Bayesian statistics for clinical research

Abstract

Frequentist and Bayesian statistics represent two differing paradigms for the analysis of data. Frequentism became the dominant mode of statistical thinking in medical practice during the 20th century. The advent of modern computing has made Bayesian analysis increasingly accessible, enabling growing use of Bayesian methods in a range of disciplines, including medical research. Rather than conceiving of probability as the expected frequency of an event (purported to be measurable and objective), Bayesian thinking conceives of probability as a measure of strength of belief (an explicitly subjective concept). Bayesian analysis combines previous information (represented by a mathematical probability distribution, the prior) with information from the study (the likelihood function) to generate an updated probability distribution (the posterior) representing the information available for clinical decision making. Owing to its fundamentally different conception of probability, Bayesian statistics offers an intuitive, flexible, and informative approach that facilitates the design, analysis, and interpretation of clinical trials. In this Review, we provide a brief account of the philosophical and methodological differences between Bayesian and frequentist approaches and survey the use of Bayesian methods for the design and analysis of clinical research.

Figure 1.

Example of a probability density distribution obtained in a hypothetical observational cohort study of mortality at one-year. In a sample of 100 patients, 30 were found to have died at one year of follow-up. These data are assumed to have a binomial likelihood. The probability density distribution computed from the likelihood function for these data, assuming a non-informative prior (constructed using a beta distribution), is shown in the plot. This data-based probability distribution represents the posterior distribution for this study. The distribution can be described in terms of its median value (solid vertical line) and 2·5^th and 97·5^th percentiles (dashed vertical lines) representing the 95% credible interval. The area under the curve in the shaded red region gives the probability that the mortality rate lies between 0·25 and 0·35.

Figure 2.

Comparison of frequentist and Bayesian statistical inference. In a hypothetical study evaluating an association between intervention and mortality, 280 patients were randomized 1:1 to intervention or control. Mortality in the control group was 38% and in the intervention group was 29%, yielding a point estimate for the odds ratio of 0·66. The sampling distribution for the null hypothesis in this study (top panel), defined based on the primary outcome and planned sample size (stopping rule), gives the expected frequency of obtaining data with a point estimate (vertical blue line) as or more extreme than the observed point estimate if the study was hypothetically repeated in precisely the same way many times (grey-shaded area). If this expected frequency is less than 5 out of 100 (“p<0·05”), the null hypothesis of no effect is generally rejected. In frequentist inference, the hypothesis is fixed (effect=null, vertical dashed line), and the expected data under the hypothesis are treated as a random variable with (hypothetical) repeated sampling (represented by the sampling distribution). In Bayesian inference, the observed data are treated as a fixed distribution (the likelihood function, middle panel) rather than as a point estimate and the hypothesis is treated as a random variable with a probability distribution (not as a point estimate). Information on the hypothesis prior to the study is represented by the prior distribution (green curve, bottom panels). Combining the prior with the likelihood yields a new random probability distribution for the updated hypothesis, the posterior distribution (red curve, bottom panels); the posterior distribution will vary according to the prior. The area under the posterior distribution may be used to estimate the probability of treatment effect below the region of practical equivalence which defines the minimum clinically relevant effect (blue-shaded region, bottom panels). The posterior distribution depends only on the prior and the likelihood; it is not determined by the analysis plan and stopping rule for the study (unlike the frequentist sampling distribution).

Figure 3.

Bayesian sequential analysis of a clinical trial. A clinical trial was simulated by randomly sampling from a population where the true mortality rate at 28 days after randomization in the Intervention and Control Groups is 0·3 and 0·4, respectively. The prior distribution for the odds ratio is neutral, with information equivalent to a sample size of 50 patients. As increasing numbers of patients are enrolled and randomized, sequential analyses reveal accumulating information about mortality in the Intervention and Control groups (left panels). This increase in information is represented by a progressive decrease in the variance of the probability distributions for mortality in each group (left panels). The increasing information about mortality allows more precise estimates of the difference in mortality between groups (quantified by the posterior odds ratio, right panels), yielding a progressively higher strength of belief that the intervention lowers mortality (posterior probability of superiority, Pr(OR<1)). Under the likelihood principle, the information in the posterior distribution is not affected by the number of interim analyses, facilitating adaptive decision-making in trial design (e.g., stop the trial once the posterior probability of superiority exceeds a pre-specified threshold).