Abstract
BackgroundWhen multiple prognostic factors are adjusted for in the analysis of a randomised trial, it is unclear (1) whether it is necessary to account for each of the strata, formed by all combinations of the prognostic factors (stratified analysis), when randomisation has been balanced within each stratum (stratified randomisation), or whether adjusting for the main effects alone will suffice, and (2) the best method of adjustment in terms of type I error rate and power, irrespective of the randomisation method.MethodsWe used simulation to (1) determine if a stratified analysis is necessary after stratified randomisation, and (2) to compare different methods of adjustment in terms of power and type I error rate. We considered the following methods of analysis: adjusting for covariates in a regression model, adjusting for each stratum using either fixed or random effects, and Mantel-Haenszel or a stratified Cox model depending on outcome.ResultsStratified analysis is required after stratified randomisation to maintain correct type I error rates when (a) there are strong interactions between prognostic factors, and (b) there are approximately equal number of patients in each stratum. However, simulations based on real trial data found that type I error rates were unaffected by the method of analysis (stratified vs unstratified), indicating these conditions were not met in real datasets. Comparison of different analysis methods found that with small sample sizes and a binary or time-to-event outcome, most analysis methods lead to either inflated type I error rates or a reduction in power; the lone exception was a stratified analysis using random effects for strata, which gave nominal type I error rates and adequate power.ConclusionsIt is unlikely that a stratified analysis is necessary after stratified randomisation except in extreme scenarios. Therefore, the method of analysis (accounting for the strata, or adjusting only for the covariates) will not generally need to depend on the method of randomisation used. Most methods of analysis work well with large sample sizes, however treating strata as random effects should be the analysis method of choice with binary or time-to-event outcomes and a small sample size.
Highlights
When multiple prognostic factors are adjusted for in the analysis of a randomised trial, it is unclear (1) whether it is necessary to account for each of the strata, formed by all combinations of the prognostic factors, when randomisation has been balanced within each stratum, or whether adjusting for the main effects alone will suffice, and (2) the best method of adjustment in terms of type I error rate and power, irrespective of the randomisation method
When randomised is not carried out within strata, it may be unnecessary to account for interactions between covariates to obtain correct type I error rates
Likewise, when a stratified analysis was used after patients were randomised using stratified permuted blocks, error rates were nominal
Summary
When multiple prognostic factors are adjusted for in the analysis of a randomised trial, it is unclear (1) whether it is necessary to account for each of the strata, formed by all combinations of the prognostic factors (stratified analysis), when randomisation has been balanced within each stratum (stratified randomisation), or whether adjusting for the main effects alone will suffice, and (2) the best method of adjustment in terms of type I error rate and power, irrespective of the randomisation method. Some randomised controlled trials (RCTs) adjust their analyses for prognostic factors which are thought to influence outcome (such as age or disease stage) This is commonly done to increase power [1,2,3,4,5,6,7], to guard against chance imbalances between treatment arms [3,8], or because the prognostic factors have been used as balancing variables in the randomisation process and it is necessary to account for them in the analysis to obtain correct type I. When randomised is not carried out within strata (e.g. when balancing factors are not used in the randomisation process, or when covariates are balanced marginally), it may be unnecessary to account for interactions between covariates to obtain correct type I error rates ( adjustment for strong interactions may lead to increased power). This implies that for minimisation, adjusting only for the covariates used in the minimisation process (and not for their interactions, or equivalently each strata) should give valid type I error rates
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