9.4 - Bayesian approach in Clinical Trials

With respect to clinical trials, a Bayesian approach can cause some difficulties for investigators because they are not accustomed to representing their prior beliefs about a treatment effect in the form of a probability distribution. In addition, there may be very little prior knowledge about a new experimental therapy, so investigators may be reluctant to or not be able to quantify their prior beliefs. In the business world, the Bayesian approach is used quite often because of the availability of prior information. In the medical field, more often than not, this is not the case.

The choice of a prior distribution can be very controversial. Different investigators may select different priors for the same situation, which could lead to different conclusions about the trial. This is especially true when the data, X, are based on a small sample size because in such situations the prior distributions are modified only slightly to form the posterior distributions. Small sample sizes only modify the prior slightly. This tends to weight the posterior distribution very closely to the prior, therefore you are basing your results almost entirely on your prior assumptions.

When there is little prior information to base your assumptions of the distribution on, Bayesians employ a reference (or vague or non-informative) prior. These are intended to represent a minimal amount of prior information. Although vague priors may yield results similar to those of a frequentist approach, the priors may be unrealistic because they attempt to assign equal weight to all values of θ. Below you can see a very flat distribution, very spread out over a wide range of values.

vague priorθ0θ1

Similarly, skeptical prior distributions are those that quantify the belief that large treatment effects are unlikely. Enthusiastic prior distributions are those that quantify large treatment effects. Let's not worry about the calculations, but focus instead on the concepts here...

An example of a Bayesian approach for interim monitoring is as follows. Suppose an investigator plans a trial to detect a hazard ratio of 2 \(\left(\Lambda = 2\right)\) with 90% statistical power \(\left(\beta = 0.10\right)\) using at least a sample size of 90 events. The investigator plans one interim analysis, approximately halfway through trial, and a final analysis. (This is the more standard approach, as opposed to the SPRT where R was calculated after each treatment.)

The estimated logarithm of the hazard ratio is approximately normally distributed with variance \(\left(\dfrac{1}{d_1}\right) + \left(\dfrac{1}{d_2}\right)\), where \(d_1\) and \(d_2\) are the numbers of events in the two treatment groups. The null hypothesis is that the treatment groups are the same, i.e., \(H_0\colon \Lambda = 1\). Note that the \(log_e\) hazard ratio is 0 under the null hypothesis and the \(log_e\) hazard ratio is 0.693 when \(\Lambda = 2\), the proposed effect size.

Suppose the investigator has access to some pilot data or the published report of another investigator, in which there appeared to be a very small treatment effect with 16 events occurring within each of the two treatment groups. The investigator decides that this preliminary study will form the basis of a skeptical prior distribution for the \(log_e\) hazard ratio with a mean of 0 and a standard deviation of \(0.35 = {\dfrac{1}{16} + \dfrac{1}{16}}^{\frac{1}{2}}\). This is called a skeptical prior because it expresses skepticism that the treatment is beneficial.

0skeptical prior

Next, suppose that at the time of the interim analysis, (45 events have occurred), there are 31 events in one group and 14 events in the other group, such that the estimated hazard ratio is 2.25 (calculations not shown). These values are incorporated into the likelihood function, which modifies the prior distribution to yield the posterior distribution for the estimated \(log_e\) hazard ratio that has a mean = 0.474 and standard deviation = 0.228 (calculations not shown). Therefore we can calculate the probability that \(\Lambda\) is \(> 2\). From the posterior distribution we construct the following probability statement:

\(Pr[\Lambda \ge 2]=1-\Phi \left(\dfrac{log_e(2)-0.474}{0.228} \right)=1-\Phi(0.961)=0.168\)

where \(\Phi\) represents the cumulative distribution function for the standard normal and is the true hazard ratio.

Conclusion: Based on the results from the interim analysis with a skeptical prior, there is not strong evidence that the treatment is effective because the posterior probability of the hazard ratio exceeding 2 is relatively small. Therefore, there is not enough evidence here to suggest that the study be stopped. What is too large? A reasonable value should be specified in your protocol before these values are determined.

In contrast, suppose that before the onset of the trial the investigator is very excited about the potential benefit of the treatment. Therefore, the investigator wants to use an enthusiastic prior for the \(log_e\) hazard ratio, i.e., a normal distribution with mean \(= log_e(2) = 0.693\) and standard deviation = 0.35 (same as the skeptical prior).

00.693enthusiastic prior

Suppose the interim data results are the same as those described above. This time, the posterior distribution for the \(log_e\) hazard ratio is normal with mean = 0.762 and standard deviation = 0.228. Then the probability for the posterior distribution is:

\(Pr[\Lambda \ge 2]=1-\Phi \left(\dfrac{log_e(2)-0.762}{.228} \right)=1-\Phi(-0.302)=0.682\)

This is a drastic change in the probability based on the assumptions that were made ahead of time. In this case, the investigator still may not consider this to be strong evidence that the trial should terminate because the posterior probability of the hazard ratio exceeding 2 does not exceed 0.90.

Nevertheless, the example demonstrates the controversy that can arise with a Bayesian analysis when the amount of experimental data is small, i.e., the selection of the prior distribution drives the decision-making process. For this reason, many investigators prefer to use non-informative priors. Using the Bayesian methods, you can make probability statements about your expected results.