Introduction to Bayesian Statistics

Part 36 of a Series on the Evaluation of Scientific Publications

Background: The analysis of a study with Bayesian statistics makes use of additional information to supplement the new study data. In this review, we explain the principles of the application of this method in clinical research.

Methods: The concept of Bayesian statistics is introduced and explained with the aid of an illustrative example from a drug approval study. Its major aspects are discussed. The existing prior knowledge is formulated as a probability distribution of an odds ratio. Multiple scenarios are shown to demonstrate how a suitable prior distribution is determined and how it can affect the final result.

Results: Bayesian statistics makes use of prior knowledge, e.g., the findings of earlier clinical trials, and combines the prior probability distribution with the findings of the current study for statistical analysis. The suitability and applicability of the prior knowledge in question must be assessed and the prior knowledge weighted accordingly, and any uncertainties must be taken into account in the analysis. The result that is derived is called the posterior distribution of the parameters of interest and is summarized in terms of point estimators and credibility intervals. In contrast to classical statistics, results of this type permit direct quantitative statements on the probability of parameter values and on the probabilities of the null and alternative hypotheses (in one-sided statistical tests).

Conclusion: Combining the current study findings with prior knowledge can enable the more precise estimation of a treatment effect, or else lessen the number of subjects needed for a clinical trial. Central elements of Bayesian statistics are the selection and weighting of prior knowledge; subjective judgements must be made. Bayesian techniques require a precise description of the methods applied, meticulous study of the available literature, and experience in the mathematical representation of the results.

Cite this as: Gerß JWO, Vonthein R: Introduction to Bayesian statistics: Part 36 of a series on the evaluation of scientific publications. Dtsch Arztebl Int 2025; 122: 271–6. DOI: 10.3238/arztebl.m2025.0035

There are two fundamental approaches to statistical analysis. Classical statistics comprises, among other procedures, statistical testing and the p-value (1). Bayesian statistics is an alternative method offering more options.

Treatment guidelines are continually being updated. In this process, it is helpful not to view each new study in isolation, but rather to take advantage of the results of previous studies, or other sources, as prior information. Two or more similar trials can be combined by means of a meta-analysis (2). Bayesian statistics is more flexible in its use of pre-existing data. In the PLUTO trial, for example, intravenous (i.v.) administration of the substance belimumab was studied in 93 children and adolescents with systemic lupus erythematosus (3). For the sake of more robust results, the data of the 93 minors were combined with prior data from two studies in adults. In contrast to a classical meta-analysis, use of the Bayesian procedure permitted weighting of the existing knowledge. In other words, the prior data were not utilized in full, but only to the extent to which they were valid in minors as well as in adults. The findings of the Bayesian analysis led to approval of the drug Benlysta for use in children and adolescents with systemic lupus erythematosus.

The exploitation of prior knowledge is not the only additional benefit offered by Bayesian statistics in comparison with classical statistical testing. If the null hypothesis is formulated as follows: “There is no difference between the two treatments with respect to their effect,” then the p-value is calculated from the observed data. “The p-value gives the probability of obtaining the present test result—or an even more extreme one—if the null hypothesis is correct” (1). Bayesian statistics permits statements such as “The probability of the null hypothesis being correct is only 10%, the probability of it being false, 90%” or “With a probability of 95%, treatment A is superior to treatment B”. A pivotal trial for the COVID-19 vaccine Comirnaty (tozinameran) was conducted using a Bayesian procedure (4). Rather than a significant p-value, the trial found a probability of over 99.99% that the rate of illness among those receiving the vaccine will be at least 30% lower than in unvaccinated persons.

Box 1

Questions and answers

Box 2

Applications of Bayesian procedures

Box 3

Basic terminology

A randomized trial explored the effect of the antibiotic doxycycline as a treatment for the rare Creutzfeldt–Jakob disease (5). The primary endpoint was overall survival. This investigation of only 12 patients found no significant treatment effect. The assessment of treatment effect was improved by using a Bayesian procedure to add to the analysis the data of 88 patients from an observational study. The result was the direct quantitative conclusion of an 84% probability of a positive treatment effect. The analysis took account of the fact that the observational data represented only a priori information and were not of equal quality. In terms of statistical power they were equivalent to 15 additional randomized study participants.

This article begins by delineating the fundamental differences between the classical statistical procedures and the Bayesian method. The section on the principles of Bayesian statistics describes the basic handling of prior data and the nature of the results of statistical analysis. The methodology and application of Bayesian statistics are then presented in detail, using the example of the PLUTO trial, and finally the key messages of the article are summarized and discussed.

Frequentist statistics and Bayesian statistics

The statistical approach described above as “classical” is also known as “frequentist statistics.” This term derives from the definition of the concept of probability, which differs from that in Bayesian statistics. In frequentist statistics, probability is the limiting value of relative incidence in an infinite number of repetitions. In a clinical trial, for example, an estimated treatment effect is ascertained from the observed data in the form of an odds ratio (6). The idea is that the same study will be conducted arbitrarily often and a new odds ratio determined each time. Given the random nature of the data, a frequency distribution would be derived from the complete set of odds ratios calculated.

In Bayesian statistics the frequency distribution of an effect indicates one’s degree of confidence in particular effect sizes. For instance, an odds ratio between 0.5 and 1.5 is more probable than an odds ratio between 5 and 6. The interpretation of probability as knowledge of the possible sizes of an effect is the essential factor behind the additional options offered by Bayesian statistics. It enables prior knowledge to be incorporated into data analysis. The precision of the current state of knowledge is determined by the data available at a given point in time. There may be pre-existing knowledge from earlier studies that has to be expressed in the form of the prior distribution before a trial is conducted. The findings of the new study provide novel information. Updating the prior knowledge by combining it with the new data yields the posterior distribution. Apart from taking advantage of prior knowledge, the particular Bayesian concept of probability is the reason why the probabilities of effect sizes and hypotheses are stated. This is seen in one-sided statistical testing of the odds ratio (OR) with the null hypothesis H₀: OR ≤ 1 and the alternative hypothesis H₁: OR > 1. The posterior probability of the alternative hypothesis can be deduced from the prior distribution as the area under the density function from 1 upwards.

Bayesian methods can therefore replace a frequentist statistical test. Instead of basing the test decision on the p-value, the null hypothesis is rejected in the event that the posterior probability of the alternative hypothesis is high enough, e.g., 97.5%. Such a procedure resembles a statistical test at a one-sided significance level of 2.5%. The inclusion of appropriate prior knowledge can lend the Bayesian procedure greater power than would be achieved by a comparable frequentist statistical test. In other words, the number of cases needed for a power of 80%, for example, may be lower with a Bayesian approach. For such statements on the power or the number of cases needed to be permissible, however, it must be ensured that the Bayesian procedure will yield the same level of significance.

Principles of Bayesian statistics

The essential steps of a Bayesian procedure are establishing the prior distribution of the model parameters and integrating the prior distribution with new data to yield the posterior distribution (7−9). There may be one model parameter or multiple parameters. In a two-armed study, the treatment effect, in the form of the odds ratio or hazard ratio compared between the two treatments, may be the sole model parameter (6, 10). In a multi-arm study, the existence of a number of pairwise comparisons between two treatments means that that are multiple model parameters.

Prior knowledge about the model parameters must always be described precisely at the study planning stage, before analysis. The sources should be transparent, and there should be a description of how the prior knowledge feeds into the analysis. The prior knowledge may be based on data from a previous study or on a meta-analysis of multiple studies. It may also be that only expert opinion is available to help assess the effect size.

Depending on how precise the prior knowledge is, the prior distribution is termed non-informative or informative. This determines to what extent the prior knowledge influences the result of Bayesian analysis. A prior distribution can be skeptical or optimistic, depending on the probability of a positive effect. If necessary, multiple analyses are conducted to cover the spectrum from a skeptical to an optimistic prior distribution. In looking for prior data it is critical to ensure they correctly match the new study. A prior-data mismatch will lead to distorted or needlessly inaccurate results. After compilation of optimally matched prior data, the further procedure should include additional safeguarding against a mismatch. Rather than using one single fixed prior distribution, the prior data are incorporated in dynamic fashion (“Bayesian dynamic borrowing”). The closer the prior data match the new study, the greater their weight. One method for dynamic weighting of the prior data is a prior distribution consisting of a blend of at least two distributions (“mixture prior”). A mixture of an informative and a non-informative distribution is a robust mixture prior; such a mixture will be used in the following example (11, 12).

Further conclusions, in addition to the graphic depiction, can be derived from the posterior distribution of the model parameters. One such is the Bayesian estimator, i.e., the value of the model with the highest posterior density, the expected value, the median, and the quantiles of the posterior distribution. The area under the posterior distribution in the area of the null hypothesis gives the posterior probability of the null hypothesis. A credible interval is an estimate of the interval of the model parameter and means, for example, that after observation of the data the probability of the odds ratio lying between 1 and 2 is 95%. All of these conclusions regarding the probability of particular values of model parameters can be reached using Bayesian procedures but not with frequentist statistics.

Methods and application

The PLUTO trial is used here to exemplify the methodology of Bayesian statistics (3). In this trial, intravenous administration of the substance belimumab was investigated in 93 young patients (age range 5 to < 18 years) with systemic lupus erythematosus. Fifty-three patients were treated in the intervention group, in 28 of whom treatment was successful in terms of the primary endpoint (success rate 28/53 = 52.8%). In the control group, comprising 39 persons, the success rate was 17/39 = 43.6%. The (unadjusted) odds ratio was therefore 1.45 with a 95% confidence interval (95% CI) of [0.63; 3.33]. Although the observed data show that the success rate was higher in the intervention group than the control group, the case numbers are too small for a confirmatory statistical test with adequate power. In order to obtain meaningful results, the PLUTO observations were supplemented with prior data from two studies of adult patients. The pooled success rate of the intervention was 285/563 = 50.6% versus 218/562 = 38.8%, yielding an odds ratio of 1.62 (95% CI [1.27; 2.06]). Figure 1a shows the probability density of the odds ratio for the adult data (violet density). The probability distribution shown here could be used as maximally informative prior distribution. However, this would presuppose that the prior data from studies on adults are completely valid for children and adolescents. Rather than relying on this assumption, violation of which would lead to distorted results, it is advisable to rule out a prior data mismatch by using a robust mixture prior, in which the maximally informative component is accompanied by a unit information prior. The latter was centered on an odds ratio of 1 and its spread selected such that its weight in the analysis is equivalent to that of a single observation in minors, to represent uncertainty (Figure 1a, green density). The proportion of the informative component in the mixture indicates to what extent the knowledge of the odds ratio in adults can be extrapolated to children and adolescents. The proportion can be assumed to be 50%, to express indecision. This yields the posterior distribution of the odds ratio depicted in Figure 1b. According to the mixture prior (Figure 1a, gray density), a priori the odds ratio has a 95% probability of being between 0.009 and 114. The posterior distribution gives a Bayesian estimator of 1.62, an expected value of 1.58, and a median of 1.60. The area under the curve corresponds to the probability that the odds ratio lies in this region. The probability of an odds ratio in excess of 1 is 97.4% (Figure 1b, pink area). With a probability of 95% the odds ratio can be found in a credible interval of 0.99 to 2.18. The confidence interval is wider, at 0.63 to 3.33. Use of the prior data has resulted in more accurate estimation of the odds ratio. The robust mixture prior enables Bayesian dynamic borrowing. The more transferable the prior data are, the more they feed into the result. To illustrate this, consider a theoretical study on children and adolescents in which the medication shows no effect. In this case the prior data on adults are not transferable to minors. Figure 1c shows the posterior distribution of the odds ratio in such a study. The observed data override the prior data, and the result is negative. The probability of an odds ratio exceeding 1 is only 89.2% (Figure 1c, pink area) and thus below the normally demanded 97.5%.

Figure 1

Bayesian analysis of the odds ratio

The guidelines for the use of Bayesian procedures stipulate that the prior distributions to be used should be defined at the study planning stage, before data analysis (13). For mixture priors, this means determining the percentages of the different components—in the case of the PLUTO trial, deciding to what extent the prior data from adults can be extrapolated to minors. Instead of this, the PLUTO authors, in consultation with the responsible licensing authority, conducted repeated analyses in which the transferability was assigned all possible values between 5% and 95%. Figure 2 shows the resulting credible intervals for the odds ratio. If the 95% credible interval is entirely over 1, i.e., the probability that the odds ratio exceeds 1 is at least 97.5%, then according to the conventional criteria a positive effect has been proved. This is the case with transferability of 55% or more. In the discussions with the licensing authority, this degree of transferability from adults to minors was deemed plausible. The medication was concluded to have a positive treatment effect in children and adolescents and was therefore licensed for use.

Figure 2

If-then diagram

Key points and discussion

Bayesian procedures offer flexible options for integration of existing knowledge into statistical analysis, provided the prior data were obtained using a similar means of evaluation of the same variables. The results of Bayesian analysis represent direct quantitative statements of the probability of unknown parameter values and of the validity of meaningful, readily interpretable hypotheses. When it comes to statistical testing of hypotheses, frequentist quality criteria such as adherence to the significance level and the power are not guaranteed and must be individually verified.

The use of prior data, providing they are capable of extrapolation, leads to improvement of a new study, e.g., more accurate assessment of the treatment effect. However, the unevaluated incorporation of unsuitable prior data into an ongoing study can lead to distorted or needlessly inaccurate results. This would be the case if informative prior distributions were used without robustification. It may therefore be viewed as beneficial for every study to stand for itself and be conducted open-mindedly with no regard to previous information. Thus, there is a fundamental argument in favor of taking advantage of prior knowledge, but also a counter-argument. These two perspectives are not incompatible. In specific situations, such as rare diseases or extremely urgent cases, trials can be brought sufficiently swiftly to a meaningful conclusion only if the study data are supplemented with existing knowledge. The guidelines of the European Medicines Agency for clinical trials involving children specifically mention pediatric extrapolation, that is the transfer of data from a reference (adult or other pediatric) population to a pediatric target population (25, 26).

Bayesian analyses often require more time and effort than frequentist statistical analyses. The large variety of options means it is not often possible to resort to standard procedures. We therefore recommend consulting an expert in Bayesian statistics.

Conflict of interest statement
RV is leader of the Bayesian Methods Study Group of the German Region of the International Biometric Society (a voluntary honorary position).

JWOG declares that no conflict of interest exists.

Manuscript received on 26 August 2024, revised version accepted on 14 February 2025.

Translated from the original German by David Roseveare.

Corresponding author
PD Dr. rer. pol. habil. Reinhard Vonthein

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