Showing papers in "Psychometrika in 1969"

A general approach to confirmatory maximum likelihood factor analysis

Abstract: We describe a general procedure by which any number of parameters of the factor analytic model can be held fixed at any values and the remaining free parameters estimated by the maximum likelihood method. The generality of the approach makes it possible to deal with all kinds of solutions: orthogonal, oblique and various mixtures of these. By choosing the fixed parameters appropriately, factors can be defined to have desired properties and make subsequent rotation unnecessary. The goodness of fit of the maximum likelihood solution under the hypothesis represented by the fixed parameters is tested by a large samplex 2 test based on the likelihood ratio technique. A by-product of the procedure is an estimate of the variance-covariance matrix of the estimated parameters. From this, approximate confidence intervals for the parameters can be obtained. Several examples illustrating the usefulness of the procedure are given.

A test of the hypothesis that cronbach's alpha or kuder-richardson coefficent twenty is the same for two tests

Abstract: An approximate statistical test is derived for the hypothesis that the reliability coefficients (Cronbach's α) associated with two measurement procedures are equal. Control of Type I error is investigated by comparing empirical sampling distributions of the test statistic with the theoretical model derived for it. The effect of platykurtosis in the test-score distribution on the test statistic is also considered.

Evaluation of factor analytic research procedures by means of simulated correlation matrices.

Abstract: In order to study the effectiveness of factor analytic methods, a procedure was developed for computing simulated correlation matrices which are more similar to real data correlation matrices than are those matrices computed from the factor analysis structural model. In the present investigation, three methods of factor extraction were studied as applied to 54 simulated correlation matrices which varied in proportion of variance derived from a major factor domain, number of factors in the major domain, and closeness of the simulation procedure to the factor analysis structural model. While the factor extraction methods differed little from one another in quality of results for matrices more dissimilar to the factor analytic model, major differences in quality of results were associated with fewer factors in the major domain, higher proportion of variance from the major domain, and closeness of the simulation procedure to the factor analysis structural model.

A monte carlo investigation of the statistical significance of Kruskal's nonmetric scaling procedure

Abstract: Recent advances in computer based psychometric techniques have yielded a collection of powerful tools for analyzing nonmetric data. These tools, although particularly well suited to the behavioral sciences, have several potential pitfalls. Among other things, there is no statistical test for evaluating the significance of the results. This paper provides estimates of the statistical significance of results yielded by Kruskal's nonmetric multidimensional scaling. The estimates, obtained from attempts to scale many randomly generated sets of data, reveal the relative frequency with which apparent structure is erroneously found in unstructured data. For a small number of points (i.e., six or seven) it is very likely that a good fit will be obtained in two or more dimensions when in fact the data are generated by a random process. The estimates presented here can be used as a bench mark against which to evaluate the significance of the results obtained from empirically based nonmetric multidimensional scaling.

Multidimensional scaling: Combining observations when individuals have different perceptual structures

Abstract: The usual methods of combining observations to give interpoint distance estimates based on interstimulus differences are shown to lead to a distortion of the stimulus configuration unless all individuals in a group perceive the stimuli in perceptual spaces which are essentially the same. The nature of the expected distortion is shown, and a method of combining individual distance estimates which produces only a linear deformation of the stimulus configuration is given.

The criterion problem in the evaluation of instruction: assessing possible, not just intended outcomes1

Abstract: This paper discusses cognitive styles and affective reactions as two major classes of criterion variables that should be taken into account in the evaluation of instruction. These variables are emphasized because of their bearing upon questions that stem from particular views about the diversity of human performance and the role of values in educational research.

A Theoretical Study of Two-Stage Testing.

Abstract: When items cannot be answered correctly by guessing, certain two-stage testing procedures are about as effective over the ability range of interest as the “best” up-and-down procedures studied previously. When answers can be guessed correctly 20 percent of the time, no two-stage procedure is found to match the “best” up-and-down procedures over this ability range. Feet-on-the-desk designs for two-stage procedures may produce poor results.

A bayesian approach to tailored testing

Abstract: The sequential design and analysis of a test consisting of dichotomously scored items is approached from a Bayesian viewpoint. For a given examinee and for given items with known parameters, the scores on the items are taken to be independently distributed. Each item characteristic curve is taken to be a weighted average of 1 and a normal ogive function (to include the case where guessing is effective). Taking a normal prior distribution on the examinee ability, explicit expressions are derived for the posterior distribution and its mean and variance. In the Decision Theoretic framework of Wald, a quadratic loss function is taken and a simple, practical and approximately locally (or stepwise) optimum procedure is derived for sequentially choosing the (difficulties, discriminating powers and guessing constants of) items and analyzing the results.

Estimating true-score distributions in psychological testing (an empirical bayes estimation problem)

Abstract: The following problem is considered: Given that the frequency distribution of the errors of measurement is known, determine or estimate the distribution of true scores from the distribution of observed scores for a group of examinees. Typically this problem does not have a unique solution. However, if the true-score distribution is “smooth,” then any two smooth solutions to the problem will differ little from each other. Methods for finding smooth solutions are developed a) for a population and b) for a sample of examinees. The results of a number of tryouts on actual test data are summarized.

A Newton-Raphson algorithm for maximum likelihood factor analysis

Abstract: This paper demonstrates the feasibility of using a Newton-Raphson algorithm to solve the likelihood equations which arise in maximum likelihood factor analysis. The algorithm leads to clean easily identifiable convergence and provides a means of verifying that the solution obtained is at least a local maximum of the likelihood function. It is shown that a popular iteration algorithm is numerically unstable under conditions which are encountered in practice and that, as a result, inaccurate solutions have been presented in the literature. The key result is a computationally feasible formula for the second differential of a partially maximized form of the likelihood function. In addition to implementing the Newton-Raphson algorithm, this formula provides a means for estimating the asymptotic variances and covariances of the maximum likelihood estimators.

A general method for analysis of covariance structures with applications: part i: general methodology*

Abstract: A general model and an associated method of data analysis are presented. It is assumed that observations on a set of response variables have a multivariate normal distribution with a general parametric form of the mean vector and the variance-covariance matrix. Any parameter of the model may be fixed, free or constrained to be equal to other parameters. The free and constrained parameters are estimated by the maximum likelihood method. Approximate standard errors and confidence intervals for the estimated parameters may be obtained by computing the inverse of the information matrix. The adequacy of any specific model contained in the general model may be tested by the likelihood ratio technique, yielding a large sample chi-square test of goodness of fit. Great generality and flexibility are obtained, in that a wide range of models is contained in the general model by imposing various specifications on the parametric structure of the general model. Part II of this paper deals with applications to various problems mainly from the field of psychology. In part I the general model is dealt with purely formally without reference to any particular specialization or application of it. Expressions for first-order derivatives and expected values of second-order derivatives of the likelihood function are derived, and the method of maximizing the likelihood function is described.

Efficient estimation in image factor analysis1

Abstract: The image factor analytic model (IFA), as related to Guttman's image theory, is considered as an alternative to the traditional factor analytic model (TFA). One advantage with IFA, as compared with TFA, is that more factors can be extracted without yielding a perfect fit to the observed data. Several theorems concerning the structural properties of IFA are proved and an iterative procedure for finding the maximum likelihood estimates of the parameters of the IFA-model is given. Substantial experience with this method verifies that Heywood cases never occur. Results of an artificial experiment suggest that IFA may be more factorially invariant than TFA under selection of tests from a large battery.

Some statistical considerations in multidimensional scaling

Abstract: Some shortcomings of current methods of estimating the magnitude of perceived difference are considered. A statistical model for perceived difference is derived which avoids these difficulties and employs judgments of ratios of differences as data. Three estimators of squared difference are developed.

Robbins‐monro procedures for tailored testing*

Abstract: In tailored testing, we try to choose for administration items at a difficulty level matching the examinee's ability, which we infer from his responses to items already administered. Robbins-Monro procedures for selecting items and for estimating the examinee's ability are evaluated. Various ideas of use for tailored testing emerge.

Estimation of true score and error variance for tests under various equivalence assumptions1

Abstract: Maximum-likelihood estimators of true score variance and error variance for mental tests are derived for six different models of equivalent measurements. Statistical properties of the estimators are examined. Main emphasis is placed upon essentiallyτ-equivalent measurements. A statistical criterion for this type of measurement is given. The solution of the comparatively simple maximum-likelihood equations is effected by means of a rapid Newton-Raphson procedure. Two different initial estimators are considered and their relative merits in terms of second moments evaluated. Four numerical examples are appended by way of illustration.

Fitting the factor analysis model

Abstract: When the covariance matrix Σ(p×P) does not satisfy the formal factor analysis model for m factors, there will be no factor matrix Λ(p×m) such that γ=(Σ-ΛΛ′) is diagonal. The factor analysis model may then be replaced by a tautology where γ is regarded as the covariance matrix of a set of “residual variates.” These residual variates are linear combinations of “discarded” common factors and unique factors and are correlated. Maximum likelihood, alpha and iterated principal factor analysis are compared in terms of the manner in which γ is defined, a “maximum determinant” derivation for alpha factor analysis being given. Weighted least squares solutions using residual variances and common variances as weights are derived for comparison with the maximum likelihood and alpha solutions. It is shown that the covariance matrix γ defined by maximum likelihood factor analysis is Gramian, provided that all diagonal elements are nonnegative. Other methods can define a γ which is nonGramian even when all diagonal elements are nonnegative.

Assumptions in making causal inferences from part correlations, partial correlations and partial regression coefficients

Abstract: Given a linear model and that X is antecedent to Y, a third variable, W, which is antecedent to both X and Y, is often used as a control variable to remove any spurious association between X and Y. Four different methods that have been used as measures of the “influence” of X on Y are described. The implicit assumptions that are made in using each of these methods to make causal inferences are stated and compared.

Factoring the multitest-multioccasion correlation matrix*

Abstract: In this paper we consider various models for analyzing multitest-multioccasion data such as obtained in growth studies where several tests have been administered to the same examinees at several occasions. A general model is described first and then some special cases of this model are studied in greater detail. It is shown that all models are special cases of an even more general model proposed and investigated by Joreskog (1969b). The estimation and testing problems may be handled by an analysis of this model. Two methods for estimating the parameters of the models are considered: the least squares method and the maximum likelihood method. One model is illustrated by means of some growth data.

On the oblique rotation of a factor matrix to a specified pattern

Abstract: This paper presents a procedure for rotating an arbitrary factor matrix to maximum similarity with a specified factor pattern. The sum of squared distances between specified vectors and rotated vectors in oblique Euclidian space is minimized. An example of the application of the procedure is given.

Best linear composites with a specified structure

Abstract: Least squares linear composites of predictors for estimating several criteria are derived, satisfying the restriction that the composites have an arbitrary specified intercorrelation matrix. These composites are compared with the usual unrestricted regression composites. An illustrative example is provided. The derivation depends on a general result, given in an appendix, about best-fitting orthonormal transformations.

Negative relationships between abilities

Abstract: Abilities are usually assumed to exist in a “positive manifold.” Experimental manipulations of physiological variables, however, suggest that negative relationships exist between certain of the neural processes contributing to simple perceptual-motor vs. perceptual-restructuring tasks. First-order correlative evidence of this phenomenon cannot be obtained because the between-individual differences in general ability level tend to exceed the behavioral effects of the intra-individual opposition between neural processes. Also, since statistical removal of the “g” variance induces bipolarity in the remaining variance, the second-order negative correlations are necessarily regarded as artifactual. A combined correlational-experimental approach is suggested to overcome this difficulty.

Univariate selection: The effects of size of correlation, degree of skew, and degree of restriction

Abstract: Pearson's formula for univariate selection was derived with the assumption of normality of variates before and after selection. This study examined the influence of skew upon estimates from Pearson's formula under certain conditions. It was found that even with essentially symmetric distributions, a large proportion of the data is necessary to obtain reasonably precise estimates of low correlations. With increasing skew, estimates become increasingly erroneous, the direction of the error depending upon which tail of the distribution is the basis of the estimates. Difficulties in applying correction for univariate selection in several studies of the predictability of college-grades for Negroes from scores on standard aptitude tests are discussed.

Derivations of learning statistics from absorbing Markov chains

Abstract: Learning-process statistics for absorbing Markov-chain models are developed by using matrix methods exclusively The paper extends earlier work by Bernbach by deriving the distribution of the total number of errors, u-tuples, autocorrelation of errors, sequential statistics, and the expectation and variance of all statistics presented The technique is then extended to latency derivations including the latencies of sequential statistics Suggestions are made for using the sequential-statistic algorithm in a maximum-likelihood estimation procedure The technique is important because statistics for very large absorbing matrices can be easily computed without going through tedious theoretical calculations to find explicit mathematical expressions

A bayesian solution for some educational prediction problems

Abstract: The paper considers a single college selecting students from a variety of high schools. It criticizes the usual statistical analysis and proposes an alternative procedure which uses more efficiently all information that is present in the system. The bulk of the paper consists of a mathematical appendix in which some general theory is developed which, it is expected, will be of value in studying other more complex prediction systems besides the one discussed here.

The learning and transfer of double-classification skills: a replication and extension

Abstract: A previous study by Jacobs and Vandeventer had shown that first-graders could learn the skill of double-classification with color and shape relationships in a half hour or less of individualized instruction. When some details of the training task were altered in an intuitively compelling way to test for transfer, it was found that the trained Ss could also handle double-classification problems with shading and size relations. Since the transfer dimensions had been selected somewhat arbitrarily, subsequent work by Jacobs and Vandeventer mapped out a universe of relations within which transfer could be more meaningfully assessed. The universe consisted of combinations of 12 basic relations. The primary purpose of the present research was to assess transfer from the training procedure of the earlier study within this universe. Subsidiary purposes were to evaluate the effects of more extensive training and to investigate the effectiveness of different trainers. Experiment 1 was concerned with certain preliminary methodological issues. Thirty-one first-grade boys and 30 first-grade girls were tested on a series of 57 double-classification problems involving 11 different kinds of relations. Learning effects within the testing series were found to be inconsequential. The type of relation had a significant effect: double-classification tasks using size relations were easiest. In Experiment 2, 57 Ss, matched for pretest score, were randomly assigned to either regular training (with shape and color the relations to be learned, as in the previous study), extended training (with shape, color, shading and addition the relations to be learned), or control (no training) groups. Within each of the training groups, Ss matched for pretest score were assigned at random to either Trainer1 or Trainer2. Posttests were administered covering a stratified random sampling of all possible pairs of the 12 basic relations. “Build-a-matrix” tasks were also administered in which S had to put four or more pieces missing from the same matrix into their correct position. As in the earlier study, regular training Ss significantly outperformed control Ss on shape and color matrices. Their superior performance also transferred throughout the universe of relations. Extended training produced significantly more transfer than regular training. Neither trained group, however, showed transfer to the build-a-matrix task with new relations. The two trainers did not differ in effectiveness. Three months later retention testing was carried out. Transfer effects for regular and extended training were found to hold up. Transfer was also found to Raven's Coloured Progressive Matrices, which was administered for the first time at this stage of the experiment. The implications for the trainability of intelligence are discussed.

A rationale for an asymptotic lognormal form of word‐frequency distributions1

Abstract: The lognormal distribution has been found to fit word-frequency distributions satisfactorily if account is taken of the relations between populations and samples. A rationale for an asymptotic lognormal distribution is derived by supposing that the probabilities at the nodes of decision trees are symmetrically distributed around .5 with a certain variance. By the central limit theorem, the logarithms of the continued products of probabilities randomly sampled from such a distribution would have an asymptotically normal distribution. Two mathematical models incorporating this notion are developed and tested; in one, the number of factors in the continued products is assumed to be fixed, while in the other, that number is dependent upon a Poisson distribution. Psycholinguistic processes corresponding to these models are postulated and illustrated with reference to two sets of data: (1) word associations to the stimulus LIGHT, and (2) the Lorge Magazine Count. Reasonable fits to observed data or to underlying lognormal distributions are obtained but there remain certain problems in estimating parameters.

Testing significance of differences between two chi-squares

Abstract: This paper presents a way to test the difference between twoX 2's. The test requires evaluating the difference with respect to theT m (x) Bessel function. Included is a table of the 5 percent and 1 percent points for the Bessel function with degrees of freedom up to 100.

Precision of prediction1, 2

Abstract: Precision of prediction in multiple linear regression is examined. Two measures of predictive precision, predictive mean square error, d2, and the squared weight validity, w2, are employed. The use of an existing estimator of e(d2) as an estimator of d2 is proposed and the mean squared error of estimation of this estimator about d2 is obtained. A significance test is given. Estimators of w2 are derived and an asymptotic approximation for their variances is given. These estimators of w2 are functions of estimators of p2, the squared multiple correlation coefficient, and of p4. The bias and mean squared error of estimation of some known estimators of p2 and of a proposed estimator of p4 are examined. Monte Carlo experiments are used to compare the proposed estimators of w2 with an estimator due to Burket [1964]. An efficient procedure for generating w2 and the estimates of w2 is described. The mean squared errors of estimation of cross-validation estimators of d2 and w2 are obtained and disadvantages of the cross-validation procedure are discussed. An example is used to illustrate relationships between predictive precision and the number of predictors. The paper is primarily concerned with a random predictor model but results for a fixed predictor model are also given.

Bayesian methods in psychological testing

Abstract: A Bayesian method of estimating ability is described that generalizes Kelley's regression estimate of true score based on a weighted average of observed score and the presumed known population mean true score. This Bayesian method uses all available data to estimate the population mean true score and incorporates this information into the estimate of each individual ability parameter. This method is superior to methods that do not include estimates of population mean values. A normal law error model is used to illustrate the method. Modal estimates of an intuitively attractive form from the joint posterior distribution of the ability parameters are identified as a solution to a set of nonlinear equations. The method is then illustrated in the context of a Poisson process model having both person-ability and item-difficulty parameters. The posterior marginal distributions for individual sets of ability and difficulty parameters are given and modal estimates are described. Modal estimates from the joint conditional distribution of the ability parameters given the item difficulty parameters are also given. The desirability of incorporating prior information is discussed and a method of accomplishing this is described. The application of these Bayesian methods to central prediction and sequential testing are discussed. It is suggested that the Bayesian method is uniquely appropriate to each of these problems. Finally, these ideas are also shown to be relevant to the problems of selecting predictor variables and multiple comparisons.