std Mean 0.002428 TzMean=0 52.47697 Pr>|TI 0.0001 Num A: 0 26 Num > 0 26 M(Sign) 13 Pr>=IMI 0.0001 Sgn Rank 175.5 Pr>=|SI 0.0001 Quantiles(Def=5) 100% Max 0.138291 99% 0.138291 75% Q3 0.13471 95% 0.137533 50% Med 0.132538 90% 0.137402 25% Q1 0.12542 10% 0.103577 0% Min 0.094064 5% 0.102313 1% 0.094064 Range 0.044228 Q3-Q1 0.00929 Mode 0.094064 Extremes Lowest Obs Highest Obs 0.094064( 27) 0.137114( 30) 0.102313( 10) 0.137176( 15) 0.103577( 7) 0.137402( 20) 0.109454( 3) 0.137533( 21) 0.113265( 4) 0.138291( 19) Missing Value . Count 4 % Count/Nobs 13.33 Stem Leaf # Boxplot 13 5777788 7 + ----- + 13 00112334444 11 * ----- * 12 59 2 +--+-—+ 12 0 1 I 11 I ll 3 1 I 10 9 1 0 10 24 2 0 9 9 4 1 * ————+—--—+--——+----+ Multiply Stem.Leaf by 10**-2 Univariate Procedure 123 Variable=RHO9 Normal Probability Plot 0.1375+ +**+* * * * I ******~k~k+** I ** * ++++ O.1225+ * ++++ I +++++ I ++++ * 0.1075+ ++++ * | ++++ * * I+++ 0.0925+ * +-———+---—+---—+----+----+----+----+——-—+-———+ -2 -1 0 +1 +2 124 RRO -0.15233 0.072743 RHO -0.093043 0.2173169 -0.152541 0.0899643 -0.14385 -0.012986 -0.201988 -0.025934 -0.157897 0.0587202 -0.18748 -0.004264 -0.144655 0.0268882 -0.029778 0.2303099 —0.150079 0.0797859 -0.151816 -0.02298 -0.076481 0.0888963 -0.196689 0.0738519 -0.l854 0.0653107 -0.182849 0.0646765 -0.153238 0.0961682 -0.11848 0.10184 -0.035922 0.1853992 -0.112992 0.1133124 -0.210056 0.0306876 -0.230823 0.0351362 -0.l38627 0.0975185 -0.215079 0.0516837 -0.103632 0.0627848 0.0225825 0.1920124 -0.102038 0.1462349 -0.20018 0.0207714 -0.006553 0.1101619 -0.214034 0.104953 -0.20148 0.0981717 -0.130321 0.0969821 -0.118188 0.1181363 Appendix6G A SAS output from program (2) -0.084628 -0.050777 0 0.130938 -0.004561 0.0225311 0.0578763 0.1328653 -0.070801 —0.030338 0.0239434 0.1342584 —0.214727 -0.088461 -0.064811 0.1094544 —O.237128 —0.235618 —0.226637 0.1132646 —0.118766 -0.093317 -0.039522 0.1322099 -0.226746 -0.228666 -0.223148 0.1204356 —0.076379 —0.057642 -0.031859 0.1035768 0.1462347 0.4329411 0.3431254 0.1330539 —0 053859 —0.014943 0.0284294 0.1254203 -0.207394 -0.101576 -0.077795 0.1023134 0.01267 0.0262203 0.0476586 0.1335608 -0 088569 —0.05954 —0.007212 0.1336305 -0 11541 —0.083249 -0.024229 0.1687808 -O.10168 —0 069988 —0 017111 0.1681896 —0.077435 —0.034343 0.0264485 0.1371757 125 0.043646 0.2526557 0.0649262 -0.039001 -0.067272 0.0184154 -0.044463 -0.001434 0.272335 0.0593437 -0.050537 0.0704985 0.0415269 0.0301886 0.0317415 0.070925 -0.192002 0.0858966 -0.216725 0.0592215 -0.127586 0.0996438 -0.145711 0.1087552 -0.067926 0.1094199 -0.154478 0.1075293 -0.194392 0.0158846 -0.181668 0.0531585 -0.055392 0.0749359 -0.157611 0.0534711 -0.084654 0.095576 -0.20465 0.0148812 -0.152127 0.103535 -0.161596 0.0485434 -0.140211 0.1181952 Variable .2167251 .2308235 .2371277 .2356177 .2266374 .0672717 .0259340 -0.091266 0.1099969 -0.208973 0.0954299 -0.05403 0.1173267 -0.089183 0.1248541 -0.00001 0.1237517 -0.113364 0.1242923 -0.090035 0.0406473 -0.l98527 0.0919801 -0.039437 0.1013587 -0.136778 0.0920938 -0.029044 0.115246 -0.227138 0.0523493 -0.108562 0.1217698 -0.l45931 0.0912364 -0.067346 0.1285706 -0.049254 0.1342313 -0.094694 0.1310436 -0.012775 0.1347104 -0.034598 0.1382911 0.0251898 0.1374021 -0.062899 0.1375327 -0.05806 0.213422 -0.202976 0.1294309 -0.024876 0.1309394 -0.11481 0.1296356 -0.007296 0.1462383 -0.226753 0.0940635 -0.060382 0.136639 -0.127747 0.1303182 -0.006049 0.1371136 .1481504 .1198988 .0830843 .0387161 0.0014760 0.0495687 0.0724791 -0.016049 -0.069018 0.0168752 0.0095697 0.0448484 -0.01334 -0.036156 -0.096187 -0.007934 -0.088274 0.0137483 -0.22218 -0.014673 -0.104224 0.0374953 Std Dev .0472468 .0738956 .0878433 .1155761 .0974320 .0741633 .0580952 126 0.026863 -0.02249 0.0534261 0.0586682 0.0720933 0.0479039 -0.013602 -0.041528 0.0215263 -0.037288 0.0466771 -0.053363 0.0431066 -0.053415 0.0805432 0.0621555 0.0246738 0.0815975 0.090647 0.0947013 0.0870594 0.0024012 0.0142268 0.0516421 0.0155642 0.0758931 -0.016234 0.0821144 0.0053668 0.105402 Minimum 127 RHOB 30 0.0998283 0.0403738 0.0207714 RHO9 30 0.1336401 0.0218096 0.0940635 Variable Maximum RHOl -0.0297784 RHOZ 0.0225825 RH03 0.1462347 RHO4 0.4329411 RH05 0.3431254 RHO6 0.2723350 RHO7 0.2303099 RHOB 0.1920124 RHO9 0.2134220 OBS _TYPE_ _FREQ_ MRHOl MRHOZ MRH03 MRHO4 l 0 30 -0.14815 -0.11990 —0.083084 -0.038716 OBS MRH05 MRHO6 MRHO7 MRHOB MRHO9 1 .0014760 0.049569 0.072479 0.099828 0.13364 OBS VRHOl VRH02 VRH03 VRHO4 VRHOS 1 .0022323 .0054606 .0077165 0.013358 .0094930 OBS VRHO6 VRHO7 VRHOB VRHO9 1 .0055002 .0033750 .0016300 .00047566 Univariate Procedure Variable=RHOl Moments N 30 Sum Wgts 30 Mean -0.14815 Sum -4.44451 Std Dev 0.047247 Variance 0.002232 Skewness 0.918047 Kurtosis 0.284406 USS 0.723192 CSS 0.064735 CV -31.8911 Std Mean 0.008626 T:Mean=0 -17.l748 Pr>|T| 0.0001 Num A: 0 30 Num > 0 0 M(Sign) -15 Pr>=|MI 0.0001 Sgn Rank -232.5 Pr>=|SI 0.0001 Quantiles(Def=5) 100% Max -0.02978 99% -0.02978 75% Q3 -0.14021 95% 50% Med -0.15289 90% 25% Q1 —0.1854 10% 0% Min -0.21673 5% 1% Range 0.186947 Q3-Q1 0.045189 Mode -0.21673 Extremes Lowest Obs Highest -0.21673( 17) -0.08465( -0.20465( 27) -0.07648( -0.20199( 4) -0.06793( -0.19669( 12) -0.05539( -0.19439( 22) -0.02978( Stem Leaf —2 0 —4 5 -6 68 -8 35 -1O -12 8 —14 884332206540 ~16 2 -18 7427532 -20 752 ---—+-———+—---+—---+ Univariate Procedure -0.05539 -0.0722 -0.19934 —0.20465 -0.21673 Obs 26) 11) 20) 24) 8) NNI—‘I-‘m: UJ\)I—'f\)l-' Multiply Stem.Leaf by 10**-2 Variable=RHOl -0.03+ * +++ I * ++++ I * *++++ -0.09+ **++++ I +++++ I ++++ * _O.15+ *********** | ++*+ I * **~k**~k -0.21+ * *+*+++ +-—--+----+-—--+—---+----+————+—-—-+----+—-—-+--—-+ —2 -1 0 +1 +2 Univariate Procedure Normal Probability Plot 128 Boxplot Variable=RH02 Moments N 30 Sum Wgts Mean -0.1199 Sum Std Dev 0.073896 Variance Skewness 0.104961 Kurtosis USS 0.589628 CSS CV -61.6316 Std Mean T:Mean=0 -8.88704 Pr>IT| Num 0: 0 30 Num > 0 M(Sign) -14 Pr>=|MI Sgn Rank -229.5 Pr>=|SI Quantiles(Def=5) 100% Max 0.022582 99% 75% Q3 -0.06735 95% 50% Med -0.11318 90% 25% Q1 -0.20018 10% 0% Min -0.23082 5% 1% Range 0.253406 Q3—Q1 0.132833 Mode -0.23082 Extremes Lowest Obs Highest -0.23082( 4) -0.03592( -0.22714( 27) -0.02904( -0.21508( 6) -0.00655( -0.21403( 12) -0.00001( -0.21006( 3) 0.022582( Stem Leaf 2 3 0 -0 70 -2 969 -4 4 -6 7 -8 109 -10 833942 —12 970 -14 6 -16 -18 9 -20 540910 -22 17 ----+—---+-—--+----+ 30 -3.59696 0.005461 -0.96073 0.158356 0.013491 0.0001 1 0.0001 0.0001 0.022582 -0.00001 -0.0178 -0.21456 -0.22714 -0.23082 Obs 1) 26) ll) 20) 8) # Boxplot l—‘UJO'NUOI—‘HUJN I—l RJOsH + I I —-| 1 1 + Multiply Stem.Leaf by 10**-2 129 Univariate Procedure Variable=RHOZ Normal Probability Plot 0.03+ +*+ I ++ I *++ I ink-+++ I *++ I +*+ I +~kir~k I ***** I *-k*+ I *++ I +++ I ++ * I *+*+**** -O.23+ * *++ +-———+--—-+————+——--+—-——+--—-+———-+-—-—+----+———-+ -2 —1 0 +1 +2 Univariate Procedure Variable=RHO3 Moments N 30 Sum Wgts 30 Mean -0.08308 Sum —2.49253 Std Dev 0.087843 Variance 0.007716 Skewness 0.034 Kurtosis 0.442815 USS 0.430867 CSS 0.223777 CV —105.728 Std Mean 0.016038 T:Mean=0 -5.18049 Pr>ITI 0.0001 Num A: 0 30 Num > 0 3 M(Sign) -12 Pr>=IM| 0.0001 Sgn Rank -197.5 Pr>=|SI 0.0001 Quantiles(Def=5) 100% Max 0.146235 99% 0.146235 75% Q3 -0.02488 95% 0.02519 50% Med -0.07359 90% 0.004054 25% Q1 —0.11877 10% -0.22074 0% Min -0.23713 5% -0.22675 1% -0.23713 Range 0.383362 Q3—Q1 0.09389 Mode -0.23713 130 Lowest -0.23713( -0.22675( -0.22675( -0.21473( -0.20739( Leaf 0 13 -0 321110 -0 998876 -1 32210 -2 433110 Extremes Obs 4) 27) 6) 3) 10) 6655 Highest -0.00605( -0.00456( 0.01267( 0.02519( 0.146235( --——+----+—--—+----+ Multiply Stem.Leaf by 10**-1 Univariate Procedure Variable=RHO3 Obs 30) l) 11) 20) 8) # Boxplot 1 0 2 I 6 + ————— + *__+_-* 5 + ----- + | 6 I Normal Probability Plot 0.175+ I *+++++ | +++++ | ++++*+* -0.025+ +++**** * I **~k***** I *****++ I ++++++ -O.225+ *+++*+* * ** +----+—--—+--——+----+---—+----+-———+—---+-—--+-—--+ -2 —1 0 +1 +2 Univariate Procedure Variable=RHO4 Moments N 30 Sum Wgts 30 Mean -0.03872 Sum -1.l6148 Std Dev 0.115576 Variance 0.013358 Skewness 2.034541 Kurtosis 9.414231 USS 0.432345 CSS 0.387377 CV -298.522 Std Mean 0.021101 131 T:Mean=0 -1.83478 Pr>IT| 0.0768 Num A: 0 30 Num > 0 8 M(Sign) —7 Pr>=|MI 0.0161 Sgn Rank -140.5 Pr>=|SI 0.0023 Quantiles(Def=5) 100% Max 0.432941 99% 0.432941 75% Q3 0.00957 95% 0.044848 50% Med -0.03525 90% 0.031858 25% Q1 -0.08846 10% -0.1632 0% Min -0.23562 5% -0.22867 1% -0.23562 Range 0.668559 Q3-Q1 0.09803 Mode -0.23562 Extremes Lowest Obs Highest Obs -0 23562( 4) 0.022531( 1) -0 22867( 6) 0.02622( 11) -0.22218( 27) 0.037495( 30) -0 10422( 29) 0.044848( 20) —0 10158( 10) 0.432941( 8) Stem Leaf # Boxplot 4 3 1 * 3 3 2 2 1 1 0 0 1122344 7 + ----- + -0 43321111 8 *--+--* -0 99987766 8 + ----- + -l 000 3 I .. l I —2 432 3 0 ---—+—-—-+---—+-——-+ Multiply Stem.Leaf by 10**—1 Univariate Procedure Variable=RHO4 Normal Probability Plot O.425+ | I 132 + I I ++++ I ++++ | ++++ I +++++ I +++**** * * * I ******~k I ****~k~k*~k* I * *++++ I ++++ -O.225+ * ++*+* +----+----+-——-+-—-—+--——+—-——+--——+——-—+————+————+ -2 -1 0 +1 +2 Univariate Procedure Variable=RH05 Moments N 30 Sum Wgts 30 Mean 0.001476 Sum 0.044279 Std Dev 0.097432 Variance 0.009493 Skewness 0.681362 Kurtosis 5.641961 USS 0.275362 CSS 0.275297 CV 6601.247 Std Mean 0.017789 T:Mean=0 0.082973 Pr>ITI 0.9344 Num A: 0 30 Num > 0 15 M(Sign) 0 Pr>=|MI 1.0000 Sgn Rank 18.5 Pr>=|SI 0.7104 Quantiles(Def=5) 100% Max 0.343125 99% 0.343125 75% Q3 0.047659 95% 0.080543 50% Med 0.007157 90% 0.065381 25% Q1 -0.03952 10% -0.0713 0% Min -0.22664 . 5% -0.22315 1% -0.22664 Range 0.569763 Q3-Ql 0.087181 Mode -0.22664 Extremes Lowest Obs Highest Obs -0.22664( 4) 0.057876( 1) -0.22315( 6) 0.058668( 19) -0.0778( 10) 0.072093( 20) -0.06481( 3) 0.080543( 30) -0.05342( 29) 0.343125( 8) 133 Stem Leaf # Boxplot 3 4 1 + 2 2 l 1 0 55556678 8 + _____ + 0 223334 6 *_-+-_* -0 444322211 9 + _____ + -0 8655 4 I -l -1 -2 32 2 0 —---+-—--+----+---—+ Multiply Stem.Leaf by 10**-1 Univariate Procedure Variable=RH05 Normal Probability Plot I I +++++ I +++++ I +++++ I ++++** * * * I *+******* I I I ********* * * **+++ +++++ I +++++ -0.225++++++* * +----+----+----+----+----+----+—---+——-—+-—--+----+ -2 -l 0 +1 +2 Univariate Procedure Variable=RHO6 Moments N 30 Sum Wgts 30 Mean 0.049569 Sum 1.487061 Std Dev 0.074163 Variance 0.0055 Skewness 1.363486 Kurtosis 3.184025 USS 0.233217 CSS 0.159506 CV 149.6171 Std Mean 0.01354 T:Mean=0 3.660828 Pr>IT| 0.0010 Num A: 0 30 Num > 0 24 M(Sign) 9 Pr>=|MI 0.0014 Sgn Rank 168.5 Pr>=|SI 0.0001 134 Quantiles(Def=5) 100% Max 0.272335 99% 75% Q3 0.081598 95% 50% Med 0.046584 90% 25% Q1 0.005367 10% 0% Min —0.06727 5% 1% Range 0.339607 Q3-Q1 0.076231 Mode -0.06727 Extremes Lowest Obs Highest -0.06727( 4) 0.090647( -0.05054( 10) 0.094701( -0.04446( 6) 0.105402( -0.039( 3) 0.252656( -0.01623( 27) 0.272335( Stem Leaf 2 57 2 1 1 1 0 566677888999 0 011222334 —0 4420 -0 75 ----+-———+--—-+-—--+ 0.272335 0.252656 0.100052 -0.04173 -0.05054 -0.06727 Obs N NubIOIQFJ Multiply Stem.Leaf by 10**-1 Univariate Procedure Variable=RHO6 19) 20) 30) l) 8) Boxplot Normal Probability Plot O.275+ * * I +++++ I +++++++ I +++++++* I ****~k**~k*** ir I *~k*~k**~k** I *+*+**++ -0.07S+ *+++*++ +----+---—+---—+---—+--——+————+-—--+————+——-—+—-—-+ —2 -1 0 +1 +2 Univariate Procedure 135 Variable=RHO7 Moments N 30 Sum Wgts Mean 0.072479 Sum Std Dev 0.058095 Variance Skewness 0.733946 Kurtosis USS 0.255473 CSS CV 80.15438 Std Mean T:Mean=0 6.833345 Pr>|TI Num A: 0 30 Num > 0 M(Sign) 11 Pr>=|MI Sgn Rank 218.5 Pr>=|SI Quantiles(Def=5) 100% Max 0.23031 99% 75% Q3 0.099644 95% 50% Med 0.074394 90% 25% Q1 0.048543 10% 0% Min -0.02593 5% 1% Range 0.256244 Q3-Ql 0.0511 Mode —0.02593 Extremes Lowest Obs Highest -0.02593( 4) 0.108755( -0.02298( 10) 0.10942( -0.01299( 3) 0.118195( -0.00426( 6) 0.217317( 0.014881( 27) 0.23031( Stem Leaf 22 0 20 7 18 16 14 12 10 048998 8 069066 6 5545 4 93399 2 7 0 56 -0 34 -2 63 ----+--—-+---—+----+ 30 .174372 .003375 .740269 .097876 .010607 0.0001 26 0.0001 0.0001 OOHON 0.23031 0.217317 0.113808 -0.00862 -0.02298 -0.02593 Obs 19) 20) 30) l) 8) # Boxplot 0 1 0 H NNNI—‘U'l-D-mm Multiply Stem.Leaf by 10**-2 136 Univariate Procedure Variable=RHO7 Univariate Procedure Variable=RH08 N Mean Std Dev Skewness USS CV T:Mean=0 Num “= 0 M(Sign) Sgn Rank 100% 75% 50% 25% 0% Max 03 Med Q1 Min Range Q3-Q1 Mode Normal Probability Plot * * ++++ +++ +++ ++++ +++ +++*** * * ****** **** ***** +*++ +++* +++-k *+++* +-———+—-—-+----+----+----+—-——+—-——+—---+----+----+ —2 -1 0 +1 +2 Moments 30 Sum Wgts 30 0.099828 Sum 2.99485 0.040374 Variance 0.00163 0.019259 Kurtosis 0.443616 0.346242 CSS 0.047271 40.44321 Std Mean 0.007371 13.543 Pr>ITI 0.0001 30 Num > 0 30 15 Pr>=|MI 0.0001 232.5 Pr>=|SI 0.0001 Quantiles(Def=5) 0.192012 99% 0.192012 0.12177 95% 0.185399 0.103156 90% 0.137403 0.091236 10% 0.037892 0.020771 5% 0.030688 1% 0.020771 0.171241 0.030533 0.020771 137 Extremes Lowest Obs Highest 0.020771( 10) 0.124854( 0.030688( 3) 0.128571( 0.035136( 4) 0.146235( 0.040647( 22) 0.185399( 0.051684( 6) 0.192012( Stem Leaf 18 52 16 14 6 12 24459 10 15003578 8 1225788 6 3 4 122 2 115 ————+—-—-+----+—-—-+ Multiply Stem.Leaf by 10**-2 Univariate Procedure Variable=RH08 0.19+ * *++++ I +++++ I ++++* I ++**** * 0.114. ******* I *****+*+ I +*+++ l ++*+** 0.03+ *+++*+* +—-—-+-———+--—-+--—-+--——+-——-+——-—+---—+-—--+—---+ -2 -1 0 +1 +2 Univariate Procedure Variable=RHO9 Moments N 30 Sum Wgts 30 Mean 0.13364 Sum 4.009202 Std Dev 0.02181 Variance 0.000476 Skewness 1.599801 Kurtosis 5.713664 USS 0.549584 CSS 0.013794 CV 16.31965 Std Mean 0.003982 Normal Probability Plot Obs 19) 30) 9) l) 8) # Boxplot 2 0 l | 5 + ————— + 8 *..+_.* 7 + ----- + 1 l 3 0 3 0 138 T:Mean=0 33.56214 Pr>|T| Num A: 0 30 Num > 0 M(Sign) 15 Pr>=|MI Sgn Rank 232.5 Pr>=ISI Quantiles(Def=5) 100% Max 0.213422 99% 75% Q3 0.137176 95% 50% Med 0.133307 90% 25% Q1 0.129431 10% 0% Min 0.094064 5% 1% Range 0.119359 Q3-Ql 0.007745 Mode 0.094064 Extremes Lowest Obs Highest 0.094064( 27) 0.138291( 0.102313( 10) 0.146238( 0.103577( 7) 0.16819( 0.109454( 3) 0.168781( 0.113265( 4) 0.213422( Stem Leaf 21 3 20 19 18 17 16 89 15 14 6 13 001123344445777788 12 059 11 3 10 249 9 4 —---+———-+----+---—+ 000000 Multiply Stem.Leaf by 10**-2 Univariate Procedure Variable=RHO9 0.215+ Normal Probability Plot 0.00 0.00 0.00 .213422 .168781 .157214 .106516 .102313 .094064 Obs 19) 26) 14) 13) 22) # Boxplot ~11: N 139 01 30 01 01 k401F101m1~ + I I I I I + ++++ I +++++ I +*+* 0.155+ +++++ | +++++ * I *~k************** I ****++++ I *+++ I *+*+*+ 0.095+ *++++ +---—+-—--+—---+----+-——-+—--—+———-+-—--+---—+---—+ -2 -l 0 +1 +2 140 W's entries are 0, AQU'IKOKO 1 .059360 .023599 .534783 .083396 .894535 .894535 1.310192 wwbmmb .841095E-01 .841095E-01 .363135E-01 .958040E-08 .903842E—01 -1.171220 -1.171220 -1.336884 -1.336884 -2.525492 -2.525492 -4.415877 -4.827520 -4.827520 -4.899200 -7.075458 -7.701774 1, 0000 00 00 Appendix6H The eigenvalues of . W(O, 1 ,2,3,4) 2, 3, 4. .000000E+00 .000000E+00 .000000E+00 .000000E+00 -1.389182 1.389182 .000000E+00 -1.732912 1.732912 .000000E+00 .000000E+00 .000000E+00 -1.956163 1.956163 -1.309866 1.309866 —1.615801 1.615801 .000000E+00 —2.857574 2.857574 .000000E+00 .000000E+00 .000000E+00 141 Appendix6l The comparison of moral levels T-Test GroupStatlstlcs Std. Std. Error W] NW2 N Mean Deviation Mean MORAL] 1.00 16 .1510794 .2560826 6.40E-02 100 8 2155950 40.5mm Independent Sample: Test Levene's?1'est for Equality of Variances t-test for Equality of Means Sig. Std. (2—tail Mean Error F 51% t df ed) Difference Difference MORAL] Equal variances 1.477 .237 -.478 22 .637 -6.5E-02 .1348340 assumed Equal xgi'ances -.411 9.891 .690 —6.SE-02 .156923] assumed T-Test GroupStatistlcs Std. Std. Error NB] 82 N Mean Deviation Mean MORAL] 1.00 18 .1092361 .2874029 6.77E-02 100 6 35763QD__3.0403.83__.12£J.Z3.L. Independent Samples Test Levene's Test for Equality of Variances t—test for Equality of Means Sig. Mean Std. Error (2-ta Differenc Differenc F Sig. t df iled) e e MORAL] Equal variances .044 .836 —].845 22 .078 .2533939 .1373046 assumed Equal :grt'ances -1.792 8.208 .110 .2533939 .1414053 assumed 142 T-Test GroupStatistics Std. Std. Error W] NW2 N Mean Deviation Mean MORAL2 1.00 16 .1496112 .2557655 6.39E-02 2-00 8 7155950 mum. Independent Samples Test Levene's Test for Equality of Variances t-test for Egualig of Means Sig. Std. Error (Z—tail Mean Differenc F Sig. t df ed) Difference e MORAL2 Equal variances 1.494 .234 -.490 22 .629 —6.60E-02 .1347570 assumed Equal ‘r’Ig’t'ance‘ -.421 9.883 .683 -6.60E-02 .1568908 assumed T-Test GroupStstlstics Std. Std. Error N8] 82 N Mean Deviation Mean MORAL2 1.00 18 .1079311 .2869526 6.76E-02 2-00 6 3526300 304.03.33.4241231. Independent Samples Test Levene's Test for Equality of Variances t—test for Equality of Means Sig. Std. (2-tail Mean Error F Sig. t df ed) Difference Difference MORAL2 Equal variances .048 .828 -1.857 22 .077 .2546989 .1371428 assumed Equal 31'3““ -1.802 8.197 .108 .2546989 .1413545 143 The racexsexxmoral table Appendix6J *1 ‘U Correlations sex " 7154' R' MORAL] MORAL2 MORAL3 Pearson SEX 1.000 -.252 .024 .026 .050 Correlation YEAR -.252 1.000 .160 .157 .126 MORAL] .024 .160 1 .000 1 .000” .988*‘ MORAL2 .026 .1 57 1 .000“ 1 .000 .991 * MORAL3 .050 .126 .988“ .991 1.000 wmwz .299 .029 .101 .104 .123 NBLIBZ .293 .099 .366 .368 .382 Sig. sex . .235 .913 .903 .815 (2-tailed) YEAR .235 . .455 .464 .556 MORAL] .91 3 .455 . .000 .000 MORAL2 .903 .464 .000 . .ooo MORAL3 .81 s .556 .000 .000 . w1 NW2 .156 .892 .637 .629 .566 NBIBZ .165 .645 .078 .077 .066 N sex 24 24 24 24 24 YEAR 24 24 24 24 24 MORAL] 24 24 24 24 24 MORAL2 24 24 24 24 24 MORAL3 24 24 24 24 24 W1NW2 24 24 24 24 24 NBIBZ 24._u__2.4___24___23_ 144 Appendix6J (Continued) The racexsexxmoral table Correlations w_1 NW2 N81 82 Pearson SEX .299 .293 Correlation YEAR .029 .099 MORAL] .101 .366 MORAL2 .104 .368 MORAL3 .123 .382 W1NW2 1.000 .816” N81 82 .816" 1.00 Sig. SEX .1 56 .165 (Z-tailed) YEAR .892 .645 MORAL] .637 .078 MORAL2 .629 .077 MORAL3 .566 .066 M NW2 . .000 N8182 .0001 N SEX 24 24 YEAR 24 24 MORAL] 24 24 MORAL2 24 24 MORAL3 24 24 w1~w2 24 24 N3] 32 2.4. **,Correlation is significant at the 0.01 level (Z-tailed). 145 Appendix6K The eigenvalues of W(0,l) W's entries are 0, 1 only. 4.051115 0.000000E+00 2.651598 0.000000E+00 1.883632 0.000000E+00 1.548894 -3.286612E-01 1.548894 3.286612E-01 7.357295E—01 0.000000E+00 3.613770E-01 —6.469532E-01 3.613770E-01 6.469532E-01 2.966202E—01 —6.964666E-02 2.966202E—01 6.964666E-02 5.441884E-07 0.000000E+00 -2.522140E-01 -7.226394E-01 -2.522140E-01 7.226394E—01 —3.105704E—01 0.000000E+00 -6.678435E-01 -1.496080E-01 -6.678435E-01 1.496080E-01 —6.870127E-01 -5.055384E-01 —6.870127E-01 5.055384E-01 -1.397442 —9.904619E-01 —1.397442 9.904619E-01 -1.533783 0.000000E+00 —1.899674 -9.174101E-02 —1.899674 9.174101E-02 —2.083131 0.000000E+00 146 llllIll Ill: IlllI REFERENCES 147 Anselin, L. (1988) Spatial Economics: Methods and Models. Kluwer Academic Publishers. Bartels, C. P. A. (1979) Operational Statistical Methods for Analyzing Spatial Data. Exploratory and Explanatory Statistical Analysis of Spatial Data. C. P. A. Bartels and R. H. Ketellapper. (Eds) Martinus Nijhoff, Boston, Mass. pp. 5-50. Cliff, AD. and Ord, J .K. (1973) Spatial Autocorrelation. London. Pion Ltd. Cressie, N. A. C. (1993) Statistics for Spatial Data (Revised Edition). John Wiley & Sons, Inc. New York. Doreian, P. (1981) Estimating Linear Models with Spatially Distributed Data. Sociolgical methodology. (Leinhardt, S. Ed.) pp. 359-3 88. SF. Jessey-Bass. Doreian, P. (1982 ). Maximum Likelihood Methods for Linear Models. Spatial Effect and Spatial Disturbance Terms. Social Methods & Research. Vol.10 No.3, Feb. 1982. pp. 245-269. Sage Publications, Inc. Doreian, P. (1989) Two Regions of Network Effects Autocorrelation. pp. 280-295. Doreian, P. and Hunmon, N. P. (1976) Modeling Social Processes. New York: Elsevier. Duke, J .B. (1993) Estimation of the Network Effects Model in a Large Data Set. Social Methods & Research. Vol.21 No.4, May 1993. pp. 465-481. Sage Publications, Inc. Frank, K. (1995) Identifying Cohesive Subgroups. Social Networks. 17 (1995), pp. 27- 56. Frank, K. (1996) Maping Interactions within and between Cohesive Subgroups. Social Networks. 18 (1996), pp. 93-119. Harman, Harry H. (1960) Modern Factor Analysis. The University of Chicago Press. Holland, Paul W. and Leinhardt, Samuel (1981) An Exponential Family of Probability Distributions for Directed Graphs. Journal of the American Statistical Association. 1981 March, Volume 78, Number 373. pp. 33-50. Kira, T. K., Ogawa, H., and Sakazaki, N. (1953) Intraspecific competition among higher plants. I: Competition-yield-density interrelationship in regularly dispersed populations. Journal of the Institute of Polytechnics, Osaka City University, Series D, 4, 1-16. Leenders, RA. (1995) Models for Network Dynamics: A Markovian Framework. Journal of Mathematical Sociology. 1995, pp. 1-21. 148 Marsden, RV. and F riedkin, NE. (1994) Network Studies of Social Influence. Social Psychology and Diffusion. Mead, R. (1967) A Mathematical Model for the Estimation of Interplant Competition. Biometrics, 23 (June 1967), pp. 189-205. Mead, R. (1971) Statistical Ecology. (Patil, G.P. etc. ed.) Vol. 2, 1971, pp. 13- 30.University Park: Pennsylvania State Univ. Press. Ord, K. (1975) Estimation Methods for Models of Spatial Interaction Journal of the American Statistical Association. 1975 March, pp. 120-126. Ripley, B. D. (1981) Spatial Statistics. John Wiley & Sons, Inc. New York. Shavelson, R. J. (1996) Statistical Reasoning for the Behavioral Sciences. (3rd ed.) ALLYN and BACON, Boston. Whittle, P. (1954) On Stationary Processes in the Plane. Biometrika, 41 (Parts 3 and 4), pp. 434-449. Xu, J. (1996) Solution Methods for Spatial Linear Models, An unpublished apprenticeship paper. College of Education, Michigan State University. 149 IIIIIIIIIIIIIIIIIIII