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Incorporating Missing Outcome Data in The Sample Size Calculation For a Future Trial: A Case Study Using a Single Trial, a Pairwise and Network Meta-Analysis

1 Background: To illustrate the advantages of using network meta-analysis (NMA) as 2 compared to a trial or a pairwise meta-analysis to estimate the amount of missing outcome 3 data (MOD) for a target comparison in order to adjust the required sample size for possible 4 participant losses in a future trial. 5 Methods: We introduced the concept of transitive risks to obtain the absolute risks of MOD 6 for all interventions of the network. We used the network of a published systematic review on 7 a binary outcome to apply the proposed concept and to calculate the required sample size in a 8 future trial for a selected target comparison. For that comparison, we also calculated the 9 required sample size using the corresponding trials separately, and after pooling these trials in 10 a random-effects meta-analysis. 11 Results: Ignoring MOD from the sample size calculation led to the smallest sample size. 12 When either trial was considered, the risk of MOD ranged from 1% to 13% in the compared 13 intervention arms, therefore, increasing the sample size from 1% to 12%. Performing a 14 pairwise meta-analysis yielded a risk of MOD equal to 6% and 9% in the active and control 15 arms, respectively, which inflated the sample size by 8%. Using NMA, the corresponding 16 risks of MOD were 10% and 13%, which increased the sample size by 13%. 17 Conclusions: Provided that the transitivity assumption holds, incorporating the absolute risks 18 of MOD in the sample size calculation for a target comparison of the network led to better 19 planning of a future trial.

demonstrated that the required sample size to achieve the desired power was dramatically 5 larger when the calculations were based on the sensitivity analysis as compared to ignoring 6 MOD altogether [11]. To the best of our knowledge, there is currently no research at the level 7 of evidence synthesis to demonstrate the implications on sample size calculations when MOD 8 from a series of relevant past trials are incorporated into the design of a future trial.

9
It is legitimate to wonder about the 'proper' synthesis framework to inform the future trial:  The present study aims to illustrate the advantages of using NMA rather than a trial or a 23 pairwise meta-analysis to estimate the amount of MOD for a target comparison in order to adjust the required sample size for possible participant losses in a future trial. For that 1 purpose, we introduce the concept of transitive risks across trials to estimate the risk of MOD 2 for each intervention within the NMA framework [12]. We use a published systematic review 3 as a motivating example to illustrate the concept of transitive risks and the implications of 4 MOD on the sample size calculation for a future trial.

6
As a motivating example, we used the systematic review of Baker et al. [13] for the treatment 7 of chronic obstructive pulmonary disease (COPD) exacerbations (Table S1 in Additional file 8 1). Figure 1A illustrates the network of five interventions alongside the extent of MOD in 9 each intervention and observed comparison: the percentage of MOD (%MOD) signified 10 moderate (more than 5% and up to 20%) and high attrition bias (> 20%) within the network.

11
According to Table 1, %MOD varied substantially in all interventions ranging from low (≤ 12 5%) to moderate in tiotropium and high in the remaining interventions. The %MOD also 13 varied considerably within the majority of the observed comparisons (Table 1). 14 [ Table 1] 15 For illustrative purposes, we focused on the comparison of tiotropium with long-acting β2-16 agonist (LABA) that was investigated in three clinical trials (Table S2 in Additional file 1).

17
For this comparison, the publication reported an odds ratio (OR) of 0.82 (95% credible To be able to introduce the concept of transitive risks, we have assumed the transitivity 3 assumption and its statistical manifestation (consistency) are plausible; otherwise, the 4 transitive risks would be invalid [12]. The concept of transitive risks builds upon the 5 transitivity assumption; namely, trials share similar clinical and methodological 6 characteristics, and differ only in the interventions compared [12]. In essence, the assumption 7 of transitive risks implies that the risk of the outcome in an intervention (hereafter, absolute 8 risk) is similar across all trials of the network. In other words, an intervention is assumed to 9 have an exchangeable absolute risk across all trials irrespective of the comparator 10 intervention(s) in each trial [12]. This notion of transitive risks stems from the interpretation 11 of the transitivity assumption in Salanti [14]: 'There are no differences between observed and 12 unobserved relative effects of AC and BC beyond what can be explained by heterogeneity'.

13
While the assumption of transitive risks might seem difficult to defend in practice, it 14 facilitates the estimation of unique absolute risks for each intervention [12]. Specifically, for 15 the calculation of the absolute risks, we only need (i) the estimated relative treatment effects 16 from NMA (e.g. log OR) for comparisons with the selected reference intervention of the 17 network, and (ii) a sensible assumption about the underlying risk in the reference intervention 18 [12]. Below we illustrate the calculation of the absolute risks following the GRADE concept 19 for binary outcomes [15]. We used the Bayesian framework to estimate the (posterior) median 20 and 95% CrI of the absolute risk for each intervention.

Absolute risks under the assumption of transitive risks
With we indicate the posterior mean of the log OR between intervention ( = 1 , , … , ) and reference intervention . With we imply the underlying risk for the 2 reference intervention that has been selected ideally from observational studies or relevant 3 randomised trials (in the absence of the former) [15]. The GRADE approach advocates the 4 risk ratio (RR) as a measure of the relative effect. However, we used the OR for its statistical 5 advantages and for not requiring constraints to ensure that the probability of an event is within intervention as a function of and : Note that equation (1) is equivalent to the predicted risk for intervention as described by   11 We plan a future two-arm trial, and we assume an event risk of 1 and 2 for the control and 12 experimental intervention, respectively. We use a Z-test to assess the null hypothesis of no power at 1 − , and type I error at , the required sample size in the absence of MOD is

Sample size adjustments for MOD in a future two-arm trial
When we expect MOD in the future trial, under the MAR assumption, the required sample 1 size in the future trial with 1:1 randomisation is inflated by 1 (1 − ) ⁄ where is the 2 probability of MOD in arm . Then the required sample size to achieve a power of 1 − after 3 adjusting for MOD is Implementation for the motivating example 5 We performed Bayesian random-effects NMA while modelling MOD under the MAR 6 assumption [7] to adjust for MOD properly, and we incorporated equation (1) into the model 7 to obtain the absolute risks 1 and 2 for LABA and tiotropium, respectively, while assuming 8 = 0.39 in placebo (median observed event risk across the placebo-controlled trials). We 9 used the estimated 1 and 2 in equation (3) to calculate the required sample size before 10 adjusting for the expected MOD in LABA and tiotropium in the future trial. We considered 11 80% power and 5% type I error. 12 Subsequently, for each trial, we calculated 1 and 2 as the ratio of MOD to the number 13 randomised in each arm. At the pairwise meta-analysis level, we pooled the three trials that tiotropium was * = 4367 which was 8% larger than = 4038 ( Figure 3A). Using NMA, 8 the posterior median of 1 and 2 were 0.13 (95% CrI: 0.11 -0.15) and 0.10 (95% CrI: 0.08 -9 0.12), respectively, for LABA and tiotropium ( Figure 3B). Then, using equation (4), the total 10 required sample size after adjusting for MOD was * = 4564 ( Figure 3A) which was 13% 11 larger than = 4038, 5% larger than * = 4367 in the pairwise meta-analysis, and 1% larger 12 than * = 4539 in trial 2.

14
This is the first study to illustrate the gains of using NMA instead of a single trial or pairwise The author declares no competing interests.    participants who took the intervention. Placebo was the reference intervention in the network. We considered an underlying risk p_A=0.39 for placebo to obtain the absolute risks for the remaining interventions in the network. ICS, inhaled corticosteroid; LABA, long-acting β2-agonist Plot A is a barplot on the total required sample size for a future trial on tiotropium versus LABA after adjusting for missing outcome data (MOD) in three different levels of evidence: individual trials, pairwise and network meta-analysis. The horizontal black line refers to the unadjusted total required sample size. On each bar, the percentage refers to the corresponding percentage in ation in the total required sample size when MOD are incorporated in the sample size calculation. Plot B is a barplot with heaped bars on the predicted number of MOD per 100 participants for each intervention of the target comparison in three different levels of evidence. LABA, long-acting β2-agonist

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