Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 33 / 5 / b / iv

The original PDF's age smooth is nearly flat, its uncertainty band includes zero across the age range, and its approximate p-value is $0.859$. The exposure smooth is increasing and convex: it is fairly flat at low exposure and rises more steeply at high exposure. Its effective degrees of freedom $1.895$ suggest a modest departure from a straight line, rather than a highly oscillatory relationship.

A suitable simpler model is therefore the grouped quasibinomial regressionwith age omitted and dispersion estimated. A positive quadratic coefficient captures the convex exposure relationship; retaining the linear term respects the usual polynomial hierarchy. This model should be refitted, rather than assigning coefficients from the smooth plot. Compare its residual pattern and dispersion-adjusted fit with the additive model to check that the quadratic approximation is adequate. An exposure-only quadratic logit model is a justified parsimonious candidate, while the graph does not justify a sharp threshold or a zero-dispersion model.