GLM hypothesis testing

Question 1

General linear hypothesis on β

Answer 1

H₀: Cβ = d
C(rxp)
d(rx1)
r: number of hypotheses we are testing
Linear independence hypothesis: all the coefficients other than β₀ are equal to 0.

Question 2

Likelihood ratio test statistic for σ² known

Answer 2

lr = (SSE_H0 - SSE) / σ² = ΔSSE/σ² ~χ²_r
Unless β^_H0 = β^, SSE_H0 > SSE because we are introducing extra restrictions.

Question 3

Constrained MLE β^_H0 and SSE H0

Answer 3

β^_H0 = β^ - (X’X)^-1 C’[C(X’X)^-1C’]^-1(Cβ^-d)
SSE_H0 = SSE + (β^ - β^_H0)’ X’X (β^ - β^_H0)

Question 3

ΔSSE_H0

Answer 3

(Cβ^-d)’[C(X’X)^-1C’]^-1(Cβ^-d)
It is not necessary to compute β^_H0.

Question 4

Likelihood ratio test statistic for σ² unknown

Answer 4

F = (n-p)/r * ΔSSE/SSE ~ F_{r, n-p} = 1/r (Cβ^-d)’cov^(Cβ^)^-1(Cβ^-d) = 1/r W
W: Wald statistics, distance between Cβ^ and the hypotesised value d, weighted by cov^(Cβ^).
cov^(Cβ^) = σ^²C(X’X)^-1C’

Question 5

Coefficient of multiple linear determination

Answer 5

R² = 1 - SSE / Σ(yi-y^_)² ∈ [0,1]

Question 6

GLM: F statistic for H₀: β_j = 0

Answer 6

F = β^_j / √ var^(β_j) | H₀ ~ t_n-p

Question 7

F statistic for H₀: linear independence

Answer 7

F = R²/(1-R²) * (n-p)/(p-1)

Question 8

Hypothesis testing in R

Answer 8

library(car)
lht(fit, C, d=NULL)
- (n-p_min) Null VS (n-p_max) Residual