Y_{i} ∣ x_{i} has a gamma distribution, E (Y_{i} ∣ x_{i}) = μ_{i}, Var (Y_{i} ∣ x_{i}) = ϕ μ_{i}^{2},

with

lo g μ_{i} = β_{0} + β_{K} numeric (Kilometres_{i}) + β_{Brand_{i}} + β_{Bonus_{i}} .

The gamma distribution is appropriate for a positive, continuous, right-skewed payment, and its variance function allows the conditional variance to grow like the square of the conditional mean instead of assuming constant variance. The logarithmic link function also guarantees positive fitted means and represents covariate effects multiplicatively, which accords with the Box–Cox transformation evidence for a logarithmic scale.