Showing papers in "Multivariate Behavioral Research in 1979"
TL;DR: In this article, a new procedure for determining the optimal number of interpretable factors to extract from a correlation matrix is introduced and compared to more conventional procedures, which evaluates the magnitude of the Very Simple Structure index of goodness of fit for factor solutions of increasing rank.
Abstract: A new procedure for determining the optimal number of interpretable factors to extract from a correlation matrix is introduced and compared to more conventional procedures. The new method evaluates the magnitude of the Very Simple Structure index of goodness of fit for factor solutions of increasing rank. The number of factors which maximizes the VSS criterion is taken as being the optimal number of factors to extract. Thirty-two artificial and two real data sets are used in order to compare this procedure with such methods as maximum likelihood, the eigenvalue greater than 1.0 rule, and comparison of the observed eigenvalues with those expected from random data.
368 citations
TL;DR: Hierachical cluster analysis is shown to be an effective method for forming scales from sets of items by testing the psychometric adequacy of each potential scale by using a new measure of internal consistency reliability, coefficient beta.
Abstract: Hierachical cluster analysis is shown to be an effective method for forming scales from sets of items. The number of scales to form from a particular item pool is found by testing the psychometric adequacy of each potential scale. Higher-order scales are formed when they are more adequate than their component sub-scales. It is suggested that a scale's adequacy should be assessed by a new measure of internal consistency reliability, coefficient beta, which is defined as the worst split-half reliability of the test. Comparisons with other procedures show that hierarchical clustering algorithms using this psychometrically based decisions rule can be more useful for scale construction using large item pools than are conventional factor analytic techniques.
334 citations
TL;DR: The results indicate that classification may lead to a substantial lack of fit of the model, i.e. an erroneous indication that more factors are needed if the variables are skewed in opposite direction and have high true loadings, but does not depend much on the number of scale steps.
Abstract: The paper is concerned with the consequences for maximum likelihood factor analysis which may follow if the observed variables are ordinal with only a few scale steps, which are assigned integer values. It is hypothesized that the observed variables are obtained through a classification of some true variables, which are multivariate normal and for which a factor model holds. Using simple formulas for the relations between true correlations and correlations based on the classified variables, we demonstrate numerically the relationships between true factor models and results obtained from classified data. This is done for several choices of thresholds, true factor loadings and numbers of variables, assuming a one-factor model. The results indicate that classification may lead to a substantial lack of fit of the model, i.e. an erroneous indication that more factors are needed. This is especially true if the variables are skewed in opposite direction and have high true loadings, but does not depend much on the number of scale steps. The classification also attenuates the loading estimates, and this effect is increased with a decreasing number of scale steps and increasing variation in skewness among the variables.
197 citations
TL;DR: A subset of high accuracy algorithms, including single, average, and centroid linkage using correlation, and Ward's minimum variance technique, was identified and all of the algorithms were significantly more accurate than a random linkage algorithm, and accuracy was inversely related to coverage.
Abstract: Due to the effects of outliers, mixture model tests that require all objects to be classified can severely underestimate the accuracy of hierarchical clustering algorithms. More valid and relevant comparisons between algorithms can be made by calculating accuracy at several levels in the hierarchical tree and considering accuracy as a function of the coverage of the classification. Using this procedure, several algorithms were compared on their ability to resolve ten multivariate normal mixtures. All of the algorithms were significantly more accurate than a random linkage algorithm, and accuracy was inversely related to coverage. Algorithms using correlation as the similarity measure were significantly more accurate than those using Euclidean distance (p < .001). A subset of high accuracy algorithms, including single, average, and centroid linkage using correlation, and Ward's minimum variance technique, was identified.
164 citations
TL;DR: It is demonstrated that Cattell's scree test and Bartlett's chi-square test for the number of factors are both based on the same rationale, so the former reflects statistical (subject sampling) variability and the latter usually involves psychometric (variable sampling) influences.
Abstract: It is demonstrated that Cattell's scree test and Bartlett's chi-square test for the number of factors are both based on the same rationale, so the former reflects statistical (subject sampling) variability and the latter usually involves psychometric (variable sampling) influences. If the alpha-level (implicit in the scree test) is set the same, the two tests should lead to the same conclusions. Analyses with some examples suggest that if the alpha-level for the Bartlett test is set (explicitly) in the neighborhood of .0003 for sample Ns of 100 to 150, the results from applications of this test will indicate approximately the same number of factors as estimated on the basis of a scree test determined on a much larger (N ≃ 600) sample. Used in this way, the Bartlett test may yield fairly good "population" estimates of the number of factors. Relationships between the Bartlett test, hence the scree test, and tests for a common factor model and for the significance of a correlation matrix are explicated.
154 citations
TL;DR: This procedure and others are applied to several previously published correlation matrices and some artificial matrices, results in the retention of too many factors, unrealistic elevation of the amount of common factor variance analyzed, and distortions in the conclusions drawn from the factor analytic investigation.
Abstract: A currently popular procedure in empirical factor analytic studies is to use unities in the main diagonal as communality estimates, extract all factors with eigenvalues of 1.0 or higher, and rotate these factors by varimax. This procedure and others are applied to several previously published correlation matrices and some artificial matrices. This procedure results in the retention of too many factors, unrealistic elevation of the amount of common factor variance analyzed, and distortions in the conclusions drawn from the factor analytic investigation. Ways of avoiding these difficulties, are discussed.
128 citations
TL;DR: If the same categories are used for two nominal scale variables, this information should be used in similarity measures between those variables, and two such similarity measures, one proposed by Goodman & Kruskal and one (kappa) by Cohen are examined.
Abstract: If the same categories are used for two nominal scale variables, this information should be used in similarity measures between those variables. Two such similarity measures, one proposed by Goodman & Kruskal and one (kappa) by Cohen, are examined. Two alternative coefficients, called C and S are further proposed. They are found to be generalizations of the G index and the phi coefficient respectively. Both of them seem to have many desirable characteristics, e.g., they are both E-coefficients. They may also be used as measures of similarity between persons classifying into categories defined beforehand.
64 citations
TL;DR: Factor analysis in several populations, covariance structure models, three-mode factor analysis, structural equation systems with measurement model, and analysis of covariance with measurementmodel are shown to be specializations of a general moment structure model published previously in this journal.
Abstract: Factor analysis in several populations, covariance structure models, three-mode factor analysis, structural equation systems with measurement model, and analysis of covariance with measurement model are all shown to be specializations of a general moment structure model published previously in this journal. Some new structured linear models are also described; they may be considered either generalizations or special cases of existing models. Simple representations are developed for complex linear models, and some applications to behavioral data are cited.
56 citations
TL;DR: A nonmetric scaling method was used to test several taxonomies of work motivation, and Alderfer's tripartite classification and Maslow's five-fold hierarchy were strongly supported by the smallest space analyses for both salesmen and repairmen.
Abstract: A nonmetric scaling method was used to test several taxonomies of work motivation. The intercorrelations among the importance ratings of 14 work goals, made by 800 salesmen and 1,800 repairmen were analyzed. A two-space solution was deemed adequate to reflect both the empirical and theoretical interrelationships among the variables. These results were consistent with an intrinsic-extrinsic grouping of the variables, but more complex relationships of motivational variables were also revealed. In particular, Alderfer's tripartite classification and Maslow's five-fold hierarchy were strongly supported by the smallest space analyses for both salesmen and repairmen.
40 citations
TL;DR: Although the overlap between the drug use and personality domains was moderately high, the ability of personality to account for variations in adolescent substance use was relatively low.
Abstract: The relationship of personality to patterns of adolescent substance use was examined in an effort to determine the extent to which these domains overlap. During the first year of a longitudinal study of adolescent drug use, self-report data were collected on personality and current use patterns of 13 substances. The data from the domains of drug use and personality were interrelated using canonical correlation analysis. Rotation of canonical variates and cross-validation were used in order to distinguish between statistically significant and theoretically meaningful results. The results revealed that a single dimension of general substance use was associated with a cluster of personality traits and attitudes. Although the overlap between the drug use and personality domains was moderately high, the ability of personality to account for variations in adolescent substance use was relatively low.
38 citations
TL;DR: A simple analytic expression which takes as arguments N, the number of points, and D, theNumber of dimensions and then yields the expected stress value is developed here, and it is shown that the error of the approximation is of the same magnitude as the standarderror of the means of the original Monte Carlo data.
Abstract: Potentially more convenient than either a table or a graph of random rankings stress values is a simple analytic expression which takes as arguments N, the number of points, and D, the number of dimensions and then yields the expected stress value. Such a function is developed here, and it is shown that the error of the approximation is of the same magnitude as the standard error of the means of the original Monte Carlo data. The approximation is highly accurate for 10 < N < 60 and 1 < D < 5. The present paper represents an extension of Spence and Ogilvie (1973).
TL;DR: It is shown that two major principles and two minor principles serve to generate a large number of multivariate models that yield solutions to problems of local identifiability, estimation, and hypothesis-testing.
Abstract: It is shown that two major principles and two minor principles serve to generate a large number of multivariate models. A general treatment based on asymptotic properties of the likelihood ratio yields solutions to problems of local identifiability, estimation, and hypothesis-testing.
TL;DR: The relationships between objective neighborhood characteristics and the wellbeing of 2421 elderly federally-assisted housing tenants were investigated and neither the racial character of the neighborhood or the consonance between race of tenant and the neighborhood racial context were associated with wellbeing.
Abstract: The relationships between objective neighborhood characteristics and the wellbeing of 2421 elderly federally-assisted housing tenants were investigated. Six factors were derived from the 1970 Census characteristics of the social areas in which 150 housing projects were located. Social-area factors accounted for significant proportions of variance in tenant activity participation, housing satisfaction, friendship behavior, and motility, but not family interaction or morale. Higher neighborhood age density was consistently associated with tenant wellbeing. Neither the racial character of the neighborhood or the consonance between race of tenant and the neighborhood racial context were associated with wellbeing, but consonance between tenant's race and the racial context of the housing was associated with wellbeing.
TL;DR: Instead of maximizing the correlation between unobserved variates, the sum of squared inter-set loadings is maximized and the shared variance between predictor variates and criterion variables is maximal.
Abstract: Although much progress has been made in clarifying the properties of canonical correlation analysis in order to enhance its applicability, there are several remaining problems. Canonical variates do not always represent the observed variables even though the canonical correlation is high. In addition, canonical solutions are often difficult to interpret. This paper presents a method designed to deal with these two problems. Instead of maximizing the correlation between unobserved variates, the sum of squared inter-set loadings is maximized. Contrary to the canonical correlation solution, this method ensures that the shared variance between predictor variates and criterion variables is maximal. Instead of extracting variates from both criterion and predictor variables, only one set of components (from the predictor variables) is constructed. Without loss of common variance, an orthogonal rotation is applied to the resulting loadings in order to simplify structure.
TL;DR: The results suggested that while underachieving and adequately achieving groups of children can be separated by subtest differences on the WISC-R, the use of derived classification equations from group statistics in individual diagnosis leads to dramatic levels of misclassifications among underach achieving students.
Abstract: This investigation analyzed the discriminatory or diagnostic utility of using WISC-R subtest scores in differentiating between groups of statistically defined underachieving children and a group of children who were adequately achieving. The sample consisted of 206 children (134 boys, 72 girls) referred for psychological evaluations by their classroom teachers. The results suggested that while underachieving and adequately achieving groups of children can be separated by subtest differences on the WISC-R, the use of derived classification equations from group statistics in individual diagnosis leads to dramatic levels of misclassifications among underachieving students.
TL;DR: Procedures for simultaneous confirmatory factor analysis in several populations are useful in a wide variety of problems, demonstrated with examples involving missing data, comparison of part correlations between groups, testing the equality of regression weights between groups with multiple indicators of each variable, and the formulation of growth models.
Abstract: Procedures for simultaneous confirmatory factor analysis in several populations are useful in a wide variety of problems. This is demonstrated with examples involving missing data, comparison of part correlations between groups, testing the equality of regression weights between groups with multiple indicators of each variable, and the formulation of growth models. Corrections for attenuation can be incorporated into analysis of covariance and analysis of variance procedures.
TL;DR: This work found that nonmetric multidimensional scaling solutions for δ matrices had relatively lower stress and solution based on D were more closely related to the underlying configurations; and determination of dimensionality by inspection of the stress plot was somewhat more difficult for solutions based on δ.
Abstract: In past research, a matrix of squared profile distances, δ, has sometimes been multidimensionally scaled rather than the matrix of original dissimilarities, D. It is thought that scaling solutions derived from δ have lower stress and enhanced interpretability when applied to data generated by sorting. Two experiments were performed to investigate the consequences of the delta transformation. First, random numbers resembling data collected by the method of sorting were simulated. Scaling solutions derived from δ matrices invariably had lower stress than solutions computed from the associated D matrices. This result suggests that the delta transformation may reduce stress irrespective of any change in interpretability. Simulated dissimilarity matrices were then generated from known stimulus configurations. It was found that: (1) nonmetric multidimensional scaling solutions for δ matrices had relatively lower stress; but under low error conditions (2) solution based on D were more closely related to the underlying configurations; and (3) determination of dimensionality by inspection of the stress plot was somewhat more difficult for solutions based on δ. These results can be understood by observing that the delta transformation tends to increase the size of large distances in the derived configurations relative to small distances.
TL;DR: The potential uses of the instrument in applied research and in the clinical setting were discussed and patterns of factor loadings indicated that the two referent factors, the two response class factors, and four of the five individual differences factors were stable across subject samples.
Abstract: The Difficulty in Assertiveness Inventory (DAI) was developed, based on a two-facet model of assertiveness for item specification—referents (interpersonal contexts or partners) x response classes (types of assertive behavior). Data from two samples of female university students were subjected to three-mode factor analysis. Two referent, two response class, and five individual differences factors were extracted. The proportion of variance accounted for by each factor matched across solutions. The patterns of factor loadings indicated that the two referent factors (distant and close), the two response class factors (positive and negative), and four of the five individual differences factors (labeled Assertiveness A, B, C, and D) were stable across subject samples. Solutions for the individual differences factor matrix and the counter-rotated and transformed core matrix were reciprocally beneficial to interpretation of the structure of the data. Correlations of DAI subscales with the Marlowe-Crowne Social Desirability Scale were not significantly different from zero. The potential uses of the instrument in applied research and in the clinical setting were discussed.
TL;DR: The procedure for hierarchical factoring suggested by Schmid and Leiman (1957) is applied within the framework of image analysis and orthoblique rotational procedures and it is shown that this approach necessarily leads to correlated higher order factors.
Abstract: The procedure for hierarchical factoring suggested by Schmid and Leiman (1957) is applied within the framework of image analysis and orthoblique rotational procedures. It is shown that this approach necessarily leads to correlated higher order factors. Also, one can obtain a smaller number of factors than produced by typical hierarchical procedures.
TL;DR: Inferential relationships of psychopathological traits were mapped by applying multidimensional scaling to judgments of the probability of joint endorsement of pairs of personality items from the Differential Personality Inventory, and correspondence between the inferential configuration and the empirical structure derived from a factor analysis of the inventory's scales was interpreted as indicating that implicit theory regarding psychopathology possessed some validity.
Abstract: Inferential relationships of psychopathological traits were mapped by applying multidimensional scaling to judgments of the probability of joint endorsement of pairs of personality items from the Differential Personality Inventory. Three bipolar dimensions of inference, identified as uncontrolled impulse expression vs. inhibition and withdrawal, cognitive dysfunctioning vs. overcontrol and denial, and resignation vs. interpersonal conflict, were obtained. The correspondence between the inferential configuration obtained in the multidimensional scaling and the empirical structure derived from a factor analysis of the inventory's scales, administered with standard self-report instructions to an independent sample, was interpreted as indicating that implicit theory regarding psychopathology possessed some validity.
TL;DR: Results indicated that a masculine factor and a feminine factor, accounting for approximately the same amount of the entire common variance, emerged for both males and females.
Abstract: Ninety-two male and 131 female undergraduate psychology students were administered the Adjective Check List (ACL), Bem Sex Role Inventory (BSRI), and Personal Attributes Questionnaire (PAQ). The masculinity (M) and femininity (F) subscale scores for each of these three measures were correlated and subjected to a principal factor analysis with varimax rotation separately for males and females. Results indicated that: (1) the ACL-F shared only a modest amount of variance with the BSRI and PAQ-F subscales, and was not responded to in the same manner by males and females; (2) the ACL-M subscale tended to be more strongly negatively related to F subscales than did the BSRI and PAQ-M subscales; (3) the BSRI and PAQ-M and F subscales shared a substantial proportion of variance between themselves, for both males and females; and (4) a masculine factor and a feminine factor, accounting for approximately the same amount of the entire common variance, emerged for both males and females.
TL;DR: The results indicated that the scaling analysis provided an effective, valid indicator of prose representation, and indicated that comprehension was a function of the passage organization mapping onto the existing memory structure superimposed upon and suppressing the prior conceptual structure.
Abstract: Multidimensional scaling was employed to study the comprehension of prose. Subjects rated the similarity between pairs of 20 nouns before reading. After reading a passage containing the nouns, the subjects re-rated the words with respect to similarity within the passage. Subjects then recalled the passage. The similarity ratings were analyzed by multidimensional scaling. The results indicated that the scaling analysis provided an effective, valid indicator of prose representation. The multidimensional structural characteristics of dimension interpretation, clustering, and centrality were interpreted in terms of the theme, episodes, and central organizing feature of the story, respectively. Theoretically, the analysis indicated that comprehension was a function of the passage organization mapping onto the existing memory structure superimposed upon and suppressing the prior conceptual structure.
TL;DR: It was concluded that D. L. Mosher's model of preservation of self-esteem best explained the behavior of high sex guilt individuals.
Abstract: Sixty males and sixty females, divided into groups of high and low sex-guilt, made all possible comparisons between fifteen descriptions of individuals. These descriptions were written to reflect characteristics felt to be important to the concept of sex guilt. Ratings of dissimilarity were analyzed by INDSCAL and an unfolding method, both of which are multidimensional scaling techniques. Results indicated that: 1) low sex guilt and high sex guilt subjects perceived the stimulus "individuals" quite similarly along the two major dimensions (sociability and positive-negative) underlying their comparisons; 2) that males and high sex guilt subjects utilized less of both the sociability and positive-negative dimensions in making their judgments concerning the interrelationships of the stimulus "individuals"; 3) that low sex guilt and high sex guilt subjects differ in evaluating themselves in relation to the stimulus "individuals"; and 4) high sex guilt subjects rated themselves as more similar to a trait sex guilty description than to descriptions dealing with trait sex anxiety, or general guilt or anxiety. It was concluded that: 1) low sex guilt and high sex guilt subjects differ in the value they place on different personal attributes; and 2) D. L. Mosher's model of preservation of self-esteem best explained the behavior of high sex guilt individuals.
TL;DR: Tucker's three-mode factor analysis is outlined, which enables the researcher to explore new taxonomic structures in three-way designs and should be easy to follow for anyone familiar with traditional two- mode factor analysis.
Abstract: As psychologists increasingly recognize the limitations of single-occasion, single-measure research designs and employ more elaborate and comprehensive data collection schemes, sophisticated analytic techniques will play an ever more important role in understanding behavioral data. This paper outlines one such promising technique, Tucker's three-mode factor analysis, which enables the researcher to explore new taxonomic structures in three-way designs. The procedure is described in step-by-step format, which should be easy to follow for anyone familiar with traditional two-mode factor analysis. Appropriate computer programs are available to interface with the widely used SPSS package to calculate Tucker's common factor model.
TL;DR: It was concluded that common items magnify an intrinsic structure already existing among scales and that the common-item problem is not likely a serious on in personality measurement.
Abstract: Procedures for analyzing common item effects on the inference of personality structure were reviewed and a study using smallest space analysis of the CPI reported. Solutions to three matrices -- intercorrelation matrix of the original CPI scales, of reduced scales (with common items removed), and of the number of common items -- were compared visually, and by interpoint distance correlations and configurational similarity analyses. Marked similarity among the original structure, the item-overlap-free structure and the built-in structure was observed. It was concluded that common items magnify an intrinsic structure already existing among scales and that the common-item problem is not likely a serious on in personality measurement.
TL;DR: Three-mode factor analysis is suggested to be more suitable, if interindividual judgment differences are to be described by means of a spatial model.
Abstract: Seventy subjects judged 9 key-politicians of the German parliamentary scene on 22 bipolar seven-point attribute scales. Profile similarities were computed for each possible stimulus pair separately for each individual. The resulting proximity matrices were analyzed with "INDSCAL." Additionally, three-mode factor analysis was applied to the original three-way data matrix. The results of the two methods are compared. Special attention is given to the problem of how interindividual judgment differences are reflected by the individual parameters of the two models; these are the INDSCAL-weights and the factor scores of an extended core matrix. The results suggest that three-mode factor analysis is more suitable, if interindividual judgment differences are to be described by means of a spatial model.
TL;DR: Two applications of Kristof's theorem on traces of matrix products are presented, including a simplified demonstration of the method for finding a set of derived orthogonal variables maximally correlated with the original set.
Abstract: Two applications of Kristof's theorem on traces of matrix products are presented. One is a simplified demonstration of the method for finding a set of derived orthogonal variables maximally correlated with the original set, and the other one is a demonstration of rotational equivalence of least squares factor analyses of a given rank. The solutions are derived algebraically, without calculus, employing the Eckart-Young decomposition.
TL;DR: The misleading character of the correlation coefficient was investigated in two studies of intuitive statistical behaviour and subjective overestimated the degree of influence on the correlation between two variables of a third, constraining variable, Z.
Abstract: The misleading character of the correlation coefficient was investigated in two studies of intuitive statistical behaviour: subjective estimation of partial correlation and subjective estimation of the minimum possible correlation between two variables given their equal correlation with a third. Fourteen Psychology Department faculty and graduate students provided estimates of partial correlations, r[SUBxy.z'] for varying positive values of rxy and rxy = ryz. Mean estimated rxy.z values were uniformly lower than theoretical (actual) values. Twenty-four statistics students and sixteen Statistics Department faculty and graduate students furnished estimates of minimum possible correlations, min rxy' for varying positive values of rxz = ryz. For both samples mean estimated min rxy was higher than the theoretical (actual) min rxy. Thus, in both studies, subjective overestimated the degree of influence on the correlation between two variables, X and Y, of a third, constraining variable, Z. The Hazards of intuitive statistical inference in the interpretation of correlational data are are discussed.
TL;DR: A cross-sectional study of maternal age-specific birthrates was conducted in an attempt to gain insight into age patterns of childbearing, finding that the late childbearing group seems best understood as attributable to religious and ethnically related differences among areas.
Abstract: A cross-sectional study of maternal age-specific birthrates was conducted in an attempt to gain insight into age patterns of childbearing. The age specific fertility rates were considered as the dependent variable set. These 6 variables consisted of 1970 live births to women in the age groups 15-19 20-24 25-29 30-34 35-39 and 40-44. The age-specific fertility rates were calculated for each of the 338 New York City health areas that are the sampling units. The canonical analysis used is a multivariate method of exploring these patterns which provides a means for dealing with some of the difficult analytical problems encountered. The relationships of fertility patterns with sets of variables reflecting housing income and occupation education and labor force participation marital patterns and ethnicity are explored for New York City neighborhoods. 3 dimensions of childbearing emerged with high consistency across the variable sets - roughly describable as early middle and late fertility. The 1st pattern - high early fertility and particularly teen-age childbearing - is identified with the poorest areas of the city. The income occupation education and unemployment figures in these areas are consistent with a minimal preceived penalty for early childbearing. The 2nd dimension - birthrates of women ages 25-34 - identifies another distinctive group of areas. Many low income women at this age continue to bear children at moderate rates; middle income women bear the children they postponed having in the earlier years. The late childbearing group seems best understood as attributable to religious and ethnically related differences among areas. The 1st dimension is the most important for the perspective of the impact on population growth.
TL;DR: Evidence is presented to indicate that the scale measures rater inconsistencies rather than patient "inconsistencies," that using inconsistency scores to eliminate unreliable ratings improves factor score reliabilities, and that increasing the number of items for random assignment of possible rating values systematically increases the inconsistency score.
Abstract: This paper describes the development and testing of an internal inconsistency scale to help identify unreliable clinical ratings. The scale consists of pairs of logically inconsistent items which have been chosen and scored using a combination of rational and statistical procedures. Evidence is presented to indicate that the scale measures rater inconsistencies rather than patient "inconsistencies," that using inconsistency scores to eliminate unreliable ratings improves factor score reliabilities, and that increasing the number of items for random assignment of possible rating values systematically increases the inconsistency score. Although the inconsistency scale was constructed specifically for the Missouri Inpatient Behavior Scale (MIBS), the methodology is potentially applicable to other rating scales.