Assessing statistical significance in variance components linkage analysis: A theoretical justification

• Proposed a threshold calculation method for testing variance components in linkage analysis.
• Studied the asymptotic distribution of the likelihood ratio test and derived a practical method to calculate the threshold.
• The new threshold calculation method yields less conservative results compared to its counterparts.
• The method applies to univariate and multivariate trait analysis.

Variance components analysis has been a standard means in family-based genetic data analysis. The variance components technique treats genetic effects as random, and tests whether variance components are zero using the likelihood ratio (LR) test. In the literature, the asymptotic distribution of the LR is claimed to follow a mixture chi-square distribution, where the mixture proportions are calculated based on the binomial coefficients, a special case in Self and Liang (1987). This threshold calculation, however, often yields conservative test results as discussed in a number of studies, especially in multi-trait analyses. In this work, we show that the LR statistic asymptotically follows a mixture chi-square distribution where the mixture proportions depend on the estimated Fisher information matrix in both univariate and multivariate trait analyses. We provide a general approximation form for the distribution of the LR under the null hypothesis of no genetic effects. We illustrate our idea with three variance components models in genetic linkage analysis. The performance of the new threshold calculation method is demonstrated via simulation studies, and its application is further illustrated via a real data analysis.