Document L8G0Ewxr8zdwppeOMgRjbrKX

P Environmental Health Perspectiws &I 90.pp. 271-27Z 1991 Recent Developmentsin the Multistage T b c m o d e H n g d c o b o r t d a t a b e r e d o o t b e A r m i g g c D o i l m u l t b f s g e d d t h e*P~n==b*- acceptanceasa methoddogyfor quantitativerisk pgcssmentforestimatingtbedose-rehtrd Fdstioashipbetween different occuptiond andenvirwuncntdcarcimgenkexposuresand cancer mortality. Tbc multist.gcmoddam beused e-f o r a r t r P p o l P t i o a t o h ~ M f W ~ ~ ~ ~ d ~ ~ ~ more thanone stage is dosprelated,which assistsin dettrminingwbethcr differnotCnrCiOOgCmPnectmerent stngcsof the cancer pnmrrs.This pspersummnripsrecmtdcvelopmentsinthem u l t i s r n g c m o d e h g o f c o b o r t d a E s a n d m r ~ practicd issuessuch as handlingdata arising fromlarge epidemiotogk fdlow-upStUdkS,the timedependentMturn of exposures and st.tlstical issues such as m- `ty in time-relatedvuiablq robustnessofpunmctcr cstlnutcs, vPUdatingdtbcnttedmodds,~I.outine~~tinlproctdura.Prob~~~toUnmfrintksofriPkatimstcsnrc discussed also. Computer pmgrpmsfor fittingmultistage modelswith oneor twodoscdnted to cohort data in- corporatingtimedependentexposure pattcras;constructingconfidence+os formodelpuuneten; and p m lifetime risksof dying fromcancer adjusting for competing causcsof death M detailed. lllust~tioo~ provided US- ing lung cancer mortality in a cohortof nonwhite male coke wen workers ex& to cool tar pitch volatiks. Introduction Recent literatureon epidemiologicstudiesof human cancer indicates that the Armitage-Doll multistage model of carcinogenesiscan be used to examine the relationshipsof excess cancer mortality to somecarcinogenicexposure and time-related variables such as age at initial exposure;duration of exposure; and time since exposure terminated (1-4). These relationships help to predict benefits of different strategies for reducing carcinogenic exposure in the workplace or environment. Because multistagemodels can be formulated to provide information regarding whether more than one stage is dose-related, thesemodels assist in determining whether different carcinogens affect different stages of the cancer process. They also are used in carcinogenicquantitative risk assessment for estimating the dose-response relationships and for extrapolatingto low doses relevant for settingenvironmentalstandards(56).The dose can be constant or time-related, and the response is usually measured by lifetime risk of dying from cancer. This paper summarizes recent developments in multistage modeling of cohort data detailing methodological as well as *kpartment of Biostatistics, Graduate School of Public Health. University o f h s b u r g h , Pittsburgh, PA 15261. &dress reprint queststo S. Mazumdar, Dcpanmentof Riostatistics,Graduate school of Public Health. University of Pittsburgh, Pittsburgh. PA 15261. computational advancements. These developmentslie mostly in dealing with inferential problems and developing computer programs. A recently developed computer program package for the direct fitting of multistage models with one or two doserelated stages and timedependent exposure patterns is described. In addition to tests of significance related to one-ata-time inference, this package provides computing algorithms b r the construction of confidence regions tbr simultaneous inferences regarding parameters of two-stage, dose-related multistage models and algorithms for computations of liktime risks adjusting for competing causes of death. Programs in this package can be grouped conveniently according to the need of researchers. Several statistical concerns that arise when cohort data are analyzed for carcinogenic risk assessment are discussed. New methodological approaches to deal with these concerns are proposed. Specific issuesaddressed include the appropriateness of statistical inferential procedures, robustness of parameter estimates, model validation,and uncertaintiesof risk estimates. Illustrations of the operationsof the program are provided, us- ing lung cancer mortality in a cohortof nonwhite, malecokeOven workers exposed to coal tar pitch volatiles. Results from the analysis of this data set are used to facilitate discussions of methodological and statisticalconcerns addressed in the paper. 272 MAZUMDARETAL. Armitage-Doll Multistage Model of Carcinogenesis The simplemultistage model of carcinogenesisassumes that a tumor results when k eventsoccur in order in a singlecell. The Occurrencerate (at age t) oftheith event given that i-1 events have occurred, is assumed to be: P = a + b D(t) 11, (1) where a, is the background rate independent of age, b, is the potency parameter for the ith cellular changes, and D(t) is the dose rate at age t. The cumulativehazard rate (orcumulativeincidence rate) by age t is then given by f -to Uk u f f fH(t) = [al+b,D(u,)l b k + b k D ( u k ) b , dUk (2) 00 0 where to is the time from initiation of tumor to clinical expression. In accordance with the above formulation, the age-specific cancer death rate [h(t)] at time t can be expressed as follows: h(t) = hJt) + hl(t) (3) where h,(t) is a baseline age-specific cancer death rate derived either from a standard population or a suitably chosen control population, and h(t) is the excessdeath rate. The expressionfor the excessdeath rate dependson the number and theorder of the dose-related stage(s)and the nature of the dose rate patterns. Recent Developments in Multistage Modeling of Cohort Data The methodology used in the present paper for multistage modeling of cohort data incorporates timedependent exposure patterns. A brief descriptionof this methodology is given below followed by recent developments in its implementation. Developments in areas of model validation and appropriateness of inferential procedures are also discussed. Direct Fitting of the Multistage Model In occupationalcohort studies, historical exposure estimates for one or more agents are usually taken as the measureof dose. Simplified expressionsfor multistage models under the assump tion of constant exposurerate are easily amenable to model fitting. Applications of this approach for occupationalarsenic and coke Oven exposuresare given by Brown and Chu (3)and Dong et al. (4). The assumption of constant exposure rate throughoutthe work history is not realistic under most occupationalsettings. The fitting of multistage models to cohort data that reflect realistic exposure experience necessitates the incorporation of timedependent exposurepatterns. A methodology to incorporateexact time-dependent exposure patterns in the calculations9f excess death rates has been provided by Crump and Howe (7).A formulation of this approach to epidemiologic data when one stage of the multistage model is dose-related is discussed in a subsequent paper (8). Our first effort toward enhancing the methodologyand com- ~ I , I CI. LUWcancer m o d i t y data ofAllegheny County nonwhite, malecoke oven workem. Data Wue n Person-years Observed lung cancerdeaths' 1100 17635 50 Expected lung cancerdeaths Standardized mortality ratio* 95% Confidenceinterval 99% Confidenceinterval 12.I 1 412.7 306.3-544.1 271.9-588.2 'Deaths arc coded tothe ScventhRevisionofthe InternationalClassification of Diseases. *Statistically significant,p < 0.01 puter softwarefor the fittingof multistagemodels tocohort data sets incorporating timedependent exposurepatterns consisted of extending the formulation provided by Crump et al. (8)to the situationwhen stages are dose-relatedand developinga computer program package for model fitting and computing liktime risks. A brief description of the formulationof the problem with exact expressionsfor the excessrisk function is provided first to facilitateour discussionof the computer package developed. To illustratethe multistage modeling programs, we used a data set from the SteelworkersMortality Study, a project conducted in the Departmentof Biostatistics, University of Pittsburgh (9). The selected cohort consists of 1100 Allegheny County black male cokeOven workerswho worked between 1951and 1953and were followed through 1970.There were 50 lung cancer deaths during thisfollow-up period (Table 1). Exposuredata consistof concentrations of coal tar pitch volatiles (CTPV)in milligrams per cubic meter (10).Exposureprofiles were constructed for the workers given the levels of CTPV atjobs Over the entire work periods divided into I-month intervals. These exposureprofiles provide the timedependent exposure rates. General Formulationof the Two-Stage --Related Multistage Model In the direct-fitting procedure of the multistage models to cohort data, a discrete version of Eq.(3)is needed. If we divide age into I-year intervals, ~0 being the age when the w r k e r starts his first job and the exposure rate in the ith interval is assumed to be a constant c,(i=1,2. . .), then for a k stage model with the rth and lth stages dose-related, Eq.(3)takes the form h = am+8Flm+B2ZIm*P,Zll (4) where h,is the age-specific cancer death rate at age x,, and CY, is the background cancer death rate at age L, In Eqs. (4-7), ,,t is the time from initiation of tumor to its clinical expressionand PI,&, and ,:?j are potency parameters. In the present formulationof the problem, only PI,&, and & need to be estimated from data.The constant b is assumed to have a MIXXISAGE MODELING OFCOHORTDATA 273 fixed value which without any loss of generality can be taken to 0. Denoting by z,, zl,, and z , ~ ,Eqs. (5-7) for the nth individual and N,, the total person-years at risk at age &, the expected number of cancer deaths at age x,,, is given by NN Assuming that the observed number of deaths at age x, has a Poisson distribution with expected value E,, the potency parameters 02,and& can beestirnatedby groupingobserved and background deaths and z values and using a suitable maximum likelihood algorithm. The grouping can be done by the age at risk variable or intervals of z values or any other variable by which deaths and person-years can be grouped suitably. Using thisfixmulation,we had developeda computerprogram package h r multistagemodelingof cohort data when hw stages are dose-related. The program package with an applicationhas been reported (IIJ2). Our most recent research deals with the methodological and statistical concerns related to the two-stage, dose-related multistage models with time-dependent exposure patterns. This includes adaptation of Armitage-Doll constraints in the general formulation of two-stage dose-related models described earlier and development of additional computer programs for the model fitting and inkrential procedures. Armitage-Doll Formulation of the Two-Stage Dose-Related Model The formulation of the two-stage, dose-related multistage model described previously provides a more generalized form than given by the Annitage-Doll theory, which requires the fbUowingconstraintson the model parameters: These constraints are usually referred to as the Armitage-Doll constraints. Detailed formulation of the two-stage, dose-related multistage model for epidemiologic data with the Armitage-Doll constraints, associated inferential procedures, and an application to coke Oven workers cohort data set are detailed by w.hwardhan (Is). Theexpressiongiven in (4) is reformulated as where, z z = z , ~(k x;-l/ad and other quantities remain the same. Computer programs have been developed for simultaneous estimation, testing, and constructionof confidence %Om for model parameters using threedistinct statistical proWUres: Wald, likelihood ratio,and score. In our initial program package for multistage modeling. Beneralized linear interactive modeling (14) was used for the a h a t i o n purpose, and the Annitage-Doll constraints could be imposed on the estimation algorithm. As part of our most recent work, the computer package has been enhanced by includingprograms for the constrained estimation methods. In addition, the new package contains several programs for inferential procedures. The package for single-stage, doserelated models is called "Multistage Modeling P r o g ~ Is" and the package for two-stage, dose-related models is called "Multistage Modeling Programs 11." Flowcharts of these two prograq packages are given in Figures 1 and 2. Liktime risks of dying from cancer adjusted for competing causes of death are calculated using the procedure developed by Gail (1.5). Using the Multistage Modeling Programs I to analyze the coke oven workers data set and specifying the number of stages (k) as 4; the dose-related stage (r) as 1; groupingsof deathsand person-years by intervalsof z values; and timedependent exposure patterns as observed, the following information was generated: a) likelihood plot (Fig. 3); b) estimateof the potency parameter, standard error, goodnessof-fit statistics, and Wald, likelihood ratio-based, and score- basedconfidenceintervds and associatedtest statistics(Table 2); and c) lifetime risks (Table 3). Using the Multistage Modeling Program I1 to evaluate the data set and specifying the number of stages (k) as 4; dose-related stages 0.r) as 3 and 1; groupings of deaths and person-years by crossclassified cells ofz value vector; and timedependent exposure patterns as observed, the tbllowing infbrmation was generated: a)likelihood surface (Fig. 4); b) estimates of potency parameters, standard errors, goodness-of-fit statistics, and Wald, likelihood ratio-based, and score test statistics (Table4); c) Wald, likelihood ratio-based, and scorebased confidence regions (Fig. 5); and d) lifetime risks (not shown). Coverage and Pbwer Study of Test Procedures Null hypotheses about the potency parameters of multistage models can be tested using standard inferential procedures suchasWald`s procedure, the likelihood ratio-based procedure, and the procedure based on score statistics. The question often arises about the appropriateness and relative merits or differences in the performances of these procedures. To address this question, one must undertake simulation studies that are conducted under conditions corresponding to those of the data set being analyzed. We have developed computer programsto perform such simulationsand used them toanalyze the coke Oven workers data set (Is). Results from the simulation study indicated that for a single-stage, dose-related multistage model, confidence intervals obtained by all three procedures have good coverage and power properties. For two-stage dose-related models, the Wald confidence region performspoorly h rextremeparameter values, whereastheother confidence regions perform satisfactorily. Robustness Assessing robustness of a regression fit is an important step in any modeling endeavor. There are hw different aspects of this assessment. The first aspect involves assessing the robustness of the m e t e r for model-fitting purposes within some fixed distributional assumptions. The second aspect 274 OCMAP FOINI MAZUMDARETAL. ?! 01 De8lh Ral8s I. ZCUT1.FOR 7.VALUEl.FOR ZV8lU.. I Groupad D88thS Fib of Daad Ralef ZAGE1.FOR I GLlU or ESTLKD1.FOR i pow Param8181 PREDICT1.FOR I FIGURE 1. Flowchart of multistage modeling programs for onc-JtSge doserelated models. FIGURE3. Likelihood function for a first-stage. dose-related, four-stage multistage model fitted to Allegheny County nonwhite d e cokc own wrkcrs' lung cancer mortality, 1953-3970. Cohort Dah in OCMAP FOIN~ 1 ZVALUE2.FOR Zvalue V8clor I D8ad Rat8r ESTLKD2.FOR Likelihood PlOlS Conlidonu Region Confidence Inle~vda PREDICT2.FOR FIGURE 2. Flowchart of multistage modeling programs for two-stage doscrelated models. Confidence intervals' Wald Likelihood ratio score Test statisticsd Wald Likelihood d o ~ Score 2.062 x 10-1.4.357 x io-: 2.171 X 10-',4.469 X 102.123 X IO-'. 4.465 X lo-' 30.057 82.945 184.843 is by grouying pmon-years, deaths, and z values in 15 intervals. b'EGstlimanadtioPnearson's X have chi-square distributions with 14 degrees Of fre.e~dom~ea~ch. 9 5 % confidence intervals. %ach test statistichas a chi-square distributionwith 1 degrees of freedom. involvesexaminingthe robustness(sensitivity)ofthe parameter estimates to the inherent assumptionsof the model, mainly the assumption about the probability distribution of the random component of the model. A new method has been proposed for investigating model robustness using the generalized linear model approach (16). Briefly, the method quantifies model robustness by defining MULXISKAGEMODELING OF COHORTMA 'Ihbk 3. Estimated lietime lung cancerrisk (througbage 85+ years) for a IypothctiCpIUS black mnkaposedt o e d tar pitch vd.tucsfor4Oycars, exposure starting at age 20. Model: four stages Exposure First stage dosc-related pattern' LLLL LLHH LMHM HLLL HMLM HHLL HHHH RiSP 0.2m 0.2566 0.2864 0.3488 0.3800 0.4013 0.4251 Ratio' 7.5 9.1 10.2 12.4 u.5 14.3 15.1 'L = 1 m%mg3for 10years; M = 2 mg/m3 for 10years; H = 3 mg/m3for 10 yean. bU.S.black male age-specific mortality for 1960 used in competing risk calculations. 'Ratio of risk for exposed individual/risk for unexposed individual (lifetime risk for unexposed, 0.0281). 275 5 43..021 0- \ t I r! '-2 0- 1 1 I , , , , II t , l FIGURE 5 . Confidence region (95%)for the parametersof a first- and thirdstage, ~ r c l a t c df,our-stage multistage model fitted to Allegheny County nonwhite male coke oven workers' lung cancer mortality, 1953-1970. FIGURE 4. Likelihood surface for a first- and third-stage,dose-related, burstage multistage model fittedto AlleghenyCounty nonwhite male cokewen workers' lung cancer mortality, 1953-1970. Table 4. Statistics for a first- and third-stage dose-rrlnted, four-stsge multistage model fitted to Allegheny County nonwhite nude coke Oven workers' lung cancer mortality data, 1953-1970. Statistics +mated potency parameters' 81 8.209 x b (SE.10.127 X 1.454 X Goodness-of-fit statistic: G' Pearson's X' T a t statisticsc Wald Likelihood ration Score (SE,1.663 x 8.463 2.990 63.936 81.026 180.219 %mation is by growing Demon-years, deaths, and z-values in 19 intervals. %'and Pearson's X' havcchi-square distributions with 17 and 4 degrees of freedom, respectively. 'Ekh test statistic has a chi-square distribution with 2 degrees of freedom. robust regions of the parameters in the variancefunctionand link function in a generalized linear model (GLM).These robust regions are defined as ranges in the spacesof variance function parameter or link function parameter for which the correspondingestimatesof the potency parameter lie within the confidence interval of the potency parameter determined under the basic model assumptionsand the model fitting is adequate. Wedderbum's quasilikelihood approach (17)and several criteria based on commonly used statistics are used to obtain the robust regions. The method applies to the robustnessinvestigationof either the variance function or the link function or both simultaneously. The GLM method has been appliedto the first of four stages dose-related multistage model fitted to the coke oven workers mortality data. Results show that the selected multistage model is sufficiently robust to the change of the relationshipbetween the first two moments of the random component (variance function) but sensitiveto the changeof the relationshipbetween the systematic componentand the mean (link function). Figure 6 displays one such variance-link robust region. Cross-Validation The coke oven workers mortality study provides a uniqueo p portunity to cross-validate models fitted to death rates for quantitativecarcinogenic risk assessment since the cohort has been followed for about 30 years (the most recent follow-up is through 1982). The extended follow-up allows fittinga model with data through a certain follow-up period and then making 216 MAZVMDAR ETAL -3.5 -3.0 -2.5 -2.0 I -1.5 -1.0 1 -01 0.0 oI s Variance Parameter p 1 IO 1.5 an 3 2.5 FIGURE 6. hiance-link robust region using four criteria.CriterionI: confidenceinterval and generalized Pcarsonianchi-square;criterion II: confidence interval and deviance; criterion lIf:test statisticand generalized Pcarsonianchi-square;criterionIv:test statisticand deviance. predictions of excess deaths for subsequent calendar periods. These predictionscan then be validated with actual observations from the extended follow-up periods. As commonly used, goodness-of-fit statistics (e.g., deviance and Pearson's chisquare)are found to be not very sensitive in choosing the best fitted model. We believe that the cross-vaiidation method is more meaningful for the modeling of cob-ort data. In the contextof multistagemodeling of cohortdata with timedependent exposure patterns, this cross-validation method requires first the grouping of Z values (vector)by specific calendar year periods and using the estimatedpotency parameters (s) and calculatingthe predicted excessdeath ratesbeyondthe calendar period used in model fitting. The predicted excessdeath rates can then be compared with the observed excess death rates obtained from the extended follow-up information. A computer program has been developed for grouping deaths and personyears by calendar year periods and was used to cross-validate a multistage model with timedependent exposurepattern fittedto the lung cancer mortality of a different cohort of coke oven workers (18). Discussion Inrecentyears, the useof multistagemodels hasbecome an integral part of environmental carcinogenic risk assessment. Multistage models provide a conceptual framework that facilitates understanding of the relationshipsbetween different timedependent variables and tumor incidence, and they also give a reasonable description of the epidemiology of many nonhormonally dependent cancers of epithelial origin. Concerns about the basis of multistage models have been raised by Moolgavkar andhiscolleaguesin a seriesof papers (19,20). A deficiency of the Armitage-Doll model is that the target tissue is not allowed to grow and that no allowanceis made for the fact that most epithelia constantly shed and replenish cells. Moolgavkar and his colleagues have developed a two-stage model to rectify this defect (18).We have developedsomeof the necessary softwaretbr the modeling of large cohortdata sets using two-stagemodels and will continueeffortsto completethe remaining work. Programs that are developed for the multistage modeling purpose will be modified suitably for two-stage modeling. We have developed methods to assess several statistical issues that arise in multistagemodeling includingthe appropriateness and merits of various procedures for testing the null hypothesis, the robustnessof parameterestimates, and the cross-validation of models. However, the methods developed so far fall short of adequately addressing uncertainties in the risk estimates. The confidence limits and the confidence regions of the potency parameters can be used in providing ranges of risk estimates. In addition, we intend to use bootstrap methods to provide alter- native estimates of the confidence limits (regions) based on asymptotic theory which then can be used to provide alternate ranges of the risk estimates. MULnSAGE MODELING OF COHORTDATA 277 This m h has been funded in part by the U.S. EnvironmentalW o n Agency under assistance agreement number CR806815 to the Center for EnvirwmentalEpidemiology, Gladuatc Schoolof Public Health, Uniwxsity of Pittsburgh. This pper does not necessarily reflect the views of the Agency and no official endorsementshould be i n f e d . We acknowledgethe Univmity of Pittcburgh Super Computing Center for allocating the time and facilitiesto use the Super Computer CRAY System. REFERENCES 1. ArmiIage, P., and Doll, R. Stochasticmodels for carcinogenesis. In: Pro- d i n g s of the Fourth Berkeley Symposiumon Mathematical Statisticsand Probability: Biology and problems of Health, 4. University of California Press, BcrkCl~,CA,1961, p ~ 1.9-38. 2. Day, N.E., and Brown. C. C Multistage modelsand primary prevention of cancer.J. Natl. Cancer Inst. 64:977-989(1980). 3. Bmwn,C.C.,andChu,K.CAncwmethodformcanalysisofcohortstudies: inlplimmofthe multi- d#or-yfo appliedto~d arsenic exposun. Environ. Health krspa3. 50: 293-308(1983). 4. Dong, M.H.,kdmnd.C. K.,Mazlundar, S., and Costantim, J. P.A multistegeapproachto thecohorianalysisoflifaime lungcancer risk among steelworkers exposed to coke wen emissions. Am. I. 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