Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 5I (Statistical Modelling)

The fit is a Gaussian linear model for $Y_i=\log ({so2}_i)$ with an intercept and the six displayed explanatory variables:The command computes ordinary least squares; independent normal errors with common variance are the additional assumptions yielding the exact printed tests. A coefficient describes an adjusted association on the log scale, holding the other variables fixed. Exponentiation gives a multiplicative effect on the conditional median of sulphur dioxide, not an additive effect on its original scale. Under the Gaussian model its conditional mean additionally includes the factor $e^{{\sigma }^2/2}$.

The estimates are the coordinates of $\hat{\beta }=(X^{T}X{)}^{-1}{X}^{T}Y$. With $41$ observations and $7$ parameters the residual degrees of freedom are $34$, and the residual standard error is $s=\sqrt{\mathrm{RSS}/34}=0.448$ to the displayed precision. The reported standard error of coefficient $j$ is $s\sqrt{[(X^{T}X{)}^{-1}{]}_{jj}}$. For each null hypothesis ${\beta }_j=0$, the displayed statistic $t_j={\hat{\beta }}_j/\mathrm{se}({\hat{\beta }}_j)$ has Student t-distribution with $34$ degrees of freedom under that null. Its two-sided p-value is $2\mathrm{Pr}(t_{34}\ge |t_j|)$.

At the $5\%$ level the intercept, temperature, manufacturing and wind coefficients differ significantly from zero in the full model. Temperature and wind have negative estimated adjusted effects; manufacturing has a positive one. The p-values for population, precipitation and precipitation days, approximately $0.136$, $0.127$ and $0.931$, do not reject their respective zero-coefficient hypotheses. This is not evidence that their effects are exactly zero, and these observational associations do not establish causation.

A sensible candidate reduced linear regression keeps an intercept, temperature, manufacturing and wind. It must be refitted: coefficients generally change when correlated predictors are removed. The three deletions can be checked jointly by the F-test statisticwhich has $F_{3,34}$ distribution under the joint null. The individual summary does not determine this joint test or establish the reduced model's adequacy.

In the final command, the uniquely matching component named by fit$df is the residual degrees of freedom. Thus the result is $\boxed{\sqrt{\mathrm{RSS}/34}=0.448\text{ to printed precision}.}$