Multivariate Likelihood Ratio Test Research Articles

Multivariate Hypothesis Testing Methods for Evaluating Significant Individual Change.

The measurement of individual change has been an important topic in both education and psychology. For instance, teachers are interested in whether students have significantly improved (e.g., learned) from instruction, and counselors are interested in whether particular behaviors have been significantly changed after certain interventions. Although classical test methods have been unable to adequately resolve the problems in measuring change, recent approaches for measuring change have begun to use item response theory (IRT). However, all prior methods mainly focus on testing whether growth is significant at the group level. The present research targets a key research question: Is the "change" in latent trait estimates for each individual significant across occasions? Many researchers have addressed this research question assuming that the latent trait is unidimensional. This research generalizes their earlier work and proposes four hypothesis testing methods to evaluate individual change on multiple latent traits: a multivariate Z-test, a multivariate likelihood ratio test, a multivariate score test, and a Kullback-Leibler test. Simulation results show that these tests hold promise of detecting individual change with low Type I error and high power. A real-data example from an educational assessment illustrates the application of the proposed methods.

A nonparametric version of Wilks’ lambda—Asymptotic results and small sample approximations

We propose a nonparametric version of Wilks’ lambda (the multivariate likelihood ratio test) and investigate its asymptotic properties under the two different scenarios of either large sample size or large number of samples. For unbalanced samples, a weighted and an unweighted variant are introduced. The unweighted variant of the proposed test appears to be novel also in the normal-theory context. The theoretical results are supplemented by a simulation study with parameter settings that are motivated by clinical and agricultural data, considering in particular the performance for small sample sizes, small number of samples, and varying dimensions. Inference methods based on the asymptotic sampling distribution and a small sample approximation are compared to permutation tests and to other parametric and nonparametric procedures. Application of the proposed method is illustrated by examples.

On the empirical size of Nielsen's multivariate likelihood ratio test of fractional integration

Nielsen (2004) provides multivariate maximum likelihood procedures to test whether several series are fractionally integrated. This note examines the finite sample size of the likelihood ratio tests applied to a bivariate system experiencing breaks in means. The results suggest that tests of a common unit root are somewhat undersized in small samples.

A dose finding procedure based on the multivariate likelihood ratio test

This paper presents a step-down procedure based on the multivariate likelihood ratio test to identify the minimal effective dose in dose-response studies. The admissibility of this procedure and some competing procedures are discussed. Simulation results show that this procedure is preferable to the other procedures in the literature.

Testing for competitive submarkets

We extend some of the methods Urban, Johnson and Hauser (1984) have developed to the problem of testing for the existence of competitive submarkets in a market. A submarket or a group of products is said to be competitive if consumers are statistically more likely to purchase again in that submarket than would be predicted using an aggregate constant ratio criterion. Unlike the ujh procedure, our method (1) identifies the specific submarkets that are competitive, (2) uses an appropriate one-sided multivariate likelihood ratio test to account for the correlation between the submarket statistics and to identify competitive submarkets over time, and (3) uses purchase probabilities estimated from behavioral data. Our application to the coffee market shows that the proposed method has substantial validity and confirms the presence of competitive submarkets.

Measures of Dependence for Cross-Lagged Panel Models

Panel studies are statistical studies in which two or more responses are measured for two or more individuals, or other sampling units, at two or more waves, or points in time. Typically the number of responses and the number of waves are small but the number of subjects is large. Cross-lagged panel studies are those in which the responses are continuous and divide naturally into two sets. One focus in such studies is to estimate and test the temporal effects with emphasis given to the effects of each set on the other. One approach to making such inferences involves casting the model as a set of regression equations and then estimating and testing the parameters by some variant of classical regression methods. Within this regression approach several authors have considered the problem of summarizing the degree of dependence by adapting the principle of “reduction in error.” We consider the same problem and develop measures from the multivariate likelihood ratio test statistics. We interpret these measures by relating them, for a stationary process, to the “feedback” measures developed for vector autoregressive processes. Finally we display the use of these measures on two sets of panel data.

Power of the likelihood-ratio test of the general linear hypothesis in multivariate analysis

1-1. General background. In a basic paper on multivariate tests of linear hypotheses, Wilks (1932) applied the method of the likelihood ratio to the problem of testing the equality of mean vectors in k multivariate populations. The general approach of this early paper has been extended and expanded in the literature until, now, a procedure is available for testing multivariate linear hypotheses in a completely general form (see, for example, Anderson, 1958). For the problem of computing the power of these tests, however, no such general procedure exists. This is due to the fact that the noncentral distribution of the test criterion has not been expressed in a numerically feasible form. A first step towards the derivation of the non-central distribution, however, was taken when Anderson & Girshick (1944) attacked the problem of finding the distribution of the Wishart matrix in the non-central case. The result led Anderson (1946) to the derivation of the moments of the criterion for testing the hypothesis of the general multivariate regression problem (or rather the equivalent Wilks-Lawley hypothesis). Anderson's results were valid for a certain class of alternatives, which he called linear and planar. The moments for the linear alternative involve an infinite sum of expressions containing gamma functions and for the planar alternative, involve a triple infinite sum of the same type of expressions. This appears to be the extent of the progress on the problem of obtaining the power of the multivariate likelihood-ratio test of the general linear hypothesis.t