Higher Order Asymptotics: Applications to Mixed Models and Bioassay
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DepartmentMathematics and Statistics
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Modified Signed Log Likelihood Ratio Test
Signed Log Likelihood Ratio Test
Likelihood based tests and confidence intervals typically require large sample sizes for their validity. For small sample problems, higher order asymptotics (or small sample asymptotics) appear to be an attractive option to handle parametric inference problems. In the thesis, we will discuss applications of a class of small sample asymptotic procedures, namely, modified signed log-likelihood ratio test (MSLRT) procedures, for the following problems: (i) interval estimation of the consensus mean in inter-laboratory studies, (ii) construction of tolerance intervals in general mixed and random effects models and (iii) inference problems in the combination of multivariate bioassays. 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, an MSLRT 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. The computation of tolerance intervals in mixed and random effects models has not been addressed at all in a general setting, when the data are unbalanced. We derive satisfactory one-sided and two-sided tolerance intervals in such a general scenario, by applying small sample asymptotic procedures. In the case of one-sided tolerance limits, the problem reduces to the interval estimation of a percentile, and accurate confidence intervals are derived using the MSLRT procedure. In the case of two-sided tolerance intervals, the problem does not reduce to an interval estimation problem; however, it is possible to derive an approximate margin of error statistic that is an upper confidence limit for a linear combination of the variance components. For the latter problem, the MSLRT procedures can once again be used in order to arrive at an accurate upper confidence limit. Here, balanced and unbalanced data situations are treated separately, and computational issues are addressed in detail. Bioassays are frequently carried out for the purpose of estimating the relative potency of a drug or material (i.e., test treatment) by comparing its effects with that of a standard treatment on a culture of living cells or a test organism. Data on the relative potencies can sometimes be obtained from several independent experiments, performed at different laboratories or locations. When this is the case, statistical inference problems of interest include testing the homogeneity of the relative potencies, and, if accepted, the interval estimation of the common relative potency. The available literature on the problem assumes a common covariance matrix for the data from the different laboratories or locations. Here we will address the problems in the setup of a MANOVA model for the multivariate bioassay data from the different studies, allowing for different covariance matrices, once again applying the MSLRT procedure. The accuracy of the proposed solutions is assessed based on simulations. Extensive numerical results show that the procedures derived based on higher order asymptotics exhibit satisfactory performance in all three problems regardless of the sample size. The results are illustrated using several examples. The overall conclusion is that the application of higher order asymptotics will result in accurate inference procedures for mixed models and for bioassay problems.