7.9 McKnight and van den Eeden (1993) analyzed a multi period, twotreatment crossover experiment to establish whether the artificial sweetener aspartame caused headaches. The trial involved randomly assigning 27 patients to different sequences of placebo and aspartame. The variables in headache.xls are Case (subject id number), Week (week of measurement), Aspartame (patient took aspartame = 1 if not = 0), Headaches (number of reported headaches), and Exp-days (the number days for which headaches counted).

(a) Many times with Poisson models, the counts reported should be analyzed as rates instead of the raw count due to unequal amounts of effort for each subject (e.g., here, some patients had headaches counted over differing numbers of days compared to other patients). The amount of time for which a count was recorded is commonly referred to as an offset and one models the count/time. For the loglink, the offset is then ln(time). Fit an appropriate random effects Poisson model with Exp-days specified as the offset variable.

(b) Does there appear to be a relationship between the number of reported headaches and whether or not a person took aspartame? Explain.

(c) Interpret the coefficients from the final model.

(d) Provide an expression for the marginal mean number of headaches and interpret.