Count data regression

Question 1

Linear Poisson model

Answer 1

η_i = λ_i = x_i‘β
- Symmetrical distribution;
- Expected large λ_i;
- Additive effects of covariates.

Question 2

Log-linear Poisson model

Answer 2

η_i = log(λ_i)
λ_i = exp(η_i) = exp(x_i‘β)
- Effect of β is linear on log-scale
- Effect of β on the original scale is exp(β)

Question 3

Handling overdispersion in count data regression

Answer 3

• Introduce φ as binary regression (with same issues): Var(y-_i) = φλ_i;
• Use Negative-Binomial distribution (more difficult interpretation);
• Use a random effects model.

Question 4

ML estimation for count data regression (loglinear)

Answer 4

l(β) = Σ y_iη_i - λ_i
s(β) = Σ x_i(y_i-λ_i)
F(β) = Σ x_ix_i‘λ_i
If offset: F(β) = Σ x_ix_i‘λ_iΔ_i

Question 5

Pearson statistics for count data regression

Answer 5

χ² = Σ (y_i-^λ_i)² / [ ^λ_i/n_i]
Remember that:
- E(y_i) = λ_i = var(y_i)
- 0log(0) = 0 in the case of y_i=0 for AIC

Question 6

Offset

Answer 6

Since λi is affected by the time frame/area, we rescale the phenomenon in the log-scale by a offset γi
- No coefficient estimated;
- η_i = x_iβ + γ_i
- γ_i = log(Δ_i)
- γi = log(ni) for grouped data;
- Gives less variability for ^β.