thfivw v A TEST OF THE HYPOTHESIS OF ADDITIVH'YaQF-sCUES IN A TWQ-CHOICE DISCRIMINATION LEARNING PROBLEM Thesis Iov Ibo chrea oI M. A. MICHIGAN STATE UNIVERSITY Thomas Robert Trabasso 1959 “£319 ’L [B R A R Y Mir‘figan State University A TEST OF THE HYPOTHESIS OF ADDITIVITY-OF-CUBS IN A TWO-CHOICE DISCRIMINATION LEARNING PROBLEM By Themes Robert Trebeeeo A THESIS Submitted to the College of Arte and Sciences of Michigan State university of Agriculture and Applied Science in partial fulfillment of the requiremente for the degree of IILSTER OF ARTS Department of Peychology 1959 Thomas R. Trabasso ABSTRACT This thesis is concerned with quantitatively formulating and testing the concept of additivity-of-cues. Previous experimentation has shown that learning is more rapid when relevant stimuli were presented in more than one modality. Qualitatively, it was known that learning is always faster in the combined-cue situation but prior to the application of mathematical models of learning to such data, no quantitative laws regarding the function of additivity had been formulated and tested. Using a theory of two-choice discrimination learning by Restle (1955), two hypotheses of additivity of the proportions of relevant cues were formulated in set-theoretic mathematics and tested. The hypotheses were additivity-cf-cuee of two kinds: direct and additivity by a function. To test additivity-of-cues, a two-choice discrimination learning problem.was used. Five groups of 16 human subjects each were tested on separate problems. Two problems had one cue relevant and one cue irrelevant. A second two problems had one one relevant, but the measure of the irrelevant cues was reduced. The remaining problem had both cues relevant. The stimuli were patterns of letters which had a fixed alphabet- ical order. but.varied in form between upper and lower case. The response was written when the subject saw a stimulus pattern. The i. Thomas R. Trabasso correct answer (an X or O) was predictable from the stimulus pattern by a consistent principle. Two hypotheses of additivity were formulated and tested. Hypothesis 1: direct additivity, where the proportion of relevant cues in a combined-cue problem was predicted by direct addition of the proportions of relevant cues in two single-cue problems. Hypothesis 2: additivity by a derived function. where the proportion of relevant cues in a combined-cue problem was predicted by a function of the preportions of relevant cues in two single-cue problems, the measure of the irrelevant cues being reduced in the latter two problems. The results may be sunnarised as to four findings: 1. The combined-cue group showed faster learning than the single-cue groups. indicating some form of additivity. One of the relevant cues was found to be stronger than the other and the reduction ,of the measure of irrelevant cues through the fixing of a letter had a small beneficial effect on learning. 2. The two methods of estimating the learning rate parameter, 9, yielded discrepant results, indicating that neither the group nor the individual learning curves were of the shape predicted by the theory. An analysis of the discrepancies between the two methods of estimation suggested a bias inherent in the methods. 3. The predictions of the mean error scores and learning rates by both hypotheses of additivity were found to be accurate in all cases. Statistical tests of these hypotheses failed to indicate that they should be rejected. ii. Thomas B. Trabasso 4. To account for individual differences in learning rate, an assump- tion of a high positive correlation as to subject.position in the groups was made. Predictions of the distributions of rates of learning in the combined-cue problem.by the application of the hypotheses of additivity to the matched rank values in the single-cue groups were made. These predictions were found to be accurate for the cumulative distribution of learning rates. the mean and the median of the combined- cue group. The discrepancies between the two methods of'estimating learning rates were discussed. Inspection of the discrepancies indicated that the discrepancies were larger with faster rates of learning. A lento- Carlo procedure was used to test this difference but did not clearly indicate the nature of the discrepancies. It is suggested that such a procedure would be fruitful for investigations of the variance and distribution of the learning rate parameter. 0. A second quantitative analysis, based on the number of stimulus patterns in the problems was discussed and found not to be consistent with the additivity-of-cues data. References Restle. F. A theory of discrimination learning. Psychol. Rev.. 1955, 62, 11-19. Approved: '3, ICU/kg) Mi] [[1 Date: )NQLLAIQJZI‘IC] VlaJor Professor (] iii. DEDICATION To IV wife 1Ve ACKNOWLEDGEMENT The author wishes to express his gratitude and sincere appreciation for the guidance and assistance in the planning and execution of this research, and the development of this manuscript to Dr. Frank J. Restle, chairman of his ccnsnittee. In addition. he wishes to convey profound thanks to Dr. ll. Ray Denny and Dr. Terrence M. Allen for their excel- lent criticism and advice, during the preparation of this filefli'e 7e INTRODUCTION. METHOD . . RESULTS . . DISCUSSION . SUMMARY . . APPENDICES . Appendix Appendix REFERENCES . I--Restle's Theory of Discrflminaticn.learning II--Tables and subject Summary Data TABLE OF CONTENTS vi. PAGE 10 17 32 4O 42 43 53 63 LIST OF TABLES AND FIGURES TABLE PAGE I. EXPERIMENTAL GROUPS USED TO TEST ADDITIVITY-OF-CUES . . 15 II. INDIVIDUAL COMPARISONS BETWEEN GROUP MEANS . . . . . . 19 III. 9 ESTIMATES FOR THE FIVE EXPERIMENTAL GROUPS AS OBTAINED BY THE MEAN TOTAL AND WEIGHTED ERROR SCORES . . . . . . 21 IV. MEAN TOTAL ERRORS IN 128 TRIALS AND PROPORTION OF REIEVANTCUES(9)FORGROUPA+E............ 24 V. MEAN WEIGHTED ERRORS IN 128 TRIALS AND PROPORTION OF RELEVANTCUES(9)FORGROUPA+E............ 24 VI. 5 AND a VALUES ESTIMATED BY THE TOTAL ERROR am) WEIGHTED ERROR METHODS OF ssrmnox FOR 128 rams . . . . . . 53 VII. SUBJECT SUMMARY'DATA, SHOWIIB NUMBER OF ERRORS PER BLOCK OF 8 TRIALS. TOTAL AND WEIGHTED ERRORS. GROUP AND INDIVIDUALS'Seaeeeeeeeeeeeeeeeeee 54" 58 IX. CORRECT RESPONSE SEQUENCE WITH PAIRw STIMULUS PATTERNS 69- 7O FIGURES 1. MEAN PROPORTION AND MEAN NUMBER OF CORRECT RESPONSES OFGROUPSA,E,A+E,A'ANDE'............ 18 2. SCATTER PLOT OF 80 INDIVIDUAL TOTAL AND WEIGHTED mon MIMTES OF 9 O O O O O O C O O O O O O O O O O 31 3. SCATTER PLOT 0F 80 HYPOTHETICAL INDIVIDUAL TOTAL AND WEIGHTED ERROR ESTIMATES OF 9, WHERE T. GROUP 8 2.25 4O Viie I NTRODUC T IO N This thesis is concerned with quantitatively formulating and testing the concept of additivity-of-cues. In a number of experiments. learning has been found to be more rapid Ihen relevant stimuli are presented in more than one modality. Eninger (1952) states that this follows from Spence's theory of dis- crimination learning. Thus. the hypothesis that cues have an additive effect in learning seems plausible. Prior to the recent application of mathematical models of learn- ing to such data. the only criterion for deciding whether or not cues were additive was whether subjects showed improved performance with increased cues. Performance increments were not algebraic in form and there was no suitable measure to apply which could reflect a rational function of additivitw. Qualitatively. it was known that learning is always faster in the combined-cue situation, but no quantitative laws had been formulated and tested (Restle. 1955). In a series of recent papers (1955, 1957, 1958). Restle has used a mathematical model of discrimination learning to quantify the analysis of additivity-of-oues. Included in his analysis have been examples of additivity-cf-cues in T-maze learning of rats (Blcdgett et al.. 1949: Sninger. l952: Galanter a: Saw, 1954) and Scharlock. 1965) and in color. form A: size discrimination learning by monkeys (Warren, 1955). Recently, (Beetle. 1959) this analysis has been extended to hI-an learning in a simple two-choice discrimination problem.- 1 2 The assumption made by Beetle (1955 and 1959) is that in simple two-choice discrimination learning. the proportion of relevant cues determines the rate of learning. Thus it follows that additivity of relevant cues will be manifested as the additivity of learning rates. which can be estimated. 2 The stimulus situation in two-choice discrimination learning experiments is represented by a set of discriminable aspects called cues. Every individual cue my be thought of as either ”relevant” or “irrelevant". A cue is relevant if it can be used by the subject to predict where or how reinforcement is to be obtained. Cues which are uncorrelated with reinforcement are irrelevant. This model contains two hypothesised processes of discrimination learning, “conditioning” and ”adaptation”. The relevant cues in the stimulus situation are conditioned to the correct response. On the other hand. the subject's responses become independent of the irrelevant cues. i.e.. irrelevant cues are adapted. Once a cue is adapted. it has no effect on response, and ciher cues contribute toward the probability of a correct response or an error accordingly as they are conditioned to be correct or wrong response. On each trial of a given problu. a constant proportion. 9. of unconditioned relevant cues becomes conditioned. The "fundamental simplifying assuszption" of his theory deals with the learning rate parameter, 9. This assumption is that 0:: r (1) r+i where 5 is the number of relevant cues in the problem and _i_ is the number of irrelevant cues (Restle. 1955. p. 12. Eq. 5). Thus. 9 is 3 the preportion of relevant cues in the problem, The rates of conditione ing and adaptation are assumed to be equal and to equal the proportion of relevant cues in the problem. The above definition of 9 implies that if one increases the number of relevant cues in the stimulus situation, the effect is to increase the learning rate. Additivity-of-cues is directly concerned with the effect of increasing the number of relevant cues in the problem. Similarly, a reduction in the number of irrelevant cues in the stimulus situation would have the effect of increasing the learning rate, 9. Both of these effecte will be treated in this thesis. Included as Appendix I to this thesis is a more detailed form of the mathematical model used in the analysis of additivity-cf-cues. Formulas and.methcds of estimating learning rates are contained therein and will be referred to when necessary. Consider a number of experhmental problems of two-choice discrimi- nation learning where the stimulus situation consists of a pattern of letters. In the present study, the letters are ABDEF which vary between capital and small from.trial to trial, but retain the same alphabetical order. The responses are X.cr 0, either of which may'be correct depending upon which pattern of letters appears. To make a given letter relevant, we make the correct response contingent on whether that letter is capital or mnall. For example, if E is relevant, when- ever E is capital the correct answer is 0 and whenever e is small the correct answer is X. To make a letter irrelevant we make the correct response independent of whether that letter is capital or small. If B is irrelevant, when B is capital, the correct answer may be either 4 I or 0. In this case, the subject cannot use B to predict reinforce- ments. We suppose that each letter gives rise to a set of cues. We shall call the set of cues arising from letter A by the name a, those from the letter B by the name 6 , etc. The measure of O. is written m(Q), and corresponds to the importance of this set of cues in controlling behavior. To test additivity-of-cues, we construct a problem where the letter A is relevant, another problem where the letter B is relevant, and a third problem where both A and E are relevant and redundant (the subject can use either A or E or both to predict reinforcement). In this third problem, the set of relevant cues 13008 , that is, the _ set of cues which are in Qor in (for comcn to both. The hypothesis of additivity states that and 8 are disjoint, having no ccmon elements. In this case, .=m+.e05 TABLE V MEAN WEIGHTED ERRORS IN 128 TRIALS AND PROPORTION OF RELEVANTCUES (9) FOR GROUP A E Mean Total Source 93 Errors ___F 2 Observed M-E .070 495.34 ----- ----- Pred. 91+ 92 .075 428.00 0.35 >.05 The four predictions of 93 shown in Tables IV and V were tested by converting 93 into a corresponding expected mean error score or mean weighted error score. The predicted mean error score or mean weighted error score was then treated as a fixed value as the variance of the predicted score was unknown. Each predicted score was then tested against the observed error score for Group AtE by means of 'a t 25 test using the variance of the mean of Group A+E. Any failure to reject THE NULL HYPOTHESOS OF No DIPFERENCB BETWEEN the predicted mean error score and the observed mean error score is favorable to the theory. Therefore, this t_test seemed more conserva- tive as it would be less likely to reject the null hypothesis than a.t test'where the two variances are unknown.but presumed equal. The re- sults indicated a failure to reject the null hypothesis in each case, and the predictions are all supported. Similarly, these predictions were tested by means of the binomial test, where the number of cases falling above and below the predicted score was hypothesized to be equal. In all four cases, these tests were statistically non-signifi- cant at the .05 level, and supportive of the predictions. Since this test assumes the predicted:median.to be fixed, whereas in fact it is a random variable depending on sampling variations in the data of groups used to make predictions, it is also overly stringent. Hypothesis 1, the prediction of direct additivity of cues, 93': 91+- 92, is fairly accurate in the predictions made using estimates obtained by both total and weighted error methods. In the first method (total errors), the tendency is to underestimate the learning rate of Group A+E. By the second method (weighted errors), the prediction tends to overestimate the observed value. When the predicted 9 values were converted to mean error scores and tested against the observed mean error score for Group A+E, the resulting differences were found to be statistically non-significant in both cases. Thus, if the total number of cues is the same in the three problems, the proportions of relevant cues will also add. Knowing the proportions of relevant cues in two single-cue problems, it is possible to predict 26 the prOportion of relevant cues in a third problem which contains the sum of the relevant cues in the first two problems. In addition, it is possible to predict additivity-of-cues of unequal strength in the combined-cue problem. Hypothesis 2, the prediction of additivity-of-cues by'a function, is perfect when mean total errors 1-9495 esthmates are used. In the case where mean weighted error estimates are used, the prediction tends to overestimate the observed value of Group AtE. By the results obtained using both total and weighted error estinntes of 9, we can demonstrate additivity of relevant cues, r3 = r44- r5, even in a situation where the 9 values do not add directly because of differences in the denonimator values of the 9's or because of the different strengths of the relevant cues. AThe peculiarities of differences betleen the two estimation.methods again are reflected in the predictions. When mean total errors are used, the predictions tend to underestimate slightly or are perfect. 'lhen mean weighted errors are used, the predictions tend to overestimate the observed values. Again the suspicion is that individual subjects did not distribute their errors as the theory predicts. However, the dis- crepancy may also be inherent in the methods of estimation. From.the differences observed in Table III, it appears that the subjects in Group A+E may have made disprOportionately more errors late in learning compared with the other four experimental groups. The difference between the Observed 9 values by the two methods for Group A+E are 27 larger than those in the other groups. The exact reason for these discrepancies remains unknown at present. As individual differences were present in terms of performance in the experimental groups, a further test of the prediction of additivity of relevant cues was carried out. Using the same two hypotheses of additivity, the question arose as to whether or not the cumulative distribution of learning rates in the combined-cue situation could be predicted. The assumption was made that there was a high positive correlation as to rank position in the different experimental groups. Acting upon this assumption, the individual 9 values were ranked from the lowest to the highest in each group. The matched rank 9 values for Groups A and E were then added directly to yield a predicted distribution (Hypothesis 1). Similarly, the matched rank 9 values for Groups A‘ and E' were combined by the described additivity function to yield a second predicted distribution.(Hypothesis 2). These calculations were carried out using both total and weighted error estimates of 9. The four resulting cumulative distributions were then tested against the observed cumulative distributions for Group A+E by means of the Kolmorgorov-Smirnov cumulative distribution test for two observed distributions. Hypothesis 1, 91‘? 92:: 93, was accurate for both total and weighted error estimates of 9 (maximum difference Observed was equal te 5, whereas for a two-tailed test, p=.05, a maximum difference of 8 is necessary for rejection of the hypothesis). Hypothesis 2, the hypothesis of additivity by a function of 94 and 95 to predict 93 was also accurate for both total and weighted 28 error estimates of 9 (maximum difference observed for total error 9's was 4 and for weighted error 9's was 3). Thus in all four cases, the prediction of additivity of cues by additivity of the proportions of relevant cues were accurate in pre- dicting the cumulative distribution of the combined-cue problem. The prediction of the whole distribution, despite the extra assumption of positive correlation, seemed appropriate since the pre- diction takes into account the large variance of the predictor groups, A a E, and A’ a E'. The predictions using means disregarded the fact that the predictions themselves are uncertain and are based on the means of the groups with skewed or bimodal distributions. 0n the other hand, the Kolmorgorov-Smirnov test is weak against any particular dif- ference between the distributions compared, making it likely that one would commit a Type-II error favorable to the theory. As the Kolmorgorov-Smirnov test is weak against any particular difference betwaen the distributions compared, it seemed desirable to make use of a more powerful test. In the tests of the mean score predictions (Tables IV and V), the variance of the predicted score was noted to be unknown. In this case, we have a predicted distri- bution of scores and a variance. It is possible to test differences betmen the predicted and observed distributions by means of a para- metric _t_ test where the variances are \mknown but presumed equal. Hypothesis 1, 93: 91 + 92, was again found to be accurate for both total and weighted error estimates of 9. For the total error prediction, t = 1.11, df = so, with .30 > p ) .20. For the weighted error prediction, t t 0.86, df= 30,.with .50 ) p ) .40. 29 Hypothesis 2, f(94,95) = 93, was also found to be accurate. For the total error prediction, t = 1.16, df a 30, with .30) p >.20. For the weighted error prediction, t = .74, df 5' 30, .50) p ) .40. Thus in all four cases, the prediction of additivity-of-cues by the additivity of individual preportions of relevant cues was accurate in predicting the distribution of learning rates of the combined-cue problem. In these tests, it was shown that the mean of the distribution, taking individual differences into account, could be accurately pre- dicted. The median, as another measure of central tendency, could also be predicted and tested by use of a binomial test. Fbr each predicted distribution and corresponding observed distribution, the median was found and a four-fold classification of subjects falling above and below the median value for the predictor group and the observed group (Id-E) was constructed. A Chi-square, corrected for continuity, was performed on the four predictions. In all four predictions, Kiwas found in be equal to 1.12, with 1 df and .30) p) .20. Again, the hypotheses of additivity are not rejected. Thus by making an additional assumption of a positive correlation as to subject position in the predictor and predicted groups, it was possible to demonstrate the accuracy of the hypothesis of additivity of cues by predicting the cumulative distribution, mean and median of the combined-cue group. The results may be summarized with respect to four points: 30 First, the general order of the means was as expected; Group ArE was best, indicating some form of additivity; the letter A produced a larger measure of cues than the letter E, and reducing the measure of irrelevant cues had a small beneficial effect.on learning. Second, the two methods of estimating 0 yielded discrepant results suggesting that neither the group nor the individual learning curves were of the shape predicted by the theory. It is possible that such discrepancies are inherent in the estimation methods but the exact nature of the discrepancies is at present unknown. Third, the quantitative predictions of additivity-of-cues were fairly accurate. The distributions of scores and the unknown distri- bution of 9 caused difficulties in making tests, but our most stringent parametric test £3§12_to indicate that the hypotheses of additivity must be rejected. Fourth, by making an additional assumption as to subject position in the distributions it was possible to make accurate predictions of the cumulative distribution, mean and median of the combined-cue group, taking into account individual differences in learning rate. DISCUSSION In this experiment, the difficulties encountered in finding comparable estimates of 9 by the two methods (total and weighted errors) suggested by the theory, forced the making of predictions of additivity with separate 9 values. An examination of the discrepancies between values of 9 estimated from the individual total and weighted error scores reveals some perculiarities inherent in the methods. Figure 2 shows a scatter-plot of individual total and weighted error estimates of 9. Insert Figure 2 Here Inspection of Figure 2 indicates that the two estimates of 9 are very close for low rates of learning (9 < .10). The discrepancies be- come larger as 9 increases. The observed discrepancy distribution .was positively skewed, with 70% of the differences being very small (from -.009 to .015). Discrepancies larger than .015 (n:23) and the corres- ponding error scores showed a negative correlation of -.58, p<.01. The range of the discrepancies was from..016 to .280, while the cor- responding range of errors was from 12 to 1. Thus, the discrepancy between the two estimation methods is largest for fast learners. A discrepancy between two estimates of the same parameter may have any of several causes: the most important possibilities are (a) one or both of the estimates may be biassed, or (b) the data may not conform to the model. If alternative (b) is the case, the model should 31 33 be rejected. To decide this, it is necessary to eliminate alternative (a), that the estimates are biased. When estimatea are computed on the group mean total errors or the group mean weighted errors, serious discrepancies can arise from.in- dividual differences among subjects in the parameter 9. As is mentioned above, the average learning curve of a heterogeneous group will tend to be flat, some subjects making almost no errors and others making errors late as well as early in learning. The weighted-error method yields a low 6‘ (indicating slow learning because of fixe numerous late errors) and the total-error method yields a higher‘6~(indicating that the group curve is above p:.50 during most of the problem). How- ever, it was also found that estimates based on individual subject's data were discrepant, and this cannot be explained by effects of averaging over heterogeneous groups. The largest individual discrepancies occurred for large values of 9. It is possible that such discrepancies arise, not because of any psychological phenomenon, but merely as a statistical artifact. This possibility was given a preliminary test‘by a Monte-Carlo com- putation as follows. The theoretical error curve, ( l-p(n) ), was computed with 9 =' .25. Using these values and a table of random numbers, data for 80 hypothetical subjects were constructed. These data conform exactly to the theoretical curve of learning. The two methods of estimating parameters, through total errors and weighted errors, were used on the data of each individual hypothetical subject. If the methods of estimation behave in a desirable fashion, the estimated values of 9 should be about .25. 34 It was found that the estimates were quite variable, with values centering in the region of .25. Figure 3 contains a scatter-plot of A 6E (9 estimates based on total errors) and 9W ( 9 estimates based on weighted errors). Inspection of Figure 3 indicates that the discre- \ pancies are larger with a higher value of 9, but the discrepancies are in the opposite direction to those observed in our data. In the hypothetical procedure, it was found that ’9“, tended to be larger than ’63 (z a 3.15, p:.0016). This difference is Opposite to that observed in our data. However, the discrepancies beheen the two estimated values, ( {SE-8W ), yielded a distribution of the same general shape as that in- Figure 2, but the distribution is shifted to the negative side of the scale. The result strongly suggests that when parameters are estimated for individual subjects, (i.e., with sample size 1) the estimtes are biased, and file bias is different for the two methods of estimationl. The Monte-Carlo computation does not explain the _A 1. An estimate is said to be biased if the average of a great many estimates, each based on a finite sample, fails to converge to the true value of fire parameter. This is distinct from. inconsistency, where a single estimate based on a very large (infinite) sample fails to converge to the true value. Many methods of estimation commonly used in statistics are biased, and this fault is usually considered less important than inconsistency or than inefficiency. In this as in most learning models (Bush & Mosteller, 1955), estimates are made by the best method available, but may be quite imperfect. Technically, both of the methods of estimation used in this thesis are extensions of the "method of moments" and are closely related to methods in common use in learning theory. Severe computational difficulties accompany attempts to use the more desirable method of maximum likelihood. 36 discrepancy found, but suggests that it may well be due to estimation rather than to a serious discrepancy between theory and facts The lento-Carlo procedure does offer a fruitful method for generating empirical distributions of parameters such as 9 and for exploring the behavior of such discrepancies further. The foregoing analysis helps to clarify the overestimation of the rate of learning of Group A+E, when the prediction was made by weighted- error 9 values of the single-cue groups. Group A+E contained all learners, most of whom solved the problem early in training. Groups having one cue problems contained learners and non-learners. It has been noted that the larger discrepancies between the two types of estimated values ( 33-3“ ) occurred when the rate of learning was faster. Secondly, Table III shows that for Group n+3, the difference between the two estimates of 9 was larger than the differences for the other four experimental groups. When the additivity computations were carried. out, it is apparent that these discrepancies remained in the prediction. It appears reasonable that the discrepancy may be due to biases in the estimation methods. The present experiment yielded a quantitative test of the concept of additivity-cf-cues used in several recent papers (Restle, 1955, 1957, and 1958). Two kinds of additivity of the proportions of relevant cues were shown, one direct and the second by a function. The pre- dictions were formulated, a priori, in terms of set-theoretic mathe- Inatics and tested on cues arising from separate letters in a stimulus pattern. The predictions were found to be reasonably accurate in all cQBCBe 37 One or both of these hypotheses of additivity has been applied in analyses of experiments involving infra-human subjects (Restle, 1955, 1957 and 1958). The first hypothesis of additivity (direct additivity of 9‘s from separateecue groups to predict the 9 of a combined-cue group) was tested against data from “place and response” experiments on rate in a T-mase (the data of Galanter a Shaw, 1954; Sherlock, 1955; Blodgett et a1., 1949; all in Restle, 1957). The second hypothesis of additivity (additivity of 9's from separate-cue groups by a function to predict the 9 of a combined-cue group) has been applied in analysis of color, form and size discrimination learn- ing by monkeys (the data of'larren, 1953; in Beetle, 1958) and T-mase learning of rats (part of the data of Scharlock, 19553 and Eninger, 19523 in Beetle, 1957 and 1955). These earlier tests of the theory were a;pggterio£i and based on data not specifically designed for the testing of the hypothesis of additivity-of-cues. Therefore, any failure of these predictions, involving data designed for the purpose of testing them, would have led to serious doubts concerning the adequacy of the theory as an analytical tool. However, the positive results obtained here fail_tc indicate that the hypothesis of additivity should be rejected and support the use of the theory as a means of quantifying and analyzing data from suitable learning experiments. In a study similar to the present one, involving human subjects, Restde (1959) used the direct additivity hypothesis to predict per- .fcrmance of a combined-cue group. The relevant cues used were letters Jin.a pattern of consonants. The letters were all capitalised and ‘thus did not change in form.over trials. In the present study, varying 38 the relevant and irrelevant letters between upper and lower case was intended to increase the number of stimulus dimensions that the sub- ject had to discriminate in order to solve the problem. This increase of irrelevant cues through the use of changing letter forms'was expected to reduce the rate of learning. VA comparison of the estimated prepor- tions of relevant cues (9's) obtained by Restle with those of this study show the effects of increasing the number of irrelevant cues. The estimates reported by Restle are consistently larger than those obtained here. Further, Restle did not find the relevant cues in his study to be of unequal strength, whereas our analysis does. The demonstration of additivity of unequal cues is of interest. It could not reasonably expect relevant cues in problems of discrimination learning to be always of equal strength. For example, lhrren (1953) found the visual component of color to be the dominant cue over form and size in the discrimination learning of monkeys. Our analysis indicates that cues of unequal strength can be adequately handled in a quantitative analysis. A second way of analyzing the additivity result is by configura- tions or stimulus patterns (Restle, 1959). Since Problems A', E' and AtE all had 15 patterns, they should be learned at the same rate, according to the pattern hypothesis. In Figure 1, it is shown that E"was slower than A' and that both E' and A. are slower than A+E in learning their problems. Also, since Problens A and E are both 30 jpattern problems, they should be learned twice as slowly as Problems .A', E' and AtE. Inspection of Figure 1 shows that Group AfE reached 90% correct in about 28 trials, while Groups A' and A reached the same 39 level of performance in 80 and 128 trials, respectively. Groups 3' and E never reached 90% correct, but are at about 82% and 73% correct at the end of 128 trials of training. These findings in no way support the hypothesis that Problems A+E, A' and E' are learned just twice as fast as Problems A and E. The failure of the configurational hypothesis may be partly due to the existence of unequal cues in the stimulus situation. Such a hypothesis does not take into account cues of differential strength ‘but deals solely with the number of patterns. The use of a theory which considers both the number and strength of relevant ones, as does the one used in this study, leads to a more accurate prediction of performance. SUMMARY To test additivity-of-cues, a two-choice discrimination learning problem was used. Five groups of 16 human subjects each were tested on separate problems. Two problems had one cue relevant and one cue irrelevant. A second two problems had one one relevant, but the measure of the irrelevant cues was reduced. The remaining problem had both cues relevant. The stimuli were patterns of letters which had a fixed alphabetical order but varied in form.between upper and lower case. The response was written when the subject saw a pattern of letters. The correct answer (an.X or O) was predictable from the pattern of letters by'a consistent principle. I Two hypotheses of additivity were formulated and tested: direct additivity (prediction of the proportion of relevant cues in a combined- cue problem.by addition of the proportions of relevant cues in two single-cue problems) and additivity by a derived function (prediction of the proportion of relevant cues in a combined-cue problem.by a function of the proportions of relevant cues in two single-cue problems, where the measure of the irrelevant cues was reduced). The results may be summarized as to four findings: 1. The combined-cue group showed faster learning than the single- cue groups, indicating some form of additivity. One of the relevant cues was found to be stronger than another and the reduction of the measure of irrelevant cues through the fixing of a latter had a small beneficial effect on learning. 40 41 2. The two methods of estimating the learning rate parameter, 9, yielded discrepant results, indicating that neither the group nor the individual learning curves were of the shape predicted by the theory. An analysis of the discrepancies between the two methods of estimation suggested a bias inherent in the methods. 3. The predictions of the mean error scores by both hypotheses of additivity were found to be accurate in all cases. Statistical tests of these hypotheses failed to indicate that they should be re- jeoted. 4. To account for individual differences in learning rats, an assumption of a high positive correlation as to subject position in the groups was made. Predictions of the distribution of rates in the combined-cue problem by the application of the hypotheses of additivity to the matched rank values in the single-cue groups were made. These predictions were found to be accurate for the cumulative distribution of learning rates, the mean and the median of the combined-cue group. The discrepancies between the two methods of estimating learning rates were discussed. Inspection of the discrepancies indicated that the discrepancies were larger with faster rates of learning. A Monte- Carlo procedure was used to test this difference but did not clearly indicate the nature of the discrepancies. It is suggested that such a procedure would be fruitful for investigations of the behavior of the learning rate parameter, 9 and its theoretical distribution. A second quantitative analysis, based on the number of stimulus patterns in the problems was discussed and found not to be consistent with the additivity-of-cues data. APPEND ICES 42 43 APPENDIX I RESTLE'S THEORY OF DISCRIMINATION LEARNING Recent mathematical formulations of learning (Bush & Mostellar, 1951 & 19503 Estes, 19503 and Restle, 1955, 1957) have described the stimulus situation as a set of elements, each of which is conditioned to (i.e., tends to evoke) exactly one response at a given time. During learning, if a certain response, A1, is reinforced, a cue may switch and become newly conditioned to A1. The probability of such a change is the rate of learning parameter, 9. The major points made in the two-choice discrimination theory are as follows: 1. The stimulus situation is represented by a set of discriminable aSpects called "cues"; 2. A cue may be ”conditioned" to either response; 3. A cue may be "adapted" and rendered nonfunctional during learning; 4. The probability of a response is the prOportion of the un- adapted cues conditioned to it. Stating these assumptions quantitatively shows that under certain limiting experimental conditions it leads to a process similar to that in Estes' theory and, in particular, with the same asymptotes. 1. This apoendix is largely a restatement of Restle's two-choice dis- crimination theory as developed in two recent papers (1955 and 1957). It is included primarily as a direct reference for those who are interested in the mathematical and logical development involved in the estimation of learning rates. Additions to the theory, in the form of definitions, examples and formulas, have been made to help clarify the theoretical formulation to the new reader. 44 Theory A) Set of cues The stimulus situation in two-choice learning experiments is represented by a set of discriminable aspects called cues, k, k', k"... The set of cues is called K and the number of cues is N. A subset:of these cues may correspond to anything to which the subject can learn to make a differential response. Such a definition assumes that the subject’has the capacity to learn a differential response. An in- dividual cue is thought of as "indivisible" in the sense that different responses cannot be learned to different parts of it. The term "cue" is also used to refer to any set of cues, all of which are manipulated in the same way during a whole experiment. Every individual cue may be thought of as either "relevant" or "irrelevant". A cue is relevant if it can be used by the subject to predict where or how reward is to be obtained. For example, Von Frisch (1955) gives summaries of experiments on determining the chemical and color senses of the bee. In one case, bees were trained to feed only on a card colored blue. This training was accomplished by rewarding the bees with sugar-water only on a blue-colored card. Then, cues ‘which were aroused by the color blue are relevant. Similarly, position of the cards was randomized, such that the bees could not gain reward from the use of position cues. Cues from position are thought to be uncorrelated with reward or irrelevant. In the two-choice experiments testing this theory, the subject has just two-choice responses and no other activities are considered. 45 B) Conditioning of Cues We assume that a cue is conditioned to one or the other response alternative at any time on an all-or-none basis. The probability that one 1: is conditioned to response A1 at trial n is called F(k,n). If a cue is conditioned to A2 and then A1 is reinforced, it may switch over. and become conditioned to A1. The probability of such a switch is a constant called 9, the "rate of learning" parameter. On this assump- tion, we get the following equation of change of F(k,n): 1) if K is reinforced on trial n (k occurring every trial), F(k, n+1): F(k,n)(l-G)+- e (1) This equation may be solved by the linear difference equation method, whereTr is l, i. e. consistent reinforcement, giving men) = (1-F(k.1)(1-e)n'1) (2) C) Adaptation of cues During learning an irrelevant one may become "adapted” and lose its effect on a response. An adapted cue is one which the subject does not consider in deciding upon his choice response. If a cue is thought of as a ”possible solution" to the problem an adapted cue is a possible solution which the subject rejects or ignores. Different cues have different probabilities of being adapted. If cue k is not adapted by the beginning of trial n, and it is irrelevant, the probability that it will be adapted by trial n+1 is 9. This consideration gives us an equation for a(k,n), the probability that cue k is adapted at the begin- ning of trial n, as follows, a(k,n+l)- a(k,n)(1-9) + 9 (3) 46 Again, solving equation 3 by use of the general linear difference solution, we find, 80931)" 0-a(k.1)(1-9)“'1) (4) It will be noted that the same constant, 9, appears in both equations 2 and 4. The fundamental simplifying assumption of this theory deals with 9, where 9 is defined as a constant prOportion of unconditioned relevant cues which become conditioned on each trial of a given problem. This assumption is that: 9 = r / r + i where r is the number of relevant cues in the problem and i is the number of irrelevant cues. Thus, 9 is defined as the preportion of relevant cues in the problem. This proportion is set equal to the fraction of unconditioned cues conditioned on each trial. 0) Probability of a Response. The probability of a response, A , is the proportion of unadapted 1 cues conditioned to it. The probability that cue k is unadapted is (1-a(k,n)) and the probability that cue k is conditioned 1:: A1 is F(k,n), thus the performance function p(n) is, p(n) = zF(k.n)(1-a(k€n)2 (5) 2(l-a k,n The 2 (sumation sign) indicates the sum over all cues in the situation. D) Some consequences Regarding Simple learning If a subject is naive at the beginning of training, so hat for any relevant cue, F(k,l) is near % (two-choice learning, where chance is about 50% for success), then from equation 2, F(k,n) = 1 - Edi-er“ (6) 47 Similarly, on trial one, the probability of a cue being adapted is O, i.e., a(k,l): 0. Then, from equation 4, a(k,n)= 1 - (1-9)""’1 (7) Under these circumstances, we can substitute equations 6 a 7 into equation 5 (performance function) and taking advantage of the simplifying effects of our definition of 9, we have, ( )=. 1 -%(l-9)n-l 1’ n ma ‘8’ The development of equation 8 is as’follows: First, in equation 5, it will be noted that the summation sign calls for summing the probabilities over all cues. We shall divide our labor into sunning over two kinds of cues, relevant and irrelevant. In other words, we are partitioning our composite sum into two sums. We first consider the numerator, F(k,n)(l-a(k,n)). For the 1: relevant cues, F(k,n) :: 1 - %(1-9)n'1 and a(k,n) : O, whence the function is just 1 - §(1-e)n'1. when an. 1. eumed over the 5 relevant cues, it being the same for each of them, we get, r(1-%(1-e)n'1). Now when we sue over the _i_ irrelevant cues, we have F'(k,n)"-= i, because by the nature of the experimmt an irrelevant cue cannot be consistently conditioned to the correct response, but a(k,n): 1 - (1-9)n-1. Summing F(k,n)(l-a(k,n)) over the irrelevant cues gives 1(%)(1-e)“'1. Suming F(k,n)(l-a(k,n)) over all cues gives, r(1-1e( HP“) + “flu-9P“. 48 The same procedure of summation is used in dealing With the denominator, which is (1-a(k,n)). Stunning over the 5 relevant cues, for each of which a(k,n) '2. 0, we get 5. Suming over the _i_ irrelevant cues gives i(l-9)n'1. The denominator term now is, r+ i(l-9)n-1. Forming the ratio, we have, pm 3 r(1-%(1-9)n'1)+ e we)“: . r.+ 1(1-e)n"1 Dividing each term in the equation by r+i, _-.-. - 1-e “'1 1 1-9 “'1 Mn) “FET— ifiré ) sari} ) r + 1 (1-9)“"1 * #1 3171' Taking advantage of the simplifying effects of G: r/r+i, and (1-9): i/rti. ph3C>C>C>#-¢-F‘C>C> 00H .086 .500 .063 .044 .035 .165 .500 .261 .055 .115 .500 .337 .261 .080 .410 .049 .043 .033 .146 .410 .146 .044 .116 .460 .213 .220 1.0001.000 1.0001.000 .086 .079 1 21211111211121: 4.9 .9... 0 0 0 0 0 1 0 0 0 0 0 0 o o 0 0 0 o 0 3 1 2 1 0 2 0 1 1 1 2 3 3 1 0 0 0 1 3 5 4 5 3 0 0 1 0 0 0 0 o 0 o 0 0 o 0 0 0 0 0 0 o 0 0 1 0 0 0 0 0 o 0 1 5 2 2 3 0 0 1 o 0 0 0 0 0 o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 o 0 0 0 0 0 0 0 0 o 0 0 o 0 o o 0 o o 0 o o o 0 0 0 0 0 0 o 0 o 0 0 §i13710711'12' 37 2' 1' 2' 96 90 92 91 91 96 96 99 96 Total Weighted Errors 358.0 4.0 1229.0 1623.5 2493.0 72.0 4.0 72.0 1511.0 134.0 3.0 25.0 23.0 0.0 0.0 374.0 495.34 Group 9‘s .100 .070 56 57 TABLE‘VII (Continued) Group A' Errors per Block of 8 Trials __..sa 121122111121111111212; .9... 0.3. 5 3 2 5 2 6 4 l 0 0 0 0 0 0 0 0 .053 .058 R.V. 4 4 3 0 2 4 0 2 0 l 0 0 0 0 0 0 .069 .067 B.R. 4 5 7 4 6 1 2 0 5 0 2 2 O 3 0 1 .038 .040 A.C. l 0 0 l 0 0 0 O 0 0 0 0 0 0 0 0 .337 .208 D.D.Ol00000000000000.500.340 V.P. l 0 0 0 0 0 0 0 l 0 0 0 0 0 0 0 .337 .151 L.F. 2 4 2 3 1 2 l 1 l 2 1 3 2 2 0 l .053 .044 Jtlh 4 6 3 4. 2 1 l 3 1 0 0 1 0 0 0 0 .057 .060 J.B. 2 l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .261 .245 IuRe 2 6 3 3 4 3 5 4 5 4 0 1 2 0 0 0 .038 .039 A.A. 5 2 2 0 0 0 0 0 O 0 0 0 0 0 0 0 .124 .136 0.8. 3 3 1 1 4 1 l 1 3 2 2 2 2 2 l 1 .051 .041 A.H. 3 3 3 3 4 3 3 4 2 2 2 4 l l 3 4 .035 .031 ll.l.2100100000101000.165.095 JkP. 3 3 5 6 4 2 1 0 0 0 l 0 0 0 3 0 .053 .053 R.K. 3 5 5 3 4 6 4 4 3 3 3 5 5 6 5 6(.005(.005 7676745 4'6 33 33' 3'4' 22 22' 20' 21 14" 12 16 1'3’ 1‘4‘ 12 13 """"‘"""'" % 6O 64 72 74 73 77 83 84 84 89 91 86 90 89 91 90 Correct 39 Total Errors Total Epighted Errors Group 9's G.E. 28.0 833.0 .060 .050 ROVI 20.0 570.0 B.R. 42.0 1824.0 A.C. 2.0 27.0 D.D. 1.0 16.0 VtP. 2.0 65.0 L.F. 28.0 1533.0 J.IL 26.0 746.0 J.B. 3.0 17.0 luR. 42.0 1919.0 A.A. 9.0 88.0 0.8. 30.0 1748.0 A.H. 45.0 2731.0 M.l. 6.0 229.0 J.F. 28.0 1011.0 R.K. 70.0 4699.0 Total 382.0 18046.0 Mean 23.88 1127.88 TABLE VII (Continued) Group E' Errors per Block of 8 Trials _s- 111.4.111.8.2121111.1111.12.11_...,_e .9... _ 2.7. 4 4 2 5 5 5 1 3 5 5 6 2 3 7 3 3 .009 .006 J.C. 2 6 2 2 3 5 2 ‘1 3 0 1 1 1 0 0 o .049 .050 9.7. 3 4 3 4 2 2 5 4 3 2 5 2 3 2 3 2 .032 .029 “no. 4 2 3 4 4 4 4 5 3 4 6 3 2 4 1 3 .024 .024 1.0. 1 0 1 0 0 0 0 0 0 0 0 0 0 0 o o .337 .206 1.2. 4 4 4 5 4 4 3 3 3 2 0 3 3 4 5 3 .027 .026 0.2. 1 0 2 2 0 0 0 0 0 0 0 0 0 o 0 0 .167 .130 J.W. 12-3 3 4 5 5 4 5 5 4 4 3 3 7 5 3 <.005<;005 J.M.3602400000000000.086.088 D.H. 1 3 1 3 1 0 0 o 0 0 0 o o 0 0 0 .124 .106 2.0. 1 0 0 0 0 0 0 0 0 0 0 0 o 0 o 0 .500 .540 0.22 2 2 5 3 2 3 5 1 3 5 3 4 1 3 0 2 .036 .034 1.0. 3 3 3 3 5 6 5 3 4 6 3 4 3 3 2 2 .022 .023 2.1. 3 1 2 4 7 4 2 3 4 3 0 3 3 3 2 2 .034 .031 J.D. 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 1.0001.000 2.1. 5 5 0 3 4 2 5 1 3 4 3 5 5 1 2 3 .030 .027 ‘mmwnammmmmmnwnnnn***' % 70 65 76 65 64 69 72 77 73 73 76 77 79 73 62 62 Correct 36 Total Errors Total'leighted Errors Group_§lg 2.7. 63.0 4039.0 .046 .039 J.C. 31.0 1152.0 9.1. . 49.0 2947.0 ‘w.0. 56.0 3419.0 1.0. 2.0 27.0 1.2. 54.0 3239.0 0.22 5.0 96.0 J.w. 64.5 4329.5 6.2. 15.0 277.0 0.2. 9.0 170.0 9.0. 1.0 1.0 0.3. 44.0 2621.0 1.c. 56.0 3488.0 2.1. 46.0 2763.0 J.D. 0.0 0.0 2.1. 51.0 3115.0 ‘2537 52275 3132375 Mean 34.26 1960.34 59 TABLE VIII STIMULUS LETTER PATTERNS, TRIAL APPEARANCES AND TOTAL NUMBER OF ERRORS MADE PER STIMULUS PATTERN Group A Errors Group E Errors Trial Appearances Response Response 0 0 1. ABDEf 7 ABDEf 16 59, 74,101,112 2. ABdEF 10 ABdEF 13 17, 49, 86,117 3. AbDEF 16 AbDEF 21 8, 45, 78,107 4. ABDeF 16 .6066 27 25, 52, 62, 99,123 5. ABdEf 16 ABdEf 20% 2, 36, 64,111 6. AbDEf 15 45061 22 26, 40, 56,113 7. ABDef 17 aBDEf 24 16, 34, 75, 95 6. AdeF 17 AdeF 23 6, 32, 57,102 9. ABdeF 22 aBdEF 25 18, 33, 73, 66,120 10. AbDeF 20 abDEF 23 13, 36, 62,110 11. Adef 13 Adef 3o 0, 29, 60, 76,116 12. ABdef 14 aBdEf 36% 15, 39, 71, 94,125 13. 450.1 11 abDEf 29 10, 20, 65,105 14. AbdeF 17 adeF 25 7, 46,100,116 15. Abdef 17 adef 26 50, 69,103,124 Response Response x x 16. .6066 - 7 ABDeF 23 55, 79,119,127 17. aBDEf 10 ABDef 21 14, 63,106,115 18. aBdEF 9 ABdeF 22 24, 53, 96,126 19. abDEF 13 450.1 35 3, 43, 66, 65 20. aBDeF 21 aBDeF 32 11, 27, 51, 92,121 21. aBdEf 12 ABdef 25 9, 30, 27, 91 22. abDEf 9 450.1 22% 21, 35, 70,104 23. aBDef 13 aBDef 27 1, 42, 58, 63 24. adeF 17 AbdeF 34 19, 47, 66, 97,114 25. abDeF 23 abDeF 35 23, 37, 54, 93,109 26. aBdeF 16 aBdeF 26 12, 31, 69, 90 27. adef 22 Abdef 24% 4, 28, 61, 61 28. aBdef 17 aBdef 26 22, 44, 60,106,122 29. abDef 15 450.1 24 5, 46, 67, 67 30. abdeF 6 abdeF 17 41, 72, 64, 96 TotaI . '4-4—0' 2&9 3&10 (nnhtkrkb &12 6&15 8&14 11&15 16&20 17&23 18&26 19&25 21&28 22&29 24&30 27 Group Response 0 aBDEf aBdEF abDEF aBDEF aBdEf abDEf adeF adef Response X ABDeF ABDef ABdeF AbDeF ABdef AbDef AbdeF Abdef .1353.— Errors 0) NIH 25 10 25 12 17 17 13 157.5 TABLE VIII (Continued) Group A' Errors Response 0 ABDEf 24 ABdEF 21 AbDEF 30 ABDEf 4 ABdEF 4 AbDEF O ABdEf 33 45061 22 AdeF ' 26 Adef 15 Response 4x aBDEF 27 aBDEf 29 aBdEF 25 abDEF 30 aBdEf 28 abDEf 28 adeF 14 adef 22 382.0 Group E! Response 0 ABDEf ABdEF AbDEF ABDEf ABdEF AbDEf ABdEf AbDEf AdeF Adef Response X ABDeF ABDef ABdeF AbDeF ABdef AbDef AbdeF Abdef Errors 39 42 4O 11 (330)01 34 26 37 35 41 39% 31 34 29 21 546.5 60 Trial Appearances 59, 16, 17, 16, 120 8. 13, 25, 25, 74,101,112 33, 45, 36, 99 52, 52,123 62 2. 15, 125 26, 10, 6. 7. 29, 50, 55, 11, 121 14, 1. 24, 12, 3. 23, 109 9. 36, 39, 40, 20, 32, 75, 95 86,117 73, 88, 78,107 62,110 82, 99,123 64,111 71, 94 56,115 65,105 57,102, 46,100,116 60, 76,118, 89,103,124 79,119,127, 27, 51, 92, 63,108,115, 42, 53, 31, 43, 37, 30, 56, 63 96, 126, 69, 90 68, 65, 54, 93, 27, 91, 22,44,60,106,122 21, 35, 70,104, 5, 46, 67, 67 19, 47, 66, 97, 114,41,72,64,96 4, 26, 61, 61 TABLE IX 61 CORRECT RESPONSE SEQUENCE'WITH PAIRED STIMULUS PATTERNS Trial Number Practice (DmQQUIDFOINH 10 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3O 31 32 33 34 35 36 37 38 39 4O 41 Correct Response Stimulus >c>c>>4c>>4c>c>c3>4>>4>>4><>4>>c3<3<3>><>>c>c>>4>4>> Pattern No. 11 23 5 19 27 29 8 14 3 21 13 20 26 10 17 12 7 2 9 24 13 22 28 25 18 4 6 20 27 11 21 26 8 9 7 22 6 26 10 12 6 3O Trial Number 42 43 44 45 46 47 46 49 50 51 52 53 54 55 56 57 56 59 60 61 62 63 64 65 66 67 66 _ 69 70 71 72 73 74 75 76 77 76 79 60 61 62 63 Correct Response >4C>P4>4>4C3>4C3C>CDC>>4C>><>4>4fi<>4C>C>>4CD>4CDC>>¢C>CD>4>4>><<3<>4C>C>64><>4 Stimulus Pattern N2: 23 19 28 3 14 24 29 2 16 20 4 18 26 16 6 8 23 1 11 27 10 17 5 13 24 29 19 26 22 12 3O 9 1 7 11 21 3 16 28 27 4 23 Trial Number 84 85 86 87 88 89 90 91 92 93 94 96 96 97 98 99 100 101 102 103 104 105 Correct Response c>c>c>c>c>c>c>>€><>4c>c>><><>4>c>>>424 TABLE IX (Continued) Stimulus Trial Pattern No. Number 30 106 '19 107 2 108 29 109 9 110 15 111 26 112 21 113 20 114 26 116 12 116 7 117 18 118 24 119 30 120 4 121 14 122 1 123 8 124 16 125 22 126 13 127 Correct Response ><>c>c3><>c>c>><>4c>c>c>c>><>4c3>4 62 Stimulus Pattern N0 0 28 3 17 25 10 5 1 6 24 17 14 2 11 16 9 20 28 4 15 12 18 16 1 REFERENCES Blodgett, H.'c., 11.0.5.5... x. a. 11.55.... 2. 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