A motivating example: how the Target Temperature Management-Trial changed the conclusion of the meta-analysis of trials with cooling of patients after out of hospital cardiac arrest

In TSA, we consider each interim-analysis result, produced after the addition of a new trial, a sequential meta-analysis. The possibility to include groups of several new trials at a time is, of course, also possible. This latter approach will decrease the number of interim-analyses in the cumulative meta-analysis [10]. However, updating the meta-analysis in a systematic review each time a new trial is published is a rational decision, and to update a systematic review before a new trial is initiated ought to become mandatory [13–15]. Previous trial results ought to be considered whenever we evaluate the cons and pros of designing new trials, as the evidence on a given intervention may already be sufficient [13–15]. It is surprising to see how little the TSA, conducted after each new trial has been interim-analysed, differs from the last TSA on groups of trials (e.g., TSA only updated every second year).

Figure 1 shows the result of a TSA of meta-analysis of four trials comparing a target temperature of 33°–34 °C versus no cooling, conducted before the initiation of the Target Temperature Management (TTM) Trial (Fig. 1a) [16–18]. The TSA shows that the four trials did not even reach half of the required information size to confirm or reject a 17% relative risk reduction which was the intervention effect indicated in a conventional meta-analysis of the trials [16]. The conventional confidence interval for the relative risk ratio of all-cause mortality in a traditional meta-analysis is 0.70 to 1.00 (P = 0.05), suggesting a reduction of mortality. The confidence interval and the P-value would not have been sufficient to claim a conclusive interim analysis stopping for benefit in a single randomised trial if analysed with Lan-DeMets’ group sequential monitoring boundaries [19]. For demonstrating a 17% relative risk reduction, the TSA-adjusted confidence interval of the relative risk is 0.63 to 1.12. This confidence interval shows that i) a target temperature of 33°–34 °C versus no cooling can either decrease or increase mortality, and ii) that definitive evidence has not yet been reached. The cumulative Z-curve in the figure does not pass through the trial sequential monitoring boundary for benefit; only the conventional and naïve P = 0.05 (Z = 1.96) level for a beneficial effect has been reached. Therefore, there is not sufficient information to document the effect, or there may not be a beneficial effect at all. Nevertheless, based on this evidence, international guidelines had recommended for ten years that the temperature of comatose cardiac arrest patients should be targeted to 33°–34 °C, calling the intervention »mild therapeutic hypothermia« [20]. No further randomised clinical trials of induced hypothermia versus no temperature control (or normothermia) in comatose cardiac arrest patients after resuscitation and admittance to intensive care units were conducted during this 10-year period. This may indicate that a P-value of 0.05 in the conventional meta-analysis was used as an unofficial »stopping boundary« for further trials within this same period.

In the TTM Trial, we compared the effect of cooling to target temperature 33 °C versus 36 °C on mortality of cardiac arrest patients [17, 18]. The updated TSA including the TTM Trial showed no statistically significant effect at the conventional level, as the Z-curve returned to the area with P > 0.05 (|Z| < 1.96) (Fig. 1b). Figure 1b shows that the cumulative Z-curve touches the futility-boundaries in the TSA diagram (see section ‘False Negative Meta-analyses’ below). Therefore, the updated TSA indicates that a 17% relative risk reduction, or an even greater reduction, most likely can be rejected, although the pre-estimated required information size of 2040 patients has not yet been reached. It is not likely that a meta-analysis will ever show a 17% statistical significant relative risk reduction of mortality, even though the continued conduct of trials until a cumulated number of patients, corresponding to the required meta-analytic information size of 2040 patients, was reached (Fig. 1b). The conclusion is that hypothermia to 33°–34 °C does not seem to have a clinical important effect on mortality compared with no cooling or targeted normothermia (36 °C), as the 17% relative risk reduction only corresponds to a median of 3 weeks’ longer survival [17, 18]. Moreover, the original conventional meta-analysis before inclusion of the TTM Trial had a false positive result; the null hypothesis was falsely rejected. Whether the avoidance of fever is actually beneficial compared with no cooling at all, remains to be tested, as the TTM trial used cooling in both the intervention (target 33 °C) and the control group (target 36 °C).

Interim-analyses during a randomised clinical trial with an accumulating number of participants

A Bonferroni adjustment of the level of statistical significance, being 5% divided with the number of tests on accumulating data, assumes that all tests are conducted on independent data. As the tests on the accumulating trial population are not statistically independent, the Bonferroni-adjusted levels of statistical significance are most often too conservative [24]. The trial participants in an early sequential analysis are also included in the subsequent later sequential analyses. Therefore, there is an increasing overlap of trial participants included in the latest sequential analysis compared to participants included in the previous sequential analyses. The closer we come to the a priori calculated sample size, the Bonferroni adjustment becomes more and more unjustified (too conservative).

Historical development of sequential analyses in a single trial with interim analyses

Methods to avoid an increased risk of a type I error due to repetitive testing on an increasing number of observations was described by Abraham Wald in 1945 in Contributions to the theory of statistical estimation and testing hypotheses [25]. Wald proposed »the sequential probability ratio test« in which the sequential testing continues until a definitive wanted or unwanted effect can be proved [26, 27]. According to this procedure, the trial continues as long as the results of the sequential tests fall within the so-called ‘zone of indifference’ amidst the two alternative hypotheses. This procedure, used as a quality assurance measure of production during the Second World War, has never achieved wide implementation in randomised clinical trials; possibly because the procedure is bound to continue infinitely as long as the true intervention effect lies between the two alternative hypotheses. Consequently, a decision to stop the trial may never become possible [28].

After the Second World War, Peter Armitage suggested more restrictive levels of statistical significance than 5% to stop a trial before the a priori calculated sample size was reached [21]. This procedure was applied in a number of interim analyses of large trials [29]. Furthermore, Stuart Pocock proposed a procedure in which the overall risk of type I error is limited to 5% by setting the statistical significance level to 0.05 divided by k, using k-1 interim analyses and a final analysis [22]. This procedure is identical to the Bonferroni procedure for interim analyses and a final analysis of a single trial [30]. Researchers might find it peculiar to only declare statistical significance if P < (0.05/k), despite the estimated sample size has been reached and the required criterion for statistical independence was not fulfilled.

In 1977, Richard Peto suggested the use of a maximal type I error risk (α-spending) in each of four interim analyses of 0.001 (1 promille) and 0.05 in the final analysis. As this would produce a summary additional type I error risk of 0.004 to the final 0.05, the total type I error risk would maximally be 5.4% [31] (Fig. 2). However, by a modest increase of the a priori estimated sample size, the summary maximal used type I error risk would remain within the usual 5%. As shown above, the rationale of statistical independence required for this procedure still lacks underlying reason as to why the trial participants in an early sequential analysis are also included in the subsequent sequential analysis.

In 1979, Peter O’Brien and Thomas Fleming proposed the group sequential design of trials with interim analyses, using exponential decreasing levels of statistical significance with the increasing number of patients in the sequentially analysed groups (Fig. 2) [32]. The recommendations of the International Conference on Harmonization – Good Clinical Practice, the U.S.A. Food and Drug Administration, and the European Medicines Agency on the design and analysis of randomised trials with interim analyses are mainly based on works from 1980s, primarily prepared by Gordon Lan, Kuyung Man Kim, and David DeMets (Fig. 2) [18, 33, 34]. Their works allow proper sequential testing at any time during the trial period, without unduly increasing the overall risk of a preset nominal type I error risk [34–36].

Avoiding the increased risk of random errors in randomised clinical trials with interim analyses

It is and should be mandatory to perform interim analyses in large randomised clinical trials addressing patient-centred outcomes. Even though the preplanned sample size has not been reached, thousands of patients might already have been randomised in a trial. Before we allow the trial to continue, there is a need to secure that no valid evidence showing superiority of one of the compared interventions exists. If one of the interventions (could also be placebo) with a sufficiently small uncertainty is superior to the other one in an interim analysis, it may be unethical to continue the trial. The explanation for this is that the superiority can be so large that it cannot be reversed even though we continue to randomise patients until the total, originally preplanned sample size is obtained. If the trial is continued despite the superiority of the intervention in one of the intervention groups, the patients in the other group will be exposed to an inferior (harmful) intervention and the trial must be stopped [37]. The use of interim analyses in a single randomised trial has to be planned at the design stage of the trial and protocolised upfront as group sequential analyses in the charter for interim analyses [33]. For the conduct of group sequential analyses, a sample size is calculated already at the design stage, based on the anticipation of a minimal important and realistic intervention effect of the primary outcome of the trial [36, 38] (see Appendix).

The sample size calculation considers the level of statistical significance at which we want to test a dichotomous or a continuous outcome when the full sample size has been reached. It is when the pre-calculated sample size has been reached, and only then, a two-sided P-value of less than 0.05, corresponding to a test-statistic Z-value of ±1.96, can be accepted as the statistical significance level when α has been set to 5% in the sample size calculation.

Interim analyses, with the potential to stop a randomised trial before the estimated (or fixed) sample size has been reached due to a positive, negative, or lack of the addressed effect, can be conducted for dichotomous and continuous outcomes by calculating the cumulative Z _i -value at the i-th analysis (see Appendix). The calculated Z _i -value is then related to the more restrictive level of statistical significance, the critical Z-value being the discrete group sequential boundary according to the actual accrued number of participants.

There is international consensus that the increase of type I error risk with sequential testing, including the risk of overestimating the intervention effect or underestimating the variance, at an interim analysis, should be outweighed by more restrictive levels of statistical significance before the a priori estimated (fixed) sample size has been reached [29, 31–37]. This is why ‘monitoring boundaries’, with significance levels much smaller than a nominal P-value of 0.05 (corresponding to much larger |Z|-values than ±1.96) are applied as criteria to stop a trial before achieving the estimated sample size [33].

Numerical integration is used to calculate the monitoring boundaries, being the critical levels of statistical significance for the Z _i -values (and P-values) of the interim analyses [39]. Most often, the O’Brien-Fleming’s α-spending–function is applied and converted to sequential boundaries (critical values) for the Z _i -values called Lan-DeMets’ sequential monitoring boundaries (Fig. 2) [18, 19]. The α-spending function allows only a small part of the total nominal type I error risk to be used initially in the sequential analyses, and with a modest increase of the estimated final (fixed) sample size, there is a full 5% type I error risk available for the final analysis when the a priori estimated sample size is reached. Lan-DeMets’ sequential boundaries allow testing whenever you want during the trial [34, 35]. If we plan, e.g., a half-way analysis in a randomised trial, we can monitor the P-value at this time point according to Lan-DeMets’ monitoring boundaries and suggest that the trial is stopped if the P-value is less than 0.003 which corresponds to a 99.7% confidence interval excluding 1.00 for a relative risk or 0.00 for a mean difference [34–36]. Therefore, sequential analyses become a theoretical decision tool to decide whether a trial should be stopped before the estimated (fixed) sample size is achieved, considering the sparse data and the repetitive testing during the trial [37].

Avoiding the increased risk of random errors in cumulative meta-analyses with sparse data and multiple meta-analytic up-dates

The majority of meta-analyses include less than the required number of randomised participants and trials in order to become conclusive [1, 3, 5, 7]. There are two reasons for this. First, most randomised trials are underpowered [1, 3, 5, 7]. Second, the estimation of the required information size in a random-effects meta-analysis ought to incorporate the heterogeneity variance (between trial variance) [1, 7, 11]. Only 22% of the meta-analyses in The Cochrane Library have 80% power to conclude whether there is an intervention effect of 30% or not when the usual maximal risks of type I error (α) of 5% and type II error (β) of 20% are applied [1]. This lack of power is primarily caused by small trials and a considerable heterogeneity variance between the estimates of the intervention effect in the included trials [1].

Meta-analyses can be conducted with a fixed-effect model or a random-effects-model [40, 41]. In the fixed-effect model, we assume one true underlying effect in all the included trials. In the random-effects model, we assume that the true underlying effects vary from trial to trial according to a normal or log normal distribution. Often, the fixed-effect assumption is unrealistic as the possible underlying effect may depend on, e.g., doses of a pharmacological intervention, duration of the interventions, timing of the outcome assessment, and differences between the trial populations. These differences between the included trials are called clinical heterogeneity. Due to these factors and possibly random variation, the included effect estimates often show considerable variation defined as statistical heterogeneity and measured as large inconsistency (I²) [42] and large diversity (D²) [11]. Considerable statistical heterogeneity leads to increased uncertainty, expressed as a wider confidence interval of the intervention effect when the meta-analytic estimate is calculated in a random-effects model. Early meta-analyses conducted before the required information size and the corresponding number of trials are achieved [43], often wrongly show unrealistic large intervention effects as well as statistical significance which cannot be reproduced when the amount of required information is adequately considered [44, 45]. The reliability in early meta-analyses is lower compared to their updated counterparts years later [2]; the estimated intervention effects, when further trials are included in the meta-analysis update, become considerably lower than previously estimated [2].

A large simulation study of random-effects meta-analyses shows that there is a considerable risk of overestimating the intervention effect when the required information size has not been reached [6]. These results were based on the assumption that the ‘true’ intervention effect was zero while the frequencies of events in the control groups and the heterogeneity variance were assumed similar to those in large cardiologic meta-analyses [6]. It has been shown empirically that approximately 25% of cardiologic meta-analyses are inconclusive because of lack of power [3]. Turner and colleagues showed that the trials and the meta-analyses of Cochrane systematic reviews have limited power [1]. In Cochrane meta-analyses, each total number of analysed participants provide only 22% of the meta-analyses with an 80% power to detect or refute a 30% relative risk reduction (which is a large intervention effect) [1] (Fig. 3). Recently, Imberger and colleaques confirmed these results in meta-analyses of anaesthesiological interventions [46]. Accordingly, four out of five meta-analyses did not have the statistical power to address even substantial intervention effects. The number of meta-analyses with sufficient power to address smaller and clinically more plausible intervention effects are, undoubtedly, even smaller.

If we test with a constant level of statistical significance (e.g., 5%) on the way towards the required information size, the risk of type I error is increased to more than 5%. The problem for cumulative meta-analyses, due to repeated updating and consecutive calculation of 95% confidence intervals, with inclusion of results from new randomised trials is, therefore, analogous to interim analyses of a single trial [8, 9]. Thus, we, as well as others, recommend that the interpretation of meta-analyses in systematic reviews is done alongside with a sequential analysis, e.g., Trial Sequential Analysis (TSA) [46, 47]. The purpose of using TSA is to avoid the risk of type I and type II errors due to sequential testing on a constant statistical significance level and with inclusion of fewer participants than the required number in order to detect or reject specified effects [7, 10, 11]. It is possible to accommodate Gordon Lan and David DeMets’ group sequential analysis for interim analysis in a single randomised trial to the updating of cumulative meta-analysis as it progresses with the addition of trials. This is done with an appropriate continuous use of type I error risk and an α-spending function of the allowed total nominal type I error risk, so that when the required information size and the required number of trials have been reached and beyond, the risk is kept below 5%. The trial sequential monitoring boundaries generated this way make it possible to test if significance is reached and to adjust the confidence intervals every time new trials are added to the meta-analysis. The latter is a prerequisite for using sequential boundaries in cumulative meta-analyses of trials with varying sample sizes [10, 12].

Besides applying the observed estimate of statistical heterogeneity—the observed statistical diversity (D²) [11, 41] in the most recently conducted meta-analysis—it may be reasonable to apply an expected heterogeneity in the calculation of the required information size, especially when the observed heterogeneity is zero [48]. As it is unlikely that diversity will stay zero when larger trials are added, an expected heterogeneity may be used in a sensitivity analysis (e.g., a diversity of 25% or the upper confidence interval of the I² (provided by the TSA program)) when the required information size is calculated [48, 49]. It may also be wise in a post hoc calculation of the required information size to apply the least likely intervention effect, i.e., the confidence limit of the summary estimate in the meta-analysis confidence interval closest to the null effect. The latter is a conservative approach facilitating the evaluation of whether a meta-analysis may show an effect of the least likely magnitude in a TSA. If a TSA with such an approach shows a statistical significant intervention effect, judged by the TSA-adjusted confidence interval, there is a very high probability that the intervention has an effect, provided that the included trials are at low risk of bias. In contrast, there will only be very low evidence of effect if the TSA-adjusted confidence interval does not exclude the null effect for an intervention effect of a magnitude indicated by the point estimate.

False positive meta-analyses

It is necessary to assume or address a specific magnitude of the intervention effect, different from zero, in order to calculate the sample size in a single trial. Therefore, when a sample size is estimated, we relate not only to the null hypothesis but also to a specific alternative hypothesis. The alternative hypothesis is the assumption or the anticipation of a specific magnitude of the intervention effect different from zero. Most often random-effects meta-analysis will be the preferred appropriate method to estimate the precision weighted average effect as it does not ignore the statistical heterogeneity variance. If statistical heterogeneity is anticipated, the information size in the conclusive meta-analysis ought to be an upward adjusted sample size of a corresponding adequately powered single trial. The upward adjustment is done with the variance expansion shifting from a ‘fixed-effect’ model to a ‘random-effects’ model, see Appendix [11].

The described example from cooling of patients after out of hospital cardiac arrest is far from being unique (Fig. 1). Among meta-analyses of interventions for neonatal patients, there were approximately 25% to 30% false positive results [5, 50]. In 2009, we showed empirically that the use of Lan-DeMets’ trial sequential monitoring boundaries eliminated 25% of the false positive traditional interim-meta-analyses. This analysis included 33 final meta-analyses with sufficient information size to detect or reject a 15% relative risk reduction [44]. In 2013, we showed that 17% of cardiovascular meta-analyses with P < 0.05 were most likely false positive [3]. In 2015, we showed that less than 12% of meta-analyses of anaesthesiological interventions had 80% power to show a 20% relative risk reduction [46].

There may be other important reasons for a traditional meta-analysis to yield a false positive result than only the increased risk of random errors. A risk of systematic error (bias) in the included trials is a frequent cause of overestimation of benefit and underestimation of harm – sequential meta-analyses do not in any way solve problems with bias [51–58]. Therefore, it is recommended that every single trial included in a systematic review with meta-analysis be evaluated for risks of bias. This evaluation should encompass the following domains: generation of the allocation sequence, allocation concealment, blinding of patients and caregivers, blinding of outcome assessment, report on attrition during the trial, report on outcomes, and industry funding. Other types of bias may also need to be considered [51–58].

False negative meta-analyses

Lack of a statistical significant intervention effect in a traditional meta-analysis is not necessarily evidence of no effect of the intervention. »Absence of evidence is not evidence of absence of effect« [59]. Nevertheless, sequential meta-analyses with the TSA software may show that the meta-analysis has sufficient statistical power to reject an intervention effect of a specific magnitude even though the estimated required information size has not yet been reached (Fig. 4).

This can be done by calculating the non-superior and non-inferior trial sequential monitoring boundaries, the socalled ‘futility boundaries’. Futility boundaries indicate when the assumed effect could be considered unachievable. Futility-boundaries are calculated using a power function analogous to the α-spending function for constructing superiority- and inferiority-boundaries with application of numerical integration [36]. The example with cooling of comatose patients after cardiac arrest shows a situation where the assumed intervention effect of 17% relative risk reduction can be rejected because the Z-curve crosses the futility-boundary (Fig. 1b). However, this is not always what happens. We found that in 25 of 56 (45%) published cardiovascular systematic reviews in The Cochrane Library, the actual accrued information size failed to reach what was required to refute a 25% relative risk reduction [3]. Only 12 of these reviews (48%) were recognised as inconclusive by the authors. Of the 33 meta-analyses not showing statistical significance, only 12 (36%) were truly negative in the sense that they were able to reject a 25% relative risk reduction [3]. This illustrates that the statistical power is also low in many cardiovascular meta-analyses, and false conclusions are imminent. Within other medical specialities, the problems are likely to be even bigger as trials and meta-analyses usually include less patients. Nevertheless, sequential meta-analyses with calculated futility-boundaries may, in some instances, contribute to adequately declare the a priori anticipated intervention effect to be unachievable, though the required information size was not reached [10].

Analogous to the false positive meta-analyses, a meta-analysis may result in a false negative result due to bias. Bias is a frequent cause for underestimation of harmful intervention effects [51–57], and therefore, the preliminary defined bias risk domains should also be evaluated for all included trials when it comes to serious and non-serious adverse events [51–58].

Trial Sequential Analysis: a position between frequentist and Bayesian thinking

TSA of meta-analysis like the sequential analysis of a single randomised trial, originates from frequentist statistics [29]. The frequentist way of thinking was initially based on testing of the null hypothesis. This applies to both the P-value and its relation to an a priori accepted maximal type I error risk (α) and the possibility of including a null effect in the corresponding (1-α)% confidence interval [29]. The anticipation of an intervention effect of a specific magnitude, the alternative hypothesis, and subsequently the calculation of a required information size enabling the conclusion whether such an effect could be accepted or rejected, is, however, intimately related to the Bayesian prior.

TSA contains an element of Bayesian thinking by relating the result of a meta-analysis to the a priori point estimate of the intervention effect addressed in the analysis [77]. Bayes’ factor (BF) for a trial result is the ratio between the probability that the trial data originates under the null hypothesis, and the probability that the trial data originates under the alternative hypothesis or even several alternative hypotheses [72, 78, 79]. The posterior odds ratio for the estimate of the intervention effect after a new trial is added is calculated given the prior odds ratio for the intervention effect before the trial as: posterior odds ratio = BF x prior odds ratio [79]. In a Bayesian analysis, the prior takes form of an anticipated probability distribution of one or more possible alternative hypotheses or intervention effects which multiplied with the likelihood of the trial, results in a posterior distribution [79].

A methodological position between the frequentist and the Bayesian thinking can be perceived both in sequential interim-analyses of a single trial and in TSA of several trials [29]. Both have a decisive anticipation of a realistic intervention effect, although a full Bayesian analysis should incorporate multiple prior distributions with different anticipated distributions of intervention effects: e.g., a sceptical, a realistic, and an optimistic prior [79]. The TSA prioritise one or a few specific alternative hypotheses, specified by point estimates of the anticipated effect in the calculation of the required information size just as in the sample size estimation of a single trial [11].

The incentive to use sequential analyses arise because the true effect is not known and the observed intervention effect may be larger than the effect addressed in the sample size estimation of a single trial as well as in the estimation of the required information size for a meta-analysis of several trials. The need to discover an early, but greater effect than the one anticipated in the sample or information size calculation, or to discard it, thereby originates. If the intervention effect, in relation to its variance, happens to be much larger during the trial or the cumulative meta-analysis, this will be discovered through the breakthrough of the sequential boundary. However, this may also be problematic as too small sample sizes (in relation to the true effect), as mentioned, increase the risk of overestimation of the intervention effect or the risk of underestimation of the variance. In other words, due to a factitious too small sample size, we may erroneously confirm an unrealistic large anticipated intervention effect due to the play of chance.

There seems to be an ancestry between the sceptical prior in a Bayesian analysis and the use of a realistic intervention effect in a sequential analysis when the sample size in a single trial or the information size in a meta-analysis should be calculated [77, 78]. The smaller the effect, the greater the demand for quantity of information, and the sequential statistical significance boundaries become more restrictive. In other words, it becomes more difficult to declare an intervention effective or ineffective, in case the required information size is not achieved.

Christopher Jennison and Bruce Turnbull, however, have shown that on average, when a small, but realistic and important intervention effect is anticipated, a group sequential design requires fewer patients than an adaptive design, e.g., re-estimating the (fixed) sample size after the first interim analysis [80]. The group sequential design seems more efficient than the adaptive design. In line with mathematical theory [72], simulation studies [6], and empirical considerations [44, 45, 81, 82], there is evidence that small trials and small meta-analyses by chance tend to overestimate the intervention effect or underestimate the variance. Early indicated large intervention effects are often contradicted in later published large trials or large meta-analyses [6, 45, 81, 82]. The reason might be that statistical confidence intervals and significance tests, relating exclusively to the null hypothesis, ignore the necessity of a sufficiently large number of observations to assess realistic or minimally important intervention effects. The early statistical significance, at the 5% level, may be a result of an early overestimation of the intervention effect or an underestimation of the variance, or both, when the required information size for a realistic effect is not achieved. In general, it is easier to reject the null hypothesis than to reject a small, but realistic and still important, alternative hypothesis [64]. The null hypothesis can never be proven, and in practice, this means that it can never be completely discarded, as this would require an infinitely large number of observations.

The reason for early spurious significant findings may be quite simple, although not self-evident. Even adequate randomisation in a small trial lacks ability to ensure the balance between all the involved, known or unknown, prognostic factors in the intervention groups [81]. When we find a statistically significant intervention effect in a small trial or in a small meta-analysis, it is often due to insufficient balance of important prognostic factors, known or unknown, between the intervention groups. Therefore, it is not necessarily intervention effects that we observe, but rather an uneven distribution of important prognostic factors between groups. In addition to the described risks of random error, the overall risk of bias which includes the risk of publication bias makes it understandable why published trials and meta-analyses often result in unreliable estimates of intervention effects [2, 83].

The power of frequentist inference in a single trial and in a meta-analysis of several trials lies in two basic assumptions. First, the only decisive difference between the intervention groups during the trial is the difference between the interventions. We conclude that ‘despite everything else’, the measured difference in the outcome is due to different properties of the interventions because everything else seems equal in the groups. In a small trial and a small meta-analysis, the assumption, that all other risk factors are equally distributed in the two intervention groups, may not be fulfilled as described above, even though adequate bias control has been exercised. Second, the power of frequentist inference depends on the correctness of applying the ‘reverse law of implication’ from mathematical logic (see Appendix): that a sufficiently small P-value, calculated as the probability that we got a specific trial result when the null hypothesis is in fact true, leads us to discard the null hypothesis itself. This assumption, which never totally excludes the possibility that the result of a trial may agree with or be a result of the null hypothesis, demands a specific a priori chosen threshold for statistical significance. That is, a sufficiently small P-value leads us to regard the trial result as virtually impossible under the null hypothesis, and, therefore, we regard the opposite to be true and discard the null hypothesis. This automatically raises the question: how small a P-value should be before we can apply the ‘reverse law of implication’. Or alternatively expressed, does a P-value less than an a priori chosen threshold of statistical significance reject the null hypothesis? Ronald A. Fisher, already in 1956, warned against using a statistical significance level of 5% in all situations [84]. Nevertheless, ever since, it seems to have broadly been implemented as a criterion for conclusion in medical research [83], and this is likely wrong [85].

Sequential interim-analyses in a single trial and TSA of meta-analyses of several trials deal systematically and rationally with the misunderstood application of a constant level of statistical significance (P < 0.05), unrelated to the accrued fraction of the pre-calculated required (fixed) sample or information size and number of trials.