Properties of the maximum likelihood solution in factor analysis regression

Algebraic properties of the normal theory maximum likelihood solution in factor analysis regression are investigated. Two commonly employed measures of the within sample predictive accuracy of the factor analysis regression function are considered: the variance of the regression residuals and the squared correlation coefficient between the criterion variable and the regression function. It is shown that this within sample residual variance and within sample squared correlation may be obtained directly from the factor loading and unique variance estimates, without use of the original observations or the sample covariance matrix.     

Bayesian Hierarchical Multivariate Formulation with Factor Analysis for Nested Ordinal Data

This article devises a Bayesian multivariate formulation for analysis of ordinal data that records teacher classroom performance along multiple dimensions to assess aspects characterizing good instruction. Study designs for scoring teachers seek to measure instructional performance over multiple classroom measurement event sessions at varied occasions using disjoint intervals within each session and employment of multiple ratings on intervals scored by different raters; a design which instantiates a nesting structure with each level contributing a source of variation in recorded scores. We generally possess little a priori knowledge of the existence or form of a sparse generating structure for the multivariate dimensions at any level in the nesting that would permit collapsing over dimensions as is done under univariate modeling. Our approach composes a Bayesian data augmentation scheme that introduces a latent continuous multivariate response linked to the observed ordinal scores with the latent response mean constructed as an additive multivariate decomposition of nested level means that permits the extraction of de-noised continuous teacher-level scores and the associated correlation matrix. A semi-parametric extension facilitates inference for teacher-level dependence among the dimensions of classroom performance under multi-modality induced by sub-groupings of rater perspectives. We next replace an inverse Wishart prior specified for the teacher covariance matrix over dimensions of instruction with a factor analytic structure to allow the simultaneous assessment of an underlying sparse generating structure. Our formulation for Bayesian factor analysis employs parameter expansion with an accompanying post-processing sign re-labeling step of factor loadings that together reduce posterior correlations among sampled parameters to improve parameter mixing in our Markov chain Monte Carlo (MCMC) scheme. We evaluate the performance of our formulation on simulated data and make an application for the assessment of the teacher covariance structure with a dataset derived from a study of middle and high school algebra teachers.     

A note on testing for homogeneity among effect sizes sharing a common control group

L. V. Hedges and I. Olkin (1985) presented a statistic to test for homogeneity among correlated effect sizes and L. J. Gleser and I. Olkin (1994) presented a large-sample approximation to the covariance matrix of the correlated effect sizes. This article presents a more exact expression for this covariance matrix, assuming normally distributed data but not large samples, for the situation where effect sizes are correlated because a single control group was compared with more than one treatment group. After the correlation between effect sizes has been estimated, the standard Q statistic for correlated effect sizes can be used to test for homogeneity. This method is illustrated using results from schizophrenia research.     

Variable error: variance-covariance heterogeneity, block size and type L error rates

The degree to which blocked (VE) data satisfies the assumptions of compound symmetry required for a repeated measures ANOVA was studied. Monte Carlo procedures were used to study the effect of violation of this assumption, under varying block sizes, on the Type l error rate. Populations of 10,000 subjects for each of two groups, the underlying variance-covariance matrices reflecting a specific condition of violation of the homogeneity of covariance assumptions, was generated based on each of three actual experimental data sets. The data were blocked in various ways, VE calculated, and subsequently analyzed by a repeated measures ANOVA. The complete process was replicated for four covariance homogeneity conditions for each of the three data sets, resulting in a total of 22,000 simulated experiments. Results indicated that the Type l error rate increases as the degree of heterogeneity within the variance-covariance matrices increases when raw (unblocked) data are analyzed. With VE, the effects of within-matrix heterogeneity on the Type l error rate are inconclusive. However, block size does seem to affect the probability of obtaining a significant interaction, but the nature of this relationship is not clear as there does not appear to be any consistent relationship between the size of the block and the probability of obtaining significance. For both raw and VE data there was no inflation in the number of Type l errors when the covariances within a given matrix were homogeneous, regardless of the differences between the group variance-covariance matrices.     

Evaluating effects in short time series: alternative models of analysis

In the applied context, short time-series designs are suitable to evaluate a treatment effect. These designs present serious problems given autocorrelation among data and the small number of observations involved. This paper describes analytic procedures that have been applied to data from short time series, and an alternative which is a new version of the generalized least squares method to simplify estimation of the error covariance matrix. Using the results of a simulation study and assuming a stationary first-order autoregressive model, it is proposed that the original observations and the design matrix be transformed by means of the square root or Cholesky factor of the inverse of the covariance matrix. This provides a solution to the problem of estimating the parameters of the error covariance matrix. Finally, the results of the simulation study obtained using the proposed generalized least squares method are compared with those obtained by the ordinary least squares approach. The probability of Type I error associated with the proposed method is close to the nominal value for all values of rho1 and n investigated, especially for positive values of rho1. The proposed generalized least squares method corrects the effect of autocorrelation on the test's power.     

Prediction of Children's Academic Competence From Their Effortful Control, Relationships, and Classroom Participation

The authors examined the relations among children's effortful control, school relationships, classroom participation, and academic competence with a sample of 7- to 12-year-old children (N = 264). Parents and children reported on children's effortful control, and teachers and children reported on children's school relationships and classroom participation. Children's grade point averages (GPAs) and absences were obtained from school-issued report cards. Significant positive correlations existed between effortful control, school relationships, classroom participation, and academic competence. Consistent with expectations, the teacher-child relationship, social competence, and classroom participation partially mediated the relation between effortful control and change in GPA from the beginning to the end of the school year. The teacher-child relationship and classroom participation also partially mediated the relation between effortful control and change in school absences across the year.     

The missing piece: a sociological autopsy of firearm suicide in the United States

Social, economic, violence, political, and gun access predictors of suicide and gun suicide were examined via sociological autopsy. The model predicting suicide rates overall had the best results, χ(2) (9, N=50)=5.279 (CMIN, the goodness of fit statistic that represents the minimum discrepancy between the unrestricted sample covariance matrix and the restricted covariance matrix) p=.809, [the goodness of fit statistic that represents the minimum discrepancy between the unrestricted sample covariance matrix and the restricted covariance matrix] indicating an excellent fit of the data and theoretical model. The model explained 76% of the variance in state suicide rates and was a significantly better predictor than one could expect by chance, F(6,43)=22.889, p<.001). All path coefficients were significant predictors of suicide with the exception of violence climate, which was not included. This study contributes to the theoretical knowledge by adding a comprehensive framework of analysis and model useful for prevention.     

A Simple Simulation Technique for Nonnormal Data with Prespecified Skewness,Kurtosis, and Covariance Matrix

We present and investigate a simple way to generate nonnormal data using linear combinations of independent generator (IG) variables. The simulated data have prespecified univariate skewness and kurtosis and a given covariance matrix. In contrast to the widely used Vale-Maurelli (VM) transform, the obtained data are shown to have a non-Gaussian copula. We analytically obtain asymptotic robustness conditions for the IG distribution. We show empirically that popular test statistics in covariance analysis tend to reject true models more often under the IG transform than under the VM transform. This implies that overly optimistic evaluations of estimators and fit statistics in covariance structure analysis may be tempered by including the IG transform for nonnormal data generation. We provide an implementation of the IG transform in the R environment.     

Multilevel modelling: Beyond the basic applications

Over the last 30 years statistical algorithms have been developed to analyse datasets that have a hierarchical/multilevel structure. Particularly within developmental and educational psychology these techniques have become common where the sample has an obvious hierarchical structure, like pupils nested within a classroom. We describe two areas beyond the basic applications of multilevel modelling that are important to psychology: modelling the covariance structure in longitudinal designs and using generalized linear multilevel modelling as an alternative to methods from signal detection theory (SDT). Detailed code for all analyses is described using packages for the freeware R.     

Using the linear mixed model to analyze nonnormal data distributions in longitudinal designs

Using a Monte Carlo simulation and the Kenward–Roger (KR) correction for degrees of freedom, in this article we analyzed the application of the linear mixed model (LMM) to a mixed repeated measures design. The LMM was first used to select the covariance structure with three types of data distribution: normal, exponential, and log-normal. This showed that, with homogeneous between-groups covariance and when the distribution was normal, the covariance structure with the best fit was the unstructured population matrix. However, with heterogeneous between-groups covariance and when the pairing between covariance matrices and group sizes was null, the best fit was shown by the between-subjects heterogeneous unstructured population matrix, which was the case for all of the distributions analyzed. By contrast, with positive or negative pairings, the within-subjects and between-subjects heterogeneous first-order autoregressive structure produced the best fit. In the second stage of the study, the robustness of the LMM was tested. This showed that the KR method provided adequate control of Type I error rates for the time effect with normally distributed data. However, as skewness increased—as occurs, for example, in the log-normal distribution—the robustness of KR was null, especially when the assumption of sphericity was violated. As regards the influence of kurtosis, the analysis showed that the degree of robustness increased in line with the amount of kurtosis.     

Information matrix estimation procedures for cognitive diagnostic models

Two new methods to estimate the asymptotic covariance matrix for marginal maximum likelihood estimation of cognitive diagnosis models (CDMs), the inverse of the observed information matrix and the sandwich-type estimator, are introduced. Unlike several previous covariance matrix estimators, the new methods take into account both the item and structural parameters. The relationships between the observed information matrix, the empirical cross-product information matrix, the sandwich-type covariance matrix and the two approaches proposed by de la Torre (2009, J. Educ. Behav. Stat., 34, 115) are discussed. Simulation results show that, for a correctly specified CDM and Q-matrix or with a slightly misspecified probability model, the observed information matrix and the sandwich-type covariance matrix exhibit good performance with respect to providing consistent standard errors of item parameter estimates. However, with substantial model misspecification only the sandwich-type covariance matrix exhibits robust performance.     

Grouping of Pupils in Secondary School Classrooms: Possible Links between Pedagogy and Learning

Pupil groupings within classrooms are a constant social pedagogic factor which effect participation and learning. The grouping of pupils provides a dynamic relationship between learning tasks and the classroom context within which learning takes place. This paper explores types of pupil groupings found in secondary school classrooms related to themes of group size and composition, learning task, group interaction and teacher presence. The paper questions how classroom groupings may be related to the circumstances that promote or inhibit school learning. Previous studies concerning classroom grouping in secondary schools have been dominated by ability or attainment level of the child. The approach and review presented in this study advances a new and more extensive understanding of social pedagogic activity within classrooms. Empirically, a novel classroom “mapping” method with follow-up questions for teachers was used to identify within-class grouping characteristics of children aged 12 and 15 years (in school Years 7 and 10), across curriculum subjects (English, mathematics, science and humanities) in secondary schools. Maps were drawn for 250 classrooms and explained by teachers from 47 schools in England. Results identified that a variety of groupings were found in most classes. Use of particular groupings related to phase of lesson and, to some extent, curriculum. The results showed: little pedagogic association with learning task or interaction; teacher presence associated with control of knowledge; and group composition dominated by stereotypical adolescent friendships. Discussion considers the social pedagogic potential of grouping (with regard to relational capital), control of knowledge and social structure of the classroom attributed to the teacher, and patterns of grouping that may be associated with polarisation in the classroom. *Research reported was part of the GRIS (The Nature and Use of Within Class Groupings in Secondary Schools) project funded by the ESRC, grant number R000238172.     

A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.     

A Review of: “Nelson,C. Ellis. (2008). Growing up Christian: A Congregational Strategy for Nurturing Disciples”

A quasi-experimental design was used to statistically compare academic achievement among 277 students in single-grade and multigrade classrooms in the Pacific Northwest. Adjustments were made using analysis of covariance to help insure comparable group means. Results indicate that academic achievement with a single exception is not related to enrollment in single grade or multigrade classrooms even after the effects of gender and length of years in classroom were held constant. Less direct instruction and more process time was observed in multigrade rooms.     

A Class of Population Covariance Matrices in the Bootstrap Approach to Covariance Structure Analysis

Model evaluation in covariance structure analysis is critical before the results can be trusted. Due to finite sample sizes and unknown distributions of real data, existing conclusions regarding a particular statistic may not be applicable in practice. The bootstrap procedure automatically takes care of the unknown distribution and, for a given sample size, also provides more accurate results than those based on standard asymptotics. But the procedure needs a matrix to play the role of the population covariance matrix. The closer the matrix is to the true population covariance matrix, the more valid the bootstrap inference is. The current paper proposes a class of covariance matrices by combining theory and data. Thus, a proper matrix from this class is closer to the true population covariance matrix than those constructed by any existing methods. Each of the covariance matrices is easy to generate and also satisfies several desired properties. An example with nine cognitive variables and a confirmatory factor model illustrates the details for creating population covariance matrices with different misspecifications. When evaluating the substantive model, bootstrap or simulation procedures based on these matrices will lead to more accurate conclusion than that based on artificial covariance matrices.     

Genetic and environmental covariance structures among WISC-R subtests: A twin study

Genetic and environmental correlations among 11 Wechsler Intelligence Scale for Children-Revised (WISC-R) subtests were estimated from a sample of 143 twin pairs using the methodology of multivariate behavioral genetics. The genetic correlations among subtests varied in magnitude and exhibited a pattern similar to corresponding phenotypic correlations. Correlations due to within-pair environmental influences were generally small, whereas those due to shared environmental influences tended to be intermediate. Several hypotheses regarding the structures of the additive-genetic, common environmental, and within-pair environmental covariance matrices were tested and evidence for differential factor structures was obtained. A model hypothesizing a three-factor structure for the genetic covariance matrix, a single factor for the common environmental matrix, and a single factor plus specifics for the within-pair environmental matrix fit the data well. Thus, the three-factor phenotypic structure typically observed in Wechsler Intelligence Scale for Children (WISC) data may be due largely to genetic influences.     

Teacher’s use of praise,clarity of school rules and classroom climate: comparing classroom compositions in terms of disruptive students

Clarity of school rules and teachers’ use of praise are strategies suggested to facilitate a positive classroom climate. Studies indicate difficulties for teachers to use such approaches in classrooms with higher levels of disruption. To study (1) if student-rated clarity of school rules, use of praise, and classroom climate differ between students in classes with lower numbers of disruptive students versus classes with higher numbers, (2) if clarity of school rules and teacher’s use of praise are longitudinally associated with classroom climate, (3) if the possible longitudinal association differs between groups, classes (n = 109) in school grades 5–7 were divided into two groups, based on head teacher ratings of disruptive students in class. Baseline and 12-month follow-up responses collected within a Swedish trial were used to perform multiple regression analysis, to compare groups and to investigate possible longitudinal associations. Students in classes with less disruption rated all variables more positively. Classroom climate deteriorated over time in both groups, even if the low disruption group perceived their climate as more positive at follow up. Clarity of school rules did not substantially contribute to classroom climate longitudinally, whereas teacher’s use of praise to some extent did. The difference in longitudinal associations between groups was marginal, hence our hypothesis on weaker associations in the high disruption group could not be confirmed. Clarity of school rules is not longitudinally associated with classroom climate, but teachers may positively influence the learning environment by giving praise, regardless of level of disruption.     

Sensitivity analysis in factor analysis: Difference between using covariance and correlation matrices

Influence curves of some parameters under various methods of factor analysis have been given in the literature. These influence curves depend on the influence curves for either the covariance or the correlation matrix used in the analysis. The differences between the influence curves based on the covariance and the correlation matrices are derived in this paper. Simple formulas for the differences of the influence curves, based on the two matrices, for the unique variance matrix, factor loadings and some other parameter are obtained under scale-invariant estimation methods, though the influence curves themselves are in complex forms.The authors are most grateful to the referees, the Associate Editor, the Editor and Raymond Lam for helpful suggestions for improving the clarity of the paper.     

The effects of didactic classroom instruction versus field exposure on career maturity

This study tested two types of career training (field exposure and didactic classroom) to determine which is more effective in bringing about the development of career maturity. Forty-six first-year Upward Bound students were assigned to either a field exposure or a didactic group by the use of a stratified random assignment technique. A pretest/post-test design was utilized with the Career Maturity Inventory as the data-collecting instrument. The data were analyzed by analysis of covariance. The results of the study indicate that field exposure career training is an effective method to increase career maturity.     

The Effect of Prolonged Implementation of Cooperative Learning on Social Support Within the Classroom

This study was conducted to determine whether cooperative learning experiences are related to social support in the classroom. The relationship between cooperative learning and social support, the impact of prolonged implementation of cooperative learning on social support, and the effect of different frequency of cooperative learning experiences on social support in the classroom were investigated. An instrument measuring social support was administered in November and in January to 91 eighth-grade students from a suburban school district in the Midwest. The results indicate that cooperative learning was highly related to social support within the classroom and that the longer and more frequently students engaged in cooperative learning, the greater the social support within the classroom.