Generalized linear mixed models + generalized additive models

Question 1

General definition for generalized linear mixed models

Answer 1

μ = h(η) = h(Xβ + Uγ), γ ~^IID N(0, G)
- y_ij indep. y_kh | γ;
- Penalized likelihood approach maximized with IWLS, approximated if G unknown;
- ^cov(β γ) (approx) F^-1;
- Marginal formulation not analytically available (^β_cond ≠ ~β_marg) unless for log-linear random intercept Poisson and probit-normal random intercept model.

Question 2

Generalized linear mixed models: binary data

Answer 2

η_ijj=log(P(y_ij=1|γ_i)/P(y_ij=0|γ_i)
η_ij=Φ^-1(P(y_ij=1|γ_i)
- n_i > 1 for identifiability.

Question 3

Generalized linear mixed models: count data

Answer 3

η_ij = log(λ_ij)
- If only intercept, λ_ij = exp(γ₀₁)exp(x_ij‘β) = ν_iexp(x_ij‘β);
- ν_i ~ LogN that can accomodate overdispersion;
- Has analytical marginal mean and var-cov.

Question 4

Generalized linear mixed models: categorical data

Answer 4

P(y_ij=r|γ_r) = exp(η_ijr) / [1+Σ exp(η_ijs)]
γ_ir ~^IID N(0,Q_r)

Question 5

Additive Model

Answer 5

y_i = f₁(z_i1) + … f_q(z_iq) + η_{i lin} + εi = η_{i add} + ε_i = Z₁γ₁ + … Z_qγ_q + Xβ + ε
- Semiparametric unless η_{i lin} removed;
- No interactions;
- Z continuous;
- Smooth functions f;
- Σ f₁(z_i1) = Σ f_q(z_iq) = 0;
- f_j = Z_jγ_j (Z linear basis expansion) estimated with penalties λ_jγ_j‘K_jγ_j;
- If generalized: E(y_i)= h(η_{i add}).

Question 6

STAR models

Answer 6

y = V₁γ₁ + … V_qγ_q + Xβ + ε = η_struct + ε
- Additive model with random effects/interactions/spatial effects modelled as non parametric part;
- Penalized least squares with tuning/smoothing parameter λ_j estimated on the data;
- Can be generalized using appropriate response function.