Law and Kaldor address the important question of clinical trials with survival outcomes in which subjects may change in a non-randomized manner from one randomized treatment to the other during the course of follow-up. These changes make the treatment experience of the two randomized groups more similar. Consequently, as the authors say, intention-to-treat analysis in this situation will tend to underestimate a 'true' treatment effect. They have therefore produced an 'adjusted' analysis which aims to correct for this underestimation. In the example they give, the intention-to-treat estimated hazard ratio was 0•82, and a corrected estimator would be expected to lie further from the null, that is, below 0•82. In fact, however, the estimated hazard ratio from the 'adjusted' analysis is 0•85, representing an adjustment in the wrong direction.
The explanation of this finding may lie in the form of the model used. Patients are divided into four groups (AA, AB, BA, BB) according to randomized treatment and whether a treatment change is observed before an event. The hazard for each patient at time t is then modelled according to group and current treatment at time t. However, group membership at time t may depend on the future: for example, a subject randomized to treatment A who has not changed treatment by time t will be in group AB only if he or she subsequently changes treatment. The use of covariates which depend on the future is dangerous in proportional hazards regression, and the present example is no exception. Subjects in group AB who have not yet changed treatment are modelled as having hazard (t)exp( a), yet their true hazard is zero because their membership of group AB means that they cannot die before changing treatment.
The effect of this model inadequacy is likely to be to bias the estimate of (the adjusted treatment effect) upwards. This bias may or may not be balanced by the corresponding model inadequacy in group BA.
A similar error occurs in the authors' criticism of the usual 'as treated' analysis. Their Figure 2(a) shows that patients randomized to radical radiotherapy who subsequently received salvage cystectomy had better survival than other patients randomized to radical radiotherapy. This finding is hardly surprising, since the former patients had to survive long enough to receive salvage cystectomy; it does not show that salvage cystectomy was allocated to patients with better prognosis. (A similar situation arises in the analysis of transplant data and is discussed in standard texts on survival analysis.) The 'as treated' analysis may still be wrong in assuming that allocation of salvage cystectomy was independent of prognosis, but at least this analysis produces a hazard ratio which has as expected been adjusted away from the null.
I performed a simulation study to investigate the performance of Law and Kaldor's estimator in a situation where treatment had no effect, treatment changes were confined to group A, and treatment changes were independent of prognosis. Specifically, I took 40,000 subjects and generated time to event and time to treatment change from independent exponential distributions with mean 1. Half the subjects were assigned to treatment A and half to B; time to treatment change was only generated for those assigned to