Binary regression

Question 1

Logit model

Answer 1

η_i = log(π_i/(1-π_i)) = log(ODDS_i)
π_i = exp(η_i)/(1+exp(η_i)
- β linear effect on η_i but not on π_i;
- β>0 : the odds increase of exp(β) for the corrisponding variable (odds ratio);
- ODDS_i = exp(η_i);
- If I increase of 1 xij while other x stay fixed: OR_i = exp(βj);
- F(β) = H(β).

Question 2

Probit model

Answer 2

η_i = Φ^-1(π)
π_i = Φ(η_i)
- Usually same results as logit when rescaled by the considered σ;
- Gives different ^βs but same ^βj/^βh and same ^π;
- β>0 : the odds increase.

Question 3

C-log-log model

Answer 3

η_i = log(-log(1-π_i))
π_i = 1-exp(-exp(η_i))
- Usually different from logit and probit;
- Uses extreme minimum value distribution;
- Gives asymmetric response.

Question 4

Latent utility model

Answer 4

We model y~_i = u_i1 - u_i0 that if > 0 gives y_i=1.
y~_i = η_i + ε_i
- ε_i ~ N: probit model, indentifiable up to 1/σ;
- ε_i ~Logistic: logit model.

Question 5

Binary data overdispersion

Answer 5

Var(y-_i) = φ π_i(1-π_i)/n_i.
We now model y-_i ~Bin(n_i, π_i)/n_i

Question 6

Log-likelihood, score and Fisher matrix for logistic regression (logit)

Answer 6

l(β) = Σ y_iη_i - log(1+exp{η_i})
s(β) = Σ x_i‘(y_i- π_i)
F(β) = Σ x_ix_i‘π_i(1-π_i) = H(β)

Question 7

Pearson statistic for logistic regression

Answer 7

χ² = Σ (y-_i - π^_i)² / [π^_i(1-π^_i)/n_i] .~ χ²_G-p