title: Adjusting for multiple prognostic factors in the analysis of randomised trials.
abstract: BACKGROUND
  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.
  
  
  METHODS
  We 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.
  
  
  RESULTS
  Stratified 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.
  
  
  CONCLUSIONS
  It 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.
authors.affiliation: MRC Clinical Trials Unit, Aviation House, 125 Kingsway, London WC2B 6NH, UK. b.kahan@ucl.ac.uk
authors.forename: Brennan C
authors.initials: BC
authors.lastname: Kahan
authors.forename: Tim P
authors.initials: TP
authors.lastname: Morris
mesh_terms: D015981:Epidemiologic Factors; D006801:Humans; D008962:Models, Theoretical; D015999:Multivariate Analysis; D011379:Prognosis; D016032:Randomized Controlled Trials as Topic; D012107:Research Design; D013223:Statistics as Topic
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