On the likelihood ratio tests in bivariate ACDE models

Abstract

The ACE and ADE models have been heavily exploited in twin studies to identify the genetic and environmental components in phenotypes. However, the validity of the likelihood ratio test (LRT) of the existence of a variance component, a key step in the use of such models, has been doubted because the true values of the parameters lie on the boundary of the parameter space of the alternative model for such tests, violating a regularity condition required for a LRT (e.g., Carey in Behav. Genet. 35:653-665, 2005; Visscher in Twin Res. Hum. Genet. 9:490-495, 2006). Dominicus, Skrondal, Gjessing, Pedersen, and Palmgren (Behav. Genet. 36:331-340, 2006) solve the problem of testing univariate components in ACDE models. Our current work as presented in this paper resolves the issue of LRTs in bivariate ACDE models by exploiting the theoretical frameworks of inequality constrained LRTs based on cone approximations. Our derivation shows that the asymptotic sampling distribution of the test statistic for testing a single bivariate component in an ACE or ADE model is a mixture of χ (2) distributions of degrees of freedom (dfs) ranging from 0 to 3, and that for testing both the A and C (or D) components is one of dfs ranging from 0 to 6. These correct distributions are stochastically smaller than the χ (2) distributions in traditional LRTs and therefore LRTs based on these distributions are more powerful than those used naively. Formulas for calculating the weights are derived and the sampling distributions are confirmed by simulation studies. Several invariance properties for normal data (at most) missing by person are also proved. Potential generalizations of this work are also discussed.

Figure 1

An elliptic cone and its polar cone . Suppose the cone is approximated by a polyhedral cone with edges g₁ =OG₁, g₂ =OG₂, etc. If the discretization is fine, the normal vector of the face OG₁G₂ can be approximated by ON_1.5, which is a generatrix on the surface of with longitude between those of g₁ and g₂. The projections of points inside the pyramid O – N_1.5G₁G₂ to the polyhedral cone lie on the face OG₁G₂; the projections of points inside the pyramid O – G₂N_1.5N_2.5 to the polyhedral cone lie on the edge OG₂.

Figure 2

Plots of the 1st–99th percentiles of the simulated sampling distribution of T against those of the χ̄² distribution in Simulation Study 2. The sample sizes (n_MZ, n_DZ) are (from left to right) (100, 100), (150, 50), (500, 500) and (750, 250). The true and null model is CE and the alternative model is ACE.

Figure 3

Plots of the 1st–99th percentiles of the simulated sampling distribution of T against those of the χ̄² distribution in Simulation Study 2. The true model is E. The null and alternative models are (from left to right) AE vs. ACE, CE vs. ACE, E vs. AE and E vs. CE. The sample sizes are n_MZ = n_DZ = 500 for the left two panels and n_MZ = 150 and n_DZ = 50 for the right two panels.

Figure 4

Plots of the 1st–99th percentiles of the simulated sampling distribution of T against those of the χ̄² distribution in Simulation Study 2. The sample sizes (n_MZ, n_DZ) are (from left to right) (100, 100), (150, 50), (500, 500) and (750, 250). The true and null model is E and the alternative model is ACE.

Figure 5

Plot of 1st–99th percentiles of the simulated sample in Simulation Study 3 against those of a χ̄² distribution based on calculated weights.