Application of Factor Analytic Procedures to Complex Evoked Potential Measurements in Psychiatric Research1 Richard A. Roemer, Charles Shagass and John J. Straumanis1 Temple University Health Sciences Center and Eastern Pennsylvania Psychiatric Institute, Philadelphia, Pa.

Key Words. Somatosensory evoked potentials • Factor analysis • Psychiatric electro physiology - Schizophrenia Abstract. Several factor analytic procedures were applied to data gathered with a modified somatosensory evoked potential (SEP) recovery function prodecure in 56 non patients and 224 psychiatric inpatients. The data were analyzed for each option reflected in three main issues. (1) Which group should be used for calculating the basic factor structure - normals only or all subjects? (2) Should measurements be preadjusted for covariance with variables known to obscure events of interest? (3) Should varimax rotation be used, or are principal components sufficient? Using two criteria, interpretability of factors in relation to experimental design, and their ability to discriminate clinical groups, it appeared that: (1) choice of either population yielded similar results; (2) when measurements were pre adjusted for covariance more interpretable factors, which also yielded clinical discrimina tions, were obtained by varimax rotations than by principal components, and (3) when data were not adjusted by covariance, principal component analysis produced a factor which, although less interpretable in relation to experimental design than the varimax factors, provided equivalent clinical discriminations.

Evoked potential (EP) recording typically provides a large amount of data, from which many variables can be derived. There is a great need for effective methods to reduce such data to a minimum number of meaningful variables, particularly when EP findings are to be correlated with individual differences of a psychological nature. Although many significant correlations or differences may be found, the fact that the EP measurements are usually not independent of one another introduces uncertainty into interpretation of the results. For

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' Research supported (in part) by Grant MH12507 from the National Institute of Mental Health, USPHS. 1 We thank /. Chung Hung, William Nixon, Judith Pressman, and Stephen Slepner for technical and computer assistance and Dr. Donald A. Overton for advice.

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example, data from recent studies of somatosensory evoked potentials (SEP) in our laboratory resulted in 41 variables in relation to which psychiatric diagnostic groups were compared (6). Univariate tests of statistical significance revealed many differences, but the extent to which these differences were independent of one another could not be evaluated with methods that handle only one variable at a time. This led us to investigate factor analytic methods as a means of resolving the uncertainty. Factor analytic methods provide a means of isolating independent factors, or ‘supervariables’, in a set of measurements. The original data set can then be compressed into a less numerous set of factor scores. For example, Donchin (1) used principal components factor analysis to demonstrate how EPs containing 80 data points could be reduced to 10 factors, which accounted for 90 % of the information in the original measures. Two major theoretical issues arise when one attempts to employ variables (factor scores) derived from factor analysis to discriminate clinical psychiatric populations. The first problem concerns selection of the appropriate population upon which to base the factor scores. Should the total subject population be used or only the normal controls? If one assumes that the ‘true’ factor structure is that of the normal control population, it would be logical to factor .analyze the control group data, and to apply the 'normal' factor structure so derived to all subjects. Differences between clinical groups in such factor scores would then reflect the patient groups’ departure from normal. Alternately, factor scores could be derived from the factor structure of the total population studied. Use of all subjects allows the factor structure to be based upon a larger sample. However, this alternative has the disadvantage that the factor structure will reflect the data of a heterogeneous population (normals, patients of various kinds), likely to vary in composition in different studies. The second issue concerns the type of factor analysis procedure to be used. Is the principal components solution sufficient, or should varimax rotations be used? The result of a principal components analysis is a unique set of unrelated (orthogonal) factors which account for progressively decreasing amounts of the variability present in the original data set (2). Further, this method requires no parametric assumptions about the data, such as normal distribution. Unfortu nately, interpretation of the factor structure is often difficult after the largest three or four factors have been extracted. Varimax rotation may provide factor structures which are more easily interpreted. However, the varimax procedure requires the assumption that the data are multivariate normally distributed; also, theoretically, there exist an infinite number of rotated solutions in contrast to the single solution provided by principal components analysis. In this investigation, we applied factor analytic procedures to SEP data previously collected by means of a modified recovery function procedure and analyzed, by univariate methods (4 -6 ). In measuring recovery functions, one

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wishes to determine how initial (conditioning) stimuli modify EPs to subsequent (test) stimuli. Usually, there is a high correlation between the EPs to condition ing and test stimuli, and we have adjusted the test EPs for their covariance with the conditioning EPs to obtain independent measures of recovery (3). The nature of these recovery function data added a third issue to the two foregoing questions. Should unadjusted or adjusted values be used in the factor analysis? The three questions stated above each involve two alternative procedures, which combine to give eight (two X two X two) options. We evaluated these eight possibilities empirically here by applying two criteria. (1) Did the factor structure reflect the experimental design in an interpretable way? (2) How well did the factor scores discriminate between clinical groups?

Methods

Data Treatment Amplitude measurements were made, automatically by computer, on four specified and overlapping time periods, or post-stimulus epochs, selected to contain events of interest: 15-30, 15-49, 31-99, and 50-199 msec post-stimulus. An amplitude value for each time epoch was obtained by calculating the mean value, in pV, of all the data points in the specific epoch, and then computing the average absolute deviation of these data points from

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Evoked Potential Data SEP modified recovery function data were available for 56 nonpatient controls and 224 psychiatric inpatients. Psychiatric diagnoses were made independently by at least two senior psychiatrists. In previous univariate analyses of the data different diagnostic groups, matched for age and sex, were compared (4 -6). For present purposes, we selected the set of matched groups with the largest number of subjects to provide a clinical discrimination criterion against which to evaluate the factor scores. This set contained 80 subjects, 20 in each of four diagnostic categories; latent schizophrenics, nonpsychotic patients (personality disorders and neurotics), chronic schizophrenics, and normals. Each group contained 11 males and 9 females. The SEP modified recovery function procedure has been described elsewhere (6). Briefly, electrical pulses of 0.1 msec duration were applied to the right median nerve at the wrist and SEP were recorded from one bipolar scalp derivation over the left somatosensory area. Stimuli were applied either singly, in pairs, or in trains of nine or ten; interval between stimuli in pairs or trains was always 10 msec. Intensity of the conditioning stimuli was varied in relation to sensory threshold (T); the five conditioning intensities used were 10, 5, 2, 0, and -0 .5 mA above T. Test stimulus intensity was kept at T + 10 mA. The net result of the procedure was, for a given subject, five sets of four evoked potential records. Figure 1 shows two such sets for one subject, with conditioning stimulus intensities of T + 10 and T. The four responses in a set, and the designations by which they will be referred to in this paper, were as follows: (a) R1T, average of 50 responses to test stimulus alone; (b) R1C, average of 50 responses to conditioning stimulus alone; (c) R2, test response after one conditioning stimulus, obtained by subtracting 50 R1C from 50 (R1C + R2) pairs; (d) R10, test response after train of nine conditioning stimuli, obtained by subtracting responses to 50 trains of nine from 50 trains of 10 stimuli.

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C o n d ition in g s tim u tu s -'O mA ab ove th re sh o ld

Fig. I. Responses of one subject to illustrate SEP records obtained with modified recovery function procedure. Conditioning stimulus intensities were 10 mA above sensory threshold and threshold; test stimulus, 10 mA above threshold. Relative positivity at poste rior electrode gives upward deflection. R1 (R1T) is average response to 50 stimuli of test intensity. RC (R1C) is average response to 50 stimuli of conditioning intensity. R2 is response to 50 paired conditioning and test stimuli (interstimulus interval 10 msec) minus 50 RC. R10 is response to 50 trains composed of 9 conditioning stimuli and 1 test stimulus (interval between stimuli 10 msec) minus 50 trains of 9 conditioning stimuli. Initial negative deflection is designated as peak 1. R2 and R10 were suppressed with 10 mA conditioning stimuli, and slightly augmented with threshold conditioning stimuli.

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the mean value. The average deviation essentially reflects the area under the curve. For each of the five conditioning intensities there were 15 average deviations: four for R1C, R1T, and R2, but only three for RIO; since the sampling time period ended before 200 msec post stimulus for RIO, the 50-199 msec epoch was unavailable. For present purposes, the data were further reduced by calculating, for each epoch of each type of response, the mean of the average absolute deviation values across the five conditioning stimulus intensities; this reduced the 75 initial average deviation values to 15 means. In addition, the relationships between EP amplitude and conditioning stimulus intensity were derived by computing, by least squares, the linear slope relating average deviations for each epoch to stimulus intensity. This produced 15 slope values which, with 15 means, resulted in 30 values for each subject. These 30 values will be referred to as the 'unadjusted values’. Additional transformations were required to determine whether or not EP values needed to be adjusted for covariates known to obscure events of interest. Both R2 and R10 measurements are usually highly correlated with R1T measurements; additionally, R10 is also correlated with R2. To compensate for these correlations, and to provide independent estimates of the effects of the conditioning stimulus on R2, and of the conditioning train on R10, the R2 and R10 values obtained above were adjusted for their covariance with R1T and R2 by means of the ‘within groups’ regression equation. The means of the adjusted values for each epoch were obtained across conditioning stimulus intensities. This resulted in ten R2 and R10 means instead of the original seven (four mean R2, adjusted for R1T; three mean R10, adjusted for R1T, and three R10 adjusted for R2. There were also 10 slopes for adjusted values. However, a further covariance adjustment was required to obtain slope values independent of the corresponding mean levels, with which they were usually highly correlated. Adjusted slopes were computed, not only for the R2 and R10 values, but also for R1T and R1C values. The final data set, referred to here as 'adjusted values’, thus contained 36 variables: four R1T and four R1C means (unadjusted); four R2 and six R10 means (adjusted); 18 adjusted slopes corresponding to the means. Principal Component and Factor Analysis We employed a principal component factor analysis, in which the derived factor load ings are the quotient of each eigenvector divided by the square root of its associated eigenvalue. We shall refer to this as a principal component analysis or solution, and to the varimax rotation of the principal component solution as rotation. To evaluate choice of population and the relative utility of rotation, we performed both types of analysis on the unadjusted data of the 56 controls and the total population of 280 subjects. The second stage of our procedure was to perform the covariance adjustments for the data of both populations and to carry out principal components and rotation analyses for each group. After each principal component or rotation analysis, the data for the four clinical groups (80 subjects) were transformed into factor scores according to the following formulae: Fx = Z R "1 S, if rotation (varimax) was used: or

if principal component was used. Where: Fx = matrix of subject factor scores; Z = normal ized (relative to the population under evaluation) subject data matrix of 30 or 36 variables per subject; V = matrix of eigenvectors obtained by the principal component analysis; L =

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F X= Z V L -0'5,

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diagonal matrix of eigenvalues obtained by the principal component analysis; S = matrix of varimax rotated factor loadings; R~‘ = inverse of original correlation matrix of dimension 30 or 36. A two-way factorial analysis of variance (ANOVA) was used to test for significant mean factor score differences across the comparison populations, using diagnostic categories as the first factor and either age or sex as the second factor.

Results

Choice o f Population If a factor derived from one population is similar to a factor derived from another population, using the same array of variables, the factor loadings across the variables should be highly correlated with one another; the converse should hold for dissimilar factors. This criterion, the degree of correlation between the loading values assigned to the same variables, was used to evaluate the similarity between the factor structures obtained for the 56 controls and for the total population of 280 subjects. Using only factors with eigenvalue of 1.0 or more, the unadjusted data yielded five factors for the total population and six factors for the controls. The adjusted data gave seven factors for each group. Productmoment correlations were calculated between factor loadings obtained by each procedure for the two populations. Since the signs associated with the loadings are arbitrary (2), only the magnitudes of the correlations are relevant for present purposes, and signs are omitted in the tables. Table la presents, for unadjusted data analyzed by principal components, the correlations between the factor loadings of the two populations. Table lb shows the results obtained by correlating the factor loadings of the total popula tion with themselves. The correlations in table I b were needed in order to evaluate the extent to which factor loadings belonging to independent factors were actually correlated with one another; it should be pointed out that, even though the factor scores were uncorrelated across subjects, the factor loadings across variables can be, and were, correlated to varying degree. In a sense, table lb provides an ‘error term’ for comparing the correlations based on the two populations. Although a number of coefficients achieved the conventional 0.05 significance level, these results are rendered more interpretable by paying atten tion only to those correlations which achieved the p < 0.005 level; these are identified in the tables by an asterisk. By applying the 0.005 criterion, it will be seen that the correlations in table la provide essentially a diagonal matrix, similar to that obtained for the total population loadings (table I b).

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The results will be presented in relation to the three issues stated in the introduction.

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Table I. Product-moment correlations between factor loading values obtained in princi pal component analysis using unadjusted data for control subjects (n = 56) and total population (n = 280)1 Control factors

Total population factors 1

2

3

4

5

a Total population versus controls 1 2 3 4 5 6

0.99* 0.12 0.19 0.04 0.02 0.19

0.87* 0.13 0.06 0.37 0.10

0.07 0.94* 0.16 0.40

0.21 0.07 0.12

0.89* 0.20

b Total population versus itself 1 2 3 4 5

1.003 0.04 0.06 0.02 0.03

1.00 0.13 0.03 0.05

1.00 0.06 0.09

1.00 0.02

1.00

Table II gives the results obtained in correlating the loadings provided by varimax rotation of the unadjusted data. The high correlations in the diagonal (table II a) indicate considerable similarity between the two sets of factors. There were also some significant (p < 0.005) correlations outside of the diagonal, but it will be seen that the incidence of these (5 of 15; 33 %) in table Ha was the same (3 of 10; 30 %) as in table IIb. The results thus support the conclusion that the factor loadings derived from the two populations were similar. Table III shows correlation matrices for the adjusted data analyzed by principal components. Table Ilia indicates that the first four factors extracted by principal component analysis of either population’s data were similar, since the correlations along the diagonal are high. The loadings for two of the three remaining factors were significantly intercorrelated, but the order of factors was not identical for the two groups. However, it should be noted that factors 5—7 accounted for only 15 and 17 % of the variance for control and total population, respectively, compared to 51 and 58 % for factors 1—4. There were no signifi cant correlations in table Ilia, apart from those near the diagonal, whereas

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* p < 0.005. 1 Number of loadings per factor is 30. J All r’s in diagonal must be 1.00.

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Table //. Product-moment correlations between factors loading values obtained by varimax rotation of principal component analysis using unadjusted data for control sub jects (n = 56) and total population (n = 280)' Control factors

Total population factors 1

2

3

4

5

a Total population versus controls 1 2 3 4 5 6

0.98* 0.67* 0.41 0.41 0.22 0.29

0.96* 0.23 0.33 0.30 0.49*

0.91* 0.24 0.49* 0.63*

0.93* 0.30 0.63*

0.84* 0.09

b Total population versus itself 1 2 3 4 5

1.00: 0.66* 0.36 0.46* 0.20

1.00 0.23 0.52 0.29

1.00 0.14 0.18

1.00 0.23

1.00

table IIIb contains one such correlation in the ‘off-diagonal’. The results in table III indicate that, at least with respect to the factors accounting for most of the variance, the factor structures obtained from the data of the two populations were similar. The correlation matrices for the adjusted data analyzed by varimax rotation are shown in table IV. Five of seven high correlations in table IVa are on the diagonal; one other, between control factor 6 and total population factor 5, suggests that the order of factors was slightly different in the two groups. Apart from these, the off-diagonals contain one high correlation in table IVa and none in table IVb. On the whole, the results support the conclusion of similarity between the factor structures. Since tables I—IV (part a) indicate that factor structures based on the two populations were reasonably similar, it seemed appropriate, for simplicity, to present here only the results derived from the total population. It can, however, be stated that the findings derived from the control population data were essen tially the same.

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* p < 0.005. 1 Number of loadings per factor is 30. ! All r’s in diagonal must be 1.00.

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Table III. Product-moment correlations between factor loading values obtained in prin cipal component analysis using adjusted data for control subjects (n = 56) and total population (n = 280)' Control factors

Total population factors 1

2

3

4

5

6

7

a Total population versus controls 1 0.94* 0.37 0.82* 2 0.20 3 0.03 4 0.27 0.19 5 0.0 0.15 0.41 6 0.40 0.14 7 0.12

0.84* 0.04 0.36 0.12 0.21

0.88* 0.06 0.08 0.05

0.24 0.67* 0.16

0.06 0.39

0.48*

b Total population versus itself 1 1.00J 2 0.34 1.00 3 0.16 0.03 4 0.50* 0.10 5 0.03 0.01 6 0.33 0.07 7 0.04 0.01

1.00 0.03 0.0 0.03 0.0

1.00 0.01 0.10 0.01

1.00 0.01 0.0

1.00 0.01

1.00

Varimax Rotation or Not? The correlations presented in the b sections of tables I—IV bear on the question concerning the need or desirability of varimax rotations. These correla tions are relevant because a matrix containing no significant off-diagonal correla tions probably reflects a ‘purer’ factor structure than one containing significant correlations. Stated differently, the degree of similarity in the patterns of the loadings for two factors will be reflected in the magnitude of the correlation. It will be seen that there are no significant off-diagonal correlations in table lb (principal components, unadjusted data) and table IVb (varimax rotation, adjusted data). In contrast, there are some high off-diagonal correlations (p < 0.005) in table IIb (rotation, unadjusted data) and table IIIb (principal com ponents, adjusted data). The correlation material suggests that either principal component analysis of unadjusted data or varimax rotation of adjusted data will yield ‘purer’ factors than the other two alternatives. This inference was supported by applying the

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* p < 0.005. 1 Number of loadings per factor is 36. 2 All r’s in diagonal must be 1.00.

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Table IV. Product-moment correlations between factor loading values obtained by varimax rotation of principal component analysis using adjusted data for control subjects (n = 56) and total population (n = 280)' Control factors

Total population factors 3

4

5

6

7

a Total population versus controls 1 0.92* 0.87* 2 0.22 3 0.22 0.19 4 0.39 0.29 5 0.33 0.08 0.17 0.22 6 0.22 7 0.26

0.90* 0.15 0.15 0.05 0.32

0.59* 0.26 0.29 0.75*

0.18 0.93* 0.39

0.05 0.20

0.43*

b Total population versus itself 1 1.00! 0.17 2 1.00 0.07 3 0.03 4 0.04 0.28 5 0.29 0.07 0.26 6 0.16 7 0.25 0.23

1.00 0.05 0.02 0.11 0.36

1.00 0.33 0.05 0.02

1.00 0.12 0.09

1.00 0.28

1.00

1

2

criterion of interpretability to the factor loadings. An example of the pertinent evidence is shown in figure 2, which compares the distributions of high factor loadings (values > 0.5), obtained for two factors using principal component and rotation procedures with adjusted data. With both procedures, factor 1 has all negative loadings, while factor 2 has all positive loadings. Note that the rotation loadings (solid bars) for factor 1 weigh highly only on variables resulting from single stimuli (R1T, R1C), while those of factor 2 load only on recovery vari ables (RIO). In contrast, the unrotated (principal component) loadings (crosshatched bars) include outlying values associated with experimental variables other than R1T, R1C, and RIO. This indicates that the high loadings derived from rotation reflect a more restricted and, therefore, more interpretable set of experimental variables than those yielded by principal components. The relatively pure factor structure, .in relation to experimental design, which is provided by applying rotation to the adjusted data is demonstrated in figure 3. Here the high (> ± 0.5) factor loadings are graphed in relation to the

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* p < 0.005. 1 Number of loadings per factor is 36. J All r’s in diagonal must be 1.00.

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I.O-i

12

12

3

1 1 5 - 3 0 M S EC 2 15- 4 9 M S E C 3 3 1 -9 9 M SEC 4 50-19 9 MSEC 1 2 3 4 1 2 3 4

R1T

R1C

R2(rit) RlO(nt) R10(r2)

means

R1C

RlOtot) R10(r2)

slope primes

Fig. J. Distribution, in terms of experimental variables, of the loadings of seven factors obtained by varimax rotation of principal component analysis using adjusted data.

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Fig. 2. Comparison of the distribution of high factor loadings (> ± 0.5) in factors 1 and 2 for adjusted data. Unrotated loadings are cross-hatched, varimax rotated loadings are solid bars. Note loadings on several experimental variables with unrotated but not rotated load ings.

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experimental variables. It will be seen that high loadings for each of the seven factors were restricted to one kind of variable, e.g., factor 1 reflects responses to single stimuli (RIT, R1C); factor 5 reflects recovery after single stimuli (R2); factor 4 reflects recovery after trains (RIO), etc. Although the correlation material (tables lb, IVb) suggested that both the principal component procedure applied to unadjusted data and the rotation procedure applied to adjusted data yielded relatively pure factor loadings, only the rotation procedure provided factors which were more clearly interpretable in terms of the experimental conditions (fig. 2 ,3 ). The next step in evaluation was to determine which method discriminated best between clinical groups. Clinical Discrimination Analyses of variance comparing the factor scores of the four clinical groups (20 subjects in each group) were performed for data derived from all four procedures. Only those two procedures which produced the relatively ‘pure’ factor loadings yielded significant group differences in factor scores (table V). In both sets of data, only the first factor provided scores which discriminated between clinical groups. F ratios (3,72 d.f.) were 3.47 and 3.64, respectively, for the unadjusted principal components and the adjusted, rotated factor scores.

The first issue raised in the introduction was the selection of the appropriate population for deriving the basic factor structures. It appears that it would be equally appropriate to use either the controls alone or the total population. With either adjusted or unadjusted data, the factor loadings derived from the control and total population results were quite similar. The choice of groups would depend upon other criteria. It could be argued that the similarity in factor structure between controls and total population may have resulted in part from the fact that the control group was included in the total. In fact, separate factor analyses of tire patient group alone gave results almost identical with those from the total population. The second and third issues, rotation or not, and covariance or not, must be considered together, and were evaluated here on the basis of the results yielded with respect to both interpretability of factor structure and the ability of the factor scores to discriminate between clinical groups. By ‘interpretability of factor structure’ we refer to the extent to which the factor structure reflects the known features of the experiment. Recall that the study reported here was a compound one, incorporating several different stimu lus procedures within the same session. If each stimulus procedure were to be specifically reflected in a factor, one would not expect to find a single (com-

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Discussion

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Table V. Mean factor scores by two procedures for four clinical groups Factor Principal components, unadjusted score Co NP CS L i 2 3 4 5 6 7

-0.31 -0.04 -0.31 0.12 -0.06

0.43 0.39 -0.41 0.21 0.06

-0.30 0.26 0.00 0.06 -0.18

-0.06* -0.05 0.07 -0.0 2 -0.05

Rotation, adjusted Co

CS

L

NP

2.24 0.59 -0.13 0.72 0.77 -0.73 -0.11

-2 .2 4 0.78 -0 .6 0 0.63 0.86 -1 .0 8 0.13

2.42 -0.33 -0 .9 3 1.44 0.02 -0 .6 0 -0.24

0.48 -0.44 -0 .1 9 0.01 0.21 0.09 -0 .5 6

mon) factor accounting for a large percentage of the total variance. In fact, the analyses of the unadjusted data did provide evidence of such a powerful com mon factor, whereas a similar factor was not yielded in analyses with adjusted data. This seems reasonable, because the major reason for adjusting data values for their covariance with a powerful common factor is to increase the prob ability that specific experimental effects will be revealed. However, even though adjusted values did not yield a large common factor, the degree to which factors corresponded to experimental conditions was different with principal compo nent analysis and rotation (fig. 2). Rotation provided ‘purer’ factors. Conse quently, the combination of covariance adjustment with varimax rotation appears to give the most interpretable set of factors (fig. 3). Conversely, with the unadjusted data, principal component analysis provided relatively pure common and specific factors (table lb), although these were not so readily interpretable in terms of the experiment. Also, when the unadjusted values were rotated, the resultant factor structure was less pure (table II b). Turning now to the criterion of discrimination between clinical groups, the data show that both procedures which provided purer factors also yielded factor scores which differed significantly between groups (table V). On the other hand, the procedures yielding less pure factors did not result in significant clinical discriminations. It is noteworthy that the two clinically discriminating factor scores differed greatly in the percentage of the total variance for which they accounted in the original data. Factor 1, with unadjusted data, principal components, accounted for 54%; factor 1, with adjusted data, varimax rotation, accounted for only 16 %. Consequently, it appears that the combination of adjustment and rotation

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Co = Nonpatient controls; CS = chronic schizophrenics; L = latent schizophrenics; NP = nonpsychotics (n = 20 in each group). * p