Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 5I (Statistical Modelling)

The command fits independent grouped binomial distributions $Y_j\sim \mathrm{Bin}(N_j,p_j)$, using the supplied totals as trial counts. The logistic regression has the additive linear predictorThe reference category is adult females. The age and gender coefficients are conditional log odds ratios: the fitted odds for a child are multiplied by $e^{-1.27079}\approx 0.281$, and those for a male by $e^{-0.37211}\approx 0.689$.

The coefficients are maximum-likelihood estimates, found by Newton's method or iteratively reweighted least squares. With design matrix $X$ and $W_{jj}=N_j{\hat{p}}_j(1-{\hat{p}}_j)$, the estimated covariance matrix is $(X^{T}WX{)}^{-1}$; the printed standard errors are square roots of its diagonal entries. Each Wald test compares a coefficient with zero using its estimate divided by its standard error, approximately a standard normal distribution under the null. The age and gender effects are strongly significant. The intercept test instead tests whether the reference probability is $1/2$.

The fitted probabilities are about $0.04476$ for adult females, $0.03128$ for adult males, $0.01298$ for female children, and $0.008981$ for male children. Adult females have the largest fitted probability, about 4.48%. The residual deviance is $0.06514$ on $4-3=1$ degree of freedom; comparison with a chi-squared distribution gives $p\approx 0.799$. There is no evidence here that an age–gender interaction is needed, under the assumed independent binomial distribution model.