British Journal of Mathematical and Statistical Psychology

Holman, Rebecca; Glas, Cees A. W.

doi: 10.1111/j.2044-8317.2005.tb00312.xpmid: N/A

A model‐based procedure for assessing the extent to which missing data can be ignored and handling non‐ignorable missing data is presented. The procedure is based on item response theory modelling. As an example, the approach is worked out in detail in conjunction with item response data modelled using the partial credit and generalized partial credit models. Simulation studies are carried out to assess the extent to which the bias caused by ignoring the missing‐data mechanism can be reduced. Finally, the feasibility of the procedure is demonstrated using data from a study to calibrate a medical disability scale.

Trendafilov, Nickolay T.

doi: 10.1111/j.2044-8317.2005.tb00313.xpmid: N/A

The well‐known problem of fitting the exploratory factor analysis model is reconsidered where the usual least squares goodness‐of‐fit function is replaced by a more resistant discrepancy measure, based on a smooth approximation of the ℓ1 norm. Fitting the factor analysis model to the sample correlation matrix is a complex matrix optimization problem which requires the structure preservation of the unknown parameters (e.g. positive definiteness). The projected gradient approach is a natural way of solving such data matching problems as especially designed to follow the geometry of the model parameters. Two reparameterizations of the factor analysis model are considered. The approach leads to globally convergent procedures for simultaneous estimation of the factor analysis matrix parameters. Numerical examples illustrate the algorithms and factor analysis solutions.

Wilcox, Rand R.

doi: 10.1348/000711005X47177pmid: 15969837

A basic property of various rank‐based hypothesis testing methods is that they are invariant under a linear transformation of the data. For multivariate data, a generalization of this property is sometimes sought (called affine invariance), but typically techniques for assigning ranks do not achieve this goal, or it is assumed that sampling is from a symmetric distribution. A rank‐based method is suggested for comparing dependent groups that is based on halfspace depth, is affine invariant in terms of difference scores, and allows sampling from asymmetric distributions.

Corte, Wilfried

doi: 10.1348/000711005X38726pmid: 15969838

A procedure is presented for determining the mean and variance of the selection differential for top‐down selections in which the candidates come from populations that have a different average score on the selection measure. Although the procedure is based on the same stochastic model and requires identical data to the currently available method for estimating the mean selection differential, it has the advantage that the resulting expressions are valid for finite‐sample selection decisions and that the variance of the selection differential can also be assessed. The difference between the two procedures is illustrated by means of an example application, and it is shown how the present results are particularly helpful in determining the expected utility of personnel selection decisions.

Bi, Jian

doi: 10.1348/000711005X38357pmid: 15969839

A commonly used method of estimating population sensitivity is the so‐called averaged d ′ method. In this method, the arithmetic mean of a set of individual d ′ is usually taken as a population sensitivity estimator. This practice ignores the fact that the individual d ′ itself is an estimator with an inherent variance. For observations with different levels of precision, the arithmetic mean is not the best estimator of a population parameter. It may lead to an estimate with a large variation. Another fact, which is often ignored, is that the variance of individual d ′ involves both between‐ and within‐subject variations in a random effects model when population sensitivity and its level of precision are estimated. Failing to account for both components of variance leads to an underestimate of variation and an overestimate of precision for the estimator. In this paper a lognormal distribution rather than a normal distribution is assumed for individual sensitivity. An iterative weighting procedure is proposed for estimating population sensitivity on the log scale on the basis of a random effects model. An ordinary weighting procedure is proposed for estimating group sensitivity on the log scale on the basis of a fixed effects model. The levels of precision of population and group sensitivity estimators are also given. Numerical examples illustrate the estimation procedures.

Raykov, Tenko; Hancock, Gregory R.

doi: 10.1348/000711005X38753pmid: 15969840

A method for examining change in maximal reliability for pre‐specified sets of congeneric measures when developing a multi‐component instrument is outlined. The approach is applicable for purposes of estimation and testing of gain or loss in the maximal reliability coefficient as a consequence of adding or dropping one or more measures from a homogeneous composite with uncorrelated errors, as well as when one is concerned with optimal component choice for highest increase or correspondingly smallest drop in maximal reliability. The method is compared with a procedure for ascertaining change in unweighted sum score reliability, and implications for instrument construction and revision are discussed. The approach is illustrated with a numerical example.

Chiang, Ching‐Yuan

doi: 10.1348/000711005X38717pmid: 15969841

This paper proposes an extension of the definition of one‐dimensional ordinal, interval, ratio, difference and absolute scales to the multidimensional case. The connection between measurement structures and statistical structures suggested in a previous paper is also extended to the multivariate case, and examples of invariance properties of multivariate functions and statistics are discussed.

Ven, Ad H. G. S.; Gremmen, Frans M.; Smit, Jan C.

doi: 10.1348/000711005X38708pmid: 15969842

A probabilistic model is presented for the phase durations in binocular rivalry experiments. The hypothetical construct of inhibition or reaction inhibition is used to account for the length of the successive phases of left‐eye dominance and right‐eye dominance. In accordance with Hull's Postulate X.B. it is assumed that the inhibition increases linearly at rate a1 during periods of left‐eye dominance and decreases linearly at rate a0 during periods of right‐eye dominance. Two different versions of the proposed model are presented: the beta and the Bessel inhibition models. Inhibition fluctuates between the boundaries 0 and 1 in the beta inhibition model and between −∞ and +∞ in the Bessel inhibition model. The transition rates λ1(t) for switches from a state of left‐eye dominance to a state of right‐eye dominance, and λ0(t) for switches from a state of right‐eye dominance to a state of left‐eye dominance depend on inhibition: , , where l1 is a non‐decreasing function and l0 is a non‐increasing function. In the beta inhibition model and . In the Bessel inhibition model and . Special attention is given to the derivation of the expectation of the stationary phase durations.

Meulders, Michel; Ip, Edward H.; Boeck, Paul

doi: 10.1348/000711005X38555pmid: 15969843

A general framework is presented for the analysis of partially ordered set (poset) data. The work is motivated by the need to analyse poset data such as multi‐componential responses in psychological measurement and partially accomplished cognitive tasks in educational measurement. It is shown how the generalized loglinear model can be used to represent poset data that form a lattice and how latent‐variable models can be constructed by further specifying the canonical parameters of the loglinear representation. The approach generalizes a class of latent‐variable models for completely ordered data. We apply the methods to analyse data on the frequency and intensity of anger‐related feelings. Furthermore, we propose a trajectory analysis to gain insight into the response function of partially ordered emotional states.

Fox, J. ‐P.

doi: 10.1348/000711005X38951pmid: 15969844

A structural multilevel model is presented where some of the variables cannot be observed directly but are measured using tests or questionnaires. Observed dichotomous or ordinal polytomous response data serve to measure the latent variables using an item response theory model. The latent variables can be defined at any level of the multilevel model. A Bayesian procedure Markov chain Monte Carlo (MCMC), to estimate all parameters simultaneously is presented. It is shown that certain model checks and model comparisons can be done using the MCMC output. The techniques are illustrated using a simulation study and an application involving students' achievements on a mathematics test and test results regarding management characteristics of teachers and principles.