For sample size calculations for randomized controlled trials (RCTs), some parameters, like the treatment difference (effect size) between the experimental and control intervention have to be estimated. The investigators also have to estimate parameters that are not of direct interest, but are required for the computations. These parameters are often termed nuisance parameters, the most common of which are the control-group event rate (CER) for binary outcomes and the standard deviation (SD) for continuous outcomes.

Determining an appropriate sample size for an RCT has always been a challenge due to these nuisance parameters being unknown and needing to be estimated [1, 2]. Usually, they can only be estimated from previous studies on the same topic, but there is always a problem when the new study is the first trial on the topic. Consequences of erroneously estimated nuisance parameters are the under-estimation or over-estimation of the required sample size, which can then lead to underpowered studies that may fail to reach a definitive conclusion [3] in the former case, or to unnecessary higher study cost and longer recruitment periods in the latter case [4].

Investigators sometimes extrapolate evidence on nuisance parameters from randomized trials in adults for sample size calculations in pediatric trials [1, 5, 6]. However, differences in clinical effects between adults and children do exist as has been previously shown by systematic empirical evaluations of the comparative effectiveness and comparative safety of medical interventions between adults and children [7–10]. A systematic empirical evaluation of nuisance parameters in pediatric RCTs, as compared to adult RCTs has not been previously performed.

We performed a meta-epidemiologic empirical evaluation to investigate whether nuisance parameters differ between pediatric and adult trials on the same topics, for the same compared interventions and for the same clinical outcomes. We also studied whether there are differences in the nuisance parameters according to the types of outcomes: binary versus continuous efficacy outcomes and mortality versus non-mortality outcomes.

We examined 208 meta-analyses, 135 with binary primary efficacy outcome data and 73 with continuous outcome data, from a total of 185 systematic reviews. All meta-analyses comprised a total of 2110 RCTs; 1515 adult RCTs (1126 with binary and 389 with continuous outcomes) and 595 pediatric RCTs (355 with binary and 240 with continuous outcomes). Each study could have contributed data to more than one meta-analysis within each systematic review (e.g. for different meta-analyses with different compared interventions or outcomes).

Binary outcome analyses

Summary pooled-CER-RRs

The summary-pooled-CER-RR of pediatric CERs versus adult CERs across all 135 meta-analyses showed that pediatric RCTs had on average a 10% smaller CERs (summary-CER-RR: 0.90; 95% CI: 0.83, 0.98). The individual CER-RRs within each meta-analysis are shown in Fig. 2 and their distribution thereof in Fig. 3. Overall, 13.3% of the examined meta-analyses had pediatric-CERs that were at least five-fold smaller than the adult-CERs; the opposite (adult RCTs with at least five-fold smaller CERs than pediatric RCTs) occurred in only 2.2% of cases. Moreover, in 60.7% of the meta-analyses, the pediatric-CERs were smaller than the adult-CERs.

I ² values were calculated for 172 of the 270 meta-analyses (135 each for pediatric and adult trials respectively); for the remaining meta-analyses, I ² was not defined as they included only single studies. The distribution of these I ² values within each meta-analysis is shown in Table 1. Since these meta-analyses do not mix pediatric and adult data, these results show high heterogeneity among studies even when the pediatric/adult factor was taken out of the analysis.

I-squared	Number of meta-analyses (%)
0%	30 (17%)
0–20	7 (4%)
20–40	5 (3%)
40–60	15 (9%)
60–80	32 (19%)
80–100	83 (48%)

Subgroup analysis: Mortality and non-mortality outcomes

There were 21 meta-analyses with mortality outcomes and 114 meta-analyses with non-mortality outcomes. Among the 21 meta-analyses with mortality outcomes the summary-pooled CER-RRs of pediatric-CERs vs. adult-CERs was 0.48 (95% CI: 0.31, 0.74). In 18 of these 21 meta-analyses the mortality CER in adult trials was larger than that in pediatric trials.

Among the 114 meta-analyses with non-mortality outcomes the summary-CER-RR of pediatric vs. adult trials was 1.00 (95% CI: 0.92, 1.09).

Pooling using magnitude (absolute CER-RR)

When we repeated the primary analysis using the “absolute-CER-RR” (Fig. 4) instead of the CER-RR the summary--absolute-CER-RR was 1.46 (95% CI; 1.37, 1.56). This indicates that CERs in pediatric trials are on average either 1.5 times larger or 1.5 times smaller than in adult trials.

Continuous outcome analyses

Summary-pooled-CE-SD-ratios

The weighted average (weighted on the log scale by the square root of the sample size) (summary-CE-SD-ratio) of the CE-SD-ratios of pediatric-CE-SDs vs adult-CE-SDs for the 73 meta-analyses with continuous efficacy outcomes was 0.74. This indicates that on average the pediatric RCTs CE-SDs were 26% smaller than their adult counterparts. The distribution of the CE-SD-ratios is shown in Fig. 5. In 27.1% of the meta-analyses the CE-SD-ratios between pediatric and adult RCTs differed by at least 2-folds in either direction. Furthermore, ignoring direction of difference, and looking only at the sizes, the weighted magnitude of CE-SD-ratio was 1.76, which means that, on average, the CE-SD in adult and pediatric RCTs differed by a factor of 1.76.

The distribution of the standardized SDs of the effect sizes in the control groups of all adult and pediatric studies within each meta-analysis is shown in Fig. 6. The SDs of the effect sizes in the control groups of pediatric vs. adult trials varied greatly both within meta-analyses and between meta-analyses.

The control-group event rate nuisance parameter in pediatric trials was on average 10% smaller than that in adult trials. In our secondary analyses, when we considered the magnitude of the control-group effect size rather than the direction of effect, pediatric trials had an average control-group event rate that was 1.5 times higher or 1.5 times smaller than that in adult RCTs. The reason for considering also the magnitude of effects, ignoring their direction thereof, is that an important issue to address is whether nuisance parameters in pediatric studies are likely to differ from those estimated from adult studies. We could have large differences in nuisance parameters (some over-estimating, some under-estimating) that average out to no difference. By analyzing magnitudes, we are pre-supposing a difference and trying to estimate how large that difference might be.

In over 60% of meta-analyses the control-group event rates in pediatric RCTs were smaller than those in adult trials and in 36% of the meta-analyses, relative differences in control-group event rates of at least 2-folds, in either direction, were identified. Specifically, for mortality outcomes, the control-group mortality rate in pediatric trials was on average 50% lower than that in adult trials.

Large variation was also seen between pediatric and adult trials when continuous efficacy outcomes were considered. The pediatric control-group SD was on average 26% smaller than that of adult trials and in 27% of the meta-analyses the relative difference in SDs between pediatric and adult trials was at least 2-fold in either direction. Moreover, when the magnitude of the control-group SD was considered, pediatric trial SDs were at least either 1.8 times larger or 1.8 times smaller than adult trial SDs.

Large differences were seen among many studies with regards to nuisance parameters. To demonstrate how erroneous estimation of nuisance parameters can affect sample size computation substantially, we will take two examples from the included meta-analyses in this study. In the review Antibiotics for the common cold and acute purulent rhinitis, for the primary meta-analysis of the persisting symptoms outcome, we had an estimated CER in the adult population of 0.48. If one wishes to conduct a pediatric trial on the same topic, with a type I error probability of 0.05, 80% power, and an assumption that a 30% reduction in number of patients with persisting symptoms would be required to demonstrate a clinically relevant antibiotic effect, the required sample size for the pediatric study would be 182 patients per arm. Under the assumption of a CER of 0.048, as was actually seen in the pediatric trials, the required sample size would be over 16 times larger at 2962. To give an example using a continuous outcome, we take the review Early emergency department treatment of acute asthma with systemic corticosteroids. For the outcome of final PEFR, we had observed a mean SD in the adult studies of 32 L/min. Suppose we plan to conduct a pediatric trial on the same topic, using a type I error probability of 0.05, a power of 80%, and a minimal clinical important difference threshold of 15 L/min. Using the adult estimated SD of 32 L/min, we could compute that we would require 643 patients in each of two groups. If we assume an SD estimate of 4.3 L/min, as was observed in the pediatric trials, we would only require a sample of 12 patients in each group, a sample size that is less than 2% of the originally computed sample size. It is clear from these examples that the erroneous estimations of these nuisance parameters can have important implications in the sample size computations, which can lead to either inappropriately powered studies that would not be able to answer the clinical question, or, on the other hand, to unnecessary waste of valuable clinical and financial resources. A third unwanted consequence might be that a proposed trial is not conducted because the erroneous estimate for the sample size is too large to be feasible.

We did observe a trend in both binary and continuous outcome data for pediatric RCTs to have smaller values of nuisance parameters (both CERs and CE-SDs) than their adult counterparts. Thus, when one does use these parameters from adult studies as surrogate for pediatric studies, the nuisance parameter is more likely to be overestimated than underestimated. This relationship has been well documented and graphed². In the case of continuous data, an overestimation of the SD will always result in an overestimation of the sample size. The situation for binary data is more nuanced, as the sample size will depend upon the ratio of the CER and the treatment-group event rate (the closer the ratio is to 1, the larger the required sample size), so an underestimation of the CER could lead to either an under- or overestimated sample size. For example, if a pediatric population had an actual CER of 0.3 with a treatment-group event rate of 0.2, then underestimating the CER (say as 0.25) would result in a larger than required sample. However; if the treatment-group event rate was 0.4, then this underestimate of the CER would result in a sample that was too small.

The discrepancies in the nuisance parameters between pediatric and adult trials were more prominent with mortality outcomes. In 86% of meta-analyses with mortality outcomes, the mortality CER in adult trials was larger than that in pediatric trials. On average, the control-group mortality rates in adult trials were two times larger than in pediatric trials. Mortality seems to be an outcome where extrapolation of adult control-group event rates for the estimation of pediatric trial sample sizes may give inaccurate results.

We should acknowledge some study limitations. Traditional meta-analysis of standard deviations was not feasible in the analysis of continuous outcomes since the systematic reviews did not provide us enough information to ascertain variances around these nuisance parameters. Meta-analyses were done on a variety of outcomes, and thus the standard deviations were all reported in different units, and therefore not comparable across meta-analyses without standardization. In both the analyses of binary and continuous outcomes we observed considerable heterogeneity in nuisance parameters, not only between meta-analyses but also within them. We assumed that studies included within the same meta-analysis of a Cochrane review would have populations sufficiently similar to use them to impute nuisance parameters. However, extremely high between-study heterogeneity (I² > 80%) was seen in more than half of the meta-analyses, which implies that even within studies of the same age-group (i.e. adult or pediatric) we cannot expect nuisance parameters to routinely be similar. This suggests that not only should we be wary of extrapolating nuisance parameters for pediatric studies from adult studies, but we should be almost equally wary of extrapolating them from other pediatric studies.

With these limitations in mind and given the results we have seen here, it would be interesting to do a further and more refined analysis as to which factors may lead to better concordance between the nuisance parameters of pediatric and adult studies. This would be a difficult endeavor, however, since these factors would likely be specific to a subject area, and not necessarily generalizable. Analysis would then have to be limited to those areas where there are enough studies to do it properly.

This study provides evidence to raise awareness among investigators planning to design trials in children, when available data on nuisance parameters are mostly from adult studies that significant differences between pediatric and adult trials do exist. Extrapolation from adult trials of nuisance parameters to guide sample size calculations for pediatric trials should be cautiously done. Inappropriate extrapolations of nuisance parameters from adult trials to pediatric trials can lead to erroneous sample size calculations. Significant over- or underestimation of the required pediatric sample sizes can occur particularly when the outcome is mortality. When there is doubt about the similarity between the population from which the estimates are derived and the prospective study population, either a blinded sample size review during the early phase of the new trial (internal pilot study [15]), a more flexible (sequential) design and analysis [1, 2], or use of a standardized effect size [16] should be considered to maximize trial efficiency.