The Bivariate Normal Mixture Distribution: A Power Study of Bootstrap Test

Abstract

Univariate analysis has been commonly used in the studies of disease-related phenotypes. The need for multivariate analysis on linkage studies of complex disease/traits has grown with the increasing use of multiple phenotypes. This research extends the model for testing a single bivariate normal distribution versus a two component bivariate normal mixture distribution. Previous research restricted the two variables to have equal means and variance. Our study considers the more general case with no restrictions on these parameter values. Simulations are used to conduct a power study of bootstrap test under different combinations of parameter values. We note that samples of sample size n = 200 or more and an average mixture effect size of 2.5 or more is needed with mixing proportions between 0.1 and 0.9 to achieve reasonable power. Regression models of LRT statistic values are also fitted to calculate the type I error rate and power. Finally the bootstrap method is shown to be a reliable approach for evaluating the LRT statistics.