It is unlikely that a crossover study ignored the fact that crossover was occurring. The reason crossover studies are employed is to allow two treatments to be administered to a single individual so that confounding effects due to an individual are accounted for equally in both treatments. For example, if one patient in a study has a confounding condition which makes the results seem worse, then both treatments automatically are adjusted by the same amount. The fact that it is a crossover study automatically accounts for these types of issues.

There are some factors that can affect crossover studies including carryover effects where the order of treatment leads to different outcomes. This could happen if an administered drug taxes a patients system and they are unable to return to their initial state of health before the second treatment is given. Another possibility is if the order the treatments are given somehow fundamentally changes the patient, for example if the first treatment involves the removal of an organ. Then the second treatment would be influenced by the first. This kind of effect will lead to bias in the data but carryover effects can be minimized by allowing sufficient time between the two treatments so that the patient returns to a "baseline" before the second treatment is given.

In a valid crossover study the statistical method to be used to analyze the data has to be detailed before the study is undertaken and the data can then be directly analyzed using that method once the data has been collected. This type of experimental design in fact can dramatically increase power due to accounting for confounding effects in individuals. The reason the statistical methodology has to be explicitly chosen before the data is taken is so that confounding factors like carryover or order can be accounted for and ideally minimized through the setup of the experiment.

Then the data can be directly analyzed using standard methods such as repeat measure Anova. No additional statistical complexity is introduced beyond the different treatment effects, so no additional changes are required in the statistical analysis. This is part of the reason that a well designed balanced block crossover experiment can achieve significant results with far fewer datapoints than are required for other methods.

Thanks for the answer, Robert. However, my question is about survival analysis, and the crossover may be discretionary, not part of the study design per se. In the BIG 1-98 trial, for example, the discretionary crossover was allowed for ethical reasons,. The original study report provided two models of overall survival; one estimate ignored crossover and the other estimate censored patients that crossed over. I am interested in reading about extensions to Cox regression that correct for discretionary crossover, if any.

Scott Got ya. I think the key statistical point you are interested in is the independence of the predictor variables and the assumed linear dependence between these variables. Since the Cox method is modelling the variability of the survival probabilities on the predictor variables, if one of the variables does have an unexpected crossover effect like you mentioned above, (ie. diabetic status and people who become diabetic midway through the study) then that confounding effect only affects that predictor variable. As you mentioned two methods of analysis can then be performed. 1) Remove those individuals for the predictor for which the crossover occurred. This only limits the power of that regression coefficient. 2) Determine if the crossover events can be categorized meaningfully into either of the two original groups or possibly be classified as an additional class possibility (diabetic, non-diabetic, converted diabetic).

The variability in the survivability probabilities due to the other predictor variables are unchanged by the crossover events. This is true by definition since the predictor variables are assumed to be independent, but this assumption can also be tested to see if there is a statistically significant interaction effect in the standard ANOVA/regression analysis. This is one reason regardless of the statistical method used it is always important to test the assumptions of the model in question. If the assumption of independence of predictor variables does not hold up then there may be additional predictor variables which would also be affected by the crossover events and those predictors would also have to be handled as described above. Fortunately this all falls under the standard statistical analyses performed in any ANOVA/regression and nothing additional needs to be done beyond testing the assumptions of the model.

How to deal with the crossover events is a choice that needs to be made, but it is even better if both methods of dealing with the confounding events are tested (as the study you described did) and if both models yield essentially identical results the "choices" don't affect the predictions of the model.

The following paper finds both methods are biased and uses IPCW as an extension of Cox regression to correct the bias. I'm still looking for other references if anyone has them.

(Colleoni M, Giobbie-Hurder A, Regan MM, et al. (2011) Analyses adjusting for selective crossover show improved overall survival with adjuvant letrozole compared with tamoxifen in the BIG 1-98 study. J Clin Oncol 29:1117–1124.)