Likelihood Ratio Test Sample Clauses

Likelihood Ratio Test. ‌ Tests on parameter estimates are performed as usual, with a likelihood ratio test (LRT) comparing the fitted model (alternative model) with the constraint H1 : A ≥ 0 to the null model with the hypothesis H0 : A = 0 (i.e., the model ACE is compared to CE, and AE is compared to E). The LRT statistic, termed as “T”, is defined as T = −2 × [A(ρ0|Y) − A(ρ1|Y)], where ρ0 and ρ1 are parameter estimates derived from the null model and the  alternative model respectively, and proven to asymptotically follow a chi-squared distribution with 1 degree of freedom, i.e., χ2 (Xxxxx, 1938). However the variance parameter A lies on the boundary of the parameter space of ρ = (A, C, E)T under the null hypothesis H0 : h2 = 0 or, equivalently, H0 : A = 0, and thus the asymp- totic sampling distribution of this LRT statistic, assuming H0 is true, is a mixture of 1 chi-squared distributions . 1 χ2 + 1 χ2Σ instead of a standard chi-square distribution (χ2) (Xxxx and Xxxxx, 1987; Xxxxxxxxx et al., 2006; Xxxxx and Xxx, 2008). Given the asymptotic null distribution of the LRT statistic, the theoretical p-value can be easily calculated. Obtaining a p-value less than a given significance level α, which is typically a small number (e.g., α = 0.05), suggests that there is a significant evidence against the null hypothesis and the null hypothesis should be rejected at level α. The p-value, denoted as p, also indicates there is a 100p% risk of incorrectly rejecting a true null hypothesis. Aside from the asymptotic theoretical p-value, the permutation-based p-value is exact, based on the empirical distribution of the LRT statistic.