Brookhart, M Alan

Curtin, François

Elbourne, Diana R.

Kesselheim, Aaron S.

Stedman, Margaret R.

RERO DOC Digital Library

Meta-analyses involving cross-over trials: methodological issuesFrom MARGARET R STEDMAN,1 FRANC¸OIS CURTIN,2 DIANA R ELBOURNE,3AARON S KESSELHEIM1,4 and M ALAN BROOKHART1,41Division of Pharmacoepidemiology and Pharmacoeconomics, Department of Medicine, Brigham and Women’s Hospital, Boston,MA, USA, 2Merck Serono SA, Geneva, Switzerland, 3Medical Statistics Unit, London School of Hygiene and Tropical Medicine,London, UK and 4Department of Medicine, Harvard Medical School, Boston, MA, USA.Corresponding author. Division of Pharmacoepidemiology and Pharmacoeconomics, Department of Medicine, Brigham andWomen’s Hospital, Boston, MA, USA. E-mail: mstedman2@partners.orgWe would like to make a correction to the descriptionof the Becker–Balagtas method found in the 2002International Journal of Epidemiology article byElbourne et al.1 The article describes methods of com-bining parallel and cross-over trials for the purposesof meta-analysis. In the following summary, we willprovide a revised description of the Becker–Balagtasapproach with an example of how to implement it.The Becker–Balagtas approach is an estimationmethod used in the meta-analysis of cross-overtrials with binary outcomes.1 It is considered amarginal method because it relies on the marginalprobabilities of the outcome to estimate the oddsratio (OR).2 Typically, a conditional approach, suchas the Mantel–Haenszel odds ratio (of the discordantpairs), is the usual effect estimate for cross-overtrials.3 However, when summarizing results acrossstudy designs the Becker–Balagtas approach is thepreferred method.3To compute the Becker–Balagtas estimates, we firstdefine the observed cell quantities from a cross-overtrial. In the cross-over design, all subjects participatein all treatments in successive periods where thesequence of treatments is determined by randomiza-tion.1 For example, some subjects are randomized toreceive placebo for the first period followed by theactive treatment in the second period, others are ran-domized to receive the active treatment in the firstperiod followed by the placebo in the second period.Each subject contributes two results for the active andplacebo treatment periods.Binary outcomes from a cross-over trial may besummarized as displayed in Table 1. Here, the out-come from the trial is either success or failure andthe treatment is either active or placebo. Let s bethe number of subjects that had a success with bothtreatments. Let v be the number of subjects that expe-rienced failure with both treatments. Let t and u rep-resent the number of subjects that had a success withone treatment, but failed with the other. Let a and crepresent the row totals, b and d the column totalsand n the total number of subjects.1According to the Becker–Balagtas method, we com-pute the marginal odds ratio and variance as follows:3ORMarginal ¼ adbcvarðInðORmarginalÞÞ ¼1aþ 1cþ 1bþ 1d 2nwhere n is the within subject covariance and is defined as:3 ¼ n2 ns ababcd Additionally, the correlation coefficient (a ratio ofthe covariance and the variance) can be estimatedfrom the table as:3¼ ns abﬃﬃﬃﬃﬃﬃﬃﬃﬃabcdpThe Becker–Balagtas estimation method is appliedwhen combining data from different designs in ameta-analysis.1 For example, consider a meta-analysisof two trials of the effect of brand name and genericanti-epileptic drugs on seizure outcomes. The firsttrial has a parallel design (independent outcomes)and the second trial has a cross-over design (corre-lated outcomes). Our objective is to combine theresults of these two trials into a single summary esti-mate of the odds ratio of uncontrolled seizures forgeneric anti-epileptic drugs compared with brand-name anti-epileptic drugs.Table 2 presents results from a trial with a paralleldesign.4 A total of 60 newly diagnosed patientswere randomized to either a brand name or genericanti-epileptic drug. Of the 45 patients randomizedto generic anti-epileptic drugs, 6 had uncontrolledseizures as did 5 of the 15 allocated to brand-namedrugs.4We compute the odds ratio and variance accordingto the usual methods. For estimates of uncontrolledseizures in generic versus brand name drugs, theTable 1 2 2 table for a cross-over trialPlacebo treatment periodSuccess FailureActive treatment periodSuccess s t aFailure u v cb d n1732 INTERNATIONAL JOURNAL OF EPIDEMIOLOGYcalculations are performed as follows:3OR ¼ 6  105  39 ¼ 0:31varðInðORÞÞ ¼ 16þ 15þ 139þ 110¼ 0:49Table 3 presents results from a different trial with across-over design.5 In this case, all 20 participantsreceived both treatments and the order of treatmentwas randomly assigned. Of the 20 participants, 2 expe-rienced uncontrolled seizures with both the genericand brand-name anti-epileptic drugs.5We apply the Becker–Balagtas method by computingthe odds ratio from the marginal quantities. The vari-ance is adjusted for the within-patient correlation(¼ 0.67).3ORMarginal ¼ 2  164  18 ¼ 0:44varðIn½ðORM arg inalÞÞ ¼ 12þ 118þ 14þ 116 2 5:5620¼ 0:31 ¼ 202 ð20  2Þ  ð2  4Þ2  4  18  16 ¼ 5:56 ¼ ð20  2Þ  ð2  4Þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2  4  18  16p ¼ 0:67Assuming there are no other methodological con-cerns or between study heterogeneity, we can thencombine these two estimates into one summary esti-mate. The estimates from the two trials may be com-bined by the usual inverse variance method formeta-analysis.6 The summary estimate (ORpool) is aweighted average of the two odds ratios where theweights (w1, w2) are determined by the precision ofthe estimates. The variance of the pooled estimate is afunction of the inverse of the sum of the weights.6ORpool ¼ expðw1  InðOR1Þ þ w2  InðOR2ÞÞ¼ expð0:39  1:18 þ 0:61  0:81Þ¼ 0:39varðInðORpoolÞÞ ¼ 1=ðvarðIn½ðOR1ÞÞÞþ 1=ðvarðIn½ðOR2ÞÞÞ¼ 10:49þ 10:31¼ 0:19InðOR1Þ ¼ 1:18InðOR2Þ ¼ 0:81w1 ¼10:4910:49 þ 10:31¼ 0:39w2 ¼10:3110:49 þ 10:31¼ 0:61A confidence interval (CI) for the pooled estimate isthen:95% CI ¼ expðInðORpoolÞ  1:96 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃvarðInðORpoolÞpÞ¼ ð0:16, 0:91ÞTo conclude this example, we report with 95%confidence that the odds of seizure for genericanti-epileptic drugs compared with brand-nameanti-epileptic drugs is between 0.16 and 0.91. Basedon these two trials, we find that the generic medica-tions in these studies show a significantly reducedodds of uncontrolled seizure compared with thebrand-name medications.Note the following corrections to the formulae for and s given in the appendix of the original article.1The formula for s may be applicable when the fre-quency is unknown but the correlation is reported.Assuming the same notation, the original formulafor  and s are: ¼ s ababcd s ¼ abþ ﬃﬃﬃﬃﬃﬃﬃﬃﬃabcdnrThe correct formulae are as follows: ¼ n2 ns ababcd s ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃabcdp þ abnCorrect application of the Becker–Balagtas method isimportant to ensure accurate estimation of summaryestimates for meta-analysis. When including variousTable 3 Results from Oles et al. (cross-over design)Brand name drugsUncontrolledseizuresControlledseizuresGeneric drugsUncontrolled seizures 2 0 2Controlled seizures 2 16 18Total 4 16 20Table 2 Results from Kishore et al. (parallel design)DrugsGeneric Brand nameSeizuresUncontrolled 6 5 11Controlled 39 10 49Total 45 15 60LETTERS TO THE EDITOR 1733trial designs in a meta-analysis, it is best to select acommon measurement that does not favour onedesign over the other.1 Of the currently availablemethods, the marginal approach (or Becker–Balagtasmethod2) introduces the least bias when themeta-analysis combines cross-over trials with paralleltrials.3References1 Elbourne DR, Altman DG, Higgins JPT, Curtin F,Worthington HV. Meta-analysis involving cross-overtrials: methodological issues. Int J Epidemiol 2002;31:140–49.2 Becker MP, Balagtas CC. Marginal modeling of cross-overdata. Biometrics 1993;49:997–1009.3 Curtin F, Elbourne D, Altman DG. Meta-analysis combin-ing parallel and cross-over trials. II: binary outcomes.Stat Med 2002;21:2145–59.4 Kishore K, Jailakhani BL, Sharma JN, Ahuja GK. Serumphenytoin levels with different brands. Indian J PhysiolPharmacol 1986;30:171–76.5 Oles KS, Penry JK, Smith LD, Anderson RL, Dean JC,Riela AR. Therapeutic bioequivalency study of brand nameversus generic carbamazepine. Neurology 1992;42:1147–53.6 Hedges LV, Olkin I. Statistical Methods for Meta-Analysis.San Diego: Academic Press Inc., 1985.doi:10.1093/ije/dyp345Advance Access publication 21 December 2009 The Author 2009; all rights reserved.When size presents problemsFrom PETER MORFELD1,2* and THOMAS C ERREN21Institute for Occupational Epidemiology and Risk Assessment (IERA) of Evonik Industries AG, Essen and 2Institute and Policlinicfor Occupational Medicine, Environmental Medicine and Prevention Research, University Hospital of Cologne, University ofCologne, Germany*Corresponding author. Institute for Occupational Epidemiology and Risk Assessment (IERA) of Evonik Industries AG,Rellinghauser Str.1-11, 45128 Essen, Germany. E-mail: peter.morfeld@evonik.comWith interest we read the Editorial by Dr Hense1headed ‘When size matters’. We wish to complimentDr Hense for pointing to several problems thatmega-projects may generate. Most importantly, wecertainly agree that mega-cohorts are not a solutionto identifying and establishing causality per se.2Moreover, we wish to complement a potential pitfallof mega-studies, namely with regard to ‘bias’, aterm which was not mentioned in the Editorial.Importantly, in view of the anticipated enormous visi-bility of ‘these lighthouse projects of epidemiology’,1it is a must to avoid the publication and dissemina-tion of possibly biased findings.In 2005, Ioannidis provided a convincing rationaleas to ‘why most published research findings arefalse’.3 Having shown that small study group sizescan be a reason for distorted results, he argued alsothat ‘claimed research findings may often be simplyaccurate measures of the prevailing bias’ (p. e124).3Now, the probability that biased results may be re-ported uncritically as valid findings actually increaseswhen precision is maximized because random errorno longer has relevant effects and is, thus, expectedto no longer mask small biases involved. In particular,observational studies of very large sizes—designed tomeasure small effects—are prone to this type of errorand have to be analysed with extra diligence via biasestimation and adjustment procedures.4,5Overall, a danger of sizeable studies is that largenumbers may be misunderstood as a sufficient toolto identify truth and as an insurance against errorsthat plague researchers in small(er) investigations.Disconcertingly, the opposite may be true: indeed,due to the lack of random error, likely involvedbut uncovered biases may become a driving force forerroneous findings and misinterpretations in mega-studies that are conducted to measure small effects.References1 Hense H-W. When size matters. Int J Epidemiol 2011;40:5–7.2 Rothman KJ, Greenland S, Lash TL. Modern Epidemiology,3rd edn. Philadelphia: Lippincott Williams & Wilkins,2008.3 Ioannidis JPA. Why most published research findings arefalse. PLoS Med 2005;2:696–701.4 Greenland S. Multiple-bias modelling for analysis ofobservational data. J R Stat Soc A 2005;168:267–306.5 Morfeld P, McCunney R. Bayesian bias adjustments ofthe lung cancer SMR in a cohort of German carbonblack production workers. J Occup Med Toxicol 2010;5(1).http://www.occup-med.com/content/5/1/23 (23 March2011, date last accessed).doi:10.1093/ije/dyr066Advance Access publication 30 March 2011 The Author 2011; all rights reserved.1734 INTERNATIONAL JOURNAL OF EPIDEMIOLOGY

Meta-analyses involving cross-over trials: methodological issues

Abstract

Similar works

Full text

Available Versions

RERO DOC Digital Library