## Bayesian Stopping Rule For Safety

The trial described in Section 2 was monitored by a data and safety monitoring board, which expressed a concern that engraftment by day 42 was not sufficient to assure the longer term safety of the patients who underwent this procedure. They requested a stopping rule for a longer term endpoint to monitor safety. We chose 100 day transplant-related mortality (TRM) as the safety endpoint. TRM encompasses multiple causes of death and so serves as a suitable safety endpoint. Thus patients who died because of failure to engraft, graft failure after engraftment, toxicity from the preparative regimen, graft versus host disease or infection would count as failures in the safety endpoint. Day 100 was chosen to include the early sequella of the transplant, but not later events, such as chronic graft versus host disease or recurrence. To monitor day 100 TRM, we adopted a Bayesian approach which formally incorporated ''prior'' expectations about the proportion of patients experiencing TRM.

Several authors have used Bayesian methods in other settings. Thall and Simon (1994) discuss Bayesian guidelines for phase II trials where comparison to a previously established standard will be made and the data are monitored continuously. Thall et al., (1995) discuss Bayesian sequential monitoring in phase II trials with multiple outcomes. Foll-mann and Albert (1999) discuss Bayesian monitoring with censored data. Thall and Russell (1998) use a Bayesian approach to assess dose and efficacy simultaneously by defining adverse, efficacious and neither as outcomes and using a cumulative odds model.

The proportion of patients experiencing TRM up through day 100 post-transplant, pTRM, was assumed to follow a binomial distribution. For the prior distribution of pTRM, we used the beta distribution. This was done for two reasons. First, it is a ''natural'' conjugate prior for the binomial distribution; that is, the likelihood functions for both the beta and the binomial distributions have the same functional form (Berger, 1985). Thus, the posterior distribution may be easily recalculated each time a patient is evaluated. Second, using a beta prior has the following attractive property: Suppose a beta prior distribution with parameters a and b is used, and so its mean is a/(a + b) and variance is ab/[(a + b)2 (a + b + 1)]. Further, suppose that among n patients enrolled, y have not engrafted (failure), and the remaining n ā y have engrafted (success). Then the posterior beta distribution has parameters a + y and b + (nāy) and mean (a + y) / (n + a+ b). This mean is the maximum likelihood estimate of pTRM based on a + y successes and (n ā y)+ b failures. Thus the prior may be thought of as contributing a ''imaginary'' failures and b ''imaginary'' successes to the posterior distribution and so is ''worth'' a + b ''patients'' compared to the n ''real'' patients that have been enrolled (Santner and Duffy, 1989).

This interpretation provides a simplified approach to specifying the prior distribution. We specify the prior mean at say r and take the worth of the prior to be a modest proportion of the total planned sample size, that is, a + b is a modest proportion of n. This assures that the prior will be influential in the early stage of the study, but that later, the data will dominate the prior.

A stopping boundary would be reached if the proportion experiencing TRM exceeds the anticipated proportion with posterior probability at some threshold, say .90 or .95. From prior experience we anticipated the mean to be .20 and we take our prior to be ''worth'' six patients. Thus the parameters of the prior distribution were 1.2 and 4.8 which also implies that the variance of the prior distribution is .0229.

We took the threshold probability for stopping as .90. That is, we would recommend stopping if the number of patients patients experiencing TRM implied the posterior distribution had .90 of its probability mass exceeding the mean of the prior, .20. For the purpose of preserving power, we did not allow for stopping after every patient, but, instead, after groups of patients. The resulting stopping boundaries are given in Table 1.

In implementing this stopping rule, it is important to be even-handed in counting those alive and dead by 100 days. Strictly speaking, we should not tally a patient as having 100 day TRM or not until their

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