A bartlett-type adjustment for the likelihood ratio statistic with an ordered alternative

For testing the equality of normal variances with an increasing alternative, under the null hypothesis the likelihood ratio test statistic is asymptotically distributed as a mixture of chi-squared distributions. In this paper a Bartlett-type adjustment is proposed to improve the approximation of the null distribution of the likelihood ratio test statistic with an ordered alternative.     

Testing homogeneity with an ordered alternative in a two-factor layout by combiningp-values

Two-factor experiments in which both factors are ordinal are considered. If it is believedapriori that the mean response is nondecreasing in each factor with the other held fixed, then one may test for a treatment effect by testing homogeneity with the appropriate ordered alternative. The likelihood ratio test has been developed in the literature, but the level probabilities needed to implement the test have only been determined in a few special cases by Monte Carlo techniques. A test obtained by combining thep-values from a test concerning the rows and a test concerning the columns is studied. Fisher's method of combiningp-values is recommended. It is shown that the likelihood ratio test is more powerful, but if one does not want to obtain Monte Carlo estimates of the level probabilities, then the procedure proposed here should be considered.This research was supported by the National Institutes of Health under Grant 1R01GM42584-01. Part of this work is taken from the first author's dissertation submitted in partial fulfillment for the Ph.D. degree at the University of Missouri-Columbia.     

A likelihood ratio test for separability of covariances

We propose a formal test of separability of covariance models based on a likelihood ratio statistic. The test is developed in the context of multivariate repeated measures (for example, several variables measured at multiple times on many subjects), but can also apply to a replicated spatio-temporal process and to problems in meteorology, where horizontal and vertical covariances are often assumed to be separable. Separable models are a common way to model spatio-temporal covariances because of the computational benefits resulting from the joint space-time covariance being factored into the product of a covariance function that depends only on space and a covariance function that depends only on time. We show that when the null hypothesis of separability holds, the distribution of the test statistic does not depend on the type of separable model. Thus, it is possible to develop reference distributions of the test statistic under the null hypothesis. These distributions are used to evaluate the power of the test for certain nonseparable models. The test does not require second-order stationarity, isotropy, or specification of a covariance model. We apply the test to a multivariate repeated measures problem.     

The likelihood ratio test for homogeneity in bivariate normal mixtures

This paper investigates the asymptotic properties of the likelihood ratio statistic for testing homogeneity in a bivariate normal mixture model with known covariance. The asymptotic null distributions of the likelihood ratio statistic and a modified likelihood ratio statistic are obtained in explicit form. The distributions are identical. The results of a small simulation study to approximate the null distribution are presented.     

The power of the likelihood ratio test for additional information in a multivariate linear model

Summary This paper deals with the likelihood ratio test for additional information in a multivariate linear model. It is shown that the power of the likelihood ratio test procedure has a monotonicity property. Asymptotic approximations for the power are also obtained.     

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.     

Repeated likelihood ratio test for the variance of normal distribution with unknown mean

This paper considers the repeated likelihood ratio test for the variance of normal distribution with un-known mean. The large deviations for Type I, and Type II error probabilities have been developed.     

Modified likelihood ratio test for homogeneity in normal mixtures with two samples

This paper investigates the modified likelihood ratio test（LRT） for homogeneity in normal mixtures of two samples with mixing proportions unknown. It is proved that the limit distribution of the modified likelihood ratio test is X^2（1）.     

On monotonicity of the modified likelihood ratio test for the equality of two covariances

For testing the hypothesis of equality of two covariances (Σ₁ and Σ₂) of two p-dimensional multivariate normal populations, it is shown that the power function of the modified likelihood ratio test increases as λ₁ increases from one and λ_r decreases from one where λ₁ > … > λ_r > 0 are the distinct characteristic roots of Σ₁Σ₂^?1, r ≤ p. As a by-product we get the unbiased result already established by Sugiura and Nagao (1968).     

On the ratio integral formula for the likelihood ratio of a maximal invariant

The usual ratio of an integral formula for the likelihood ratio of a maximal invariant in a group model is shown to be correct under assumption that the denominator integral is finite almost everywhere. The limitation of this assumption is discussed, and an application to invariant suffciency is given.     

Modified likelihood ratio test for homogeneity in bivariate normal mixtures with presence of a structural parameter

This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ22 null limiting distribution.     

Power analysis for the bootstrap likelihood ratio test for the number of classes in latent class models

Latent class (LC) analysis is used to construct empirical evidence on the existence of latent subgroups based on the associations among a set of observed discrete variables. One of the tests used to infer about the number of underlying subgroups is the bootstrap likelihood ratio test (BLRT). Although power analysis is rarely conducted for this test, it is important to identify, clarify, and specify the design issues that influence the statistical inference on the number of latent classes based on the BLRT. This paper proposes a computationally efficient ‘short-cut’ method to evaluate the power of the BLRT, as well as presents a procedure to determine a required sample size to attain a specific power level. Results of our numerical study showed that this short-cut method yields reliable estimates of the power of the BLRT. The numerical study also showed that the sample size required to achieve a specified power level depends on various factors of which the class separation plays a dominant role. In some situations, a sample size of 200 may be enough, while in others 2000 or more subjects are required to achieve the required power.     

Near-exact distributions for the likelihood ratio test statistic to test equality of several variance-covariance matrices in elliptically contoured distributions

The exact distribution of the likelihood ratio test statistic to test the equality of several variance-covariance matrices has a non-manageable form. On the other hand, the existing asymptotic approximations do not exhibit the necessary precision for many applications. For these reasons, the development of near-exact approximations to the distribution of this statistic, arising from a different method of approximating distributions, emerges as a desirable goal. These distributions, while being manageable are much closer to the exact distribution than the usual asymptotic distributions and opposite to these, are also asymptotic for increasing number of variables and matrices involved. Computational modules to implement the near-exact distributions are made available on a web-site.     

Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

Annals of the Institute of Statistical Mathematics - Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing...     

Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses

The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.     

An approach to defining the pattern of interaction effects in a two-way layout

Summary A method is given to classify rows and columns into subgroups so that additivity holds within each of the subtables made of the grouped rows or the grouped columns. The least squares estimators of the cell means are easily obtained for the resulting linear model together with their variances. An estimator of the error varianceσ ₂ is given when there is only one observation per cell. A treatment of an ordered table is also given.     

Restricted one way analysis of variance using the empirical likelihood ratio test

Empirical likelihood (EL) ratio tests are developed for testing for or against the hypothesis that k-population means μ₁,μ₂,…,μ_k are isotonic with respect to some quasi-order ? on {1,2,…,k}. The null asymptotic distributions are derived and are shown to be of chi-bar squared type. The asymptotic power of the proposed test for testing for equality of these means against the order restriction is derived under contiguous alternatives and a simulation study is carried out to investigate the finite sample behaviors of this test. In addition, an adjusted EL test is used to improve the small size performance of our test and an example is also discussed to illustrate the theoretical results.     

An extension of the conditional likelihood ratio test to the general multiparameter case

In a set-up, where both the interest parameter and the nuisance parameter are possibly multi-dimensional and global parametric orthogonality may not hold, we suggest a test that is superior to the usual likelihood ratio test with regard to second-order local maximinity. The test can be motivated from the principles of conditional and adjusted likelihood.