A parametric empirical Bayes method for cancer screening using longitudinal observations of a biomar

14 Pages·0.129 MB·English

by  Mcintosh M. W.

Biostatistics(2003),4,1,pp.27–40
PrintedinGreatBritain
A parametric empirical Bayes method for cancer
screening using longitudinal observations of a
biomarker
∗
MARTINW.MCINTOSH ,NICOLEURBAN
FredHutchinsonCancerResearchCenter,1100FairviewAvenueMP-900,Seattle,WA98109,USA
[email protected]
SUMMARY
A revolution in molecular technology is leading to the discovery of many biomarkers of disease.
Monitoring these biomarkers in a population may lead to earlier disease detection, and may prevent
death from diseases like cancer that are more curable if found early. For markers whose concentration
isassociatedwithdiseaseprogressiontheearliestdetectionisachievedbymonitoringthemarkerwithan
algorithmabletodetectverysmallchanges.Onestrategyistomonitorthebiomarkersusingalongitudinal
algorithmthatincorporatesasubject’sscreeninghistoryintoscreeningdecisions.Longitudinalalgorithms
that have been proposed thus far rely on modeling the behavior of a biomarker from the moment of
disease onset until its clinical presentation. Because the data needed to observe the early pre-clinical
behavior of the biomarker may take years to accumulate, those algorithms are not appropriate for
timelydevelopmentusingnewbiomarkerdiscoveries.Thismanuscriptpresentsacomputationallysimple
longitudinal screening algorithm that can be implemented with data that is obtainable in a short period
of time. For biomarkers meeting only a few modest assumptions our algorithm uniformly improves the
sensitivity compared with simpler screening algorithms but maintains the same specificity. It is unclear
what performance advantage more complex methods may have compared with our method, especially
when there is doubt about the correct model for describing the behavior of the biomarker early in the
disease process. Our method was specifically developed for use in screening for cancer with a new
biomarker, but it is appropriate whenever the pre-clinical behavior of the disease and/or biomarker is
uncertain.
Keywords:CA125;Change-pointmodel;Ovariancancer;PEBalgorithm.
1. INTRODUCTION
A revolution in molecular technology is leading the discovery of a large number of genes that are
highly expressed or amplified in cancer. If a gene’s protein product, a tumor marker, can be found
elevated in the blood of diseased subjects then that tumor marker may be useful for cancer screening.
Thus, with numerous gene discoveries come numerous candidate tumor markers. Translating a tumor
marker discovery from the laboratory to use in a population requires stating a screening algorithm. The
mostcommonlyusedalgorithm,thesingle-threshold(ST)rule,classifiesasubjectpositive(likelytohave
disease) when his or her marker concentration exceeds a common population-wide threshold. A better
approach may be to use a longitudinal screening algorithm that makes use of a subject’s accumulated
∗
Towhomcorrespondenceshouldbeaddressed
(cid:2)c OxfordUniversityPress(2003)
28 M.W.MCINTOSHANDN.URBAN
screeninghistoryinordertoachieveevenearlierdetection.Thismanuscriptdescribesaformalprocedure
forgeneratinglongitudinalalgorithmsappropriatefornewmarkerdiscoveries.
Tumormarkersareusefulforearlydetectionifaftercanceronset,andpriortoclinicalpresentation,
their concentration in serum or other fluids deviates from their normal levels. Previously proposed
longitudinal algorithms work by monitoring their concentration over time and give a positive screen to
subjectswhosetrajectoriesappearmoresimilartothatsuspectedinpre-clinicaldiseasethannormal.To
makethisassessment,however,thosealgorithmsmustmodelthepre-clinicalmarkertrajectoryfromthe
moment of disease onset through its clinical presentation. For example, Slate and Cronin (1997); Slate
and Clark (1998), and Morrell et al. (1995) screen for prostate cancer with the PSA tumor marker and
SkatesandPauler(2001)screenforovariancancerwiththeCA125tumormarker.Iftheearlypre-clinical
behavior of the tumor marker is modeled correctly, these algorithms yield optimal screening decisions.
Unfortunately, obtaining sufficient data to estimate the pre-clinical marker trajectory is impractical for
manydiseases,especiallythosewithlowprevalenceandrapidprogression.
Because longitudinal algorithms are intended to detect cancer earlier than the ST rule they must do
sobygivingapositivescreenwhenmarkerconcentrationsarelowerthantheSTrules’threshold.Thus,
the behavior after marker trajectory exceeds the ST rule is not relevant for earlier detection, and so any
approachthatmodelsmarkertrajectorymustbeparticularlyaccurateinitsrepresentationoftheveryearly
pre-clinicalperiod.Ofcourse,observingtheearlytrajectoryrequireshavingaccesstoasufficientnumber
ofserialspecimenscollectedduringthistime.Unfortunately,suchobservationsareextremelyrarebecause
bydefinitionascreenedpopulationhaslowdiseaseprevalence(screenedpopulationsareasymptomatic).
Evenserastoredaspartofalargecohortstudymaynotbesufficient.Forexample,arecentovariancancer
screeningtrial(Jacobsetal.,1999)collectedserumspecimensateachofthreeannualscreensfrom10958
women.SpecimenswerealsocollectedeverythreemonthsforuptoayearfromwomenwhoseCA125
concentration was found above its ST rule of 30 U ml−1. Only six women were found with ovarian
cancerduringthescreeningperiod,withtenmorefoundduringthesevenyearfollowup.Becausethepre-
clinical period of ovarian cancer is thought to be between 10 months (Bast et al., 1985) and 18 months
(Zurawski et al., 1988), few of those ten cases are likely to contain any serial samples collected during
the pre-clinical period. Although serial samples may have been captured from the other six cases, the
screening interval may have been too short to observe the trajectory of this rapidly developing disease.
Moreover,althoughthesixscreen-detectedcasescontributeserialspecimensfollowinginitialelevation,
thatobservedtrajectoryisnotinformativeabouttheearlierbehaviorwhenitwasbelow30Uml−1,the
part most relevant if earlier detection is to be achieved. Finally, the information contained in any stored
specimens,eveniftheyareabundantenough,maynotbeaccessiblefornewmarkersbecausespecimen
needs(i.e.plasmaorurineinsteadofsera)and/orspecimen-processingrequirementsofnewdiscoveries
cannotbeanticipated.
Becausemarkerbehaviourduringtheearlypre-clinicalperiodmaybeunobservable,algorithmsthat
do not rely on knowing it may offer a valuable alternative. One simple screening algorithm that uses
screening history but does not depend on the assumed marker trajectory has been proposed by Urban
et al. (1997). Using a microsimulation model, they found that detection rates for ovarian cancer were
improvedbygivingapositivescreenwhenCA125haseitherdoubledsincethemostrecentscreenorhas
reachedathreshold.Becausenoformal(statistical)frameworkwasusednogeneralstatementsaboutits
efficiencycanbemade,andtheirconclusionscannotbegeneralizedtoothermarkersordiseases.Indeed
undersomecircumstancestheSTrulemayoutperformadoublingrule(seeSection4).
Our method uses screening history to decide if a general, rather than a specific (i.e. exponential or
linear), change in marker behavior has occurred, and does not require stating a specific marker growth
model. Our method is motivated by the need for an algorithm that is robust to a variety of growth
behaviors, and to have specimen demands that are practical. Indeed, our approach uses specimens that
arequiteabundantbecauseitsattentionisfocusedonthehealthysubjects.
Longitudinalscreeningwithcancerbiomarker 29
Our screening rule uses parametric empirical Bayes (PEB) theory to generalize the ST rule to be
conditional on screening history. Each subject’s screening history is used to set an individually tailored
threshold that achieves an arbitrary false-positive rate (FPR). Controlling the FPR is particularly appro-
priateforcancerscreeningbecausecosts(socialandeconomic)aredominatedbytherateofpositivetests
which,becausediseaseprevalenceislowinscreening,isdominatedbytheerrorrateinthehealthysubjects
(FPR).Thus,anoverarchingconstraintofanyscreeningprogramistolimittheerrorrateinthehealthy
subjects,andanyscreeningalgorithm,whateveritsdetectionrateorapproach,mustaccommodateit.
WiththePEBscreeningrulewefindthatagreatmajorityofallsubjectscanhavetheircancerdetected
with marker concentration at levels far lower than with the ST rule, and so cancer is detected earlier.
Importantly, this earlier detection is achieved while maintaining the same population FPR and so the
overall cost of a screening program is controlled. The PEB rule uniformly dominates the ST rule, and
can be used with any marker having some mathematical transformation that gives the healthy subjects
ahierarchicalnormalmarkerdistribution.Theseassumptionsappearwidelyappropriateformanytumor
markers(Crumpetal.,2000;Pauleretal.,2001).Nospecificrestrictionsaremadeonthebehaviorofthe
markers in the cases other than that their concentrations deviate (elevate or fall) from normal following
canceronset.Specimenrequirementsaremodest,asserialmeasurementsoncasesarenotneeded.
Section 2 outlines the notation and statistical model assumed throughout the text. Section 3 uses
thestatisticalmodeltoderivetheSTscreeningrulemostcommonlyusedincancerscreening,andthen
generalizesittoperson-specificthresholdsusingPEBtheory.AllscreeningrulesinSection3aredefined
to have comparable performance in healthy subjects, and the performance of the algorithm in diseased
subjects is discussed by Section 4. Section 5 presents an example using CA 125 to screen for ovarian
cancer.Section6providesadiscussionoftherobustnessofouralgorithm,andpresentstechnicaldetails
thatmaybeskipped.AsummaryisgiveninSection7.
2. BACKGROUNDANDNOTATION
We use Y to represent the marker concentration at a person’s jth screen, and D to represent the
j j
disease status at that time; D = 0 when disease free, and D = 1 otherwise. We assume cancer does
j j
not regress, so Dj = 1 implies Dj(cid:3) = 1 for all j(cid:3) > j. We use lower case letters to denote observed
quantities,andsothescreeninghistoryfromasubjecthavingcompletednscreensisdenoted y ,...,y .
1 n
Section 2.1 represents the statistical model used to represent the behavior of the marker when without
cancer (when D = 0), and Section 2.2 represents the behavior of the marker after cancer onset (when
j
D =1).WhenwithoutcancerourmarkerbehavesinamannersimilartothatusedbySkatesandPauler
j
(2001), but whereas their method is for only a specific pre-clinical marker behavior, ours requires only
thatthemarkerdeviates(elevatesorfalls)followingcanceronset.
2.1 Markerbehaviorinsubjectswithoutcancer
We represent the marker concentration at the jth screen for a healthy subject with the following
hierarchicalmodel:
Y |µ,σ ∼ N(µ,σ2) (1)
j
µ|µ¯,τ ∼ N(µ¯,τ2) (2)
whichimpliesthatthemarginaldistributionofthemarkerinthepopulationofhealthysubjectsis
Y |µ¯,τ,σ ∼ N(µ¯,τ2+σ2). (3)
j
30 M.W.MCINTOSHANDN.URBAN
Notethatthedistributionsaboveareimplicitlyconditioningon D =0.
j
The individuals in a population are given person-specific normal marker levels denoted by µ (we
suppressperson-levelsubscriptsforµandY forclarity),butwithinsubjectvariances,σ2,areassumed
j
equal.µ¯ andτ denotethemeanandstandarddeviationofµinthepopulation,respectively.
Screening rules are invariant to monotonic transformation so generality is not lost by assuming
distribution (3) for any continuous marker because then some transformation to normality is assured.
Althoughourhierarchicalmodelissimilartothatusedbyothermorecomplexmethods(SlateandCronin,
1997; Pauler et al., 2001; Skates and Singer, 1991), the existence of suitable transformation for those
methodsisnotassuredbecausetheyrequirethetransformationtobecompatiblewiththeirspecificmarker
growthmodel.
Wecanfurtherreducethetotalnumberofparametersinthemodelbyre-centeringandrescaling(both
monotonic transformations) so that the marginal distribution (3) has zero mean and unit variance. This
reducedmodelisgivenby
Y |µ∼ N(µ,1−B ) (4)
j 1
µ|B ∼ N(0,B ) (5)
1 1
where B =τ2/(σ2+τ2).Themarginaldistribution(3)becomes
1
Y ∼ N(0,1). (6)
j
The reduced model (4)–(6) has only a single parameter, B , which represents the fraction of total
1
variability accounted for the between-woman variance. The reason for the subscript becomes apparent
below. Because it has fewer parameters we will prefer it to the full model (1)–(3) when deriving our
screeningrulesinSection3.
The model above and the screening rules below can be extended to accommodate fixed and time
varying covariates (e.g. ‘race’ for the CA 125 ovarian cancer marker and ‘age’ for the prostate cancer
markerPSA),andserialcorrelation.Timevaryingcovariatesandserialcorrelationmustbeaccommodated
if their effects are substantial in order to avoid bias. Including fixed covariates improves the overall
precision of the screening rule. Accommodate fixed covariates by replacing µ¯ with a linear predictor
µ¯ + Xβ (seeMorris,1983bandPauleretal.,2001).Accommodatetimevaryingcovariatesbymodeling
theresiduals,ordeviations,fromthepopulationexpectedvalue.Specifically,ifX representsthecovariate
t
valueattimet,andµ¯ + X β predictsthemeanmarkerconcentrationforallsubjectswithcovariate X
it t
(i.e.age),thentheresidual R =Y −X β followsthehierarchicalmodel(1)–(3).Accommodateserial
it it it
correlationbyinflating(deflating)thewithin-personvariancein(1)or(4)basedonthetime-dependence
modelused.Fortheremainderofthispaperweassumethattheseeffectshavebeenaccommodatedsothat
thehierarchicalmodel(4)–(6)isanadequaterepresentationofthemarkertrajectoryinhealthysubjects.
2.2 Markerbehaviorwiththeonsetofcancer
Aftercanceronsetthelevelofamarkermaydeviatefromitspre-cancermeanµbyanamountδ ,where
t
t (cid:1)0representsthetimesincecanceronset.Weassume0<δt <δt(cid:3) whent <t(cid:3)becausemosttypically
markersgrowaftercanceronset(seeBaronetal.,1999forarareexception;ourmethodisalsoappropriate
formarkersthatdescendafteronset).Thus,whenasubjecthascancertheexpectedvalueoftheirmarker
isgivenbyµ+δ .
t
3. SCREENINGRULES
TheSTruledefinesasinglevalueandgivesapositivescreentoanysubjectfoundtohaveamarkerthat
exceedsit.Thethresholdischosentoachieveaspecifiedpopulation-wideFPR,denoted f .Formally,the
0
Longitudinalscreeningwithcancerbiomarker 31
STrulegivesapositivescreenwheneverY >c(f )wherec(f )satisfies f = P(Y >c(f )|D =0).
j 0 0 0 j 0 j
Incontrast,alongitudinalalgorithmdeterminesathresholdthatdependsonthesubject’sscreeninghistory
y1,...,yn. Formally, a positive screen is given to screen n +1 if Yn+1 > c(y1,...,yn; f0), where the
function c(y ,...,y ; f ) is determined by a particular choice of algorithm. This section shows how
1 n 0
a longitudinal algorithm can be generated by naturally generalizing the ST rule while maintaining the
overallFPRat f0 = P(Yn+1 >c(y1,...,yn; f0)|Dn =0,y1,...,yn).
Wegeneratescreeningrulesusingthelogicofhypothesistests(seeMcIntoshandPepe,2002forthe
equivalenceofhypothesistestingandscreeningtestgeneration).Allscreeningrulesbelowusethereduced
model(4)–(6)andassumethatthemodelparametersµ¯,τ,σ andthetransformationtomarginalnormality
are known, or estimated with high precision. The actual number of specimens required to estimate the
transformation and model parameters depends on the intended specificity of the screening algorithm;
highspecificityimpliesthatthetailofthedistributionmustbeaccurate,andwillincreasedatademands.
However,because,serialsamplesareneededonlyonhealthysubjectssufficientdatacanbeaccumulated
inashortperiodoftime.Ifneeded,themodelscanbeextendedtoallowforuncertaintyinthem(perhaps
usingafullyBayesianframework,orwithamixedlinearmodelandusingbestlinearunbiasedpredictors).
3.1 Single-thresholdscreeningrule
TheSTruleisgeneratedbythemarginaldistribution(6)bycarryingoutthehypothesistestH : E(Y )=
0 j
µ¯ versus H : E(Y )>µ¯ withtypeIerrorrateequalto f .TheSTrulegivesapositivescreenwhenever
a j 0
Yj >z1−f0 wherez1−f0 isthe100×(1− f0)percentileofthenormaldistribution.
3.2 Anindividuallytailoredscreeningrule
If µ were known perfectly the ST rule could be tailored to the individual by performing the hypothesis
test H : E(Y |µ) = µ, versus H : E(Y ) > µ. Because the between-subject source of variation is
0 j a j
eliminated(equals B inthereducedmodel)thislimitingrulescreenspositivewhenever
1
(cid:1)
Yj (cid:1)µ+z1−f0 1−B1. (7)
Intuitive√lyweseethatthislimitingruleoutperformstheSTrulebecauseitdetectsdeviationsfromnormal
thatare 1−B ×100%thesizebutwiththesameFPR.WeshowthisformallyinSection4.
1
3.3 Ana¨ıvesequentialrule
Because µ is not known we must find a compromise rule than uses information about µ obtained over
time.Asubjectund(cid:2)ergoingan(n+1)thscreeningeventhasahistoryoflengthn,andwesummarizethat
historywithy¯ = n y /n.Asimpleapproachtocontrolforthishistoryreplacesµin(7)withy¯ ,but
n j=1 j n
adjustsfortheuncertaintyin y¯ asanestimateofµinordertomaintainFPR= f .Inthereducedmodel
n 0
thisna¨ıvesequential(NS)rulegivesapositivescreenwhenever
(cid:1)
Yn+1 > y¯n +z1−f0 (1−B1)(1+1/n). (8)
N√ote that the NS rule has the same form as the ST and limiting rule but with the factor
(1−B )(1+1/n).TheSTrule(whenn = 0)andthelimitingrule(whenn = ∞)arespecialcases
1
oftheNS√rule.IntuitivelyweseethattheNSrulecandobetterorworsethantheSTruledependingon
whether (1−B )(1+1/n) is less than or greater than unity, which is not assured when n or B are
1 1
small.Forexample,if B =1/4thennotuntiln =5willthena¨ıverulemeetorexceedtheperformance
1
oftheSTrule.
32 M.W.MCINTOSHANDN.URBAN
3.4 PEBscreeningrule
TheNSrulereplacesµin(7)byitsusualunbiasedestimate, y¯ .HereweproposereplacingµbyitsPEB
n
estimate.PEBreferstotheclassofstatisticalproceduresforestimatingagroupofindividualmeanswhen
drawnfromacommonpopulation(forexampleseeMorris,1983b;Casella,1985andEfronandMorris,
1997 for technical and non-technical overviews of PEB methods). The PEB estimator, denoted µˆ , is a
n
functionoftheobservedsamplemean y¯ andthepopulationparameters.Theestimatorisexpressedas
n
µˆ =µ¯(1−B )+y¯ B (9)
n n n n
wheretheshrinkagefactor B equals
n
τ2
B = .
n σ2/n+τ2
Settingµ¯ = 0in(9)givesthePEBestimateforthereducedmodel.Althoughtheexpression(9)hasthe
formofthefamiliarBayesestimatorwecallitanempiricalBayesestimatorbecausethe‘prior’parameters
areestimatedempirically,andsothepropertiesweattributetoitarefrequentistinnature.
Becausetheshrinkagefactor, B ,liesbetweenzero(whenn =0)andone(whenn islarge)thePEB
n
estimatorisacompromiseestimatorthatfallsbetweenthepopulationmeanandtheindividual’ssample
average. Because the shrinkage factor increases with n the PEB estimate comes into closer agreement
with the individual average when history is plentiful. With small n the PEB estimator is closer to the
populationaverage.Thus,thePEBestimatorallowstheindividualhistorytocarryagreatervoicewhen
historyissubstantial,butanticipatesregressionforsubjectshavinglittlescreeninghistory.
ThePEBestimatorhastwopropertiesthatmakeitusefulforscreening:
(i) itisanunbiasedestimateofafutureobservationjustlike y¯ ,
n
E(Yn+1|y¯n)=µˆn
(ii) butithassmallervariance
σ2 σ2
Var(µˆ |y¯ )=Var(Y¯ )B = B (cid:2) =Var(Y¯ ).
n n n n n n
n n
Inthereducedmodelweexpressthisvarianceas
1
Var(µˆ )| y¯ )= (1−B )B .
n n 1 n
n
The expectations in (i) and (ii) are conditioned on Y¯ = y¯ and the population parameters. See Morris
n n
(1983a) for derivations, but note that the definition of B and B used here differs slightly. When no
1 n
covariates are used these properties may be derived using normal regression formulae as well. For
example,deriveproperty(i)bynotingCov(Yn+1,Y¯n)=τ2 andVar(Yn)=τ2+σ2/n,andsoregressing
Yn+1 on Y¯n gives the coefficient τ2/(τ2 +σ2/n) = Bn. The advantage of the PEB viewpoint over the
regressionviewpointisthatgeneralizationtocovariatesisstraightforward.
WegeneralizetheSTrulewiththePEBestimatorusingthehypothesistest H0 : E(Yn+1|µˆn) = µˆn
versus Ha : E(Yn+1 | µˆn) (cid:1) µˆn,andrejectwhenYn+1 exceedsitsconditionalexpectationbytoomuch.
ThedistributionofYn+1−µˆn underthenullisnormal,withvariancegivenby
Var(Yn+1−µˆn | y¯n,Dn+1 =0)=Var(Yn+1|y¯n)+Var(µˆ | y¯n) (10)
1
=(1−B )+ (1−B )B
1 1 n
n
=1−B B . (11)
1 n
Longitudinalscreeningwithcancerbiomarker 33
Expression (10) follows from the independent sampling of Y and property (ii), and (11) follows by
j
usingtheidentity 1(1− B )B = (1− B )B andsimplifying.Thesimplicityof(11)isaconsequence
n 1 n n 1
of defining the shrinkage factor slightly differently than Morris (1983a). Expression (11) can also be
derived simply from regression formulae when covaria(cid:3)tes are not use(cid:4)d by simplifying the expression
Var(Yn+1 |Y¯n)=Var(Yn+1)−Bn2Var(Yn)=Var(Yn+1) 1−Bn2VVaarr((YY¯nn)) .
Thus,thePEBrulegivesYn+1apositivescreenwhenever
(cid:1)
Yn+1 >µˆn +z1−f0 1−B1Bn. (12)
NotethatwhennohistoryisavailablethePEBrulehas B = B = 0,andthePEBruleequalstheST
n 0
rule. Subjects with long screening history have B∞ = 1 and the PEB rule equals the limiting rule (7).
BetweentheseextremesthePEBruleprovidesasmoothgeneralizationoftheserules.Aspecificexample
ofthePEBruleisgiveninSection5.
The derivations above assume the population parameters are estimated with high precision. If the
model parameters are estimated with substantial error then (11) must be adjusted to reflect this added
uncertainty.Perhapsthesimplestapproachwouldbetoaddadditionaltermsto(11)toinflatethevariance
untildatabecomemoreabundant(forexampleseeSection5ofMorris,1983b).
4. PERFORMANCEOFSCREENINGRULESINCANCERCASES
ThissectioninvestigatesthebehaviorofthePEBruleindiseasedsubjectsbyprovidingexpressions
foritssensitivity,ortruepositiverate(TPR).TheexactTPRofanyparticularscreeningruledependson
howsoonaftercanceronsetthescreentakesplace,andonaspectsofthetumor’sgrowthbehaviorwhich
arenotknown.TheTPRoftheNSandPEBrulesdependalsoontheabundanceofscreeninghistoryat
thetimeofthescreen.BecausewewishtoavoidstatingagrowthmodelweexpressestheTPRofeach
rulewhenthemarkerhaselevatedafixedbutarbitraryamountδ >0;thatis,themarkerYn+1nowhasan
expectationequaltoµ+δinsteadofµ,buthasthesamevariance.
With screening history summarized by y¯ and cancer onset occurring after the nth screen, the PEB
n
screeningrulewithnhistoricalscreensisexpressedas
(cid:3) (cid:1) (cid:4)
tn = P Yn+1 >µˆ +z1−f0 1−B1Bn | y¯n,E(Yn+1)=µ+δ (13)
andfromexpression(11)thisbecomes
(cid:3) (cid:1) (cid:4)
= P Z >z1−f0 −δ/ 1−B1Bn
where Z representsastandardnormaldistribution,z1−f0 itsupper1− f0quantile.
Moreover,bychoosingn = 0orn = ∞expression(13)representstheTPRoftheSTandlimiting
rules, respectively. Because t0 (cid:2) tn (cid:2) t∞ the PEB always outperforms the ST rule and will eventually
meet, but never e√xceed, the perf√ormance of the limiting rule. The PEB rule always outperforms the NS
ruletoobecause 1−B B (cid:2) (1−B )(1+1/n).
1 n 1
ThePEBTPR(13)iscomputedassumingthatthecanceronsetoccursfollowingthenthscreen,which
assures that screening history is uncontaminated with any false negative measurements. False negatives
mustbeaddressedbecauseamarkerthatelevatesbutnotenoughtocauseapositivescreenwillcause(12)
to deviate from optimal on subsequent screens. It is not immediately apparent that the PEB rule should
stillbepreferredtotheSTruleinthepresenceoffalsenegatives.Section6showsthatindividualsandthe
populationshouldpreferthePEBruletotheSTruleevenwiththeriskofafalsenegative.Wefirstgive
anexampleofthePEBruleusingtheCA125ovariancancertumormarker.
34 M.W.MCINTOSHANDN.URBAN
5. EXAMPLEOFTHEPEBRULEFOROVARIANCANCER
Mostwomendiagnosedwithovariancancerarediagnosedatalatestage,andnearlyalldiefromtheir
disease.However,mostofthosefewdiagnosedwithearlystagediseasesurvive.Thissuggeststhatearly
detection of ovarian cancer may have a dramatic impact on mortality. Because the disease is relatively
rare,itisdifficulttoobtainserialspecimenscollectedduringthesuspectedpre-clinicalperiod.However,
many longitudinal cohort studies have very large stores of serum for asymptomatic women that can be
usedtoestimatethePEBalgorithm.Forexample,atenyearstudyofover1200high-riskwomen(Karlan
et al., 1993) was used by Crump et al. (2000) to determine the information sufficient to implement the
PEB algorithm for five different ovarian cancer tumor markers. Here we present the algorithm for the
CA125tumormarker.
5.1 Implementingthealgorithm
Many studies report that log(CA 125) in healthy subjects has a normal distribution, with B falling
1
between0.6inahigh-riskpopulation(Crumpetal.,2000)and0.9inageneral-riskpopulation(Pauleret
al.,2001).Thepopulationmeanlog(CA125)level,µ¯,hasbeenfoundtodependonthetypeofCA125
assayused(Davelaaretal.,1988),andthemenopausalstatusandraceofthewomen(Crumpetal.,2000;
Karlanetal.,1993;Pauleretal.,2001).However,thevalueof B isunaffectedbythesecovariates(e.g.
1
pre-andpost-menopausalwomenappeartohavethesame B eventhoughtheyhavedifferentµ¯;Crump
1
etal.,2000).
ThePEBscreeningalgorithmpresentedhereusesaFPR f =0.01,andB =0.6,thelowestreported
0 1
value B forlog(CA125).Highervaluesof B willimplybetterperformanceofthePEBrulecompared
1 1
to the ST rul√e. For convenience the mean and marginal variance of log(CA 125) are selected as µ¯ =
log(10) and σ2+τ2 = 0.5, respectively. This mean is similar to that observed for a population of
post-menopausal women (Crump et al., 2000), and a different mean should be used if applied to a pre-
menopausal population. Because serial correlation between subsequent log(CA 125) values cannot be
detected when observed more than one month apart (Crump et al., 2000), we may safely ignore serial
correlationinourmodelforanyscreeningprogramwithintervalsexceedingthatamountoftime.
Forafixedamountofhistory,n,thePEBrule(12)islineariny¯ ,withslopeequalingB ,andsoitcan
n n
bedisplayedgraphically,asinFigure1a.Thehorizontalaxisrepresentsthemeanofawoman’shistorical
log(CA125)levels,andeachlineinthefigureislabeledbyn,thenumberofscreensusedtocomputethe
mean.Thelinewithlabeln =0representstheSTruleandhasinterceptz1−f0 = z0.99 =2.33,andslope
B0 =√0. Subsequent screens have slopes Bn that increase to B∞ = 1, and an intercept that reduces to
2.33 1−B1.Wecantransform(cid:3)backtotheraw√CA125s(cid:4)caletoo,asinFigure1b,whosehorizontalline
givesaCA125thresholdofexp µ¯ +z1−f0 × σ2+τ2 =exp(log(10)+2.33×0.50)=32.
Figure1ashowsthatthePEBruledivergesfromtheSTruleashistoryaccumulates.Thefewwomen
whohavenaturallyhighCA125concentrationswillhavethresholdsabovetheSTrule,butthemajority
willhavelowerthresholds,somedramaticallyso.Becauseanaveragehealthywomanhasy¯ =0(reduced
n
model), we see that far more than half of all women will have a lower threshold if using the PEB rule.
Indeed, average women ha√ve a ST rule threshold of 2.33, but this becomes 80% of it (or 20% smaller)
√afteronlyasinglescreen, 1−0.62 = 1.86,andonly20%(or80%smaller)ofitafterseveralscreens,
1−0.6=1.47.
Figure 1a shows that all women having histories with y¯ less than approximately y¯ = 1 are given
n n
lowerthresholds.Theexactfractionwithalowerthresholdisgivenby P(Y¯ < 1)whichdependsonthe
n
specificvalueof B andn;largervaluesofeitherincreasethefractionwithloweredthresholds.Forour
1
example84,88,89,and95%ofwomenaregivenlowerthresholdsforn =1,2,3,∞,respectively.Thus,
whenusingthePEBruletheoverwhelmingmajorityofwomenwillhavealowerthresholdthantheST
Longitudinalscreeningwithcancerbiomarker 35
44444 4040404040
A 125A 125A 125A 125A 125
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Fig.1. Graphic representation of the PEB screening rules on the log scale (left) and raw scale (right) with false
population and individual positive rates equal to f0 = 0.01. (a) Reduced model PEB algorithm with B1 = 0.6;
(b)PEBalgorithmforrawCA125.
rule gives them, and so their cancers can be detected earlier. A key point is that the PEB rule gives this
lowerthresholdwhilemaintainingthesamepopulation-wideFPR.
Of course, lowering the threshold will increase an individual’s FPR compared with the ST rule. In
ordertomaintainthepopulation-wideerrorratetheincreaseexperiencedbythemajoritymustbeoffset
byaconcomitantdecreasefoundbytheminority.Thefactthatincreasingthethresholdinasmallfraction
of women permits drastic reductions in the thresholds of the majority is a consequence of the fact that
whenmarkersbehaveheterogeneouslytheSTthresholdsaredeterminedbyafewextremecases.Indeed,
inourexample5%ofwomenexperienceoverhalfthefalsepositiveswhentheSTruleisused.ThePEB
rule spreads the burden of false positives evenly throughout the population so that all women have the
same burden, or better, it can even permit each woman to choose her own FPR, depending on her own
screeninghistory.Aconsequenceisadramaticincreaseinthecancerdetectionability,asweseenext.
5.2 Thebehaviorincases
TheTPRofthePEBruletodetectanelevationδ >0isgivenbyt ,in(13).Thepracticalsignificanceof
n
anyelevationmustbemadeonacase-by-casebasis.Fordemonstrationpurposesonlywechooseδ =1as
aconvenientandarbitrarymarkerelevationofonemarginalstandarderror.ThesolidcurvesinFigure2
relatetheTPR(verticalaxis)versusFPR(horizontalaxis)overalowrangeofFPRrelevantforscreening
(i.e. low FPRs). Such curves are often referred to as receiver-operating characteristic (ROC) curves and
may be used to characterize the performance of diagnostic tests (Hanley and McNeil, 1982). Each plot
representstheROCcurvewithdifferentvaluesofn.
Whennohistoryisavailable(Figure2)thePEBandSTrulesdetect26%ofthecasesat f = 0.05.
0
Whenonlyasinglehistoricscreenisavailablenearly34%ofsuchcasescanbedetectedbythePEBrule,
andwithoneadditionalscreenthePEBrulefinds38%ofthem.Withsubstantialhistory(n =∞)thePEB
rulefinds47%ofcases,whichrepresentsarelative77%improvementovertheSTrule.With f = 0.01
0
thesepercentagesare10,15,17,and23,respectively.Thegaininperformancewillbegreaterforlarger
valuesof B ,butforanyfixedelevation,andany B > 0,thePEBROCcurveswilluniformlydominate
1 1
theSTrules.
36 M.W.MCINTOSHANDN.URBAN
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0.0.0.0.0.
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Fig.2. SequenceofcurvesrelatingtheFPRtotheTPRofthePEBscreeningrule.Numbernexttothecurveindicates
thenumberofscreenscontrolledfor.Curvelabelled0representstheROCcurvefortheSTrule.
Note that for even this modest B , controlling for only a single historical screen (n = 1) achieves
1
a 30% increase in TPR, which is nearly half the potentially achievable amount. This suggests that only
afewrecentscreensneedtobecontrolledforinordertoobtainthemajorityofbenefitofthePEBrule.
Screensinthedistantpastmaybeignoredwithoutsubstantiallossofperformance.
5.3 Theperformanceinapopulation
A comprehensive ovarian cancer microsimulation model has been used to evaluate the behavior of the
PEB algorithm in a large cohort of women in the US (McIntosh et al., 2002; Urban et al., 2002). The
microsimulationmodel,explainedindetailbyUrbanetal.(1997,1998),isintendedtoexaminethecost
andhealthimpactofovariancancerscreeningprograms.Thesesimulationsarequitecomprehensive.The
lifehistories(cancerincidence,cancerdeath,treatment,andothercausedeath)fortheUSpopulationare
modeledtoreflectUSCensusBureaulifetablesandtheage-adjustedUScancerincidence.
Urban et al. (2002) used the microsimulation model and the PEB algorithm to estimate the
performanceofscreeningexpectedwhenusingascreeningprotocolsimilartothatoutlinedbySkatesand
Singer(1991),whorecommendtwolevelsofpositivityforascreen:positivescreenforextremeelevations
andearlyrecallformodestones.Notethatforovariancancerapositivescreenfromamarkerresultsin
referraltoultrasound,notsurgery.ThePEBalgorithmusedaFPRof2%todefinetheextreme,and15%
for modest (i.e. 13% of healthy women were recalled early). With annual screening this configuration
of the PEB algorithm detects 70% of cancers before their clinical diagnosis, whereas the ST rule finds
only 46% of them. Importantly, the PEB rule finds over 50% when at early stage compared with a
mere30%bytheSTrule,andonly20%withoutscreening.ThePEBrulefoundanexpected31%drop
incausespecificmortalitycomparedwiththe18%oftheSTrule.Thissimulationstudyused B = 0.6,
1
which is lower than what is now reported in general-risk populations with a modern assay (Skates and
Pauler,2001;Pauleretal.,2001)usinghighervaluesof B wouldfindevengreaterimprovement.
1
The predictions of Urban et al. (2002) are very close to that of the complex algorithm described by
SkatesandPauler(2001):theypredict60%ofcancersdetectedatearlystage.Thesesimulationsarenot
exactlycomparablebecauseSkatesandPauler(2001)assumealarger B andalongerdurationofearly
1