Question

Question 3 20 marks (a) (i) In a study of osteoporosis, the spines of patients undergoing treatment were X-rayed periodically, and counts of the number of crush fractures were made. The results were typically small numbers, most often 0, but with some counts of 1, 2, or occasionally more. Several other variables were measured on each patient, including measures of the amount of exercise they took (graded on a three-point scale: none, some, a lot) and of their milk consumption (pints per week) and cheese consumption (ounces per week). If you were to attempt to build a generalised linear model for these data, with number of crush fractures as the response, what would be a reasonable distribution to assume for the response, and what link function would you adopt? 2 (ii) In a study of moth-proofing agents, different concentrations of the agents were tested on batches of 20 moth larvae, and the number of larvae killed per batch was observed. What distribution might one assume in a generalised linear model for such data, and what linlk function would be appropriate? 2

Answer 1

(i) Here in this case, Truncated Poisson distribution truncated for values 3 and more will be a reasonable distribution to assume.

Writing the poisson distribution in the form of one parameter exponential family yields us the (canonical) link function to be 'log (base e)' link.

(ii) For this, it is safe to assume the distribution to be a Binomial distribution for the response variable denoting the number of larvae killed per batch.

Similarly as before, writing the binomial distribution in the form one-parameter exponential family yields us the (canonical) link to be 'logit link'.