= Solution
Write $Y_i$ for the request count on day $i$. Both fits are <Poisson regression> models with independent
$$
Y_i\sim\operatorname{Poisson}(\lambda_i).
$$
In model 1,
$$
\log\lambda_i=\beta_0+\beta_WW_i+\beta_EE_i+\beta_BB_i
+\beta_LL_i+\beta_MM_i,
$$
where the variables are <indicator variables> for winter, weekend, bank holiday, and low or medium pollution; non-winter weekdays that are not bank holidays and have high pollution form the baseline. Model 2 imposes
$$
H_0:\quad\beta_B=\beta_L=\beta_M=0,
$$
retaining only winter and weekend effects.
The models are nested, and the <likelihood-ratio test statistic> is the decrease in residual deviance,
$$
316.39-304.97=11.42.
$$
There are three restrictions. By <Wilks theorem>, under $H_0$ this statistic is asymptotically a <chi-squared distribution> with three degrees of freedom. Since
$$
11.42>\chi^2_{3,0.99}=11.34487,
$$
\b[we reject model 2 at the $1\%$ level in favour of model 1.]
Back to article page
