A typology of parent rated child behavior for a national U.S. sample.

The purpose of the current study was to build on the emerging effort to produce a meaningful typology of child behavior for school-aged children. The Behavior Assessment System for Children (BASC) Parent Rating Scales for Children (PRS-C) norming data were collected for 2029 6- to 11-year-old children at 116 sites representing various regions of the United States. The PRS-C has 130 items that are rated by the parent on a 4-point scale of frequency, ranging from "never" to "almost always". The Ward method of cluster analysis was used to identify the initial centroids or cluster seeds in this norming sample. An iterative clustering method, a K-means procedure, was used to refine the Ward cluster solution. A nine-cluster solution was selected based on both rational and empirical considerations. The resulting clusters were labeled adapted, physical complaints/worry, average, well-adapted, minimal problems, attention problems, internalizing, general psychopathology-severe, and disruptive behavior problems. The nine-cluster solution is similar to those of Achenbach (1991), Thompson, Kronenberger, and Curry (1989), and other researchers. At the same time, some significant differences exist. The resulting typology points the way toward future cluster studies of child psychopathology and normal behavioral development by delineating additional research and theoretical directions.

Behavioral Clustering of School Children.

The intent of this article is to illustrate how a cluster analysis might be conducted, validated, and interpreted. Data normed for a behavioral assessment instrument with 14 scales on a sample drawn from a nationally representative pool of U.S. school children were utilized. The analysis discussed covers the cluster method, cluster typology, cluster validity, cluster structure, and prediction of cluster membership.

A typology of teacher-rated child behavior for a national U.S. sample.

The purpose of the current study was to build on the emerging effort to produce a meaningful typology of classroom behavior for elementary school age children. The Behavior Assessment System for Children (BASC) Teacher Rating Scales for Children (TRS-C) norming data were collected for 1,227 six- to eleven-year-old children at 116 sites representing various regions of the United States. The TRS-C has 148 items that are rated by the teacher on a 4-point scale of frequency, ranging from Never to Almost always. The Ward method of cluster analysis was used to identify the initial centroids or cluster seeds. An iterative clustering method, a K-means procedure, was used to refine the Ward cluster solution. A seven-cluster solution was selected based on both rational and empirical considerations. The resulting clusters were named well-adapted, average, learning disorder, disruptive behavior disorder, physical complaints and worry, severe psychopathology, and mildly disruptive. The seven-cluster solution resembles those of Achenbach (1991), Curry and Thompson (1985), and other researchers. The resulting typology points the way toward future cluster studies of child psychopathology by delineating additional research and theoretical questions.

Assessing Predictive Accuracy in Discriminant Analysis.

The estimation of probabilities of correct classification is a primary concern in predictive discriminant analysis. Three such probabilities are: (a) the optimal hit rate, that obtained when the classification rule is based on known parameters; (b) the actual hit rate, that obtained by applying a rule based on a particular sample to future samples; and (c) the expected actual hit rate. Methods of estimating these hit rates include formulas (in the two-group case), resubstitution, and external analyses. The methods are tentatively compared via Monte Carlo sampling from two real data sets.

Selecting a Two-Group Classification Weighting Algorithm.

The cross-validated classification accuracies of three predictor weighting strategies (least squares, ridge regression, and reduced rank) were compared under varying simulated data conditions for the two-group classification problem. Results were somewhat similar to previous findings with multiple regression when absolute rather than relative cross-validation accuracy was the criterion. The alternate weighting strategies performed better than the usual least squares algorithm in many of the data conditions, but the margin by which they surpassed least squares was in many cases quite small. Also, the alternate algorithms were poorer than least squares in some data configurations, and in some cases, by an appreciable margin. Verification of these simulation findings was obtained by documenting agreement with theory, and with an analytic sample-specific procedure that was used with some "real" data sets. The results of the analyses with real data sets yielded results that were similar to those from the simulation. The routine and uncritical use of biased weighting algorithms in classification as advanced by P. J. DiPillo was not supported.

Using Categorical Variables in Discriminant Analysis.

Three methods of transforming unordered categorical response variables are described. One is a method using dummy variables. The second method, in which all categorical variables are analyzed simultaneously, is based on an eigenanalysis of frequency patterns scaled relative to within-groups variance, jointly developed by J. E. Overall and J. A. Woodward. With the third method, independently developed by R. A. Fisher and H. O. Lancaster, each categorical variable is analyzed separately with scale values generated so that the grouping variable and the categorical variable are maximally correlated. Results from analyzing two real data sets are used to illustrate the application of the three methods.

The Study Of Effects In Manova.

A particular strategy for investigating effects resulting from a MANOVA is proposed. The strategy involves multiple two-group multivariate analyses. The two groups result from considering multivariate pairwise group contrasts or multivariate complex group contrasts. Assuming a given two-group analysis yields real effects, the resultant single linear discriminant function (LDF) may be studied. A rationale based on a transformation of LDF weights, due to V. Y. Urbakh, is recommended for assessing variable relative contribution. The analysis strategy is described in detail and illustrated with real data sets.

Linear Versus Quadratic Multivariate Classification.

A linear classification rule (used with equal covariance matrices) was contrasted with a quadratic rule (used with unequal covariance matrices) for accuracy of internal and external classification. The comparisons were made for seven situations which resulted from combining conditions (equal and unequal covariance matrices, and two and three criterion groups) for different sets of real data. For the internal analysis the quadratic rule was superior to the linear rule in all seven situations. For the external analysis the linear rule was nearly as good as or superior to the quadratic rule in all seven situations.

AN EMPIRICAL COMPARISON OF THE ACCURACY OF SELECTED MULTIVARIATE CLASSIFICATION RULES.

This study involved two phases: first, when classification was based on the calibration sample; and second, in a cross-validation setting. Computer generated data were used. Results obtained from rules based on probabilities of group membership were compared for accuracy when classifying in the discriminant space and in the predictor variable spaces. In the first phase accuracy was greater in the predictor variable spaces, while the reverse was true in the second phase. In general, rules based on probabilities of group membership were approximately equally accurate and more accurate than a rule related to a multiple regression analysis. Other findings are also discussed.

MULTIVARIATE INDICES OF STRENGTH OF ASSOCIATION.

Multivariate extensions of univariate measures of association that are employed in comparative experiments are reviewed. Two of these are generalizations of the correlation ratio, and the third is a generalization of Hays' omega. The numerical values of these three indices are empirically compared along with a fourth index, the proportion of correct classifications, using two sets of data and various numbers of predictor variables. As in the univariate case, the values of the correlational indices are nearly the same in all situations investigated. The use of a correlational index versus the proportion of correct classifications depends upon the intent of the researcher.

BRIEF REPORT: THE EQUIVALENCE OF TWO MULTIVARIATE CLASSlFICATION SCHEMES.

The equivalence of two multivariate classification schemes is shown when the sizes of the samples drawn from the populations to which assignment is required are identical. One scheme is based on posterior probabilities determined from a Bayesian density function; the second scheme is based on likelihood ratio discriminated scores. Both of these procedures involve prior probabilities; if estimates of these priors are obtained from the identical sample sizes, the equivalence follows.