The following examples are intended to show the advantages of Bayesian reporting of treatment efficacy analysis, as well as to provide examples contrasting with frequentist reporting. As detailed here, there are many problems with p-values, and some of those problems will be apparent in the examples below. Many of the advantages of Bayes are summarized here. As seen below, Bayesian posterior probabilities prevent one from concluding equivalence of two treatments on an outcome when the data do not support that (i.e., the “absence of evidence is not evidence of absence” error).
Suppose that a parallel group randomized clinical trial is conducted to gather evidence about the relative efficacy of new treatment B to a control treatment A. Suppose there are two efficacy endpoints: systolic blood pressure (SBP) and time until cardiovascular/cerebrovascular event. Treatment effect on the first endpoint is assumed to be summarized by the B-A difference in true mean SBP. The second endpoint is assumed to be summarized as a true B:A hazard ratio (HR). For the Bayesian analysis, assume that pre-specified skeptical prior distributions were chosen as follows. For the unknown difference in mean SBP, the prior was normal with mean 0 with SD chosen so that the probability that the absolute difference in SBP between A and B exceeds 10mmHg was only 0.05. For the HR, the log HR was assumed to have a normal distribution with mean 0 and SD chosen so that the prior probability that the HR>2 or HR<1⁄2 was 0.05. Both priors specify that it is equally likely that treatment B is effective as it is detrimental. The two prior distributions will be referred to as p1 and p2.
Example 1: So-called “Negative” Trial (Considering only SBP)
- Incorrect Statement: Treatment B did not improve SBP when compared to A (p=0.4)
- Confusing Statement: Treatment B was not significantly different from treatment A (p=0.4)
- Accurate Statement: We were unable to find evidence against the hypothesis that A=B (p=0.4). More data will be needed. As the statistical analysis plan specified a frequentist approach, the study did not provide evidence of similarity of A and B (but see the confidence interval below).
- Supplemental Information: The observed B-A difference in means was 4mmHg with a 0.95 confidence interval of [-5, 13]. If this study could be indefinitely replicated and the same approach used to compute the confidence interval each time, 0.95 of such varying confidence intervals would contain the unknown true difference in means.
- Assuming prior distribution p1 for the mean difference of SBP, the probability that SBP with treatment B is lower than treatment A is 0.67. Alternative statement: SBP is probably (0.67) reduced with treatment B. The probability that B is inferior to A is 0.33. Assuming a minimally clinically important difference in SBP of 3mmHg, the probability that the mean for A is within 3mmHg of the mean for B is 0.53, so the study is uninformative about the question of similarity of A and B.
- Supplemental Information: The posterior mean difference in SBP was 3.3mmHg and the 0.95 credible interval is [-4.5, 10.5]. The probability is 0.95 that the true treatment effect is in the interval [-4.5, 10.5]. [could include the posterior density function here, with a shaded right tail with area 0.67.]
Example 2: So-called “Positive” Trial
- Incorrect Statement: The probability that there is no difference in mean SBP between A and B is 0.02
- Confusing Statement: There was a statistically significant difference between A and B (p=0.02).
- Correct Statement: There is evidence against the null hypothesis of no difference in mean SBP (p=0.02), and the observed difference favors B. Had the experiment been exactly replicated indefinitely, 0.02 of such repetitions would result in more impressive results if A=B.
- Supplemental Information: Similar to above.
- Second Outcome Variable, If the p-value is Small: Separate statement, of same form as for SBP.
- Assuming prior p1, the probability that B lowers SBP when compared to A is 0.985. Alternative statement: SBP is probably (0.985) reduced with treatment B. The probability that B is inferior to A is 0.015.
- Supplemental Information: Similar to above, plus evidence about clinically meaningful effects, e.g.: The probability that B lowers SBP by more than 3mmHg is 0.81.
- Second Outcome Variable: Bayesian approach allows one to make a separate statement about the clinical event HR and to state evidence about the joint effect of treatment on SBP and HR. Examples: Assuming prior p2, HR is probably (0.79) lower with treatment B. Assuming priors p1 and p2, the probability that treatment B both decreased SBP and decreased event hazard was 0.77. The probability that B improved either of the two endpoints was 0.991.
One would also report basic results. For SBP, frequentist results might be chosen as the mean difference and its standard error. Basic Bayesian results could be said to be the entire posterior distribution of the SBP mean difference.
Note that if multiple looks were made as the trial progressed, the frequentist estimates (including the observed mean difference) would have to undergo complex adjustments. Bayesian results require no modification whatsoever, but just involve reporting the latest available cumulative evidence.