Composite Likelihood: Multiple Comparisons and Non-Standard Conditions in Hypothesis Testing
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Computational intensity in using full likelihood estimation of multivariate and correlated data is a valid motivation to employ composite likelihood as an alternative that eases the process by using marginal or conditional densities and reducing the dimension. We study the problem of multiple hypothesis testing for multidimensional clustered data. The problem of multiple comparisons is common in many applications. We propose to construct multiple comparisons procedures based on composite likelihood statistics. The simultaneous multivariate normal quantile is chosen as the threshold that controls the multiplicity. We focus on data arising in four cases: multivariate Gaussian, probit, quadratic exponential models and gamma. To assess the quality of our proposed methods, we assess their empirical performance via Monte Carlo simulations. It is shown that composite likelihood-based procedures maintain good control of the familywise type I error rate in the presence of intra-cluster correlation, whereas ignoring the correlation leads to invalid performance. Using data arising from a depression study and also kidney study, we show how our composite likelihood approach makes an otherwise intractable analysis possible. Moreover, we study distribution of composite likelihood ratio test when the true parameter is not an interior point of the parameter space. We approached the problem looking at the geometry of the parameter space and approximating it at the true parameter by a cone under Chernoff's regularity. First, we established the asymptotic properties of the test statistic for testing continuous differentiable linear and non-linear combinations of parameters and then we provide algorithms to compute the distribution of both full and composite likelihood ratio tests for different cases and dimensions. The proposed approach is evaluated by running simulations.