Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis

It is shown that for any full column rank matrix X ₀ with more rows than columns there is a neighborhood $\mathcal{N}$ of X ₀ and a continuous function f on $\mathcal{N}$ such that f(X) is an orthogonal complement of X for all X in $\mathcal{N}$ . This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ ² distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ ². The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions.     

Sensitivity of Fit Indices to Misspecification in Growth Curve Models

This study investigated the sensitivity of fit indices to model misspecification in within-individual covariance structure, between-individual covariance structure, and marginal mean structure in growth curve models. Five commonly used fit indices were examined, including the likelihood ratio test statistic, root mean square error of approximation, standardized root mean square residual, comparative fit index, and Tucker-Lewis Index. The fit indices were found to have differential sensitivity to different types of misspecification in either the mean or covariance structures with severity of misspecification controlled. No fit index was always more (or less) sensitive to misspecification in the marginal mean structure relative to those in the covariance structure. Specifying the covariance structure to be saturated can substantially improve the sensitivity of fit indices to misspecification in the marginal mean structure; this result might help identify the sources of specification error in a growth curve model. An empirical example of children's growth in math achievement (Wu, West, &; Hughes, 2008 Wu, W., West, S. G. and Hughes, J. N. 2008. Effect of retention in first grade on children's achievement trajectories over four years: A piecewise growth analysis using propensity score matching.. Journal of Educational Psychology, 100: 727–740. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) was used to illustrate the results.     

Resampling-Based Inference Methods for Comparing Two Coefficients Alpha

The two-sample problem for Cronbach’s coefficient \(\alpha _C\), as an estimate of test or composite score reliability, has attracted little attention compared to the extensive treatment of the one-sample case. It is necessary to compare the reliability of a test for different subgroups, for different tests or the short and long forms of a test. In this paper, we study statistical procedures of comparing two coefficients \(\alpha _{C,1}\) and \(\alpha _{C,2}\). The null hypothesis of interest is \(H_0 : \alpha _{C,1} = \alpha _{C,2}\), which we test against one-or two-sided alternatives. For this purpose, resampling-based permutation and bootstrap tests are proposed for two-group multivariate non-normal models under the general asymptotically distribution-free (ADF) setting. These statistical tests ensure a better control of the type-I error, in finite or very small sample sizes, when the state-of-affairs ADF large-sample test may fail to properly attain the nominal significance level. By proper choice of a studentized test statistic, the resampling tests are modified in order to be valid asymptotically even in non-exchangeable data frameworks. Moreover, extensions of this approach to other designs and reliability measures are discussed as well. Finally, the usefulness of the proposed resampling-based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications.     

Asymptotically Correct Standardization of Person-Fit Statistics Beyond Dichotomous Items

The \(l_z\) statistic (Drasgow et al. in Br J Math Stat Psychol 38:67–86, 1985) is one of the most popular person-fit statistics (Armstrong et al. in Pract Assess Res Eval 12(16):1–10, 2007). Snijders (Psychometrika 66:331–342, 2001) derived the asymptotic null distribution of \(l_z\) when the examinee ability parameter is estimated. He also suggested the \(l^*_z\) statistic, which is the asymptotically correct standardized version of \(l_z\). However, Snijders (Psychometrika 66:331–342, 2001) only considered tests with dichotomous items. In this paper, the asymptotic null distribution of \(l_z\) is derived for mixed-format tests (those that include both dichotomous and polytomous items). The asymptotically correct standardized version of \(l_z\), which can be considered as the extension of \(l^*_z\) to such tests, is suggested. The Type I error rate and power of the suggested statistic are examined from several simulated datasets. The suggested statistic is computed using a real dataset. The suggested statistic appears to be a satisfactory tool for assessing person fit for mixed-format tests.     

The mixed model for multivariate repeated measures: validity conditions and an approximate test

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate correction factor is derived. The correction factor generalizes Box's and can be used to construct an adjusted MMM test.I am grateful to an anonymous referee for carefully attending to the mathematical details of this paper.     

When can we trust theF-approximation of the box-test?

Consider a multivariate context withp variates andk independent samples, each of sizen. To test equality of thek population covariance matrices, the likelihood ratio test is commonly employed. Box'sF-approximation to the null distribution of the test statistic can be used to computep-values, if sample sizes are not too small. It is suggested to regard theF-approximation as accurate if the sample sizesn are greater than or equal to 1+0.0613p ²+2.7265p-1.4182p ^0.5+0.235p ^1.4* In (k), for 5p30,k20.This research was supported by the Deutsche Forschungsgemeinschaft through Ste 405/2-1.     

On testing hypotheses regarding a class of covariance structures

Let x be ap-component random variable having a multivariate normal distribution with covariance matrix . In this paper, we consider the problem of testing hypotheses of the formH ₀: =b ₁₁ + +b _{m m} , whereb _i 's are unknown scalars, and _i 's are a set of known and simultaneously diagonalizable matrices. This problem has both psychometric and statistical interest, and its basic theory is developed here. Besides, the problem of obtaining likelihood-ratio statistic for testingH ₀ is studied, and the statistic obtained in a special case.     

Robust step‐down tests for multivariate independent group designs

A composite step‐down procedure, in which a set of step‐down tests are summarized collectively with Fisher's combination statistic, was considered to test for multivariate mean equality in two‐group designs. An approximate degrees of freedom (ADF) composite procedure based on trimmed/Winsorized estimators and a non‐pooled estimate of error variance is proposed, and compared to a composite procedure based on trimmed/Winsorized estimators and a pooled estimate of error variance. The step‐down procedures were also compared to Hotelling's T² and Johansen's ADF global procedure based on trimmed estimators in a simulation study. Type I error rates of the pooled step‐down procedure were sensitive to covariance heterogeneity in unbalanced designs; error rates were similar to those of Hotelling's T² across all of the investigated conditions. Type I error rates of the ADF composite step‐down procedure were insensitive to covariance heterogeneity and less sensitive to the number of dependent variables when sample size was small than error rates of Johansen's test. The ADF composite step‐down procedure is recommended for testing hypotheses of mean equality in two‐group designs except when the data are sampled from populations with different degrees of multivariate skewness.     

The many null distributions of person fit indices

This paper deals with the situation of an investigator who has collected the scores ofn persons to a set ofk dichotomous items, and wants to investigate whether the answers of all respondents are compatible with the one parameter logistic test model of Rasch. Contrary to the standard analysis of the Rasch model, where all persons are kept in the analysis and badly fittingitems may be removed, this paper studies the alternative model in which a small minority ofpersons has an answer strategy not described by the Rasch model. Such persons are called anomalous or aberrant. From the response vectors consisting ofk symbols each equal to 0 or 1, it is desired to classify each respondent as either anomalous or as conforming to the model. As this model is probabilistic, such a classification will possibly involve false positives and false negatives. Both for the Rasch model and for other item response models, the literature contains several proposals for a person fit index, which expresses for each individual the plausibility that his/her behavior follows the model. The present paper argues that such indices can only provide a satisfactory solution to the classification problem if their statistical distribution is known under the null hypothesis that all persons answer according to the model. This distribution, however, turns out to be rather different for different values of the person's latent trait value. This value will be called ability parameter, although our results are equally valid for Rasch scales measuring other attributes.As the true ability parameter is unknown, one can only use its estimate in order to obtain an estimated person fit value and an estimated null hypothesis distribution. The paper describes three specifications for the latter: assuming that the true ability equals its estimate, integrating across the ability distribution assumed for the population, and conditioning on the total score, which is in the Rasch model the sufficient statistic for the ability parameter.Classification rules for aberrance will be worked out for each of the three specifications. Depending on test length, item parameters and desired accuracy, they are based on the exact distribution, its Monte Carlo estimate and a new and promising approximation based on the moments of the person fit statistic. Results for the likelihood person fit statistic are given in detail, the methods could also be applied to other fit statistics. A comparison of the three specifications results in the recommendation to condition on the total score, as this avoids some problems of interpretation that affect the other two specifications.The authors express their gratitude to the reviewers and to many colleagues for comments on an earlier version.     

Derivation of likelihood-ratio tests for Guttman quasi-simplex covariance structures

In this paper, maximum-likelihood estimates have been obtained for covariance matrices which have the Guttman quasi-simplex structure under each of the following null hypotheses: (a) The covariance matrix , can be written asTT + where and are both diagonal matrices with unknown elements andT is a known lower triangular matrix, and (b) the covariance matrix *, is expressible asT*T + I where is an unknown scalar. The linear models from which these covariance structures arise are also stated along with the underlying assumptions. Two likelihood-ratio tests have been constructed, one each for the above null hypotheses, against the alternative hypothesis that the population covariance matrix is simply positive definite and has no particular pattern. A numerical example is provided to illustrate the test procedure. Possible applications of the proposed test are also suggested.Adapted from portions of the author's dissertation under the same title submitted to the Department of Psychology, University of North Carolina, in partial fulfillment of the requirements for the Ph.D. degree. The author wishes to express his gratitude to his thesis chairman Dr. R. Darrell Bock and to his committee members Professors Samarendra Nath Roy, Lyle V. Jones, Thelma G. Thurstone, and Dorothy Adkins. Indebtedness is also acknowledged to Dr. Somesh Das Gupta who was quite helpful during the initial stage of the study.Formerly at the Department of Psychology, Indiana University. The author is grateful both to Indiana University and University of North Carolina for the support extended to him during his doctoral studies.     

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

An approximate statistical test is derived for the hypothesis that the reliability coefficients (Cronbach's ) 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.     

On the Likelihood Ratio Tests in Bivariate ACDE Models

The ACE and ADE models have been heavily exploited in twin studies to identify the genetic and environmental components in phenotypes. However, the validity of the likelihood ratio test (LRT) of the existence of a variance component, a key step in the use of such models, has been doubted because the true values of the parameters lie on the boundary of the parameter space of the alternative model for such tests, violating a regularity condition required for a LRT (e.g., Carey in Behav. Genet. 35:653–665, 2005; Visscher in Twin Res. Hum. Genet. 9:490–495, 2006). Dominicus, Skrondal, Gjessing, Pedersen, and Palmgren (Behav. Genet. 36:331–340, 2006) solve the problem of testing univariate components in ACDE models. Our current work as presented in this paper resolves the issue of LRTs in bivariate ACDE models by exploiting the theoretical frameworks of inequality constrained LRTs based on cone approximations. Our derivation shows that the asymptotic sampling distribution of the test statistic for testing a single bivariate component in an ACE or ADE model is a mixture of χ ² distributions of degrees of freedom (dfs) ranging from 0 to 3, and that for testing both the A and C (or D) components is one of dfs ranging from 0 to 6. These correct distributions are stochastically smaller than the χ ² distributions in traditional LRTs and therefore LRTs based on these distributions are more powerful than those used naively. Formulas for calculating the weights are derived and the sampling distributions are confirmed by simulation studies. Several invariance properties for normal data (at most) missing by person are also proved. Potential generalizations of this work are also discussed.     

Examining Factor Score Distributions to Determine the Nature of Latent Spaces

Similarities between latent class models with K classes and linear factor models with K ? 1 factors are investigated. Specifically, the mathematical equivalence between the covariance structure of the two models is discussed, and a Monte Carlo simulation is performed using generated data that represents both latent factors and latent classes with known amounts of overlap. It is shown that, under certain conditions, the distribution of factor scores can be related to the continuity of the latent space via tests of multimodality as suggested by McDonald (1967) McDonald, R. P. 1967. Non-linear factor analysis. Psychometric Monographs, 15 Psychometric Society [Google Scholar].     

Abstract: Local Influence and Robust Methods for Mediation Models

Mediation analysis investigates how certain variables mediate the effect of predictors on outcome variables. Existing studies of mediation models have been limited to normal theory maximum likelihood (ML) or least squares with normally distributed data. Because real data in the social and behavioral sciences are seldom normally distributed and often contain outliers, classical methods can result in biased and inefficient estimates, which lead to inaccurate or unreliable test of the meditated effect. The authors propose two approaches for better mediation analysis. One is to identify cases that strongly affect test results of mediation using local influence methods and robust methods. The other is to use robust methods for parameter estimation, and then test the mediated effect based on the robust estimates. Analytic details of both local influence and robust methods particular for mediation models were provided and one real data example was given. We first used local influence and robust methods to identify influential cases. Then, for the original data and the data with the identified influential cases removed, the mediated effect was tested using two estimation methods: normal theory ML and the robust method, crossing two tests of mediation: the Sobel (1982) Sobel, M. E. 1982. “Asymptotic confidence intervals for indirect effects in structural equation models”. In Sociological methodology, Edited by: Leinhardt, S. 290–312. Washington, DC: American Sociological Association. [Crossref] , [Google Scholar] test using information-based standard error (z _I ) and sandwich-type standard error (z _SW ). Results show that local influence and robust methods rank the influence of cases similarly, while the robust method is more objective. The widely used z _I statistic is inflated when the distribution is heavy-tailed. Compared to normal theory ML, the robust method provides estimates with smaller standard errors and more reliable test.     

A Bibliometric History of the Journal of Psychology Between 1936 and 2015

The Journal of Psychology: Interdisciplinary and Applied is a leading international journal in psychology dating back to 1935. This study examines its publications since its creation utilizing a bibliometric analysis. The primary objective is to provide a complete overview of the key factors affecting the journal. This analysis includes such key issues as the publication and citation structure of the journal, its most cited articles, and the leading authors, institutions, and countries referenced in the journal. The work uses the Scopus database to classify the bibliographic material. Additionally, the analysis provides a graphical mapping of the bibliographic data by using visualization of similarities viewer software. This software uses several bibliometric techniques including co-citation, bibliographic coupling and co-occurrence of keywords. The Journal of Psychology is strongly connected to most of the current leading journals in psychology, and currently has a 5-year impact factor of 1.77 (Thomson Reuters, 2015 Reuters, T. (2015). InCites? Journal Citation Reports®. Web of Science. [Google Scholar] Journal Citation Reports).     

Comparing the Effect of Thinking Maps Training Package Developed by the Thinking Maps Method on the Reading Performance of Dyslexic Students

The present study aimed to develop the thinking maps training package and compare its training effect with the thinking maps method on the reading performance of second and fifth grade of elementary school male dyslexic students. For this mixed method exploratory study, from among the above mentioned grades’ students in Isfahan, 90 students who met the inclusion criteria were selected by multistage sampling and randomly assigned into six experimental and control groups. The data were collected by reading and dyslexia test and Wechsler Intelligence Scale for Children-fourth edition. The results of covariance analysis indicated a significant difference between the reading performance of the experimental (thinking maps training package and thinking maps method groups) and control groups (\(p< .01\)). Moreover, there were significant differences between the thinking maps training package group and thinking maps method group in some of the subtests (\(p< .01\)). It can be concluded that thinking maps training package and the thinking maps method exert a positive influence on the reading performance of dyslexic students; therefore, thinking maps can be used as an effective training and treatment method.     

Adaptation et validation francophone d’un questionnaire sur les conduites parentales à caractère violent (PC-CTS)

Introduction

The Parent-Child Conflict Tactics Scale (Straus et al., 1998) is the most commonly used instrument to document violence against children in the general population. It was translated into French and adapted to be used in several population-based surveys over the past 15 years.

Two simple approximations to the distributions of quadratic forms

Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to with z_i being independent and following N(0,1). Two approximations to the distribution of T have been implemented in popular software and are widely used in evaluating various models. It is important to know how accurate these approximations are when compared to each other and to the exact distribution of T. The paper systematically studies the quality of the two approximations and examines the effect of the λ_i and the degrees of freedom d by analysis and Monte Carlo. The results imply that the adjusted distribution for T can be as good as knowing its exact distribution. When the coefficient of variation of the λ_i is small, the rescaled statistic is also adequate for practical model inference. But comparing T_R against will inflate type I errors when substantial differences exist among the λ_i, especially, when d is also large.     

Fitting correlated residual error structures in nonlinear mixed-effects models using SAS PROC NLMIXED

Nonlinear mixed-effects (NLME) models remain popular among practitioners for analyzing continuous repeated measures data taken on each of a number of individuals when interest centers on characterizing individual-specific change. Within this framework, variation and correlation among the repeated measurements may be partitioned into interindividual variation and intraindividual variation components. The covariance structure of the residuals are, in many applications, consigned to be independent with homogeneous variances, $ {\sigma}^2{\mathbf{I}}_{n_i} $ , not because it is believed that intraindividual variation adheres to this structure, but because many software programs that estimate parameters of such models are not well-equipped to handle other, possibly more realistic, patterns. In this article, we describe how the programmatic environment within SAS may be utilized to model residual structures for serial correlation and variance heterogeneity. An empirical example is used to illustrate the capabilities of the module.     

On a new property of partially balanced association schemes useful in psychometric structural analysis

In an earlier paper [Psychometrika,31, 1966, p. 147], Srivastava obtained a test for the HypothesisH ₀ : = ₀₀ + ... + _ll, where i are known matrices,i are unknown constants and is the unknown (p ×p) covariance matrix of a random variablex (withp components) having ap-variate normal distribution. The test therein was obtained under (p ×p) covariance matrix of a random variablex (withp components) the condition that ₀, ₁, ..., l form a commutative linear associative algebra and a certain vector, dependent on these, has non-negative elements. In this paper it is shown that this last condition is always satisfied in the special situation (of importance in structural analysis in psychometrics) where ₀, ₁, ..., l are the association matrices of a partially balanced association scheme.This research was partially supported by the U. S. Air Force under Grant No. AF33(615)-3231, monitored by the Aero Space Research Labs.Now at Colorado State University.