Model 3 directly specifies the conditional mean and variance of the positive response on its observed scale. Its coefficients give multiplicative effects on mean record time, its prediction intervals concern time itself, and its likelihood can be compared directly with other original-scale models.

Model 2 instead models the mean of $\log T$. Back-transformation does not give the mean time without a <retransformation bias> correction; for a log-normal model, $\mathbb E(T\mid x)=\exp(x^T\alpha+\tau^2/2)$. The Gamma model also represents the increasing original-scale variance seen in the diagnostic plot, so it is preferable here.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 3 / f / Solution

In general, $\mathbb E[Y\mid X]$ is not obtained by exponentiating $\mathbb E[\log Y\mid X]$. For log-normal errors of variance $\sigma^2$, the correction factor is $e^{\sigma^2/2}$.


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 Incoming links: Retransformation bias