Bayes factors for superiority, noninferiority, and equivalence designs
Abstract
Background
In clinical trials, study designs may focus on assessment of superiority, equivalence, or noninferiority, of a new medicine or treatment as compared to a control. Typically, evidence in each of these paradigms is quantified with a variant of the null hypothesis significance test. A null hypothesis is assumed (null effect, inferior by a specific amount, inferior by a specific amount and superior by a specific amount, for superiority, noninferiority, and equivalence respectively), after which the probabilities of obtaining data more extreme than those observed under these null hypotheses are quantified by pvalues. Although ubiquitous in clinical testing, the null hypothesis significance test can lead to a number of difficulties in interpretation of the results of the statistical evidence.
Methods
We advocate quantifying evidence instead by means of Bayes factors and highlight how these can be calculated for different types of research design.
Results
We illustrate Bayes factors in practice with reanalyses of data from existing published studies.
Conclusions
Bayes factors for superiority, noninferiority, and equivalence designs allow for explicit quantification of evidence in favor of the null hypothesis. They also allow for interim testing without the need to employ explicit corrections for multiple testing.
Keywords
Bayes factors Clinical trials Statistical inference Noninferiority designsAbbreviations
 BF
Bayes factor
 CBT
Cognitive behavior therapy
 CI
Confidence interval
 ICBT
Internetdelivered cognitive behavior therapy
 MADRS
MontgomeryAsberg depression rating scale
 NHST
Null hypothesis significance test
Background
The first, and by far most common, type of design is the superiority design (see top two rows). In the superiority design, the null hypothesis is that the true population effect size is exactly zero. The test can typically be conceived as being onetailed, even though in practice superiority analyses often employ a twotailed test. In other words, the null hypothesis states that a new medicine or other intervention being tested does not work better than an existing placebo or active control. The first row in Fig. 1 provides an example of a superiority design in which the null hypothesis was rejected and the second row in Fig. 1 provides an example of a superiority design in which the null hypothesis was not rejected.
The second type of design is the noninferiority design (see middle two rows). In the noninferiority design, the null hypothesis is that the true population effect size is lower than −c. This amounts to a onetailed test in which a pointnull hypothesis of effect size =−c is compared to an alternative hypothesis of effect size >−c. In other words, the relevant test is that a new medicine or other intervention being tested works better than an existing placebo or medication minus an apriori determined amount c. The third row in Fig. 1 provides an example of a noninferiority design in which the null hypothesis was rejected and the fourth row in Fig. 1 provides an example of a noninferiority design in which the null hypothesis was not rejected. Note that it is possible for an intervention to be deemed noninferior, but simultaneously lower than zero (in this case, the constructed confidence interval would fall between −c and zero in its entirety).
The third type of design is the equivalence design (see bottom two rows). In the equivalence design, one essentially carries out two NHSTs. In this design, the null hypotheses are that the true population effect size is lower than −cand higher than c. This amounts to a onetailed test in which a pointnull hypothesis of effect size =−c is compared to an alternative hypothesis of effect size >−cand a onetailed test in which a pointnull hypothesis of effect size =c is compared to an alternative hypothesis of effect size <c. If both of these null hypotheses are rejected, equivalence is established. Graphically speaking, equivalence is established if the confidence interval falls in its entirety between the borders of −c and c. The fifth row in Fig. 1 provides an example of an equivalence design in which both null hypotheses were rejected and the sixth row in Fig. 1 provides an example of an equivalence design in which at least one of the two null hypotheses was not rejected. Analogous to noninferiority designs, it is possible for an intervention to be deemed equivalent, but simultaneously different from zero (in this case, the constructed confidence interval would either fall between −c and zero in its entirety or between zero and c in its entirety). Study results can be interpreted very differently depending on whether the original design was a superiority or an equivalence design [3].
Each of these designs seeks to answer important questions. Unfortunately, the NHSTs employed to carry out statistical inference do not allow researchers to quantify evidence in favor of the null hypothesis. The desire to quantify evidence in favor of the null hypothesis is perhaps most relevant in equivalence designs. We quote Greene, Concato, and Feinstein [4], who say: “…Methodological flaws in a systematic review of 88 studies claiming equivalence, published from 1992 to 1996. Equivalence was inappropriately claimed in 67% of them, on the basis of nonsignificant tests for superiority. Fiftyone percent stated equivalence as an aim, but only 23% were designed with a preset margin of equivalence. Only 22% adopted appropriate practice: a predefined aim of equivalence, a preset Δ, consequent sample size determination, and actually testing equivalence.” A nonsignificant pvalue (any p>.05) can result from (1) the null hypothesis being true or (2) the null hypothesis being false combined with an underpowered trial (that is, if we would have collected more data, the results of our inference would have been statistically significant see [5]. In medical research, it is important to distinguish between these two scenarios. Quantifying evidence in favor of the null hypothesis potentially leads to a reduction in the waste of scarce research resources, as research into ineffectual interventions can be discontinued [6].
Another problem with NHST emerges when there is multiple testing in interim analyses. In biomedicine, a range of methods exists that are employed to account for sequential testing and interim analyses, and they all basically change the level of statistical significance by asking for more stringent statistical thresholds to reject the null hypotheses when multiple analyses due to sequential testing or interim reassessments are performed. However, these correction methods are not always applied. Furthermore, the number of participants tested in clinical trials often changes relative to the number decided upon apriori based on interim analysis results [7]. Both of these practices lead to an overestimation of the evidence in favor of an effect.
Bayesian methods are an alternative to NHST that allow quantification of evidence in favor of the null hypothesis, sequential testing, and comparison of strength of evidence across different studies [8, 9]. Bayesian methods are increasingly considered for more widespread use in clinical trials (see e.g., [10]; for an overview of different fields, see [11]) and their advantages have been argued many times (e.g., [12, 13, 14, 15, 16, 17], but see [18]). Several approaches to carrying out Bayesian inference exist, but for the remainder of this manuscript we will focus on the Bayes factor [19, 20]. The Bayes factor allows for explicit quantification of evidence in favor of the null hypothesis, which means that the interpretational pitfalls associated with noninferiority and equivalence designs naturally disappear.
In the case of equivalance designs, traditional methods require specification of a potentially arbitrary band around zero, even if clear theoretical grounds for the width of this band are lacking. Bayes factors can quantify evidence in favor of a point null hypothesis or in favor of an interval null hypothesis, depending on which one is theoretically appropriate.
Bayes factors also allow for sequential testing without having to correct for multiple testing (see e.g. the simulation results reported in [21]). It is “…entirely appropriate to collect data until a point has been proven or disproven, or until the data collector runs out of time, money, or patience” [22], but see [23]. To put this quote into perspective, NHST has essentially one decision criterion (i.e., p<α). As such, if one employs sequential testing, every additional test increases the chance that this criterion is reached, even if the null hypothesis is true (see Table 1 in [24]). Bayesian testing does not require a fixed n in the sampling plan because the decision criterion is symmetrical. If one were to decide, for instance, to test until the relative evidence for one hypothesis over the other is at least ten, one would stop when the evidence provided by the data is ten over one in favor of the alternative hypothesis or ten over one in favor of the null hypothesis, and one would be wrong once for every ten times one were correct. The Bayes factor will provide progressively stronger relative support for the hypothesis that is true when data continues to be collected.
In what follows, we will describe how to implement Bayes factors for the three types of study design mentioned above.
Methods
The Bayes factor
In this equation, p(yM) is the marginal likelihood of the data, a constant that does not involve θ. The posterior p(θy,M) is a mathematical product of prior knowledge p(θM) and the information coming from the data p(yθ,M); hence, the posterior contains all that we know about θ (under model M) after observing the data y.
In the same way, one can calculate the posterior probability of H_{1}, p(H_{1}y). These quantities require specification of the null hypothesis H_{0} and the alternative hypothesis H_{1}. A common choice is to specify the hypotheses in terms of effect size [27]. The null hypothesis then becomes H_{0}:δ=0 and the alternative hypothesis becomes H_{1}:δ≠0 (or, alternatively, H_{1}:δ<0 or H_{1}:δ>0).
which shows that the change from prior odds of the hypotheses p(H_{1})/p(H_{0}) to posterior odds of the hypotheses p(H_{1}y)/p(H_{0}y) is given by the ratio of marginal likelihoods p(yH_{1})/p(yH_{0}), a quantity known as the Bayes factor, or BF [19, 20].
To see how Bayes factors may be obtained for point null hypotheses, it is illustrative to first consider the calculation of a Bayes factor for interval hypotheses. Let H_{0} be that the population effect size falls in an interval around zero: −c<δ<c and let H_{1} be that the population effect size does not fall in that interval: δ<−c or δ>c. We obtain the Bayes factor by calculating (p(H1data)/p(H1))/(p(H0)/p(H0data)). The smaller one chooses c (and therefore the interval around zero), the more p(H0data)/p(H0) will dominate in the calculation of the Bayes factor, as p(H1data)/p(H1) will tend to 1. In the limit of a point null hypothesis, one can get the Bayes factor by calculating p(H0)/p(H0data), or by evaluating the ratio of the density of the prior and the posterior, evaluated at δ=0. This way of calculating the Bayes factor for point null hypotheses is known as the SavageDickey procedure, see [28] for a mathematical proof. Alternatively, one could calculate Bayes factors for a point null hypothesis over a point alternative hypothesis (say δ=0.25), based on prior study results or theoretical grounds [23].
Bayes factors represent “the primary tool used in Bayesian inference for hypothesis testing and model selection” [29, p. 378]; Bayes factors allow researchers to quantify evidence in favor of the null hypothesis vis à vis the alternative hypothesis. For instance, when a Bayes factor BF_{10}=10, with the subscript meaning the alternative hypothesis over the null hypothesis, the observed data are 10 times more likely to have occurred under the alternative hypothesis than under the null hypothesis. When BF_{10}=1/10=0.1, the observed data are ten times more likely to have occurred under the null hypothesis than under the alternative hypothesis. As for interpreting the strength of evidence as quantified by a Bayes factor, an oftenused standard is described in [30]. The authors classify a Bayes factor between 1 and 3 (or, conversely, between 1/3 and 1) as ‘not worth more than a bare mention’, a Bayes factor between 3 and 20 (or, conversely, between 1/20 and 1/3) as ‘positive’, and a Bayes factor between 20 and 150 (or, conversely, between 1/150 and 1/20) as ‘strong’.
Foundational work on choosing appropriate priors for calculating Bayes factors has been done by Jeffreys [19] and the resulting ‘default’ Bayes factor remains to this day one of the most popular approaches to obtaining Bayes factors. We will briefly describe the default Bayes factor, then discuss more recent extensions to this work [27, 31].
The default Bayes factor and implementations
Jeffreys’ [19] work applies to situations where the two hypotheses to be compared break down into a hypothesis that assigns a single value to the parameter of interest and a hypothesis that specifies a range of values to the parameter of interest. In biomedicine, the practical analogue of this is a point null hypothesis that specifies δ=0, where δ is an effect size parameter, and an alternative hypothesis that may specify δ<0,δ>0, or δ≠0.
Jeffreys [19] chose a Cauchy prior distribution with location parameter 0 and scale parameter 1 for the effect size δ parameter. This choice was motivated by the fact that it led to a Bayes factor of exactly 1 in case of completely uninformative data, and on the fact that the Bayes factor would tend to infinity or 1/infinity when the data are overwhelmingly informative. Mathematically, this Cauchy prior corresponds to a normal prior with a mean μ_{δ} of zero and a variance g that itself follows a scaled inverse chisquare distribution with one degree of freedom, in which the variance is integrated out [32, 33]. It is important to note that Jeffreys’ choice of prior was largely motivated by practical reasons, he had no philosophical objections to more informed priors. An extensive discussion of desiderata related to the choice of objective prior distributions may be found in [34].
The impact of Jeffreys’ default Bayes factor had been mostly theoretical until quite recently. An online tool was developed to calculate default Bayes factors for diverse ttest designs ([27], available at http://pcl.missouri.edu/bayesfactor). This same group also created the BayesFactor package for the statistical freeware program R [35]. An alternative group, focusing more on informative hypothesis testing, developed the Bain package for the statistical freeware program R [36]. Specialized point–and–click computer software was created for the explicit purpose of doing Bayesian analyses [37] which incorporates many features from the BayesFactor and Bain packages.
where n is the sample size, μ_{δ} and g are the mean and standard deviation of the original effect size prior distribution, t is the ttest statistic, Γ denotes the Gamma function, _{1}F_{1} denotes the confluent hypergeometric function, and r denotes the scale parameter of the Cauchy distribution. This expression allows making modifications to the prior distribution, such as increasing (decreasing) the scale parameter r for fields in which high effect sizes are more (less) frequent and shifting the center of the prior distribution away from zero for the implementation of Bayes factors in noninferiority designs.
Results
In the next subsections, we discuss calculating Bayes factors specifically for superiority and equivalence designs (for which the procedure is essentially identical) and noninferiority designs. We provide worked examples of reanalyses of real data from publications of clinical trials for each of these to highlight the calculation of these Bayes factors, as well as to provide insight into the merits of this approach over more conventional analyses. Annotated code for conducting these reanalyses is available at https://osf.io/8br5g/.
Bayes factors for superiority designs
For superiority designs, the null hypothesis is defined as δ=0. In order to evaluate this null hypothesis, we can use the Cauchy prior distribution for effect size δ, centered on zero. Ample examples of this approach have been reported elsewhere [12, 38]. Here, we will illustrate this approach with a reanalysis of data reported in [39].
Superiority of racemic adrenaline and ondemand inhalation with acute bronchiolitis
In Skjerven et al. [39], the authors examine the comparative efficacy of adrenaline inhalation by means of bronchodilators versus control (saline inhalations). Specifically, they test for superiority of racemic adrenaline over inhaled saline. In a separate hypothesis, the authors examine whether administration on a fixed schedule is superior to administration on demand. In both cases, the primary outcome is the length of stay in the hospital in hours. The authors conclude that “In the treatment of acute bronchiolitis in infants, inhaled racemic adrenaline is not more effective than inhaled saline. However, the strategy of inhalation on demand appears to be superior to that of inhalation on a fixed schedule.” The authors support their first conclusion with a pvalue of.42 and their second conclusion with a pvalue of.01. Note that the pvalues reported by the authors suggest the performed tests were twosided, although the study goals are more consistent with a onesided test. In what follows, we report both a one and twosided reanalysis.

Obtain the standard error, SE_{treat}, from the 95% confidence interval reported in Table 2 of Skjerven et al. [39]: SE_{treat}=11/1.966≈5.6

Calculate the tstatistic for the nullhypothesis that the difference in estimated length of stay between patients that inhaled racemic adrenaline and patients that inhaled saline is zero: \(t = \frac {63.668.1}{5.6} = 0.80\) (which yields a twosided pvalue of.42).

We use Eq. 4 to calculate a onesided Bayes factor quantifying the relative likelihood of the onesided alternative of superiority, d<0, versus the null hypothesis of no effect, d=0, given the data (BF_{−0}). This leads to BF_{−0}=0.24 (or BF_{0−}=4.23), indicating that the nullhypothesis is over 4 times more likely than the onesided alternative, given the data. The corresponding Bayes factor for a twosided test is BF_{10}=0.15 (or BF_{0−}=6.64), indicating that the nullhypothesis is over 6 times more likely than the twosided alternative, given the data.
These and other superiority Bayes factors can be obtained by providing values for the confidence interval margin, sample size n, and group means to the script: CI_{mar}=(15.5−(−6.5))/2,n_{1}=203,n_{2}=201,M_{1}=63.6, and M_{2}=68.1 (further details can be found in the annotated code). This reanalysis corroborates the finding of the original authors, who found no significant difference between racemic adrenaline and inhaled saline. The Bayes factor indicates that the null hypothesis is a little over four times more likely than the onesided alternative of superiority, given the data.
A similar reanalysis for the superiority of fixed schedule inhalation over inhalation on demand yields a onesided Bayes factor BF_{0−}=31.48 indicating that the null hypothesis is over 31 times more likely than superiority of fixed schedule inhalation over inhalation on demand, given the data. In the sample data, the trend is actually in the direction indicating superiority of inhalation on demand over fixed schedule inhalation. Given that the onesided test compares two inappropriate hypotheses, we consider the results of a twosided test more appropriate here. The Bayes factor in favor of a twosided alternative, BF_{10}, equals 2.24 (recall that [30] classify Bayes factors lower than 3 as not worth more than a bare mention). This finding tempers the conclusion of the original authors: although the data is slightly more consistent with the twosided alternative hypothesis than with the null hypothesis, the Bayes factor suggests that the evidence is ambiguous and that more study is needed.
In sum, we have seen that Bayes factors can augment interpretation of the statistical evidence for superiority designs in important ways: we can quantify the strength of evidence of one hypothesis relative to another one; and we can explicitly quantify evidence in favor of the null hypothesis. The latter is particularly important for the evaluation of equivalence designs, to which we now turn.
Bayes factors for equivalence designs
The objective of equivalence designs is to show that “the new treatment is at least as good as (no worse than) the existing treatment” [1]. Under a classical NHST approach, it is not possible to test for equivalence directly (the null hypothesis cannot be confirmed). As a result, equivalence needs to be tested by proxy by constructing a band around δ=0 of 2c and evaluating two null hypotheses: δ=−c and δ=c.
From a Bayesian perspective, the procedure is similar to that of the procedure for superiority designs. Instead of examining the Bayes factor’s strength of evidence in favor of H_{1}, we now examine the strength of evidence in favor of H_{0}. This removes the ambiguity associated with the traditional approach to equivalence testing. Examine for instance the example where equivalence was demonstrated in Fig. 1 (the fifth row). Equivalence was established, because both δ=−c and δ=c are rejected (i.e., the confidence interval lies fully between these two boundaries). However, the confidence interval does not overlap with δ=0, suggesting that the effect size is not zero, which is a counterintuitive conclusion to draw simultaneously with the conclusion of equivalence.
Note that it is possible to calculate a Bayes factor for the same band around δ=0 of 2c, but there is no need as the evidence in favor of δ=0 can be quantified directly. Because of this, the Bayes factor approach simplifies testing for equivalence, such that no arbitrary band needs to be established. Furthermore, one is allowed to make claims about the absence of an effect, something that is not possible with the conventional NHST approach. We will illustrate this approach with a reanalysis of data reported in [40].
Equivalence between short and longterm storage of redcells on the Multiple Organ Dysfunction Score
In Steiner et al. [40], the authors examine the properties of the duration of storage for redcells intended for transfusion. The authors assert that there is considerable uncertainty about potentially deleterious effects of longterm storage of redcells before transfusion. In this study, the authors examine whether there are differences on the Multiple Organ Dysfunction Score (MODS) between patients that receive redcells for transfusion that have been stored a short time (10 days or less) versus a long time (21 days or more). Although the authors do not explicitly conduct an equivalence design, the implicit goal seems to be to test whether or not longer storage of red cells is harmful. The authors conclude that “duration of redcell storage was not associated with significant differences in the change in MODS”. The authors support this claim with a pvalue of 0.44.
The application of the conventional NHST does not allow us to make any definite claims about the absence of a difference. In this demonstration, we reanalyze these data and calculate a Bayes factor to quantify the strength of evidence for equivalence provided by the data. For the analyses, we make use of the data presented in Table 2 of Steiner et al. [40]. In this table, the means are rounded to one decimal. To approximate the original analysis as accurately as possible, we work with means of 8.516 and 8.683 (reported means are 8.5 and 8.7, respectively) to approximate the reported pvalue as closely as possible. For calculation of the Bayes factor, we assume a Cauchy prior centered on δ=0.

Calculate the tstatistic for the nullhypothesis that the difference in MODS scores between patients that were administered redcells that were stored short versus long is zero: \(t = \frac {8.516  8.683}{3.6\sqrt {1/538+1/560}} = 0.77\).

We use Eq. 4 to calculate a twosided Bayes factor quantifying the relative likelihood of the hypotheses d=0 versus d≠0 given the data (BF_{01}). This leads to BF_{01}=11.04, indicating that the nullhypothesis is over 11 times more likely than the twosided alternative, given the data.
These and other equivalence Bayes factors can be obtained by providing values for the sample size n, group means, and group sds to the script: n_{1}=538, n_{2}=560, M_{1}=8.516, and M_{2}=8.683, sd_{1}=3.6, and sd_{2}=3.6 (further details can be found in the annotated code). This reanalysis corroborates the finding of the original authors, but allows us to go beyond the original claim by stating we have found evidence in favor of equivalence between shortterm and longterm storage of redcells as far as MODS scores are concerned. The Bayes factor lies between 3 and 20, which may be interpreted as positive evidence in favor of equivalence.
Steiner et al. [40] do not provide an equivalence margin, but it is important to stress that if they had, a Bayes factor for the relative likelihood of the population parameter being inside versus outside of this equivalence band can easily be calculated as well. Say, for instance, that c=0.05, then the twosided Bayes factor quantifying the relative likelihood of the nul hypothesis −c<d<c versus the alternative hypothesis d<−c or d>c given the data is 19.09.
Bayes factors for noninferiority designs
The NHST approach for noninferiority houses some unfortunate inconsistencies. Take for instance the top confidence interval in the left panel. This is an example of a situation where the nullhypothesis of inferiority gets rejected. From an NHST perspective, there is nothing wrong with this conclusion, as the confidence interval overlaps with zero, making the conclusion “noninferior” warranted within that framework.
By contrast, examine the middle confidence interval in the left panel. Again, the nullhypothesis of inferiority gets rejected. This time, the implications are a bit less clear, because the confidence interval does not overlap with zero. From an NHST perspective, one would simultaneously reject the inferiority hypothesis and a classical onesided nullhypothesis, reaching opposite conclusions. This makes it somewhat unclear if the conclusion “noninferior” is really warranted here.
Finally, the bottom confidence interval in the left panel shows the scenario where the “inferior” nullhypothesis cannot be rejected. From an NHST perspective, we are unable to draw any further conclusions: is the drug/treatment inferior, or was the trial underpowered?
The Bayesian approach is not hampered by these pitfalls in interpretation. A Bayesian is concerned with the two hypotheses depicted in the right panel of Fig. 2. In [41] Bayesian approaches for noninferiority trials are discussed (see also [42, 43, 44]), but discussion of the implementation of Bayes factors is limited to dichotomous data [45]. Here, we propose to calculate Bayes factors for continuous data, using the same principle as for superiority and equivalence designs illustrated above. The Bayes factor in this case quantifies the relative likelihood of the data having occurred given inferiority versus the likelihood of the data having occurred given noninferiority.
Analogous to the Bayes factor for superiority/equivalence designs, we use the Cauchy prior distribution for effect size δ, centered on zero. The classical z or tstatistics were evaluated against δ=−c. In order to maintain the theoretical property of the prior being centered on zero as specified in Jeffreys work, we shift the center of the Cauchy prior distribution to c. The easiest way to see why this is so is by imagining adding c to all datapoints, all hypotheses, and all distributions, so that we evaluate the ttest for null hypothesis δ=0. The resulting test statistic will not change, as the data and the hypotheses have shifted by the same amount, but the prior distribution is now centered at c. Equation 4 allows for different specifications (for instance, a prior centered on the noninferiority margin), but for these examples, we will keep the prior consistent across the design types. We will illustrate this approach with two examples. We first reanalyze dichotomous data published in [46], and then reanalyze continuous data published in [47].
Noninferiority of betalactam
In [46], the authors examine antibiotic treatments for patients with clinically suspected communityacquired pneumonia (CAP). Specifically, guidelines recommend supplementing administration of betalactam with either macrolides or fluoroquinolones. The authors state that there is limited evidence that macrolides and/or fluoroquinolones have added benefits over the administration of just betalactam. In this study, the authors “tested the noninferiority of the betalactam strategy to the betalactammacrolide and fluoroquinolone strategies with respect to 90day mortality using a noninferiority margin of 3 percentage points and a twosided 90% confidence interval.” The authors conclude that “the risk of death was higher by 1.9 percentage points (90% confidence interval [CI], 0.6 to 4.4) with the betalactammacrolide strategy than with the betalactam strategy and lower by 0.6 percentage points (90% CI, 2.8 to 1.9) with the fluoroquinolone strategy than with the betalactam strategy. These results indicated noninferiority of the betalactam strategy.”

The critical noninferiority tests compare two proportions. Like the original authors, we use the normal approximation for the sampling distribution of proportions. In all three groups, sample sizes are sufficiently large to make this a safe assumption.

The Bayes factor approach requires specifying the noninferiority margin in terms of effect size Cohen’s d. Cohen’s h for proportions has similar properties to Cohen’s d for continuous data. Converting the 3 percentage points yields a Cohen’s h of \(2*\arcsin (\sqrt {\frac {59+82}{656+739}})2*\arcsin (\sqrt {\frac {59+82}{656+739}.03}) = 0.11\). Going forward, we will refer to this value as h.

The equation we use to calculate the relevant Bayes factors, Equation 4, assumes a ttest statistic. For these sample sizes, the tstatistic is virtually indistinguishable from the Zstatistic provided by the normal approximation.

Calculate the Zstatistic for the nullhypothesis that the difference in proportions of mortality in the betalactam group and the betalactammacrolide group is.03: \(Z = \frac {59/65682/739.03}{\sqrt {(59+82)/(656+739) \times (1(59+82)/(656+739)) \times (1/656+1/739)}} = 3.16\).

We use Eq. 4 to calculate a onesided Bayes factor quantifying the relative likelihood of the hypotheses h<0.11 versus h=0.11 given the data (BF_{−h}), and to calculate a onesided Bayes factor quantifying the relative likelihood of the hypotheses h=0.11 versus h>0.11 given the data (BF_{h+}).

Finally, we use the principal of transitivity, BF_{−+}=BF_{−h}×BF_{h+}. BF_{−+} quantifies the relative evidence for noninferiority (difference in mortality rate is lower than 3 percentage points) versus inferiority (difference in mortality rate is higher than 3 percentage points), given the data. For these data, BF_{−+}=1307.76, indicating that the noninferiority hypothesis is over 1300 times more likely than the inferiority hypothesis, given the data.
These and other noninferiority Bayes factors for proportions can be obtained by providing values for the sample size n, mortality count k, and the noninferiority margin to the script: n_{1}=656,n_{2}=739,k_{1}=59,k_{2}=82, and NI_{mar}=0.03 (further details can be found in the annotated code). A similar reanalysis for the betalactam versus betalactamfluoroquinolone groups yields BF_{−+}=39.07, indicating that the noninferiority hypothesis is almost 40 times more likely than the inferiority hypothesis, given the data. Thus, our results corroborate those of the original authors, we find noninferiority for betalactam versus betalactammacrolide and betalactamfluoroquinolone. The Bayes factors allow us to make claims about the strength of evidence, with support for noninferiority of betalactam compared to betalactamfluoroquinolone being strong, and support for noninferiority of betalactam compared to betalactammacrolide being overwhelming.
The above example demonstrates calculation of the Bayes factor for noninferiority trials with dichotomous outcome measures. We now turn to a second example of our approach that showcases the application of our method for outcome data that is measured on a continuous scale.
Noninferiority of internetdelivered cognitive behavior therapy
In [47], the authors examine the efficacy of internetdelivered cognitive behavior therapy (ICBT) in the treatment of mild to moderate depression symptoms, specifically by comparing its effectiveness to the ‘regular’ groupbased cognitive behavior therapy (CBT). Depression symptoms are measured with the selfrated version of the MontgomeryAsberg Depression Rating Scale (MADRS). The authors define inferiority as a twopoint difference on the MADRS between CBT and ICBT. The authors assess noninferiority directly posttreatment and in a threeyear followup and conclude that “Results on the selfrated version of the MontgomeryAsberg Depression Scale showed significant improvements in both groups across time indicating noninferiority of guided ICBT.”
Sample sizes in the ICBT and CBT groups are 32 and 33 respectively posttreatment and 32 and 30 respectively in the three year followup. In this demonstration, we reanalyze these data and do two Bayesian tests for noninferiority. For the analyses, we make use of the data presented in Table 2 of Andersson et al. [47].

The Bayes factor approach requires specifying the noninferiority margin in terms of effect size Cohen’s d. Converting the 2 point difference yields a Cohen’s d of \(d_{post} = 2/\sqrt {\frac {31*9.8^{2}+32*8^{2}}{63}} \approx 0.22\) for the posttreatment group and \(d_{3} = 2/\sqrt {\frac {31*7.6^{2}+29*8.7^{2}}{60}}\approx 0.25\) for the three year followup group.

Calculate the tstatistic for the nullhypothesis that the difference in MADRS scores in the ICBT group and the CBT groups is 2: \(t_{post} = \frac {13.6  17.1  2}{\sqrt {\frac {31*9.8^{2}+32*8^{2}}{63}} \times \sqrt {1/32+1/33}} = 2.48\).

We use Eq. 4 to calculate a onesided Bayes factor quantifying the relative likelihood of the hypotheses d_{post}<0.22 versus d_{post}=0.22 given the data (BF_{−d}), and to calculate a onesided Bayes factor quantifying the relative likelihood of the hypotheses d_{post}=0.22 versus d_{post}>0.22 given the data (BF_{d+}).

Finally, we use the principal of transitivity, BF_{−+}=BF_{−d}×BF_{d+}. BF_{−+} quantifies the relative evidence for noninferiority (difference in depression scores is lower than 2 points) versus inferiority (difference in depression scores is higher than 2 points), given the data. For these data, BF_{−+}=90.52, indicating that the noninferiority hypothesis is over 90 times more likely than the inferiority hypothesis, given the data.
These and other noninferiority Bayes factors for continuous data can be obtained by providing values for the sample size n, group means, group sds, and the noninferiority margin to the script: n_{1}=32,n_{2}=33,M_{1}=13.6, and M_{2}=17.1,sd_{1}=9.8,sd_{2}=8, and NI_{mar}=2 (further details can be found in the annotated code). A similar reanalysis for the three year followup noninferiority test yields BF_{−+}=353.61. Thus, our results corroborate those of the original authors, we find noninferiority for ICBT versus CBT directly after treatment and in a threeyear followup. Note that despite the relatively small sample size, the Bayes factors quantifying strength of evidence in favor of noninferiority are substantial, highlighting one of the advantages of quantifying evidence with Bayes factors: a clear measure of the strength of evidence for one hypothesis relative to another that can be used to compare evidence across studies.
Discussion
In this paper, we showed worked examples of the application of default Bayes factors to superiority, noninferiority, and equivalence designs. In each of these cases, we believe that application of Bayes factors brings significant advantages. For superiority and equivalence designs alike, it is possible to explicitly quantify evidence in favor of the null hypothesis. For equivalence studies, specification of a potentially arbitrary band of equivalence is no longer necessary. For noninferiority and equivalence designs alike, the interpretational hazard of simultaneously claiming noninferiority/equivalence on one hand, but rejecting the null hypothesis of an effect size of zero on the other hand, disappears. The Bayes factor offers a way to quantify each of these types of evidence in a compelling and straightforward way.
The two panels demonstrate a twosided Bayes factor (left) and a onesided Bayes factor (right), calculated based on the same hypothetical prior (N(0,1)) and posterior (N(1,0.75)) distributions for effect size. In both cases, the Bayes factor is obtained by dividing the density of the prior, evaluated at zero, by the density of the posterior, evaluated at zero (i.e., the black dot divided by the red dot). For the onesided Bayes factor, both distributions are truncated at zero. Because both distributions are normalized to have a density of 1, the effect of this truncation is especially strong for a distribution that falls almost entirely inside the truncated region, such as in the posterior distribution of our example data here and in the [39] superiority design data we reanalyzed. As a result, the Bayes factors in Fig. 3 lead to opposite conclusions, depending on whether the test was designed to be onetailed or twotailed. This example demonstrates that it is crucial to think about the hypotheses one wishes to test and the direction of testing before one obtains the data. Similar considerations apply when testing within the classical NHST framework.
Secondly, in NHST the status of α=0.05 is well established as a cutoff for significance (but see citeBenjaminEtAl2018). Bayesian inference does not have such universally agreed upon decision thresholds. Although different suggestions are offered in the literature [19, 30, 48], the authors caution against too rigid interpretation of these labels. We would argue that every cutoff value one chooses is to some extent arbitrary. With Bayes factors, one can at least choose a symmetrical cutoff score (for instance, we test until one hypothesis is 20 times more likely than the other given the data, so BF_{10}=20 or BF_{10}=1/20=0.05), whereas no such symmetry can be obtained with a pvalue.
Thirdly, there are different ways to calculate Bayes factors [45]. Arguably the most important determinant for differences in Bayes factors stem from the choice of the underlying prior. Taking as an example the category of Bayes factors that assume a prior distribution on effect size, a prior that places a relatively high weight on an effect size of zero (i.e., is tightly peaked around zero), will lead to a relatively large Bayes factor in favor of the alternative hypothesis if the sample effect size is relatively different from zero. For reasonable priors, the effect of the choice of prior on the Bayes factor appears to be mostly quantitative and unlikely to alter the qualitative conclusions [31]. Nevertheless, in specific applications, these default prior analyses can be supplemented by substantive knowledge based on earlier experience. With a more informative prior distribution, the alternative hypothesis will make different predictions, and a comparison with the null hypothesis will therefore yield a different Bayes factor. The more informed the prior distribution, the more specific the model predictions, and the more risk the analyst is willing to take. Highly informed prior distributions need to be used with care, as they may exert a dominant effect on the posterior distribution, making it difficult to “recover” once the data suggest that the prior was illconceived. With informed prior distributions, it is wise to perform a robustness analysis to examine the extent to which different modeling choices lead to qualitatively different outcomes.
Conclusions
Our paper offers an easy way of calculating Bayes factors for superiority, equivalence, and noninferiority designs that is consistent across methods and scale of the outcome measure. With increasing accessibility of software aimed to conduct Bayesian inference [37], the absence of tools necessary to obtain Bayes factors is no longer a reason to refrain from using Bayesian analyses. We recommend standard consideration of Bayesian inference in clinical trials for obtaining strength of evidence that is consistent across studies.
Notes
Acknowledgements
Not applicable.
Funding
This research was supported by a Dutch scientific organization VIDI fellowship grant to DvR, which supported the writing of the manuscript (016.Vidi.188.001).
Availability of data and materials
Code for reproducing the analyses reported in the manuscript may be obtained from https://osf.io/8br5g/.
Authors’ contributions
DvR drafted the original manuscript and conducted the formal analyses. DvR and JI conceptualized the project. JT, RM, and JI were major contributors in writing the manuscript. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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