Bayesian Estimation and Model Choice in Item Response Models

Item response models are essential tools for analyzing results from many educational and psychological tests. Such models are used to quantify the probability of correct response as a function of unobserved examinee ability and other parameters explaining the difficulty and the discriminatory power of the questions in the test. Some of these models also incorporate a threshold parameter for the probability of the correct response to account for the effect of guessing the correct answer in multiple choice type tests. In this article we consider fitting of such models using the Gibbs sampler. A data augmentation method to analyze a normal-ogive model incorporating a threshold guessing parameter is introduced and compared with a Metropolis-Hastings sampling method. The proposed method is an order of magnitude more efficient than the existing method. Another objective of this paper is to develop Bayesian model choice techniques for model discrimination. A predictive approach based on a variant of the Bayes factor is used and compared with another decision theoretic method which minimizes an expected loss function on the predictive space. A classical model choice technique based on a modified likelihood ratio test statistic is shown as one component of the second criterion. As a consequence the Bayesian methods proposed in this paper are contrasted with the classical approach based on the likelihood ratio test. Several examples are given to illustrate the methods.     

Some robustness issues for comparing multiple logistic regression slopes to a control for small samples

For comparing several logistic regression slopes to that of a control for small sample sizes, Dasgupta et al. (2001) proposed an "asymptotic" small-sample test and a "pivoted" version of that test statistic. Their results show both methods perform well in terms of Type I error control and marginal power when the response is related to the explanatory variable via a logistic regression model. This study finds, via Monte Carlo simulations, that when the underlying relationship is probit, complementary log-log, linear, or even non-monotonic, the "asymptotic" and the "pivoted" small-sample methods perform fairly well in terms of Type I error control and marginal power. Unlike their large sample competitors, they are generally robust to departures from the logistic regression model.     

Assessing the performance of variational methods for mixed logistic regression models

We present a variational estimation method for the mixed logistic regression model. The method is based on a lower bound approximation of the logistic function [Jaakkola, J.S. and Jordan, M.I., 2000, Bayesian parameter estimation via variational methods. Statistics & Computing, 10, 25–37.]. Based on the approximation, an EM algorithm can be derived that results in a considerable simplification of the maximization problem in that it does not require the numerical evaluation of integrals over the random effects. We assess the performance of the variational method for the mixed logistic regression model in a simulation study and an empirical data example, and compare it to Laplace's method. The results indicate that the variational method is a viable choice for estimating the fixed effects of the mixed logistic regression model under the condition that the number of outcomes within each cluster is sufficiently high.     

Asymptotic approximations to the distribution of Kendall's sample τ

This paper provides a saddlepoint approximation to the distribution of the sample version of Kendall's τ, which is a measure of association between two samples. The saddlepoint approximation is compared with the Edgeworth and the normal approximations, and with the bootstrap resampling distribution. A numerical study shows that with small sample sizes the saddlepoint approximation outperforms both the normal and the Edgeworth approximations. This paper gives also an analytical comparison between approximated and exact cumulants of the sample Kendall's τ when the two samples are independent.     

Simulating random variables using moment-generating functions and the saddlepoint approximation

When we are given only a transform such as the moment-generating function of a distribution, it is rare that we can efficiently simulate random variables. Possible approaches such as the inverse transform using numerical inversion of the transform are computationally very expensive. However, the saddlepoint approximation is known to be exact for the Normal, Gamma, and inverse Gaussian distribution and remarkably accurate for a large number of others. We explore the efficient use of the saddlepoint approximation for simulating distributions and provide three examples of the accuracy of these simulations.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

Modified Wald statistics for generalized linear models

Summary: Wald statistics in generalized linear models are asymptotically ² distributed. The asymptotic chi–squared law of the corresponding quadratic form shows disadvantages with respect to the approximation of the finite–sample distribution. It is shown by means of a comprehensive simulation study that improvements can be achieved by applying simple finite–sample size approximations to the distribution of the quadratic form in generalized linear models. These approximations are based on a ² distribution with an estimated degree of freedom that generalizes an approach by Patnaik and Pearson. Simulation studies confirm that nominal level is maintained with higher accuracy compared to the Wald statistics.     

Evaluation of trace evidence in the form of multivariate data

Summary.  The evaluation of measurements on characteristics of trace evidence found at a crime scene and on a suspect is an important part of forensic science. Five methods of assessment for the value of the evidence for multivariate data are described. Two are based on significance tests and three on the evaluation of likelihood ratios. The likelihood ratio which compares the probability of the measurements on the evidence assuming a common source for the crime scene and suspect evidence with the probability of the measurements on the evidence assuming different sources for the crime scene and suspect evidence is a well-documented measure of the value of the evidence. One of the likelihood ratio approaches transforms the data to a univariate projection based on the first principal component. The other two versions of the likelihood ratio for multivariate data account for correlation among the variables and for two levels of variation: that between sources and that within sources. One version assumes that between-source variability is modelled by a multivariate normal distribution; the other version models the variability with a multivariate kernel density estimate. Results are compared from the analysis of measurements on the elemental composition of glass.