British Journal of Mathematical and Statistical Psychology

Oertzen, Timo

doi: 10.1348/000711009X441021pmid: 19527562

Implementing large‐scale empirical studies can be very expensive. Therefore, it is useful to optimize study designs without losing statistical power. In this paper, we show how study designs can be improved without changing statistical power by defining power equivalence, a relation between structural equation models (SEMs) that holds true if two SEMs have the same power on a likelihood ratio test to detect a given effect. We show systematic operations of SEMs that maintain power, and give an algorithm that efficiently reduces SEMs to power‐equivalent models with a minimal number of observed parameters. In this way, optimal study designs can be found without reducing statistical power. Furthermore, the algorithm can be used to drastically increase the speed of power computations when using Monte Carlo simulations or approximation methods.

Yuan, Ke‐Hai; Bentler, Peter M.

doi: 10.1348/000711009X449771pmid: 19793410

Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to with zi 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 λi 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 λi is small, the rescaled statistic is also adequate for practical model inference. But comparing TR against will inflate type I errors when substantial differences exist among the λi, especially, when d is also large.

Molenaar, Dylan; Dolan, Conor V.; Verhelst, Norman D.

doi: 10.1348/000711009X456935pmid: 19796474

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.

Ng, Marie; Wilcox, Rand R.

doi: 10.1348/000711009X456845pmid: 19807946

In this study, we explore the effects of non‐normality and heteroscedasticity when testing the hypothesis that regression lines associated with two independent groups have the same slopes. Our results indicate that some recently proposed methods that allow heteroscedasticity and perform well in extant simulation studies do not perform well for the situation at hand. Two of the methods studied here are recommended for general use.

Kreuzpointner, Ludwig; Simon, Patricia; Theis, Fabian J.

doi: 10.1348/000711009X465647pmid: 20298645

The ad coefficient was developed to measure the within‐group agreement of ratings. The underlying theory as well as the construction of the coefficient are explained. The ad coefficient ranges from 0 to 1, regardless of the number of scale points, raters, or items. With some limitations the measure of the within‐group agreement of different groups and groups from different studies is directly comparable. For statistical significance testing, the binomial distribution is introduced as a model of the ratings' random distribution given the true score of a group construct. This method enables a decision about essential agreement and not only about a significant difference from 0 or a chosen critical value. The ad coefficient identifies a single true score within a group. It is not provided for multiple true score settings. The comparison of the ad coefficient with other agreement indices shows that the new coefficient is in line with their outcomes, but does not result in infinite or inappropriate values.

Li, Yong; Wang, Hai‐Zhong

doi: 10.1348/000711009X466367pmid: 19719904

This paper considers finite mixtures of structural equation models with non‐linear effects of exogenous latent variables and non‐recursive relations among endogenous latent variables. A Bayesian approach is developed to analyse this kind of model. In order to cope with the label switching problem, the permutation sampler is used to choose an appropriate identification constraint. Furthermore, a hybrid Markov chain Monte Carlo method that combines the Gibbs sampler, Metropolis–Hastings algorithm, and Langevin–Hastings algorithm is implemented to produce the Bayesian outputs. Finally, the proposed approach is illustrated by a simulation study and a real example.

Wilcox, Rand R.

doi: 10.1348/000711009X467618pmid: 20021728

This paper considers the problem of estimating the overall strength of an association, including situations where there is curvature. The general strategy is to fit a robust regression line, or some type of smoother that allows curvature, and then use a robust analogue of explanatory power, say η2. When the regression surface is a plane, an estimate of η2 via the Theil–Sen estimator is found to perform well, relative to some other robust regression estimators, in terms of mean squared error and bias. When there is curvature, a generalization of a kernel estimator derived by Fan performs relatively well, but two alternative smoothers have certain practical advantages. When η2 is approximately equal to zero, estimation using smoothers has relatively high bias. A variation of η2 is suggested for dealing with this problem. Methods for testing H0: η2=0 are examined that are based in part on smoothers. Two methods are found that control Type I error probabilities reasonably well in simulations. Software for applying the more successful methods is provided.

Ip, Edward Haksing

doi: 10.1348/000711009X466835pmid: 19840494

Multidimensionality is a core concept in the measurement and analysis of psychological data. In personality assessment, for example, constructs are mostly theoretically defined as unidimensional, yet responses collected from the real world are almost always determined by multiple factors. Significant research efforts have concentrated on the use of simulated studies to evaluate the robustness of unidimensional item response models when applied to multidimensional data with a dominant dimension. In contrast, in the present paper, I report the result from a theoretical investigation that a multidimensional item response model is empirically indistinguishable from a locally dependent unidimensional model, of which the single dimension represents the actual construct of interest. A practical implication of this result is that multidimensional response data do not automatically require the use of multidimensional models. Circumstances under which the alternative approach of locally dependent unidimensional models may be useful are discussed.

Raykov, Tenko

doi: 10.1348/000711009X468004pmid: 19793411

An interval estimation procedure is outlined that can be used for evaluating the proportion of observed variance in a response variable, which is due to the third level of nesting in a hierarchical design. The approach is also useful when it is of concern to address the necessity of including a third level in analyses of data from a multi‐level study, relative to an alternative of proceeding with two‐level modelling. The proposed method is illustrated with an empirical example.

Ferrando, Pere J.; Lorenzo‐Seva, Urbano

doi: 10.1348/000711009X470740pmid: 19840493

In personality and attitude measurement, the presence of acquiescent responding can have an impact on the whole process of item calibration and test scoring, and this can occur even when sensible procedures for controlling acquiescence are used. This paper considers a bidimensional (content acquiescence) factor‐analytic model to be the correct model, and assesses the effects of fitting unidimensional models to theoretically unidimensional scales when acquiescence is in fact operating. The analysis considers two types of scales: non‐balanced and fully balanced. The effects are analysed at both the calibration and the scoring stages, and are of two types: bias in the item/respondent parameter estimates and model/person misfit. The results obtained theoretically are checked and assessed by means of simulation. The results and predictions are then assessed in an empirical study based on two personality scales. The implications of the results for applied personality research are discussed.