Heteroscedastic one‐factor models and marginal maximum likelihood estimation

In the present paper, a general class of heteroscedastic one‐factor models is considered. In these models, the residual variances of the observed scores are explicitly modelled as parametric functions of the one‐dimensional factor score. A marginal maximum likelihood procedure for parameter estimation is proposed under both the assumption of multivariate normality of the observed scores conditional on the single common factor score and the assumption of normality of the common factor score. A likelihood ratio test is derived, which can be used to test the usual homoscedastic one‐factor model against one of the proposed heteroscedastic models. Simulation studies are carried out to investigate the robustness and the power of this likelihood ratio test. Results show that the asymptotic properties of the test statistic hold under both small test length conditions and small sample size conditions. Results also show under what conditions the power to detect different heteroscedasticity parameter values is either small, medium, or large. Finally, for illustrative purposes, the marginal maximum likelihood estimation procedure and the likelihood ratio test are applied to real data.     

Testing and modelling non‐normality within the one‐factor model

Maximum likelihood estimation in the one‐factor model is based on the assumption of multivariate normality for the observed data. This general distributional assumption implies three specific assumptions for the parameters in the one‐factor model: the common factor has a normal distribution; the residuals are homoscedastic; and the factor loadings do not vary across the common factor scale. When any of these assumptions is violated, non‐normality arises in the observed data. In this paper, a model is presented based on marginal maximum likelihood to enable explicit tests of these assumptions. In addition, the model is suitable to incorporate the detected violations, to enable statistical modelling of these effects. Two simulation studies are reported in which the viability of the model is investigated. Finally, the model is applied to IQ data to demonstrate its practical utility as a means to investigate ability differentiation.     

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.     

Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis: An Orthogonal Case

The problem of penalized maximum likelihood (PML) for an exploratory factor analysis (EFA) model is studied in this paper. An EFA model is typically estimated using maximum likelihood and then the estimated loading matrix is rotated to obtain a sparse representation. Penalized maximum likelihood simultaneously fits the EFA model and produces a sparse loading matrix. To overcome some of the computational drawbacks of PML, an approximation to PML is proposed in this paper. It is further applied to an empirical dataset for illustration. A simulation study shows that the approximation naturally produces a sparse loading matrix and more accurately estimates the factor loadings and the covariance matrix, in the sense of having a lower mean squared error than factor rotations, under various conditions.     

The team‐level model of climate for innovation: A two‐level confirmatory factor analysis

The level structure of West's (1990) four‐factor model of team climate for innovation was assessed by means of multi‐level confirmatory factor analysis (MCFA). The sample consisted of 1,487 individuals (195 teams) from a wide range of professions. Results showed that a considerable portion of the variance in the data was explained on the team level with intra‐class correlations ranging from .30 to .39. Furthermore, the results demonstrated that the overall measurement model fitted the data well at both the team and individual levels, while the factor loadings were slightly different across the levels with item loadings showing partial invariance. Results from confirmatory factor analyses conducted on separate levels, however, showed that the four‐factor model displayed the best fit to the data for both individual and team levels. A second‐order one‐factor model also fitted the data well on both levels. The results indicate that the team climate for innovation model can be used as a team‐level consensus model of team climate for innovation.     

Asymptotic comparison of missing data procedures for estimating factor loadings

Large sample properties of four methods of handling multivariate missing data are compared. The criterion for comparison is how well the loadings from a single factor model can be estimated. It is shown that efficiencies of the methods depend on the pattern or arrangement of missing data, and an evaluation study is used to generate predictive efficiency equations to guide one's choice of an estimating procedure. A simple regression-type estimator is introduced which shows high efficiency relative to the maximum likelihood method over a large range of patterns and covariance matrices.     

Effect of non-normality on test statistics for one-way independent groups designs

The data obtained from one‐way independent groups designs is typically non‐normal in form and rarely equally variable across treatment populations (i.e. population variances are heterogeneous). Consequently, the classical test statistic that is used to assess statistical significance (i.e. the analysis of variance F test) typically provides invalid results (e.g. too many Type I errors, reduced power). For this reason, there has been considerable interest in finding a test statistic that is appropriate under conditions of non‐normality and variance heterogeneity. Previously recommended procedures for analysing such data include the James test, the Welch test applied either to the usual least squares estimators of central tendency and variability, or the Welch test with robust estimators (i.e. trimmed means and Winsorized variances). A new statistic proposed by Krishnamoorthy, Lu, and Mathew, intended to deal with heterogeneous variances, though not non‐normality, uses a parametric bootstrap procedure. In their investigation of the parametric bootstrap test, the authors examined its operating characteristics under limited conditions and did not compare it to the Welch test based on robust estimators. Thus, we investigated how the parametric bootstrap procedure and a modified parametric bootstrap procedure based on trimmed means perform relative to previously recommended procedures when data are non‐normal and heterogeneous. The results indicated that the tests based on trimmed means offer the best Type I error control and power when variances are unequal and at least some of the distribution shapes are non‐normal.     

The use of item parcels in structural equation modelling: Non‐normal data and small sample sizes

Maximum likelihood estimation in confirmatory factor analysis requires large sample sizes, normally distributed item responses, and reliable indicators of each latent construct, but these ideals are rarely met. We examine alternative strategies for dealing with non‐normal data, particularly when the sample size is small. In two simulation studies, we systematically varied: the degree of non‐normality; the sample size from 50 to 1000; the way of indicator formation, comparing items versus parcels; the parcelling strategy, evaluating uniformly positively skews and kurtosis parcels versus those with counterbalancing skews and kurtosis; and the estimation procedure, contrasting maximum likelihood and asymptotically distribution‐free methods. We evaluated the convergence behaviour of solutions, as well as the systematic bias and variability of parameter estimates, and goodness of fit.     

Obtaining maximum-likelihood estimates of factor loadings and communalities using an easily implemented iterative computer procedure

This paper considers the problem of computing estimates of factor loadings, specific variances, and communalities for a factor analytic model. The equations for maximum-likelihood estimators are discussed. Iterative formulas are developed to solve the maximum-likelihood equations and a simple and efficient method of implementation on a digital computer is described. Use of the iterative formulas and computing techniques for other estimators of factor loadings and communalities is also considered to provide a very general approach for this aspect of factor analysis.     

Modelling non‐normal data: The relationship between the skew‐normal factor model and the quadratic factor model

Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew‐normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew‐normal factor are equivalent to those under a quadratic model up to third‐order moments. The reverse only holds if the quadratic loadings are equal to each other and within certain bounds. We illustrate that observed data which follow any skew‐normal factor model can be so well approximated with the quadratic factor model that the models are empirically indistinguishable, and that the reverse does not hold in general. The choice between the two models to account for deviations of normality is illustrated by an empirical example from clinical psychology.     

Generalized latent variable models with non‐linear effects

Until recently, item response models such as the factor analysis model for metric responses, the two‐parameter logistic model for binary responses and the multinomial model for nominal responses considered only the main effects of latent variables without allowing for interaction or polynomial latent variable effects. However, non‐linear relationships among the latent variables might be necessary in real applications. Methods for fitting models with non‐linear latent terms have been developed mainly under the structural equation modelling approach. In this paper, we consider a latent variable model framework for mixed responses (metric and categorical) that allows inclusion of both non‐linear latent and covariate effects. The model parameters are estimated using full maximum likelihood based on a hybrid integration–maximization algorithm. Finally, a method for obtaining factor scores based on multiple imputation is proposed here for the non‐linear model.     

The structure of self‐reported emotional experiences: A mixed‐effects Poisson factor model

Multivariate count data are commonly analysed by using Poisson distributions with varying intensity parameters, resulting in a random‐effects model. In the analysis of a data set on the frequency of different emotion experiences we find that a Poisson model with a single random effect does not yield an adequate fit. An alternative model that requires as many random effects as emotion categories requires high‐dimensional integration and the estimation of a large number of parameters. As a solution to these computational problems, we propose a factor‐analytic Poisson model and show that a two‐dimensional factor model fits the reported data very well. Moreover, it yields a substantively satisfactory solution: one factor describing the degree of pleasantness and unpleasantness of emotions and the other factor describing the activation levels of the emotions. We discuss the incorporation of covariates to facilitate rigorous tests of the random‐effects structure. Marginal maximum likelihood methods lead to straight‐forward estimation of the model, for which goodness‐of‐fit tests are also presented.     

Sensitivity to initial values in full non‐parametric maximum‐likelihood estimation of the two‐parameter logistic model

Parameters of the two‐parameter logistic model are generally estimated via the expectation–maximization (EM) algorithm by the maximum‐likelihood (ML) method. In so doing, it is beneficial to estimate the common prior distribution of the latent ability from data. Full non‐parametric ML (FNPML) estimation allows estimation of the latent distribution with maximum flexibility, as the distribution is modelled non‐parametrically on a number of (freely moving) support points. It is generally assumed that EM estimation of the two‐parameter logistic model is not influenced by initial values, but studies on this topic are unavailable. Therefore, the present study investigates the sensitivity to initial values in FNPML estimation. In contrast to the common assumption, initial values are found to have notable influence: for a standard convergence criterion, item discrimination and difficulty parameter estimates as well as item characteristic curve (ICC) recovery were influenced by initial values. For more stringent criteria, item parameter estimates were mainly influenced by the initial latent distribution, whilst ICC recovery was unaffected. The reason for this might be a flat surface of the log‐likelihood function, which would necessitate setting a sufficiently tight convergence criterion for accurate recovery of item parameters.     

Properties of hypothesis testing techniques and (Bayesian) model selection for exploration‐based and theory‐based (order‐restricted) hypotheses

In this paper, the performance of six types of techniques for comparisons of means is examined. These six emerge from the distinction between the method employed (hypothesis testing, model selection using information criteria, or Bayesian model selection) and the set of hypotheses that is investigated (a classical, exploration‐based set of hypotheses containing equality constraints on the means, or a theory‐based limited set of hypotheses with equality and/or order restrictions). A simulation study is conducted to examine the performance of these techniques. We demonstrate that, if one has specific, a priori specified hypotheses, confirmation (i.e., investigating theory‐based hypotheses) has advantages over exploration (i.e., examining all possible equality‐constrained hypotheses). Furthermore, examining reasonable order‐restricted hypotheses has more power to detect the true effect/non‐null hypothesis than evaluating only equality restrictions. Additionally, when investigating more than one theory‐based hypothesis, model selection is preferred over hypothesis testing. Because of the first two results, we further examine the techniques that are able to evaluate order restrictions in a confirmatory fashion by examining their performance when the homogeneity of variance assumption is violated. Results show that the techniques are robust to heterogeneity when the sample sizes are equal. When the sample sizes are unequal, the performance is affected by heterogeneity. The size and direction of the deviations from the baseline, where there is no heterogeneity, depend on the effect size (of the means) and on the trend in the group variances with respect to the ordering of the group sizes. Importantly, the deviations are less pronounced when the group variances and sizes exhibit the same trend (e.g., are both increasing with group number).     

Non‐parametric three‐way mixed ANOVA with aligned rank tests

Research problems that require a non‐parametric analysis of multifactor designs with repeated measures arise in the behavioural sciences. There is, however, a lack of available procedures in commonly used statistical packages. In the present study, a generalization of the aligned rank test for the two‐way interaction is proposed for the analysis of the typical sources of variation in a three‐way analysis of variance (ANOVA) with repeated measures. It can be implemented in the usual statistical packages. Its statistical properties are tested by using simulation methods with two sample sizes (n = 30 and n = 10) and three distributions (normal, exponential and double exponential). Results indicate substantial increases in power for non‐normal distributions in comparison with the usual parametric tests. Similar levels of Type I error for both parametric and aligned rank ANOVA were obtained with non‐normal distributions and large sample sizes. Degrees‐of‐freedom adjustments for Type I error control in small samples are proposed. The procedure is applied to a case study with 30 participants per group where it detects gender differences in linguistic abilities in blind children not shown previously by other methods.     

A unified approach to exploratory factor analysis with missing data,nonnormal data,and in the presence of outliers

Factor analysis is regularly used for analyzing survey data. Missing data, data with outliers and consequently nonnormal data are very common for data obtained through questionnaires. Based on covariance matrix estimates for such nonstandard samples, a unified approach for factor analysis is developed. By generalizing the approach of maximum likelihood under constraints, statistical properties of the estimates for factor loadings and error variances are obtained. A rescaled Bartlett-corrected statistic is proposed for evaluating the number of factors. Equivariance and invariance of parameter estimates and their standard errors for canonical, varimax, and normalized varimax rotations are discussed. Numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures.This project was supported by a University of North Texas Faculty Research Grant, Grant #R49/CCR610528 for Disease Control and Prevention from the National Center for Injury Prevention and Control, and Grant DA01070 from the National Institute on Drug Abuse. The results do not necessarily represent the official view of the funding agencies. The authors are grateful to three reviewers for suggestions that improved the presentation of this paper.     

A generalized estimating equations approach to mixed‐effects ordinal probit models

Clustered ordinal responses, which are commonplace in behavioural and educational research, are often analysed using mixed‐effects ordinal probit models. Likelihood‐based inference for these models can be computationally burdensome, and may compromise the consistency of estimators if the model is misspecified. We propose an alternative inferential approach based on generalized estimating equations. We show that systems of estimating equations can be specified for mixed‐effects ordinal probit models that avoid the potentially heavy computational demands of maximum likelihood estimation, and can also provide inferences that are robust with respect to some forms of model misspecification—particularly serial effects in longitudinal data.     

Human values and non‐adherence to doctors' instructions across Europe

This study examines the relationships between human values and patient non‐adherence. Two types of non‐adherence are studied: non‐adherent views and non‐adherent behavior in response to doctor's instructions. The study uses data from the European Social Survey Round 2 from 14 countries: Austria, Belgium, Denmark, Estonia, Finland, France, Germany, Iceland, Luxembourg, Netherlands, Norway, Poland, Sweden, and Switzerland. Correlation analyses and multiple logistic regression analyses were conducted both using the pooled data from these 14 countries and within countries. The sample size ranged from n = 20,012 to n = 21,600 in the analyses of pooled data. Human values were found to be associated with non‐adherence. As hypothesized, endorsing openness‐to‐change values (vs. conservation values) was positively related to non‐adherent views and behavior.     

The Infinitesimal Jackknife with Exploratory Factor Analysis

The infinitesimal jackknife, a nonparametric method for estimating standard errors, has been used to obtain standard error estimates in covariance structure analysis. In this article, we adapt it for obtaining standard errors for rotated factor loadings and factor correlations in exploratory factor analysis with sample correlation matrices. Both maximum likelihood estimation and ordinary least squares estimation are considered.     

Asymptotic sampling variances for factor analytic models identified by specified zero parameters

Formulas are derived for the asymptotic variances and covariances of the maximum likelihood estimators for oblique simple structure models which are identified by prior specification of zero elements in the factor loading matrix. The formulas are expressed in terms of the various submatrices of the inverse of the required variance-covariance matrix. A numerical example using artificial data is given and problems in the application of the formulas discussed.Now at The Pennsylvania State University.