The likelihood ratio test for the two-component normal mixture problem: power and sample size analysis

We find, through simulation and modeling, an approximation to the alternative distribution of the likelihood ratio test for two-component mixtures in which the components have different means but equal variances. We consider the range of mixing proportions from 0.5 through .95. Our simulation results indicate a dependence of power on the mixing proportion when pi less than .2 and pi greater than .80. Our model results indicated that the alternative distribution is approximately noncentral chi-square, possibly with 2 degrees of freedom. Using this model, we estimate a sample of 40 is needed to have 50% power to detect a difference between means equal to 3.0 for mixing proportions between .2 and .8. The sample size increases to 50 when the mixing proportion is .90 (or .1) and 82 when the mixing proportion is .95 (or .05). This paper contains a complete table of sample sizes needed for 50%, 80%, and 90% power.