Nonnegative continuous data regression

Question 1

Options for nonnegative continuous data

Answer 1

• Far from 0 with small variance: could use Gaussian;
• Lognormal model;
• Gamma regression.

Question 2

Lognormal model

Answer 2

We model y~ = log(y) = η_i ~ N(x_i‘β, σ²)
However we cannot go back, because then y_i ~ LogN(x_i‘β + σ²/2, exp{2x_i‘β+σ²}(exp{σ²}-1))
Which means we are introducing a prominent bias since we don’t know σ² but only ^σ²

Question 3

Gamma regression’s distribution

Answer 3

We model μi from y_i~Gamma(μ_i shape, ν scale) with E(y_i) = μ_i and Var(y_i) = μ_i²/ν

Question 4

Multiplicative exponential model

Answer 4

η_i = log(μ_i)
μ_i = exp(η_i)
- Coefficients have a multiplicative effect on μ_i.

Question 5

Canonical Gamma model

Answer 5

η_i = - 1/μ_i
μ_i = - 1/η_i
- Does not guarantee non-negative values for μ_i.