Gaussian linear mixed models (GLMMs)

Question 1

Two-stage sampling scheme

Answer 1

1: m groups
2: n_i units ∀ i=1,…m

Question 2

Two-level hierarchical units and regressors

Answer 2

1: n_i observations ∀ i=i,..m
2: m groups, regressors that are common for all observations in the same group

Question 3

IntraClass Correlation (ICC) for intercept-only GLMM

Answer 3

cor(Y_ij, Y_lh) = τ₀² / (τ₀² + σ²)

Question 4

Random intercept and slope GLMM model

Answer 4

Y_ij = β₀ + γ_0i + β₁ x_ij + γ_1ix_ij+ ε_ij = β_0i + β_1i x_ij + ε_ij
∀ j=1,…n_i, i=1,…m
- E[Y_ij] = β₀ + β₁ x_ij
- E[Y_ij|γ_0i] = β_0i + β_1i x_ij
- Var(Y_ij) = σ² + τ₀² + τ₁²x_ij² + 2τ₀₁x_ij
- Cov(Y_ij, Y_lh) = τ₀² + + τ₁²x_ijx_lh + τ₀₁(x_ij+x_lh) if i=l, 0 otherwise

Question 5

GLMM general definition

Answer 5

Y = Xβ + Uγ + ε
- X_nxp
- β_p=k+1
- ε_nx1 ∼ Nn(0, σ²In)
- U_nxm(q+1) = blockdiag(U1,…Um)
- γ_m(q+1)x1 ∼ Nm(q+1)(0, G=blockdiag(Q,…Q)) independent of ε
- E[Y] = Xβ
- E[Y|γ] = Xβ + Uγ
- cov(Y) = V = UGU’ + σ²In
- cov(Y|γ) = σ²In

Question 6

Marginal and conditional formulations of GLMMs

Answer 6

Y ∼ Nn(Xβ, V=UGU’+R)
Y|γ ∼ Nn(Xβ + Uγ, R) and γ ∼ N_q(0, G)

Question 7

GLMMs seen as GLMs

Answer 7

ε* = Uγ + ε ∼ Nn(0, V = UGU’+σ²In)
Y = Xβ = ε* ∼ Nn(Xβ, V)

Question 8

Extension of GLMMs to heteroschedasticity and error dependence

Answer 8

Y = Xβ + Uγ + ε
- ε ∼ Nn(0, R)
- R_nxn = blockdiag(R1,…Rm)
Heteroschedasticity: Ri = σ²Ini, stratification of variances
Error correlation: Ri for autocorrelation (ARMA)
To avoid overparametrization it is important to put emphasis either on R or UiQUi’ (one emphasised and the other simple since now V = UGU’ + R)

Question 9

GLMMs fitting in R

Answer 9

lme(fixed=y~lvl1+lvl2, random=~lvl1|group, method=”REML”)
- random = pDiag(~lvl1) defines a diagonal Q
- weights = varIdent(form=~1|strata) defines heteroschedasticity between strata

Question 9

GLMMs coefficients in R

Answer 9

fixef(fit) #Beta
ranef(fit) #Gamma
coef(fit) #Conditional formulation, beta_i + gamma_i

Question 10

Hypothesis testing on Betas in R

Answer 10

• H₀: Beta_i = 0
  summary(fit) #Check t
  anova(fit, type=”marginal”) #t² from summary
• H₀: CBeta = 0
  anova(fit, L=C) #Valid only for one level regressors at a time, not mixed!
• Nested models:
  anova(fitmin, fitmax) #Valid only if method=”ML” for both!

Question 11

Hypothesis testing on Gammas in R

Answer 11

• H₀: uncorrelated gammas
  anova(fit_pDiag, fit_Default)
• H₀: excluding gamma_i
  L = anova(fit_excluded, fit_not)$L.Ratio[2]
  p = 0.5*(1-pchisq(L, r-1))+0.5(1-pchisq(L, r))
• H₀: strata heteroschedasticity is as valid as group heteroschedasticity
  anova (fit_strata, fit_group)
• H₀: homoschedasticity is as valid as heteroschedasticity
  anova(fit_homo, fit_het)

Question 12

Number of parameters of matrix Q

Answer 12

• DProportional to identity matrix: 1
• Compound symmetry: 2
• Diagonal: q
• Identical covariance, different variance: q+1
• Unstructured: 1/2 q(q+1)
  Total number of variance parameters: parameters of Q + 1 (σ²)