Gaussian linear models (GLMs)

Question 1

Assumptions of GLMs

Answer 1

• Linearity: E[Y_i|x_1i, … x_ki] = μ_i = β₀ + β₁x_1i + β_kx_ki ∀ i
• Homoschedasticity: Var(Y_i|x_1i, … x_ki) = σ² ∀ i
• Conditional (linear) uncorrelation: cov(Y_i|x_1i, … x_ki, Y_h|x_1h, … x_kh = 0 ∀ i ≠ h
• Normality: Y_i|x_1i, … x_ki ~N(μ_i, σ² which, together with uncorrelation, makes for the conditional independence assumption.

Question 2

GLM definition

Answer 2

Y | X ~ N_n (Xβ, σ²In)
Or, alternatively: Yi = β₀ = β₁x_1i + … β_kx_ki + ε_i, with ε_i|x_1i, … x_ki ~ N_n(0, σ²)

Question 3

Fitting GLMs in R

Answer 3

lm(y~x1+x2) #Based on OLS
glm(y~x1+x2, family=gaussian)
- Std. Error = [σ²^ (X’X)^-1]^1/2_jj
- Residual standard error = sqrt(σ²^) not MLE
- glm shows Null deviance (model only with β₀, n-1) and Residual deviance (n-p)

Question 4

Linearity assumption checking in R

Answer 4

plot(fit, which=1)
- Residuals VS Fitted
- Red line (local average) straight on 0: no systematic difference between fitted values and observed ones if we focus on units sharing about the same covariance pattern

Question 5

Normality assumption checking in R

Answer 5

plot(fit, which=2)
- QQ-Plot
- Points on the bisetrix: same distribution between theoretical and empirical quantiles of the standardized residuals

Question 6

Homoschedasticity assumption checking in R

Answer 6

plot(fit, which=3)
- Scale-Location plot
- Red line (local average) straight on 1: the variability pattern does not change