Integrated likelihood functions for non-Bayesian inference

Consider a model with parameter = (, ), where is the parameterof interest, and let L(, ) denote the likelihood function. Oneapproach to likelihood inference for is to use an integratedlikelihood function, in which is eliminated from L(, ) by integratingwith respect to a density function (|). The goal of this paperis to consider the problem of selecting (|) so that the resultingintegrated likelihood function is useful for non-Bayesian likelihoodinference. The desirable properties of an integrated likelihoodfunction are analyzed and these suggest that (|) should be chosenby finding a nuisance parameter that is unrelated to and thentaking the prior density for to be independent of . Such anunrelated parameter is constructed and the resulting integratedlikelihood is shown to be closely related to the modified profilelikelihood.     

Conditional properties of unconditional parametric bootstrap procedures for inference in exponential families

Higher-order inference about a scalar parameter in the presenceof nuisance parameters can be achieved by bootstrapping, incircumstances where the parameter of interest is a componentof the canonical parameter in a full exponential family. Theoptimal test, which is approximated, is a conditional one basedon conditioning on the sufficient statistic for the nuisanceparameter. A bootstrap procedure that ignores the conditioningis shown to have desirable conditional properties in providingthird-order relative accuracy in approximation of p-values associatedwith the optimal test, in both continuous and discrete models.The bootstrap approach is equivalent to third-order analyticalapproaches, and is demonstrated in a number of examples to givevery accurate approximations even for very small sample sizes.     

Likelihood analysis for the ratio of means of two independent log-normal distributions

Existing methods for comparing the means of two independent skewed log-normal distributions do not perform well in a range of small-sample settings such as a small-sample bioavailability study. In this article, we propose two likelihood-based approaches-the signed log-likelihood ratio statistic and modified signed log-likelihood ratio statistic-for inference about the ratio of means of two independent log-normal distributions. More specifically, we focus on obtaining p-values for testing the equality of means and also constructing confidence intervals for the ratio of means. The performance of the proposed methods is assessed through simulation studies that show that the modified signed log-likelihood ratio statistic is nearly an exact approach even for very small samples. The methods are also applied to two real-life examples.     

Plausibility Inference for Generalized Poisson Distribution

The two parameter generalized Poisson distribution (GPD) has been usefully applied to many biological and genetic problems and important interpretations have been given to both the parameters; however, reliable interval estimates for them are not available. A number of results on the plausibility inference for the GPD have been provided. Likelihood intervals and likelihood regions are also considered for the two parameters of the GPD. Expressions needed to compute the intervals and regions are given. These results are then applied to two biological problems to derive the relevant regions and intervals for the parameters of interest.     

Higher order inference for the consensus mean in inter-laboratory studies

In inter-laboratory studies, a fundamental problem of interest is inference concerning the consensus mean, when the measurements are made by several laboratories which may exhibit different within-laboratory variances, apart from the between laboratory variability. A heteroscedastic one-way random model is very often used to model this scenario. Under such a model, a modified signed log-likelihood ratio procedure is developed for the interval estimation of the common mean. Furthermore, simulation results are presented to show the accuracy of the proposed confidence interval, especially for small samples. The results are illustrated using an example on the determination of selenium in non-fat milk powder by combining the results of four methods. Here, the sample size is small, and the confidence limits for the common mean obtained by different methods produce very different results. The confidence interval based on the modified signed log-likelihood ratio procedure appears to be quite satisfactory.     

Testing for group effect in a 2 x k heteroscedastic ANOVA model

It is natural to want to relax the assumption of homoscedasticity and Gaussian error in ANOVA models. For a two-way ANOVA model with 2 x k cells, one can derive tests of main effect for the factor with two levels (referred to as group) without assuming homoscedasticity or Gaussian error. Empirical likelihood can be used to derive testing procedures. An approximate empirical likelihood ratio test (AELRT) is derived for the test of group main effect. To approximate the distributions of the test statistics under the null hypothesis, simulation from the approximate empirical maximum likelihood estimate (AEMLE) restricted by the null hypothesis is used. The homoscedastic ANOVA F -test and a Box-type approximation to the distribution of the heteroscedastic ANOVA F -test are compared to the AELRT in level and power. The AELRT procedure is shown by simulation to have appropriate type I error control (although possibly conservative) when the distribution of the test statistics are approximated by simulation from the constrained AEMLE. The methodology is motivated and illustrated by an analysis of folate levels in the blood among two alcohol intake groups while accounting for gender.     

The use of score tests for inference on variance components

Whenever inference for variance components is required, the choice between one-sided and two-sided tests is crucial. This choice is usually driven by whether or not negative variance components are permitted. For two-sided tests, classical inferential procedures can be followed, based on likelihood ratios, score statistics, or Wald statistics. For one-sided tests, however, one-sided test statistics need to be developed, and their null distribution derived. While this has received considerable attention in the context of the likelihood ratio test, there appears to be much confusion about the related problem for the score test. The aim of this paper is to illustrate that classical (two-sided) score test statistics, frequently advocated in practice, cannot be used in this context, but that well-chosen one-sided counterparts could be used instead. The relation with likelihood ratio tests will be established, and all results are illustrated in an analysis of continuous longitudinal data using linear mixed models.     

Practical use of modified maximum likelihoods for stratified data

Stratified data arise in several settings, such as longitudinal studies or multicenter clinical trials. Between-strata heterogeneity is usually addressed by random effects models, but an alternative approach is given by fixed effects models, which treat the incidental nuisance parameters as fixed unknown quantities. This approach presents several advantages, like computational simplicity and robustness to confounding by strata. However, maximum likelihood estimates of the parameter of interest are typically affected by incidental parameter bias. A remedy to this is given by the elimination of stratum-specific parameters by exact or approximate conditioning. The latter solution is afforded by the modified profile likelihood, which is the method applied in this paper. The aim is to demonstrate how the theory of modified profile likelihoods provides convenient solutions to various inferential problems in this setting. Specific procedures are available for different kinds of response variables, and they are useful both for inferential purposes and as a diagnostic method for validating random effects models. Some examples with real data illustrate these points.     

The Residual Likelihood Ratio Test for the Variance Ratio in a Linear Model with Two Variance Components

Residual maximum likelihood has proved to be a successful approach to the estimation of variance components. In this paper, its counterpart in testing, the residual likelihood ratio test, is applied to testing the ratio of two variance components. The test is compared with the Wald test and the locally most powerful invariant test.     

A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures

The problem of testing the separability of a covariance matrix against an unstructured variance‐covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first‐order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ² distribution. The tests are implemented on a real dataset from medical studies.