Showing papers in "Psychometrika in 1967"
TL;DR: A useful correspondence is developed between any hierarchical system of such clusters, and a particular type of distance measure, that gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data.
Abstract: Techniques for partitioning objects into optimally homogeneous groups on the basis of empirical measures of similarity among those objects have received increasing attention in several different fields. This paper develops a useful correspondence between any hierarchical system of such clusters, and a particular type of distance measure. The correspondence gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data. In an explicitly defined sense, one method forms clusters that are optimally “connected,” while the other forms clusters that are optimally “compact.”
4,560 citations
TL;DR: In this paper, a new computational method for the maximum likelihood solution in factor analysis is presented, which takes into account the fact that the likelihood function may not have a maximum in a point of the parameter space where all unique variances are positive.
Abstract: A new computational method for the maximum likelihood solution in factor analysis is presented. This method takes into account the fact that the likelihood function may not have a maximum in a point of the parameter space where all unique variances are positive. Instead, the maximum may be attained on the boundary of the parameter space where one or more of the unique variances are zero. It is demonstrated that suchimproper (Heywood) solutions occur more often than is usually expected. A general procedure to deal with such improper solutions is proposed. The proposed methods are illustrated using two small sets of empirical data, and results obtained from the analyses of many other sets of data are reported. These analyses verify that the new computational method converges rapidly and that the maximum likelihood solution can be determined very accurately. A by-product obtained by the method is a large sample estimate of the variance-covariance matrix of the estimated unique variances. This can be used to set up approximate confidence intervals for communalities and unique variances.
837 citations
TL;DR: It is pointed out that the statement that “coefficientα is equal to the mean of the split-half reliabilities” is true only under the restricted condition assumed by Cronbach in the body of his derivation of this result.
Abstract: Following a general approach due to Guttman, coefficientα is rederived as a lower bound on the reliability of a test. A necessary and sufficient condition under which equality is attained in this inequality and hence thatα is equal to the reliability of the test is derived and shown to be closely related to the recent redefinition of the concept of parallel measurements due to Novick. This condition is then also shown to be closely related to the unit rank assumption originally adopted by Kuder and Richardson in the derivation of their formula 20. The assumption later adopted by Jackson and Ferguson and the one adopted by Gulliksen are shown to be related to the necessary and sufficient condition derived here. It is then pointed out that the statement that “coefficientα is equal to the mean of the split-half reliabilities” is true only under the restricted condition assumed by Cronbach in the body of his derivation of this result. Finally some limitations on the uses of any function ofα as a measure of internal consistency are noted.
496 citations
TL;DR: A large-sample test for the significance of the difference between two detection data points is developed based upon the assumptions of a one-parameter signal detectability model.
Abstract: A large-sample test for the significance of the difference between two detection data points is developed based upon the assumptions of a one-parameter signal detectability model. In essence, the null hypothesis tested is that two observed data points belong to the samed' function.
227 citations
TL;DR: Estimates of test size (probability of Type I error) were obtained for several specific repeated measures designs for configurations where the underlying covariance matrices exhibited varying degrees of heterogeneity.
Abstract: Estimates of test size (probability of Type I error) were obtained for several specific repeated measures designs. Estimates were presented for configurations where the underlying covariance matrices exhibited varying degrees of heterogeneity. Conventional variance ratios were employed as basic statistics in order to produce estimates of size for a conventional test, an e-adjusted test, and e-adjusted test and a conservative test. Indices for empirical distributions of two estimators of e, a measure of covariance heterogeneity, were also provided.
151 citations
147 citations
TL;DR: In this paper, it was shown that constructed factor scores cannot be simultaneously univocal and orthogonal, unless we choose the special basis in factor space given by Canonical Factor Analysis.
Abstract: Four least-squares methods for constructing factor scores have been described in the literature. The formal properties of these scores are developed, and they are compared in terms of four generally desirable properties of constructed factor scores. In particular, it is shown that two of the methods yield scores that are conditionally unbiassed, and univocal in the sense of Guilford and Michael, though not orthogonal, while one of the other methods yields orthogonal scores. It is shown that constructed factor scores cannot be simultaneously univocal and orthogonal, unless we choose the special basis in factor space given by Canonical Factor Analysis. The general problem of choosing between the methods is discussed, on the basis of the theoretical relations obtained.
137 citations
TL;DR: It is found that by examining manifest categories it is possible to reconstruct information about the latent partition and about its relation to the manifest partitions.
Abstract: Latent partition analysis has been formulated to study the relationships between two or more partitions of the same set of items. The major structural hypothesis is that a latent partition underlies the manifest partitions; that is, it is assumed that each item belongs to a latent category and that the manifest categories are derived by dividing and combining the latent categories. We have found that by examining manifest categories it is possible to reconstruct information about the latent partition and about its relation to the manifest partitions.
68 citations
TL;DR: In this article, the main difficulty in the use of maximum likelihood estimation in factor analysis has been the lack of satisfactory methods of obtaining numerical solutions, and this defect has now been remedied, and new rapid methods of finding maximum likelihood estimates.
Abstract: Until recently the main difficulty in the use of maximum likelihood estimation in factor analysis has been the lack of satisfactory methods of obtaining numerical solutions. This defect has now been remedied, and in this paper we describe new rapid methods of finding maximum likelihood estimates.
62 citations
TL;DR: In this article, an effective iterative method for obtaining the exact solution is developed by algebraic manipulation by expressing the set of equations in the form of one polynomial equation in one unknown Newton's method.
Abstract: The equations involved in the rotation of an arbitrary factor matrix to a least squares fit to a specified factor structure have been known for many years These equations, in general, cannot be solved by purely algebraic means, and an approximate solution has previously been used in practical applications In this paper an effective iterative method for obtaining the exact solution is developed By algebraic manipulation the set of equations is expressed in the form of one polynomial equation in one unknown Newton's method is suggested for solving this equation Practical applications of the procedure indicate that convergence within small tolerance limits is generally attained after few iterations
61 citations
TL;DR: The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II, which produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions.
Abstract: Two related orthogonal analytic rotation criteria for factor analysis are proposed. Criterion I is based upon the principle that variables which appear on the same factor should be correlated. Criterion II is based upon the principle that variables which are uncorrelated should not appear on the same factor. The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II. An example is presented in which this procedure produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions. A computer program is provided.
TL;DR: In this article, the basic concepts of nonlinear factor analysis are introduced and some extensions of the general theory are developed, using more elementary algebraic methods than have been employed in earlier accounts of this theory.
Abstract: The basic concepts of nonlinear factor analysis are introduced and some extensions of the general theory are developed. An elementary account of the class of multiple-factor polynomial models is presented, using more elementary algebraic methods than have been employed in earlier accounts of this theory. Working formulas are developed for the multiple-factor polynomial model without product terms. Some empirical results are presented.
TL;DR: In this paper, the authors designed a study to test two major hypotheses: (1) that game-like contexts will induce higher creativity levels than will test-like context, and (2) that in gamelike contexts creative and intellective performance will be unrelated to each other.
Abstract: The present study was designed to test two major hypotheses: (1) that game-like contexts will induce higher creativity levels than will test-like contexts, and (2) that in game-like contexts creative and intellective performance will be unrelated to each other, whereas in test-like contexts, these two kinds of performance will be positively related. A total of 104 fifth-grade children (52 males and 52 females) in four separate classrooms participated in the study. Two of the classrooms were assigned to the game-like condition and the other two to the test-like condition. Two creativity tasks of an associative type and a vocabulary test were administered. Kuhlman-Anderson IQ scores were available for almost all subjects. No clearcut superiority for test- or game-like contexts in reference to creativity level was observed. Rather, effects varied dependent upon the task, the criterion of creativity, anxiety and defensiveness levels of subjects, and the sex of the subject. With regard to the second hypothesis, overall correlational differences were not found. However, when test anxiety was employed as a moderator variable, results consistent with the hypothesis were obtained. Implications for future creativity research with children are spelled out in the paper.
TL;DR: In this article, the logistic model of a mental test was introduced by the present writer in Chapters 17 through 20 of Lord and Novick, Statistical Theories of Mental Test Scores, where statistical inference methods were developed without assumption of a prior distribution of ability.
Abstract: The logistic model of a mental test was introduced by the present writer in Chapters 17 through 20 of Lord and Novick, Statistical Theories of Mental Test Scores, where statistical inference methods were developed without assumption of a prior distribution of ability. Such a distribution is assumed in this paper, and corresponding statistical inference methods and distribution theory are developed. Results include detailed investigation of posterior distributions, given observed response patterns; point-estimation of ability, and its mean-squared error; and distributions of test scores and their moments. Some computational methods and detailed examples are given.
TL;DR: If misclassification in the row direction is independent of that along the column variable then the size of the usual chi-square test is unchanged and it is shownHow to calculate loss of power in this case and also how to calculate the change in size of this test if the errors are not independent.
Abstract: The misclassification process is represented by a stochastic matrix containing the probabilities that an individual who belongs in one cell is counted as belonging to another (or perhaps the same) cell. These probabilities are supposed known. If misclassification in the row direction is independent of that along the column variable then the size of the usual chi-square test is unchanged. It is shown how to calculate loss of power in this case and also how to calculate the change in size of the test if the errors are not independent. A modified test criterion is suggested when errors are not independent and a numerical example is included.
TL;DR: A Monte Carlo approach was employed in determining whether or not certain variables produce systematic effects on the sampling variability of individual factor loadings by generating a number of sample correlation matrices from a specified population.
Abstract: A Monte Carlo approach is employed in determining whether or not certain variables produce systematic effects on the sampling variability of individual factor loadings. A number of sample correlation matrices were generated from a specified population, factored, and transformed to a least-squares fit to the population values. Influences of factor strength, communality and loading size are discussed in relation to the statistics summarizing the results of the above procedures. Influences producing biased estimators of the population values are also discussed.
TL;DR: Variance analyses are presented for two data layouts—each corresponding to the class of all ordered pairs from a single finite set and a joint treatment of concurrently observed dominance and composition layouts is suggested and illustrative data are presented.
Abstract: Variance analyses are presented for two data layouts—each corresponding to the class of all ordered pairs from a single finite set. The analysis of the dominance layout is in terms of a fixed effects linear model which includes parameters representing the scale values of the elements of the set, response bias, and pairwise interactions. A parallel parametrization is carried out for the composition layout for which corresponding point estimates and hypothesis tests are given. A joint treatment of concurrently observed dominance and composition layouts is suggested and illustrative data are presented.
TL;DR: In this paper, a maximum likelihood solution for the method of successive intervals is presented together with an iterative procedure for obtaining estimates for stimulus scale values, boundary scale values and stimulus dispersions.
Abstract: A maximum likelihood solution for the method of successive intervals is presented together with an iterative procedure for obtaining estimates for stimulus scale values, boundary scale values, and stimulus dispersions. The method applies without modification to incomplete data matrices. The method was programmed for ILLIAC and a brief description of the program and a numerical illustration are included.
TL;DR: In this paper, the authors investigated the relationship of a set of factor-analytically derived ability measures to performance at different stages of practice in a learning set task employing concept problems, and to consider possible constructs to explain these relationships.
Abstract: : Theories of learning have typically failed to give an account of the role of human abilities, while accounts of mental abilities have not explained their relationship to learning in any rigorous way. The purpose of this study was to investigate the relationship of a set of factor-analytically derived ability measures to performance at different stages of practice in a learning set task employing concept problems, and to consider possible constructs to explain these relationships. Twenty-six problems, each of which required the identification of a four-dimensional conjunctive concept, were administered to a sample of 145 Princeton University undergraduates. To account for the complex learning involved in this task, a conceptual model involving information-processing constructs was developed. Three higher-order processes were postulated: a problem analysis process, which defined gaps between information available and information needed for solution; a search process, which looked among available responses for operators to reduce or to remove these gaps; and an organization process, which integrated operators thus found into a terminal program for solving the problems. A battery of 30 mental tests was administered to the same sample of students. The tests were selected as measures of factors chosen for their relevance to the model. The intercorrelations among the test scores were factor analyzed by a maximum-likelihood procedure. This procedure extracted factors in accordance with a simple-structure hypothesis set up in advance. A statistical test of this hypothesis showed it to be tenable. (Author)
TL;DR: A complete algebraic treatment is given for the case of a product of two latent variables, and certain more general cases are briefly outlined.
Abstract: Factor interaction models are defined as those cases in nonlinear factor analysis in which the specification equation contains products of two or more latent variables or functions of latent variables. A complete algebraic treatment is given for the case of a product of two latent variables, and certain more general cases are briefly outlined.
TL;DR: The scaling model is applied to scaling of a set of nine “immoral” actions and the results agree rather well with those from conventional successive intervals scaling of the same stimuli.
Abstract: In this paper a method is developed for analyzing data resulting from the use of a modification of the usual paired comparisons procedure which allows for ratings of size of difference. The scaling model is essentially an extension of the Thurstonian successive intervals model. The method is applied to scaling of a set of nine “immoral” actions and the results agree rather well with those from conventional successive intervals scaling of the same stimuli.
TL;DR: In this article, a solution is presented to the problem of determining a proper correction for spuriousness in correlation coefficients, where the general case developed is the estimate of correlation betweenu andv, both being linear functions of the same set of variables.
Abstract: A solution is presented to the problem of determining a proper correction for spuriousness in correlation coefficients. The general case developed is the estimate of correlation betweenu andv, both being linear functions of the same set of variables. Special formulae relate to overlapping scales correlations, part-whole correlations, and item-total test correlations.
TL;DR: Group members under the relevant acquaintance condition achieved consistently good validity for all three cognitive areas—with the best validity for ratings of math ability and inter- and intrarater reliability.
Abstract: Unacquainted Ss worked in three-man groups under relevant (mathematical tasks) and irrelevant (socializing) acquaintance conditions. The Ss rated one another on scales that defined several cognitive skills. They were also rated on these same scales by observers who were dependent on visual information, exclusively, and were unacquainted with the group members or the specific nature of the tasks being performed. As hypothesized, group members under the relevant acquaintance condition achieved consistently good validity for all three cognitive areas—with the best validity for ratings of math ability. Validity under the irrelevant acquaintance condition was nil on all scales. Observers, surprisingly, achieved significant validity (although at lower levels than participating group members) for ratings under the relevant acquaintance condition. Levels of inter- and intrarater reliability were not associated with levels of validity under the various rating conditions.
TL;DR: This article investigated the usefulness of desirability judgments of personality items as an indirect measure of the judge's personality and found that self-reports and judgments correlated with response styles and intelligence.
Abstract: This study investigated the usefulness of desirability judgments of personality items as an indirect measure of the judge's personality. Self-reports and judgments were compared on separate forms of five personality scales. With a few striking exceptions, the two kinds of measures were generally unrelated, and they had markedly different patterns of correlations with behavioral measures of conformity. In general, self-reports only correlated with conformity for subjects who were unsuspicious about the conformity procedures, but judgments correlated with conformity for unsuspicious and suspicious subjects. Both self-reports and judgments correlated with response styles and intelligence, but the self-reports were more extensively involved with the response styles. The differential functioning of the two kinds of measures could not be attributed to the operation of response styles or intelligence, nor did these latter variables moderate the relationship between self-reports and judgments.
TL;DR: A theory is presented which unifies and interprets two paired comparison models that have proved useful in fitting experimental data and which interpret the various parameters in terms of sensation in a natural manner.
Abstract: The Thurstone and the Bradley-Terry models, both initially advanced on intuitive grounds, have proved useful in the analysis of paired comparisons. The psychological meaning of these models and their relation to one another is unclear, but they fit data. Stevens has observed that there may be two basic mechanisms of discrimination 1) additive and 2) substitutive. We advance two corresponding mathematical models: that experienced sensation is 1) the sum of a large number of independent signals and 2) the maximum of a large number of independent signals. These assumptions yield 1) Thurstone's model and 2) the model of Bradley-Terry. Psychological interpretations of the various parameters, in terms of sensation, present themselves in a natural manner. Thus this paper presents a theory which unifies and interprets two paired comparison models that have proved useful in fitting experimental data.
TL;DR: It has been observed that the number of different ways in which a graph withp points can be labelled isp! divided by thenumber of symmetries, and that this holds regardless of the species of structure at hand.
Abstract: It has been observed that the number of different ways in which a graph withp points can be labelled isp! divided by the number of symmetries, and that this holds regardless of the species of structure at hand. In this note, a simple group-theoretic proof is provided.
TL;DR: In this paper, the authors present a method of analysis for obtaining inter-battery factors and battery specific factors for two sets of tests when the complete correlation matrix including communalities is given.
Abstract: It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.
TL;DR: Interpretation of the creativity-risk taking relationship stressed the importance of self-confidence as a mediating link and significant relationships were obtained for boys between number of associates generated to creativity tasks administered under gamelike conditions and preferred level of risk taking on a shuffleboard task.
Abstract: The present study explored the interrelationships among selected measures of creativity, intelligence, and risk taking in a sample of fifth-grade children. Significant relationships were obtained for boys between number of associates generated to creativity tasks administered under gamelike conditions and preferred level of risk taking on a shuffleboard task. An IQ index, unrelated to creative ability, showed a marginal association with risk taking. Interpretation of the creativity-risk taking relationship stressed the importance of self-confidence as a mediating link. Consistent with that interpretation, the foregoing relationship was not found in high defensive boys but was significant and quite substantial in low defensive boys.
TL;DR: In this article, the authors found that the larger the individual's own risky shift, the stronger his tendency to attribute a risky shift to the group of which he is a member.
Abstract: If groups of male adults are systematically composed in terms of the field dependence-independence of their members, they exhibit a number of differences in risk-taking phenomena. Although field-dependent and independent groups both shift on the average toward greater risk taking after discussion, longer discussion among field dependents enhances the risky shift, while longer discussion among field independents reducesor even reverses the risky shift. Further, among field dependents but not among field independents, the larger the individual's own risky shift, the stronger his tendency to attribute a risky shift to the group of which he is a member. These together with other findings support the view that responsibility diffusion causes enhanced risk taking by field-dependent groups.
TL;DR: The effects of enforced rehearsal, model-observation and model-plus-voluntary rehearsal on private charitable behavior, both immediately following the experimental manipulations and after an interval of several days, were examined in this article.
Abstract: The effects of enforced rehearsal (ER), model-observation (O), and model-observation plus voluntary rehearsal (OVR) on private charitable behavior, both immediately following the experimental manipulations and after an interval of several days, were examined.
Obtaining a winning score on a preprogrammed miniature bowling game provided the rationale for furnishing 210 fourth- and fifth-grade children with five-cent gift certificates. Located near the game was an open charity box. The number of certificates which a subject (S) deposited in the box, while alone, constituted his altruism score. All Ss were given identical game instructions by a male experimenter (E), who also served as the model (M) in those conditions demanding one. Each S was run individually.
ER Ss were explicitly instructed to donate one gift certificate each time they won two, and E remained in S's presence while S rehearsed the giving rule twice. All other Ss were given permissive instructions concerning the charity box.
OVR Ss observed M donate twice and played the game in M's presence, also winning twice. If OVR Ss donated, the act went unnoticed and unrewarded. O Ss also observed M donate; however, no opportunity was provided to rehearse in M's presence.
The results indicate that where the motivational stakes are reasonably high, observation alone is not a sufficient eliciting agent. O Ss failed to evidence significantly more altruism than controls. This finding is interpreted as applying a needed corrective to the modeling research area, imposing a boundary condition on the efficacy of modeling.
For girls, OVR resulted in significantly more altruism than obtained among controls. This result held for both sessions, indicating that the OVR effects were powerful enough to bridge an interval of several days. By all available indicants, OVR girls consistently internalized the altruistic norm, which they had observed and then complied with in M's presence, even though the rehearsal had received no external positive sanctions. These results are interpreted as supporting those theoretical formulations which assign motivational properties to internalized standards.
The results for OVR boys are directionally the same as for OVR girls; however, they fail to reach statistical significance.
ER Ss, evidencing a rule-obedience component of socialization, engaged in extensive giving during Session 1, giving more than Ss in any other condition While their Session 2 performance dropped markedly, it still remained higher than for all other Ss, except OVR girls. In general, the explicit, though benign, giving instructions had a powerful effect upon ER Ss' behavior; however, some interesting complications were suggested by an internal analysis of the ER condition.
In general, girls gave more and were significantly more consistent in their giving than boys.
Results from three additional experimental conditions, employed to allow a comparison of Ss who were exposed to the experimental manipulations hut were not tested for altruism until Session 2 with Ss described above, indicate that the experimental effects on altruistic behavior tended to decrease as a function of time regardless of whether S practiced the charitable behavior in E's absence, during Session 1.