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.     

Robustness of Parameter Estimation to Assumptions of Normality in the Multidimensional Graded Response Model

A central assumption that is implicit in estimating item parameters in item response theory (IRT) models is the normality of the latent trait distribution, whereas a similar assumption made in categorical confirmatory factor analysis (CCFA) models is the multivariate normality of the latent response variables. Violation of the normality assumption can lead to biased parameter estimates. Although previous studies have focused primarily on unidimensional IRT models, this study extended the literature by considering a multidimensional IRT model for polytomous responses, namely the multidimensional graded response model. Moreover, this study is one of few studies that specifically compared the performance of full-information maximum likelihood (FIML) estimation versus robust weighted least squares (WLS) estimation when the normality assumption is violated. The research also manipulated the number of nonnormal latent trait dimensions. Results showed that FIML consistently outperformed WLS when there were one or multiple skewed latent trait distributions. More interestingly, the bias of the discrimination parameters was non-ignorable only when the corresponding factor was skewed. Having other skewed factors did not further exacerbate the bias, whereas biases of boundary parameters increased as more nonnormal factors were added. The item parameter standard errors recovered well with both estimation algorithms regardless of the number of nonnormal dimensions.     

Random effects and extended generalized partial credit models

In this paper it is shown that under the random effects generalized partial credit model for the measurement of a single latent variable by a set of polytomously scored items, the joint marginal probability distribution of the item scores has a closed-form expression in terms of item category location parameters, parameters that characterize the distribution of the latent variable in the subpopulation of examinees with a zero score on all items, and item-scaling parameters. Due to this closed-form expression, all parameters of the random effects generalized partial credit model can be estimated using marginal maximum likelihood estimation without assuming a particular distribution of the latent variable in the population of examinees and without using numerical integration. Also due to this closed-form expression, new special cases of the random effects generalized partial credit model can be identified. In addition to these new special cases, a slightly more general model than the random effects generalized partial credit model is presented. This slightly more general model is called the extended generalized partial credit model. Attention is paid to maximum likelihood estimation of the parameters of the extended generalized partial credit model and to assessing the goodness of fit of the model using generalized likelihood ratio tests. Attention is also paid to person parameter estimation under the random effects generalized partial credit model. It is shown that expected a posteriori estimates can be obtained for all possible score patterns. A simulation study is carried out to show the usefulness of the proposed models compared to the standard models that assume normality of the latent variable in the population of examinees. In an empirical example, some of the procedures proposed are demonstrated.     

当结构假设和分布假设不满足时的验证性因子分析：稳健极大似然法估计和贝叶斯估计的比较研究

结构方程模型已被广泛应用于心理学、教育学、以及社会科学领域的统计分析中。结构方程模型分析中最常用的估计方法是基于正 态分布的估计量,比如极大似然估计法。这些方法需要满足两个假设。第一, 理论模型必须正确地反映变量与变量之间的关系,称为结构假 设。第二,数据必须符合多元正态分布,称为分布假设。如果这些假设不满足,基于正态分布的估计量就有可能导致不正确的卡方指数、不 正确的拟合度、以及有偏差的参数估计和参数估计的标准误。在实际应用中,几乎所有的理论模型都不能准确地解释变量与变量之间的关系, 数据也常常呈非多元正态分布。为此,一些新的估计方法得以发展。这些方法要么在理论上不要求数据呈多元正态分布,要么对因数据呈非 正态分布而导致的不正确结果进行纠正。当前较为流行的两种方法是稳健极大似然估计和贝叶斯估计。稳健极大似然估计是应用 Satorra and Bentler (1994) 的方法对不正确的卡方指数和参数估计的标准误进行调整,而参数估计和用极大似然方法得出的完全等同。贝叶斯估计方法则是 基于贝叶斯定理,其要点是：参数的后验分布是由参数的先验分布和数据似然值相乘而得来。后验分布常用马尔科夫蒙特卡洛算法来进行模拟。 对于稳健极大似然估计和贝叶斯估计这两种方法之间的优劣比较,先前的研究只局限于理论模型是正确的情境。而本研究则着重于理论模型 是错误的情境,同时也考虑到数据呈非正态分布的情境。本研究所采用的模型是验证性因子模型,数据全部由计算机模拟而来。数据的生成 取决于三个因素：8 类因子结构,3 种变量分布,和3 组样本量。这三个因素产生72 个模拟条件（72=8x3x3）。每个模拟条件下生成2000 个 数据组,每个数据组都拟合两个模型,一个是正确模型、一个是错误模型。每个模型都用两种估计方法来拟合：稳健极大似然估计法和贝叶 斯估计方法。贝叶斯估计方法中所使用的先验分布是无信息先验分布。结果分析主要着重于模型拒绝率、拟合度、参数估计、和参数估计的 标准误。研究的结果表明：在样本量充足的情况下,两种方法得出的参数估计非常相似。当数据呈非正态分布时,贝叶斯估计法比稳健极大 似然估计法更好地拒绝错误模型。但是,当样本量不足且数据呈正态分布时,贝叶斯估计在拒绝错误模型和参数估计上几乎没有优势,甚至 在一些条件下,比稳健极大似然法要差。     

Asymptotic biases in exploratory factor analysis and structural equation modeling

Formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the Wishart maximum likelihood estimation for normally and nonnormally distributed variables. When multivariate normality is satisfied, considerable simplification is obtained for the models of unstandardized variables. Formulas for the models of standardized variables are also provided. Numerical examples with Monte Carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data. Some relationships between the asymptotic biases and other asymptotic values are discussed.The author is indebted to the editor and anonymous reviewers for their comments, corrections, and suggestions on this paper, and to Yutaka Kano for discussion on biases.     

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.     

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.     

A one-way random effects model for trimmed means

The random effects ANOVA model plays an important role in many psychological studies, but the usual model suffers from at least two serious problems. The first is that even under normality, violating the assumption of equal variances can have serious consequences in terms of Type I errors or significance levels, and it can affect power as well. The second and perhaps more serious concern is that even slight departures from normality can result in a substantial loss of power when testing hypotheses. Jeyaratnam and Othman (1985) proposed a method for handling unequal variances, under the assumption of normality, but no results were given on how their procedure performs when distributions are nonnormal. A secondary goal in this paper is to address this issue via simulations. As will be seen, problems arise with both Type I errors and power. Another secondary goal is to provide new simulation results on the Rust-Fligner modification of the Kruskal-Wallis test. The primary goal is to propose a generalization of the usual random effects model based on trimmed means. The resulting test of no differences among J randomly sampled groups has certain advantages in terms of Type I errors, and it can yield substantial gains in power when distributions have heavy tails and outliers. This last feature is very important in applied work because recent investigations indicate that heavy-tailed distributions are common. Included is a suggestion for a heteroscedastic Winsorized analog of the usual intraclass correlation coefficient.     

Checking the Assumptions of Rasch's Model for Speed Tests

Two new tests for a model for the response times on pure speed tests by Rasch (1960) are proposed. The model is based on the assumption that the test response times are approximately gamma distributed, with known index parameters and unknown rate parameters. The rate parameters are decomposed in a subject ability parameter and a test difficulty parameter. By treating the ability as a gamma distributed random variable, maximum marginal likelihood (MML) estimators for the test difficulty parameters and the parameters of the ability distribution are easily derived. Also the model tests proposed here pertain to the framework of MML. Two tests or modification indices are proposed. The first one is focused on the assumption of local stochastic independence, the second one on the assumption of the test characteristic functions. The tests are based on Lagrange multiplier statistics, and can therefore be computed using the parameter estimates under the null model. Therefore, model violations for all items and pairs of items can be assessed as a by-product of one single estimation run. Power studies and applications to real data are included as numerical examples.     

Limited‐information goodness‐of‐fit testing of item response theory models for sparse 2P tables

Bartholomew and Leung proposed a limited‐information goodness‐of‐fit test statistic (Y) for models fitted to sparse 2^P contingency tables. The null distribution of Y was approximated using a chi‐squared distribution by matching moments. The moments were derived under the assumption that the model parameters were known in advance and it was conjectured that the approximation would also be appropriate when the parameters were to be estimated. Using maximum likelihood estimation of the two‐parameter logistic item response theory model, we show that the effect of parameter estimation on the distribution of Y is too large to be ignored. Consequently, we derive the asymptotic moments of Y for maximum likelihood estimation. We show using a simulation study that when the null distribution of Y is approximated using moments that take into account the effect of estimation, Y becomes a very useful statistic to assess the overall goodness of fit of models fitted to sparse 2^P tables.     

Pairwise Likelihood Ratio Tests and Model Selection Criteria for Structural Equation Models with Ordinal Variables

Correlated multivariate ordinal data can be analysed with structural equation models. Parameter estimation has been tackled in the literature using limited-information methods including three-stage least squares and pseudo-likelihood estimation methods such as pairwise maximum likelihood estimation. In this paper, two likelihood ratio test statistics and their asymptotic distributions are derived for testing overall goodness-of-fit and nested models, respectively, under the estimation framework of pairwise maximum likelihood estimation. Simulation results show a satisfactory performance of type I error and power for the proposed test statistics and also suggest that the performance of the proposed test statistics is similar to that of the test statistics derived under the three-stage diagonally weighted and unweighted least squares. Furthermore, the corresponding, under the pairwise framework, model selection criteria, AIC and BIC, show satisfactory results in selecting the right model in our simulation examples. The derivation of the likelihood ratio test statistics and model selection criteria under the pairwise framework together with pairwise estimation provide a flexible framework for fitting and testing structural equation models for ordinal as well as for other types of data. The test statistics derived and the model selection criteria are used on data on ‘trust in the police’ selected from the 2010 European Social Survey. The proposed test statistics and the model selection criteria have been implemented in the R package lavaan.     

Testing Manifest Monotonicity Using Order-Constrained Statistical Inference

Most dichotomous item response models share the assumption of latent monotonicity, which states that the probability of a positive response to an item is a nondecreasing function of a latent variable intended to be measured. Latent monotonicity cannot be evaluated directly, but it implies manifest monotonicity across a variety of observed scores, such as the restscore, a single item score, and in some cases the total score. In this study, we show that manifest monotonicity can be tested by means of the order-constrained statistical inference framework. We propose a procedure that uses this framework to determine whether manifest monotonicity should be rejected for specific items. This approach provides a likelihood ratio test for which the p-value can be approximated through simulation. A simulation study is presented that evaluates the Type I error rate and power of the test, and the procedure is applied to empirical data.     

Extensions of the partial credit model

The partial credit model, developed by Masters (1982), is a unidimensional latent trait model for responses scored in two or more ordered categories. In the present paper some extensions of the model are presented. First, a marginal maximum likelihood estimation procedure is developed which allows for incomplete data and linear restrictions on both the item and the population parameters. Secondly, two statistical tests for evaluating model fit are presented: the former test has power against violation of the assumption about the ability distribution, the latter test offers the possibility of identifying specific items that do not fit the model.The authors are indepted to professor Wim van der Linden and Huub Verstralen for their helpful comments.     

A dynamic generalization of the Rasch model

In the present paper a model for describing dynamic processes is constructed by combining the common Rasch model with the concept of structurally incomplete designs. This is accomplished by mapping each item on a collection of virtual items, one of which is assumed to be presented to the respondent dependent on the preceding responses and/or the feedback obtained. It is shown that, in the case of subject control, no unique conditional maximum likelihood (CML) estimates exist, whereas marginal maximum likelihood (MML) proves a suitable estimation procedure. A hierarchical family of dynamic models is presented, and it is shown how to test special cases against more general ones. Furthermore, it is shown that the model presented is a generalization of a class of mathematical learning models, known as Luce's beta-model.     

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.     

A comparison of two‐stage procedures for testing least‐squares coefficients under heteroscedasticity

This study explores the performance of several two‐stage procedures for testing ordinary least‐squares (OLS) coefficients under heteroscedasticity. A test of the usual homoscedasticity assumption is carried out in the first stage of the procedure. Subsequently, a test of the regression coefficients is chosen and performed in the second stage. Three recently developed methods for detecting heteroscedasticity are examined. In addition, three heteroscedastic robust tests of OLS coefficients are considered. A major finding is that performing a test of heteroscedasticity prior to applying a heteroscedastic robust test can lead to poor control over Type I errors.     

Regression among factor scores

Structural equation models with latent variables are sometimes estimated using an intuitive three-step approach, here denoted factor score regression. Consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest. In this simple example estimation of the structural parameter proceeds as follows: First, common factor models areseparately estimated for each latent variable. Second, factor scores areseparately assigned to each latent variable, based on the estimates. Third, ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter. We investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables. It is demonstrated that the conventional approach to factor score regression performs very badly. Revised factor score regression, using Regression factor scores for the explanatory latent variables and Bartlett scores for the response latent variables, produces consistent estimators for all parameters.     

Maximum likelihood estimation for Luce's choice model

A terminology for general choice models based on the choice axiom is given. It applies to all kinds of choice experiments, such as confusion choice experiments, paired comparisons, triadic comparisons, directional rankings, scores on binary test items, and others. Maximum likelihood estimation for such general choice models is considered. Conditions for the uniqueness of maximum likelihood estimates are given, and it is shown that the estimates can be derived by iterative proportional fitting. This offers the opportunity of a general test of the choice axiom for all kinds of choice experiments using the likelihood ratio. The estimation and testing procedure is applied to data from a form recognition experiment, reported by W. A. Wagenaar (Nederlands Tijdschrift voor de Psychologie, 1968, 23, 96–108).     

Factorial invariance in multilevel confirmatory factor analysis

This paper presents a procedure to test factorial invariance in multilevel confirmatory factor analysis. When the group membership is at level 2, multilevel factorial invariance can be tested by a simple extension of the standard procedure. However level‐1 group membership raises problems which cannot be appropriately handled by the standard procedure, because the dependency between members of different level‐1 groups is not appropriately taken into account. The procedure presented in this article provides a solution to this problem. This paper also shows Muthén's maximum likelihood (MUML) estimation for testing multilevel factorial invariance across level‐1 groups as a viable alternative to maximum likelihood estimation. Testing multilevel factorial invariance across level‐2 groups and testing multilevel factorial invariance across level‐1 groups are illustrated using empirical examples. SAS macro and Mplus syntax are provided.