Maximum likelihood estimation for GLMMs

Question 1

How to estimate through MLE unknown quantities in GLMMs

Answer 1

• β: marginal or conditional formulation
• γ: conditional formulation
• θ: marginal formulation

Question 2

MLE for GLMMs with θ known

Answer 2

β^= (X’V^-1X)^-1X’V^-1y ∼ Np(β, (X’V^-1X)^-1)
- BLUE, equivalent to generalised/weighted least squares that gives more weight to less variable observations
γ = GU’V^-1(y-Xβ^)
- BLUP

Question 3

MLE for GLMMs with θ unknown

Answer 3

• Profile likelihood: biased (asymptotically unbiased) estimator
  lP(θ) = - 1/2 ln|V(θ)| - 1/2 (y-Xβ^(θ))’V(θ)^-1(y-Xβ^(θ))
• Restricted likelihood: unbiased estimator
  lR(θ) = ln(∫ L(β,θ)dβ) = lP(θ) - 1/2 ln|X’V(θ)X|