Analysis with Categorical

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Discriminant Analysis with Categorical DataJohn E. Overall and J. Arthur WoodwardThe University of Texas Medical Branch, Galveston

A method for studying relationships amonggroups in terms of categorical data patterns is de-scribed. The procedure yields a dimensional rep-resentation of configural relationships among mul-tiple groups and a quantitative scaling of cate-gorical data patterns for use in subsequent assign-ment of new individuals to the groups. Two ex-

amples are used to illustrate potential of themethod. In the first, profile data that were pre-viously analyzed by metric multiple discriminantfunction analysis are reanalyzed by the nonmetriccategorical data pattern technique with highlysimilar results. The second example examines re-lationships among psychiatric syndrome groups interms of similarities in patterns of categoricalbackground variables. Results appear consistentwith other available information concerning theepidemiology of psychiatric disorders.

The investigation of group differences in

multivariate categorical data patterns has re-ceived less attention than has the multivariatenormal case. This paper examines a method for

dimensional analysis of group differences incategorical data patterns which can be em-ployed whenever N; individuals in each of ggroups have observations recorded on n cate-

gorical variables. The analysis results in thedefinition of multiple scale dimensions thattend to reflect maximally the differences incategorical data patterns of the several groups.

This method of analysis is developed by anal-ogy to multiple discriminant function analysis(MDA), with certain simplifying assumptionsto facilitate use with large numbers of categori-cal variables. It is essentially a principal com-ponents analysis of frequency patterns acrossthe multiple categorical variables; however,the scaling of category proportions relative towithin-groups variance renders the analysis dif-ferent from a simple factor analysis or com-ponents analysis of group profiles. This differ-ence, which is easily overlooked, is an impor-tant one in producing category weights whichhave discriminatory value.

In multiple discriminant function analysis,the solution vectors for the matrix equation (B- AW)a = 0 define weighting coefficients

which, when applied to the original measure-ment variables, produce linear combinationsthat maximally account for group differences.In the case of multicategory nominal data,weighting coefficients which, when summedover the specific categories in which each indi-vidual belongs, will produce composite scoreshaving reasonably good normal distributions

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within groups and reflecting maximum meandifferences relative to the within-groups disper-sion. To accomplish this, matrices similar tothe between-groups B and the within-groups Ware required; however, in view of the fact thateach category of a multicategory variable as-sumes the status of a variable, simplification bydisregarding error covariances is proposed,which renders W a diagonal matrix. This ap-pears to be more appropriate because optimalproperties of the MDA solution are based onassumptions that are not tenable with cate-

gorical data. The evaluation of the proposedmethod will thus rest upon pragmatic criteria.

Description of the Method

Consider each category of the several multi-

category variables to represent a variable inthe analysis. Let p; (1 x m ) be a vector contain-ing the proportion of subjects in the i’&dquo; groupthat falls in each category of the several multi-

category variables. The number of elements in

p; is equal to the total number of categories inall of the multicategory variables. Note that thenumber of categories for each original multi-category variable need not be the same.

Let z; = p;’- p, represent the deviation of thecategory proportion vector for the i‘&dquo; groupabout the unweighted mean of the categoryproportion vectors for all g groups, and let Z’(gx) represent a matrix containing those mean-corrected category proportion vectors for the ggroups.The individual category variates have bi-

nomial distributions so that the variance asso-ciated with the j’&dquo; element in the i‘h groupvector is

where p is the,j’&dquo; element in the category pro-portion vector p;(1 x m). The mean of thebinomial variances for then element across all

groups is

Let D-’ be a diagonal matrix containing the re-ciprocals of square roots of the mean within-groups -,-aria n ces 7 .The analysis of group differences in multi-

category data patterns can now be developedby direct analogy to the computation of mul-tiple discriminant analysis as described byOverall and Klett (1972, pp. 281-285). To sum-marize the notation, let

Z’ (g x m ) be a matrix containing categoryproportion vectors for the g groups expressedas deviations about the unweighted mean ofthe g-group vectors.V (m x m ) be a diagonal matrix containing

the pooled within-groups variances of the mcategory variates.

D-’ (m X m ) be a diagonal matrix containingreciprocals of the square roots of the cor-

responding elements in V, so that V-’ = D-’D-1.The function to be maximized is

Introducing a Lagrange multiplier to imposethe restriction a&dquo;,Vai = 1 for i = 1, 2, ---, r, the

following familiar matrix equation is obtained:

Multiplying on the left by D-’ and insertingD-’D = I on the right of Z’Z, the equation istransformed to

and then



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where a; = Dai.The solution vectors a; are principal

components of the symmetric matrixD-’D’ZD-1.The vectors of weighting coefficients that

satisfy the criterion function are obtained bymultiplying ii = Z’ZD-1

and

The elements in A (m x r) are standardized

category weights. They should be consideredfor interpretation of the relevance of each

category variable to the composite dis-

criminant functions. The elements in A (m x r)are the raw-score category weights that definethe r composite discriminant variates in termsof category membership. To obtain the dis-criminant variate scores for any individual, onesimply sums the elements of A (m x r) cor-responding to categories in which he/she be-longs. Mean scores for groups can be obtainedas the means of individual discriminant variate

scores, or the elements in each of the r

columns of A (m x r) can be applied directly tothe group category proportion vectors p;’(1 xm ) to obtain the group mean values on the dis-criminant dimensions. The discriminant. vari-ates derived from categorical data patterns canbe employed for assignment of individuals

among multiple groups in the same manner asare discriminant function scores derived frommultivariate profile data.

Figure 1

Brief Psychiatric Rating Scale Profile Viewed asQuantitative Measurement Profile



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Figure 2

Configural Relationships Among Diagnostic Groups Derivedfrom Analysis of Quantitative Measurement Profiles

(see Table 1 for variable names)

Application to Data AlternativelyAnalyzed as Multivariate Normal

The analysis of category proportions impliesno consideration of an ordering among thecategories of any variable. This method is in-tended for use in situations where the attri-

butes of individuals within several groups are

entirely categorical; however, an important in-sight into the efficiency and validity of the

method can be gained through comparison ofresults in cases where the data can alter-

natively be treated as (reasonably) multivariatenormal.An opportunity for such a comparison is pro-

vided by a study of diagnostic concepts of Ger-man-speaking psychiatrists in which Overalland Hippius (1974) obtained symptom ratingprofiles descriptive of 12 psychiatric diagnosticgroups from 87 to 108 experienced psychia-trists. In the original data, the severity of each



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of 16 symptoms was rated on a 7-point scale ofseverity as shown in Figure 1. The data in thisform were originally analyzed as if they werecontinuous measurements with multivariate

normal distributions. A computer program for

assignment of individuals to diagnostic groupsbased on a multivariate normal probabilitydensity model resulted in the proportions ofagreement shown in Table 1. The configura-tion of diagnostic-group means in the plane de-fined by the first two dimensions from a mul-tiple discriminant function analysis, in whichthe data were treated as quantitative measure-ments, is shown in Figure 2.

For analysis as categorical data, the originalsymptom ratings were categorized into threebroader intervals, which will be designated&dquo;mild,&dquo; &dquo;moderate,&dquo; and &dquo;severe.&dquo; The profileshown in Figure 1 was recoded to yield thecategorical data pattern shown in Figure 3. Thecollapsing of ratings into three categories in-stead of seven is not the point of this example.It is the comparison of results from a totallynonmetric analysis with the results from a

metric discriminant function analysis that is ofinterest. In the form shown in Figure 3, it is ob-vious that the data cannot be considered as

quantitative measurements and that they

Figure 3

Brief Psychiatric Rating Scale Profile Viewedas Categorical Data Pattern



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should be analyzed by a method appropriatefor categorical observations. Although there isstill a logical order among the mild, moderate,and severe categories, it is emphasized that theconfigural analysis based on categorical datapatterns takes no cognizance of that order. Thecategories could just as easily be &dquo;married,&dquo;&dquo;single,&dquo; and &dquo;divorced,&dquo; or &dquo;Caucasian,&dquo;&dquo;Negro,&dquo; and &dquo;other..&dquo;The data in the form shown in Figure 3 were

analyzed by the method of scaling for categori-cal data described herein. The configuration ofdiagnostic groups obtained by plotting groupmeans on the first two dimensions of differencein probabilities of various categorical patternsis shown in Figure 4. Comparison of the rela-tionships and distances among the various

groups with those previously derived from mul-tiple discriminant analysis of the same data re-veals a very high degree of consistency.

Figure 4

Configural Relationships Among Diagnostic Groups Derived fromMultidimensional Scaling of Categorical Data Patterns

(see Table 1 for variable names)



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The analysis of categorical data patterns(such as Figure 3) resulted in scale values forthe individual patterns. These transformedscale scores were analyzed by a computer pro-gram for assignment of individuals based on anormal probability density model. The result-ing correct and incorrect assignments are tabu-lated in Table 2. Comparison of the classifica-tion results with those previously obtained bytreating the original data as multivariate quan-titative measurements (Table 1) reveals exactlythe same proportion of correct classifications.Moreover, the pattern of misclassifications canbe seen to be quite similar in the off-diagonalelements of Tables 1 and 2. It is concluded thatfor this example, at least, very little was lost bytreating quantitative measurement profiles ascategorical data, even though they mightotherwise be analyzed by maximally efficientmultivariate normal methods.

Application to Epidemiologic Datafor Psychiatric Groups

Although comparison with multivariate nor-mal statistical methods is important for evalua-tion of relative efficiency, the real importanceof the method discussed here is for problems inwhich observations cannot be considered to be

quantitative measurements. To illustrate onesuch application, consider the demographicand social history characteristics of psychiatricpatients in eight distinct symptom profiletypes. A sample of 589 psychiatric outpatientswas classified among eight phenomenologicalprofile types according to consistencies of theirsymptom profile patterns with eight empiri-cally derived cluster prototypes (Overall,1974). The eight phenomenologically distinctsyndromes were designated florid thinking dis-order, withdrawn-disorganized thinking dis-

turbance, hostile-suspiciousness syndrome,anxious depression, agitated depression, hos-tile depression, retarded depression, and agita-tion-excitement syndrome.

Fifteen categorical background variablesidentified in the left margin of Table 3 were

entered into the analysis in the form of 43 ele-ments of an expanded category data vector foreach of the 589 individuals in the eightphenomenologically distinct groups. Principalcomponents analysis of the standardized cate-gory proportion matrix resulted in definition ofthree primary dimensions of difference in

background data patterns separating the eightpsychiatric syndromes. The category weightingcoefficients, which are important for inter-

pretation of the nature of the primary dimen-sions, are shown in Table 3.

Mean scale values for the eight groups arepresented in Table 4. The group means wereobtained by applying the category scale

weights to the original (unstandardized) cate-gory proportion vectors for the eight groups. Adistance function model showing the con-

figuration of the eight groups in the 3-space de-fined by the primary dimensions of differencesin categorical background data patterns isshown in Figure 5.

It is important to examine the interpretivevalue of the results. The first primary dimen-sion of epidemiologic difference separates thedepressive types from the thinking disturbanceand agitation syndromes. The depression endof the continuum (neg ) is associated withmiddle age, female, no previous psychiatrichospitalization, married and having children.The thinking disturbance, agitation end of thecontinuum (pos) is associated with young,male, previous psychiatric hospitalization, bet-ter education, single, and no children. With ex-ception of the dubious implication of educa-tional achievement, these patterns appear

quite consistent with the extensive literatureon epidemiology of depression and schizo-

phrenia. With regard to the diagnosticterminology, however, it should be remem-

bered that the positive end of the first con-

tinuum is perhaps better understood as &dquo;non-

depressive&dquo; than as schizophrenic, because themanic-like agitation-excitement syndrome ac-tually occupies the extreme position on thepositive end of the scale.



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The second primary dimension clearly corre-sponds to a psychomotor activation-retarda-tion continuum, which contrasts the with-

drawn-disorganized thinking disturbance andwithdrawn-retarded depression groups with theagitated depression and (manic-like) agitation-excitement syndromes. The activation-agita-tion end of the second continuum (pos) is asso-ciated with white, recurrent episodes, bettereducation, skilled work level, married, chil-

dren, alcohol use, and strong positive religiousattitude. In general, the psychomotor activa-tion-retardation continuum in psychopatho-logic manifestations appears related to a gen-eral factor of social class or social achieve-ment.

The third primary dimension of epidemio-logic difference separates the hostile sus-

piciousness and hostile depression groups fromthe others. The hostile end of the continuum

(pos) is primarily associated with age-relatedvariables. Age less than 30 years, early age ofonset, longer duration of illness, and middlelevel of educational achievement characterizesthe hostile patients. Conversely, patients whoare older, with onset of symptoms after 30,short duration or slowly declining level of com-petence, and either low or high educationalachievement are less prone to the hostile-extra-

punitive syndrome.The analysis of the categorical background

variables confirmed underlying epidemiologic

Figure 5

Configural Model of Group Relationships in ThreeDimensional Epidemiologic Measurement Space



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distinctions between depression versus think-ing disorder syndromes, withdrawal-retardationversus agitation-excitement, and hostility-ex-trapunitiveness versus groups without such ag-gressive orientation. The combinations of

background variables defining subpopulationsin which these various phenomenological typesare more likely to be encountered were eluci-dated. The results suggested that certain vari-ables, such as marital history (divorces), haveless relevance than other variables for definingsubpopulations in which the different syn-dromes have differential likelihood of occur-

rence. Used as an efficient screening tech-

nique, the analysis would lead to selection ofage, ethnicity, sex, and history of previous hos-pitalization as providing the most discriminat-ing patterns. It is interesting that &dquo;age, race,

and sex&dquo; are generally considered the epi-demiological variables in biomedical researchand that the history of previous hospitalizationis an obviously relevant variable when con-sidering the differences in psychopathology.

References

Overall, J. E. The brief psychiatric rating scale inpsychopharmacology research. Modern Prob-

lems of Pharmacopsychiatry, 1974, 7, 67-68.Overall, J. E., & Hippius, H. Psychiatric diagnostic

concepts among German-speaking psychiatrists.Comprehensive Psychiatry. 1974, 15, 103-117.

Overall, J. E., & Klett, C. J. Applied MultivariateAnalysis. New York: McGraw-Hill, 1972.

Author’s Address

John E. Overall, Department of Psychiatry, Univer-sity of Texas Medical Branch, Galveston, TX 77550.



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Analysis with Categorical