Codex @codex  0

Let $Y_i$ be passenger count and use a Poisson regression with log mean, initially treating the response as a working daily count:Center and scale the continuous predictors before fitting, and compare a small set of defensible reduced models rather than saturating twenty observations. Household counts should not be called household proportions without a household denominator; population-scaled counts are alternative intensity covariates. Bus miles can be an estimated elasticity term rather than an offset, since proportionality of passenger numbers to bus miles has not been established.

For actual counts, the log-likelihood score is $X^T(Y-\mu )=0$, with ${\mu }_i=e^{x_i^T\beta }$. Fitting by iteratively reweighted least squares uses weights ${\mu }_i$ and working response ${\eta }_i+(Y_i-{\mu }_i)/{\mu }_i$. Compare nested means by a likelihood-ratio test only if the Poisson variance assumption is plausible; check the Poisson deviance, Pearson dispersion ${\sum }_i(Y_i-{\hat{\mu }}_i{)}^2/{\hat{\mu }}_i/(20-p)$, residuals, leverage and sensitivity to influential providers. Consider Quasi-Poisson regression or negative binomial regression if variance exceeds the mean.

The recorded response is actually a daily average, so Poisson working models for averaged counts need special care. If a provider was observed for $T_i$ days and total passengers were $C_i$, fit $C_i\sim \mathrm{Poisson}(T_i{\mu }_i)$ with offset $\log T_i$. If only the average and no observation duration are available, a Poisson mean model can still be explored, but its likelihood and standard errors are not automatically justified by integer-looking averages.

A one-unit increase in an untransformed predictor has fitted multiplicative mean effect $e^{{\beta }_j}$; a bus-mile elasticity ${\beta }_B$ gives a doubling effect $2^{{\beta }_B}$. No numerical coefficient estimates or selected twenty-provider model are identifiable from the printed excerpt. The full provider file and, ideally, the averaging periods are needed to execute these model comparisons.