2. A study was carried out to investigate the effect of a new drug in reducing operative mortality following major abdominal surgery. Patients were assigned to treatment or control groups within each of three categories of surgical risk: high, medium and low. The results were as follows: Surgical Risk Treatment Outcome Low Medium High Control Died 3 7 6 Survived 12 6 3 Treated Died 2 3 2 Survived 10 10 8 l for treatment and 331 = 0 1 for high risk and 333 = 0 Denote by 71' the probability of death following surgery. Let 3:1 for control; 1:2 2 1 for medium risk and 332 = 0 otherwise; :53 otherwise. (a) Express each of the following hypotheses as a logistic regression model for 71' in terms of the appropriate linear predictor and determine the degrees of freedom for the deviance statistic. Do not impose any additional constraints besides those explicitly stated in words and dene any additional covariates you require. 1. H1: There is no association between treatment and outcome in any category of risk. ii. Hg: The log odds ratios associating treatment and outcome are constant across categories of surgical risk. iii. H3: The log odds ratios associating treatment and outcome are linearly related to a continuous covariate, 3:4, which expresses surgical risk. iv. H4: The logit transform of the 11' is linearly related to 3:4, with a common slope for treatment and control groups