Hypothesis testing for GLMMs

Question 1

Testing for fixed effects with θ known

Answer 1

Wald test statistic:
W = (Cβ^-d)’[C(X’V^-1X)^-1C’]^-1(Cβ^-d)
W|H0 ∼ χ²r

Question 2

Testing for fixed effects with θ unknown

Answer 2

• Wald test statistic:
  W = (Cβ^-d)’[C(X’V^-1X)^-1C’]^-1(Cβ^-d)
  W|H0 -d-> χ²r
• F = 1/r W |H0 -d-> Fr,g
• g = n - m - (#level 1 parameters)
• g = m - 1 - (#level 2 parameters)
• Likelihood ratio test statistics, profile ok for nested models but restricted not ok since lacks invariance

Question 3

Testing for random effects

Answer 3

Means checking if their variance is 0 (since their mean is 0 by definition), with a number of constraints that changes whether Q is diagonal or not. Since we are testing close to a boung, Wilks theorem is violated and we need to correct the p-value under H0:
P(- ≥ -_obs|H0) ≈ [P(- ≥ -_obs|χ²_r-1)+P(- ≥ -_obs|χ²_r)]/2
Or, if H0 is the exclusion of 1 random effect and Q is diagonal:
P(- ≥ -_obs|H0) ≈ [P(- ≥ -_obs|χ²₁)]/2
r = #variances put to 0 + #covariances becoming 0

Question 4

H0: β_j = 0 in GLMM

Answer 4

W = F = {t_n-1}²
β^_j / √(X’V^^-1X)^-1_jj | H0 -d-> t_g ≈ N(0,1)