1. No category

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 [email protected].
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