Small Sample Properties of Some Estimators of a Common Hazard Ratio Author(s): Alexander M. Walker Source: Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 34, No. 1 (1985), pp. 42-48 Published by: Blackwell Publishing for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2347883 . Accessed: 17/01/2011 09:32 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=black. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Blackwell Publishing and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series C (Applied Statistics). http://www.jstor.org Appi. Statist. (1985) 34, No. 1, pp. 42-48 Small Sample Propertiesof some Estimators of a CommonHazard Ratio By ALEXANDER M. WALKER InternationalAgencyforResearchon Cancer,France [Received October 1983. RevisedApril1984] SUM MARY The simulatedsmall-samplebehavioursof severalestimatorsof a common hazard ratioare compared. In most circumstances,a modifiedversion of the empirical logit (Haldane 1955; Anscombe 1956) seems to be the analytictechnique of choice. The performanceof the StandardizedMortalityRatio (SMR) is indistinguishable fromthat of the ML estimatorwhen the denominatorexperienceis much largerthan that of the numeratorexperience. When many cells have small expectations,then the empirical logit becomes biased, and a variant of the Mantel-Haenszel estimator (Mantel and Haenszel, 1959) is the mostwidely reliablenon-iterative estimator. Keywords: Hazard ratio;Small samples 1. Introduction Accumulated times at risk (person-yearsof experience) in cohort studies are commonlyassigned to strata within which a reasonable approximationis to assume a uniformhazard (or incidence) function.The stratifying variablesmay be fixed (such as gender)or time dependent(such as age); they determinethe baseline stratum-specific hazards and thusdefinethe contextwithinwhichthe effectof any furthercharacteristics(such as an industrialexposure) can be examined. Often the ratio of hazard rates in the presence vs. absence of the additional characteristicunder study is taken to be fixed, conditionallyon stratifying variables,and the analytic goal is to estimatethe common hazard ratio over strata.Maximum likelihood estimatesof common hazard ratioscan be readily obtained with mathematicalmodelling programmesthat take into account the Poisson structureof the likelihood function(Clayton, 1982; Berry,1983) but these may be unavailable, or cumbersomewhen the object is simplyto analyse stratifieddata fromsimple tables. A review of currentjournals of epidemiology and industrialmedicine, where cohort studies appear most often,suggeststhat non-iterativeestimatesare used farmore commonlyin practicethan are maximum likelihood procedures,and that there are many studies in which at least some cells contain few or no observed events. The purpose of this note then is to compare the theoreticaland observed variancesof severalsimple, consistentestimatorsof the common hazard ratio, with the intent of identifyingsituationsin which such estimatorsare most useful,and of signallingsome data configurationsin which theymightprovidemisleadingresults. 2. Estimators For each stratumi = 1, 2, ... , k let therebe ai events in a total observationtime Ci (including both censored and non-censoredindividuals) and bi events in a total observationtime Di. The two observed incidence or hazard rates are Ili = ailCi and I2i = bilDi. Let Xli and X2i be the Presentaddress: Unit of AnalyticalEpidemiology,InternationalAgencyfor Researchon Cancer, 150 cours AlbertThomas, 69372 Lyon Cedex 08, France. ? 1985 Royal StatisticalSociety 0035-9254/85/34042 $2.00 SMALL SAMPLES 43 with X1/X242i= for all i, and let I=in (). The stratum-specific corresponding paramneters estimatesof 4 are 4i = I1,/I21. AlthoughC1 and Di are randomvariables,theai and bi can be analysedas if theywereindependently Poissonwithparameters Oli= Xi1C1and ,Bi X21Difortwo reasons:the firsttwomoments ofthedistributions ofI1i andI2i areidenticalto thosethatwould be obtainedif indeedC1 and Di werefixed,and thekernelof thelikelihoodforthecountand observations is identicalto thatforthecountalonewithfixedperson-time person-time (Breslow, 1977, 1978,Clayton,1982; Berry,1983). The moststudiednon-iterative estimator whichcouldbe appliedto suchdata is theempirical logit,coupledwithan offsetto accountforvarying levelsof Ci andDi overstrata.The empirical logitforstratumi, modifiedso as to avoidundefined values(Haldane,1955; Anscombe,1956; Cox, 1970),is oi = In [(ai + 0.5)/(b1+ 0.5)] - ln (C1/D1), whosevariance(GartandZweifel,1967) is estimated by = (ni + 1) (n1+ 2)! [ni(ai+ 1) (bi + 1)], V?,,. whereni = ai + bi. An estimator ofthecommonhazardratioacrossstratais then =exp (LGT) = exp [(E2V-'i)/I V-'], with summations (here and below) takenoveri= 1 to k. An estimateof the varianceof the logitis empirical hLGT VLGT = ( 1 (1) Use oftheempirical in contemporary logitis infrequent epidemiologic analysis. andtwonewones,canbe derivedas information-weighted Commonlyusedestimators, averages of thestratumspecifichazardratios.Nurminen (1981) has shownthata commonratioestimate obtainedas a weightedaverageof stratum is asymptotically specificestimates efficient, provided thatthe weightsare asymptotically to the inverseof the varianceof the stratum proportional The varianceoftheweighted specificestimates. lowerbound, averageapproachestheCram6r-Rao givenby theinverseof thesumof stratum-specific Fisherinformations. derivedby this Estimators devicehave the attractive propertyof yielding4= 4' whenthe 4, are identicalacrossstrata. By a firstorderTaylorseriesexpansion,it can be shownthatthe varianceof the stratumspecificestimate4i is approximately equal to V4 = 0K ( (o +jp-l). (2) of observedcountsforthe parameters Aftersubstitution in thisexpression, an averageofthe4i, ofthevariances, to theinverse weighted according to simplifies Y2a3D1/(Cini) A The varianceof thenaturallogarithm of 4'i, foundagainby a firstorderTaylorseriesexpansion, iS A VAl fIV 2a2(a. + 4bi) Di/(niCQ2) [a[2DI/(Cini)] 2 + 2aibi(a? + bi)/n"4 22a3b,(a. - 2bi)D,/(n?Ci) + )2 (aibi/n,)2 (2atD,/C,ni)(Zaib/lni) 3 The numericvalue of this cumbersomeexpressionis in most instancesvery close to [Z(a[' + bf')] -1 The stratum-specific varianceofequation(2) canbe reparameterized to Vqi = 42 (4 1 Di + Ci)/(iCi). (4) 44 APPLIED STATISTICS When(4) is used as a basisforderiving theleadingterm(42) cancelsout. Substituting weights, bi forj3i and 1 forthe 4 withintheparentheses, one obtainsa simpleestimateof 4 firstproposed by Rothmanand Boice (1979) (RB hereafter), as an analogueof theMantel-Haenszel (Mantel andHaenszel,1959) estimateofthecommonoddsratioovera seriesof 2 x 2 tables: ^ 4,RB = -aiDilTi YbiClTi whereTi = Ci + Di. The varianceof the naturallogarithmof the expressionis approximately A _ VRB Sa1(Da/Tj)2 (2a1D/Tj)2 Y2bi(CI/T)2 (2b5C/T)) WhentheDi areall muchlargerthantheCi, thenexpression (4) becomes Vv,i= ,lDi (3iCi)'. (6) as above,bi for,i and weighting the 4i accordingto theinverseof(6), one obtains Substituting, thestandardized ratio mortality lai A h,MR = ____ Y,bilCiDi thevarianceofthelogarithm ofwhichis V?kSMR= (Ea{) -+ C/D+)2 (b (7) intoequation(4) willlead withinverse-variance Any consistentestimate ; of 4 substituted to an asymptotically weighting efficient "two-step"estimator, , S 2aiDi1/(,'D1+ CQ) (8) + CQ) b1Ci1/(_7'Di of 4TS willbe superiorto thatof 4RB as a conseGenerally4 is takenas 4RB. The efficiency quenceofusinga moreaccurateestimateof 4 in theweighting function. solutionof equation(8) leads to themaximum Clayton(1982) has pointedout thatrecursive likelihoodestimate,4ML. It followsimmediately thatwhenCi/Diis a constant, H, acrossstrata, thenthepooledestimate {'p = (lai) (H'Zbi)-l is the maximum-likelihood estimateof 4. The pooled estimateis also, forconstantH, equal to 4RB, 4TS, and 4SMR. Because it does not allow discrimination amongthevariousestimators, constantH willbe considered a degenerate ofthefollowing case and avoidedin the comparisons section. 3. Simulated Behaviour In orderto testtheperformance ofthemeasures in Section2, a seriesofsimulations presented have been carriedout. Each analysisconsistedof 10 000 randomreplications of a two-stratum, theincidenceratesshownin Table 1. The completeseriesconsisted two-sample studycomparing of 10 distinctdistributions of "large" and "small" expectednumbersoverthe fourcells,large forthispurposebeingtakenas 30 andsmallas 3, exceptwhereadjustment was necessary to avoid a constantCq/D1acrossstrata.For each study,a setof expectedcountswas chosenfora 1,b1, a2, and b2; persontimesat risk(C1, D1, C2,D2) werechosenso as to producetheexpectedincidence ratesof Table 1. Each randomreplication was generated usingtheexpectedcountsas theparametersof fourseparatePoisson distributions. strata(ai +bi = 0) were reAll uninformative SMALL SAMPLES 45 TABLE 1 ratesunderlying Inzcidence therandomizations in Table2, in eventsper1000 person-years expressed lpii Stratum 1 2 p=3 0.3 1.2 X2i 0.1 0.4 i 3.0 3.0 In (p) = 1.099 randomized.Replicationsfor which laibi = 0 were replacedwithnew ones; replacement was on thestudy,andis necessary by thiscriterion forfrom0 to 1 percentofall samples,depending therefore unlikely to haveaffected thesimulations in anyimportant way.The randomcountswere combinedwiththefixedtimesat riskandanalysedby eachofthetechniques ofSection2. Table 2 presentsthe randomization parameters chosen,theCram6r-Raolowerboundto the varianceof ( (i.e. ln (4)), calculatedfromthe parameters, and (for each technique)the mean valueof , theexpectedvariance(as derivedfromsubstituting parameter valuesintoequations(1), of skewness(the third (3), (5) and (7)), the observedvarianceof 5, the observedcoefficient momentabout the mean dividedby 1.5thpowerof the variance),and the mean squareerror estimators (observedvarianceplus the squareof the observedbias). Of the fournon-recursive on the log scale,OLGT appearsto be theleastbiasedoverthe rangeof circumstances examined; it has furthermore smallobservedvariances, and uniformly thelowestmeansquareerror.Of some interestis thatthevarianceof kLGT iS frequently less thantheasymptotic estimateprovidedby the Crarn6r-Rao lowerbound,mostnotablywhensmallcellspredominate. qRB is onlyslightly morebiasedon theaverage, andperforms betterthanOLGT in somecircumstances. Overtherange of simulations, the directionand magnitudeof theobservedbias in ORB correlates veryclosely withitscoefficient ofskewness, suggesting thatthevariations in theaverageobservedestimates are of q"ORB transformation the distribution closelytied to efficacyof the logarithmic in rendering symnmetrical. as thesmallcellspredominate. 'iv lhasa positivebias whichbecomesnon-negligible and by and largea moreskewedsampling OSMR has largestvariancein most circumstances, oflargebi and thantheotherlog estimates, distribution althoughin theimportant circumstances fromtheMLE. is,as expected,indistinguishable Di, itsperformance The recursive estimators, the OTS and theMLE, are closelysimilarwithobservedvariances close to the lowerbound,and meanvalueswhichtendslightly than closerto thelog parameter As with deviations in the mean estimate to be a function of skewness appear principally ORB ORB, in thelogdistributions. fromlog symmetry Within therange Departures followa regularpatternforall theestimators. of examplesstudied,smallcountsin ai coupledwithlargecorresponding bi resultin distributions skewednegatively on the log scale,and largeai together withsmallcorresponding bi givedistributionsskewedpositively. In mixeddata setstheeffectof a positiveskewnessseemsto predominate,and in data setswithai and bi similarforall i, theobservedsampling distributions arevery nearlysymmetric. Whencountsfromcellswithsmallexpectedvaluesplayan important rolein theestimates, the distribution of thelattercan becomenotonlyskewbutalso polymodal, particularly in thelonger of the smallcell distributions tail, as the discreteness becomesdominant.As an example,in the most commonlyencountered estimationproblemfromTable 2 (all ai small,all bi large),two and secondarypeaks containing over3.0 per centof the probability massfor TRB,qSMR, I to the null hypothesis, VML lie beyondthe estimatecorresponding despitethe factthat = 2.58 standard is ln(3)/(0.181)1/2 errors belowtheparameter. Whenrandomizations wereextendedto largernumbersof strataand proportionately smaller countsin the smallcells,the relativeperformance of thevariousestimators remainedthesame, withtheexceptionoftheempirical logit.Although optimalwhentheexpectedcountsin thesmall TABLE2 Performance of estimators ofa commonhazardratioin 10 000 randomrepetit Randomizationparameters Lower bound Method Observed mean Expected variance Observed variance Coeff of ske et B1 2 B2 4 3 3 3 0.311 IV RB StMR LGT TS MLE 1.300 1 112 1.114 1.082 1.111 1.111 0.311 0.312 0.313 0.273 0.413 0.408 0.410 0.258 0.406 0.406 0.219 0.018 0.030 -0.021 0.006 0.006 30 3 3 3 0.237 IV RB SMR LGT TS MLE 1 275 1 199 1.266 1.037 1 158 1.170 0 237 0.249 0.309 0.194 0.302 0.335 0.543 0.196 0.277 0.282 0.626 0.963 1.577 0.422 0.676 0.668 3 30 3 3 0 237 IV RB SMR LGT TS MLE 1.225 1.042 1.053 1.161 1.020 1.028 0 .237 0.243 0.258 0.194 0.262 0.295 0.312 0.193 0.274 0.278 -0.235 -0.542 -0.440 -0.387 -0.634 -0.637 40 30 3 3 0 054 IV RB SMR LGT TS MLE 1.131 1 103 1.103 1 099 1 103 1.103 0.054 0.054 0 054 0.052 0.057 0.057 0.057 0.053 0.057 0.057 0.135 0.133 0.131 0.121 0.134 0.133 40 3 3 0.181 IV RB SMR LGT TS MLE 1.199 1.184 1.186 0.972 1.184 1.184 0.181 0.181 0.184 0.143 0.235 0.235 0.239 0.155 0.235 0.235 0.847 0.847 0.859 0.531 0.847 0.847 30 TABLE 2 (continued) Randomizationparameters Lower bound Method Observed mean Expected variance Observed variance Coeffici of skewn aI I1 2 62 30 3 3 30 0 183 IV RB SMR LGT TS MLE 1.212 1.169 1.270 1.098 1.062 1.098 0.183 0.214 0.306 0 145 0.189 0.255 0.528 0.156 0.167 0.174 0.008 0.427 1.297 -0.039 0.035 -0.038 3 40 3 30 0 181 IV RB SMR LGT TS MLE 1.179 1.008 1 008 1 221 1 008 1.008 0.181 0 181 0.181 0.143 0.202 0.234 0.234 0.154 0.234 0.234 -0 524 -0.843 -0.843 -0.526 -0.844 -0.844 3 30 30 30 0 0 6 IV RB SMR LGT TS MLE 1 138 1.093 1.094 1.129 1.091 1.092 0.056 0.057 0.058 0.054 0 .061 0.059 0.060 0.056 0.059 0.059 -0 .047 -0.074 -0.062 -0.078 -0.089 -0.087 30 3 30 30 0.056 IV RB SMR LGT TS MLE 1 113 1.109 1.141 1.066 1.102 1.104 0.056 0.059 0.108 0.054 0 .056 0.061 0.115 0.054 0.057 0.057 0.100 0.130 0.261 0.089 0.097 0.096 40 30 30 30 0.031 IV RB SMR LGT TS MLE 1.117 1.102 1.102 1.098 1.102 1.102 0.031 0.031 0.031 0.031 0.032 0.032 0.032 0.030 0.032 0.032 0.047 0.038 0.041 0.042 0.041 0.038 APPLIED STATISTICS 48 logit cells wereon the orderof three,and the numberof stratalimitedto two, the empirical becamealmostalwaysmorebiasedon thelog scalethanORB, OSMR, or kTS with10 strataand expectedcountsof 0.6 in thesmallestcells. 4. Acknowledgements providedadviceforthisreport. N. Day, J. Kaldorand J. Wahrendorf Drs R. Brookmeyer, estimator. Dr J.R. Anderson suggested thetwo-step References F. J.(1956) On estimating Anscombe, binomialresponse relations. Biometrika, 43, 461-464. G. (1983) Theanalysisofmortality method. Berry, bythesubject-years Biometrics, 39, 173-184. Breslow,N. (1977) Some statistical modelsusefulin the studyof occupationalmortality. In Environmental Health:Quantitative Methods.(A. Whittemore ed.), pp. 88-103. Philadelphia: SocietyforIndustrial and AppliedMathematics. Commun.in Statist.,A7, -(1978) The proportional hazardsmodel: Applications to epidemiology. 315-332. studiesof diseaseaetiology.Commun.in Statist.,All, Clayton,D. G. (1982) The analysisof prospective 2129-2155. Cox,D. R. (1970) Analysis ofBinaryData, Section3.2. London:ChapmanandHall. of thelogitand itsvariancewithappliGart,J.J.and Zweifel,J. R. (1967) On thebias of variousestimators cationto quantalbioassay.Biometrika, 54, 181-187. ofa ratiooffrequencies. Haldane,J. B. S. (1955) The estimation andsignificance ofthelogarithm Ann.Human Genetics, 20, 309-311. Mantel,N. and Haenszel,W. (1959) Statistical aspectsof the analysisof data fromretrospective studiesof disease.J.Nat.CancerInst.,22,719-748. M. (1981) Asymptotic of commonrelative Nurminen, efficiency of generalnon-iterative estimators risk.Biometrika, 68, 525-5 30. Rothman,K. J. and Boice, J.D. (1979) Epidemiologic Analysiswitha Programmable Calculator.Washington D.C. NIH Publication 79-1649. Appendix An editor,in additionto offering manyhelpfulsuggestions whichhavebeenincorporated into the text,has pointedout thattheestimators 4B, "'PSMR, and 4TS can all be derivedfromlikelion ni,ai - Binomial(ni,H1(1 + Hi)1). Setting hood functions. Let Hi = Ci/Di.Thenconditional of theconditional thefirstderivative likelihoodequal to zero,one obtains ;(ai - bi'Hi) (1 + "Hi)1 =0 whichsuggests therecursive relation =a1 (1 + i/*Hi)l [Yb1H1(1+ p*H.)y] -1* 4* = Taking4* = 0 givestheSMR, 4* = 1 givestheRothman-Boice estimator, stepestimator. 4RB givesthetwo