A General formulation for standardization of rates as a method to control confounding by measured and unmeasured disease risk factors

The Annals of Applie d Statistics 2008, V ol. 2, No. 3, 1103–112 2 DOI: 10.1214 /08-A OAS170 c  Institute of Mathematical Statistics , 2 008 A GENERAL F ORMULA TION FO R ST ANDARDIZA TION OF RA TES AS A METHOD TO CONTRO L CONFOUN DING BY MEASURED AND UNMEASURE D DISEASE RISK F A C TORS By Steven D. Mark 1 University of Color ado Scho ol of Public He alth Standardization, a common approac h for controlling confounding in popu lation- studies or data from diseas e registries, is defined to be a w eigh ted av erage of stratum sp ecific rates. T y pically , discussio ns on the construction of a particular stand ardized rate regard the strata as fixed, and focus on the considerations that affect th e sp eci- fication of w eights. Eac h year the data from the SEER cancer registries are analyzed using a weig hting pro cedure referred to as “direct standardization for age.” T o eva luate the p erformance of d irect standardization, w e define a general class of standardization operators. W e rega rd a particular standardized rate to b e the output of an opera- tor and a given data set. Based on the fun ctional form of th e op erators, we defin e a sub class of standardization op erators that controls for confounding b y measured risk factors. Using the fundamen tal disease probability paradigm for inference, we establish the conclusions that can b e drawn from year-to-year contrasts of stand ardized rates prod uced by these op erators in t he presence of unmeasured cancer risk factors. These conclusions take the form of falsifying sp ecific assumptions ab out the cond itional prob- abilities of disease given all the risk factors (b oth measured and un measured), and t he conditional probabilities of the unmeasured risk factors given the measured risk factors. W e sho w th e one- to-one corresp ondence b etw een these falsifications and the inferences made from the con trasts of directly stand ardized rates rep orted eac h year in the A n- nual Re p ort to the Nation on the Status of Canc er . W e further show th at the “direct standardization for age” p roced u re is not a member of the class of u nconfounded stan- dardization operators. Consequently , it can, and u sually will, introduce confounding when confounding is not present in the d ata. W e prop ose a particular standardization operator, the SCC op erator, that is in the class of unconfounded op erators. W e contra st the mathematical properties of th e SCC and the SEER op erator (SCA), and present an analysis of SEER cancer registry data that demonstrates the consequences of these differences. W e further p ro ve that the SCC op erator is a pro jection op erator. W e dis- cuss how this property can enable the SCC op erator to b e developed as a metho d for comparing nested cond itional ex p ectations in the same manner as is currentl y d one with regression metho ds that con trol for confounding. Received January 2008; revised January 2008. 1 Supp orted by th e Universit y of Colorado Health Sciences Cen ter at Denv er. Key wor ds and phr ases. Cancer registry, cancer t rends, causal inference, confoundin g, direct standardization, fund amental disease probabilit y , SEER, standardization. This is a n ele c tronic reprint of the o riginal a rticle published b y the Institute of Mathematical Statistics in The Annals of Applie d St atistics , 2008, V o l. 2, No. 3, 1 103–1 122 . This r eprint differs from the or iginal in pag ination and typo graphic detail. 1 2 S. D. MARK 1. In tro d uction. Eac h y ear the NCI’s Surv eillance, Epidemiology , and End Resu lts (SEER ) program compiles data (henceforth called SEER data) on cancer incidence and mortalit y from (currentl y) 17 p opulation-based can- cer registries in the United S tates [Ho w e et al. ( 2006 )]. Since 1998 the Na- tional Cancer I nstitute, the American Cancer So ciet y , the Cen ters for Dis- ease Con trol, and the North American Asso ciation of Cent ral C ancer Reg- istries ha v e analyzed the SEER data to p ro duce an Annual R ep ort to the Nation on the Status of Canc er in the Unite d States (subsequently referred to as the Annual R ep orts ). Th ese reports con tain estimates of the o ve rall ann ual cancer incidence and mortalit y , as w ell as incidence/mortali t y by cancer site, and incidence/mortalit y within p opulation su bgroups d efined b y gender, race, ethnicit y , and geo graphic location of the cancer registry . Some of the stated goals of these rep orts are to: (1) rep ort on the cancer burd en as it relates to cancer incidence and mortalit y and p atien t su rviv al; (2) iden tify u n usual c hanges and differences in the p atterns of o ccurrence of sp ecific forms of cancer in p opulation subgroups defined by geog raphic, de- mographic, and so cial c h aracteristics; (3) describ e temp oral changes in can- cer inciden ce, mortalit y , exten t of disease at diagnosis (stage) , therapy , and patien t sur v iv al that might impact cancer pr ev entio n and control strategies; (4) monitor the o ccurrence of p ossib le iatrogenic cancers; and (5) attribute c hanges in cancer rates to temp oral c hanges in diagnostic criteria, screening, prev en tive measures, cancer treatmen ts, or environmen tal exp osures [SEER ( 2005a ), W ard et al. ( 20 06 )]. In addition to the goals common to all of the Annual R ep orts , eac h rep ort h as a sp ecial su b-fo cus. S ince 2001 these re- p orts hav e stated conclusions regarding: (1) absolute p opulation rates and c hanges in cancer rates [Ho we et al. ( 2001 ), Edwards et al. ( 20 02 ), W eir et al. ( 2003 ), Jemal et al. ( 2004 ), Edwards et al. ( 2005 ), How e et al. ( 2006 )]; (2) the impact of screening and treatmen t on sp ecific cancers [Ho we et al. ( 2001 )]; (3) d ifferences in cancer rates b y gend er , race, ethn icit y , and geo- graphic lo cation [How e et al. ( 2001 ), W eir et al. ( 2003 ), Jemal et al. ( 2004 ), Edwa rds et al. ( 2005 )]; (4) causes of the difference in rates within the sub- groups listed in (3) [Jemal et al . ( 2004 ), Edw ard s et al. ( 2005 ), Ho we et al. ( 2006 )]; and (5) the futu re public p olicies and exp enditures that sh ould b e undertake n to in crease cancer p rev ention and improv e access to medical care [W eir et al. ( 2003 ), Edwards et al. ( 200 5 ), Ho we et al. ( 2006 )]. In order to mak e meaningful statemen ts ab out the yea r-to-y ear c hanges in cancer incidence/mortalit y as a function of one set of c haracteristics, it is n ecessary to con trol for d ifferences in the frequency of cancer risk f actors that are not in the set of interest. W e refer to an y statistical pro cedure that attempts to separate the effect on cancer r ates of one set of m easured co- v ariates from another set as p ro cedures that cont rol for confounding. The common metho ds of con trolling for confounding are as follo w s: (1) multi- v ariate regression; (2) stratification; and (3) s tandardization. Standard iza- tion is virtually alw ays the metho d of c hoice w hen inference is made from A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 3 p opulation-studies, or data from disease r egistries. It is the p ro cedure used in the A nnual R e p orts [Klein and Sc ho en b orn ( 2001 ), Ries and K osary ( 2005 )]. The particular standard ization metho d used to analyze S EER data is de- signed to control for ye ar-to-y ear differences in age distr ib utions. W e refer to th is metho d as S tandardization Controlli ng for Age (SCA ) . In this pa- p er w e present a new pr o cedure that allo ws r esearc hers to con trol for any set of measured co v ariates. W e refer to this pro cedur e as Stand ardization Con trolling f or C o v ariates (SCC) . The pap er is organized as follo ws. In Section 1 w e defin e n omenclature for a completel y general data stru cture, and describ e the SE E R data in terms of th is nomenclature. In Section 2 we giv e form u lae for the usual represen- tations of standardized rates as w eigh ted a ve rages of a given set of str atum sp ecific rates. W e d etail the rationale for the sp ecific choic e of weig hts used in SCA standardization. W e then define SCA and SCC standardized rates as the output of S C A and SCC op erators. These oper ators are fun ctionals of the empirical distribution of a giv en set of data, and a u ser-defined we igh ting distribution. Using the op erator form ulation, we define a general class of all standardization op erators. In Section 3 w e formalize our p revious discuss ion of the goals of standard ization. W e d efine cr iteria that sp ecify when con- trasts of cru de-cancer rates are “not confounded.” W e extend these criteria to define the sub class of standardization op erators that pro d uce contrasts of standardized rates that are not confounded . The SCC op erator falls within this su b class; the SCA op erator do es n ot. W e show that if one b egins with crude rate differences that are not confoun ded, the SCA op erator in tro du ces confounding. W e discus s h o w the d ifferences in prop erties of the SCA and SCC op erators relate to the differences in the fu nctionals. In Section 4 w e present analyses of th e SEER 13 data that demonstrate the prop erties de- scrib ed in Section 3 . Up until S ection 5 w e discuss confounding in terms of measured risk fac- tors. In Section 5 w e pro vide a formal framew ork for examining what infer - ences can b e made from the s tand ardized rate d ifferen ces pro d u ced by the SCC operator in the presence of unm easur ed risk factors. Suc h inferences re- quire assumptions that can b e neither completely f alsified nor confirmed by examination of the observed distribution functions. W e sho w that nonzero b et w een-yea r differences in standardized r ates allo w one to reject certain assumptions ab out unm easur ed r isk f actors, and that violations of these as- sumptions corresp ond directly to the in f erences made by SEER inv estigato rs in the A nnual R ep orts [W ard et al. ( 2006 )]. In Section 6 w e change fo cus from b et wee n-y ear inferences to w ith in -y ear inferences. W e define “nested” stand ardized rates, deriv e the p r op erties of nested rates pro duced by the SCA and SCC op erators, and discuss implica- tions for within-y ear mo d el bu ilding of standardized rates. 4 S. D. MARK In the discussion section we summarize our results, suggest h o w the stan- dardization op erators w e p rop ose can b e us ed for nonparametric, semipara- metric, or p arametric estimation of conditional m eans, and discuss the di- rection of our curr ent work on dev eloping softw are that w ill implemen t the op erators w e describ e. 2. Data structure, crude-cancer rates and finest-crude-cancer rates. Let ( D y , Z y ) b e any vec tor of real v alued r an d om v ariables, and P y ( D , Z ) an y set of p robabilit y distributions defined on the su p p ort of ( D y , Z y ); y ∈ Y ; Y ≡ { 1 , 2 , . . . , n } ; n finite. S ince the supp ort of P y ( D , Z ) do es not v ary with y , w e frequently drop the sup erscripts for random v ariables and write ( D , Z ). In SEE R data D is a v ector of indicator v ariables denoting the presence or absence of a sp ecific form of cancer t yp e; Z is the set of all other measured co v ariates; P y ( D , Z ) is the empirical distrib ution of ( D y , Z y ) for year y ; Y is the set of y ears for which we hav e data on ( D , Z ). T hus, th e S EER data for y ear y consists of an observ ation of P y ( D , Z ). Sin ce in the Annua l R ep orts an ind ividual is classified as either ha ving or not ha ving a sp ecific cancer (or a cancer in a defined set), and the joint distrib u tion of cancers is not of in terest, we will regard D to b e a binary random v ariable: D = 1 when an individu al is in the set of cancers of interest, D = 0 otherwise. W e assume that our in terest is in the d istribution P y ( D , E ) , wher e E is the (p ossibly improp er ) subset of Z w h ic h inv estigat ors b eliev e conta in all the “measured cancer risk factors.” Without loss of generalit y we adhere to the structure of the SEER data, and regard the supp ort of ( D , E ) as discrete. W e assume that in SEE R the measured cancer risk factors, E , consist en tirely of information ab out an ind ivid ual’s age, gender, r ace, ethnicit y , and catc hm en t area of cancer registry (henceforth called place): E = (age, gender, race, ethnicit y , p lace) . (1) These are, in fact, the only co v ariates requir ed to pr o duce the stand ardized rates giv en in the Annual R ep orts . Let E = ( E 1 , E 2 ) b e an y f actorizat ion of E su c h th at E 1 ∩ E 2 = ∅ ; E 1 ⊆ E . W e use notation suc h as P y ( D | E ), P y ( D | E 1 ), P y ( E 2 | E 1 ) to denote the conditional distributions of P y ( D , E ). F or any E † ⊆ E , we refer to P y ( D | E † ) as the crude-cancer rate for E † . When E † = E , we r efer to P y ( D | E ) as the finest-crude-cancer rate . An y crude-cancer r ate is related to the fin est-crude-cancer rate b y the in tegral give n in (2). P y ( D | E 1 ) = Z E 2 P y ( D | E 1 , E 2 ) dP y ( E 2 | E 1 ) . (2) The region of inte gration in ( 2 ) is E 2 , the supp ort of E 2 . Throughout the pap er calligraphic letters indicate the supp ort of rand om v ariables. When as A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 5 in the SEER data, ( D y , E y ) ha ve discrete su pp ort, we can express the righ t- hand side of ( 2 ) as a su m of the pro du ct of discrete conditional probabilities, P y ( D | E 1 ) = X E 2 P y ( D | E 1 , E 2 ) × P y ( E 2 | E 1 ) . The cru de-cancer rate on the left-hand s ide is the f r equency of disease in sub jects with a giv e v alue of E 1 . 2.1. Gener al definition and formulae for standar dize d r ates. W e d efine s ∗ y [ D | E 1 ] to b e a stand ard ized cancer rate giv en E 1 = e ∗ 1 if it can b e expressed in the form of the inte gral giv en in (3), s ∗ y [ D | e ∗ 1 ] ≡ Z E † 2 P y ( D | e ∗ 1 , e † 2 ) dP ∗ ( e † 2 ) . (3) Here E † 2 ⊆ E , e † 2 ∈ E † 2 , and P ∗ ( E † 2 ) is any us er -d efined measure that has the same sup p ort as E † 2 and is consisten t with a probability measure. Under th e restriction that ( D , E ) has discrete supp ort, w e can write ( 3 ) as s ∗ y [ D | e ∗ 1 ] ≡ X e † 2 ∈E † 2 P y ( D | e ∗ 1 , E † 2 = e † 2 ) dP ∗ ( e † 2 ) . (4) Equation ( 4 ) is the w eigh ted sum of stratum sp ecific we ights: E † 2 define th e strata; dP ∗ ( e † 2 ) is the w eigh t for stratum e † 2 ; P y ( D | e ∗ 1 , E † 2 = e † 2 ) is the crud e- cancer rate within stratum E 1 = e ∗ 1 , E † 2 = e † 2 . Equ ation ( 4 ) is equiv alen t to the usual algebraic definition of a standardized rate [Rothman ( 1986 )]. T yp ically , discussions ab out standardization assume the s tr ata are fixed and fo cus on th e c h oice of w eights [Rothman ( 1986 )]. The general advice is that the c hoice of w eight s should dep end up on the inte rpretation one desires to ascrib e to the standard ized r ates [Rothman ( 1986 )]. Th e weigh ts used in the S EER imp lemen tation of the SCA metho d are the age-frequency of the US p opulation in y ear 2000 [Klein and S choenb orn ( 2001 ), Ries, Eisn er, and Kosary ( 2005 ), W ard et al. ( 2006 )]. Th is is referr ed to as direct stan- dardization [Klein and Sc ho enborn ( 2001 ), Rothman ( 1986 )]. In particular, the SCA pro cedure of S EER is d escrib ed as: “Age adju stmen t, u s ing the direct metho d, is the application of observed age-sp ecific r ates to a stan- dard ag e distribution to eliminate differences in cru de rates in p opulations of in terest that resu lt from differences in the p opulations’ age distribution [Klein and S c ho en b orn ( 2001 )].” The j u stification for d irect standardiza- tion of the cancer rates in y ear y is that the stand ardized rate for year y will represent th e cancer rates that would ha ve b een observed in y ear y had the age distribution in yea r y b een iden tical to the age distribu- tion in y ear 2000 [Klein and Sc ho en b orn ( 2001 ), Rothman ( 198 6 ), Anderson 6 S. D. MARK and Rosenberg ( 1998 )]. T he adv an tage of exp r essing standardized rates in terms of “what wo uld h av e b een seen in some y ear y ” is that s uc h w eighting pro du ces standardized r ates whic h preserve the magnitude of the crud e- cancer rates. Since the magnitude of the y ear-to-y ear differences in cancer rates are of imp ortance to th e inferences m ade in the A nnu al R ep orts , it is desirable to choose a standardization pro cedur e that preserv es these v al- ues. 2.2. Defining the SCA and SCC op er ators. T o con trast the prop erties of SCA and S CC standard ized rates, it is b est to regard them as the outpu t of SCA and SCC op erators. W e defi n e a standardization op erator, S ∗ y [ D | E 1 ], to b e any functional of P y ( D , E ), E 1 , E † 2 and a us er-defined probabilit y distribution, P ∗ ( E † 2 ), that can b e expr essed by the inte gral in ( 5 ): S ∗ y [ D | E 1 ] ≡ Z E † 2 P y ( D | E 1 , E † 2 ) dP ∗ ( E † 2 ) . (5) F or a particular E 1 = e ∗ , the standardized rate is denoted by the left-hand side of ( 3 ). Let P ∗ ( E ) b e a user-d efined prob ab ility distribution w ith the same su p- p ort as E . Let A b e any rand om v ariable in E , and P ∗ ( A ) the marginal probabilit y of A from d istr ibution P ∗ ( E ). W e denote the p oint s of sup p ort of A as ( a 1 , a 2 , . . . , a N ). Using “ \ ” as the set d ifferen ce op erator, w e define E a = E \ A , E a 1 = E 1 \ A ; E a 2 = E 2 \ A . W e define th e S CA op erator , S ca y [ D | E a 1 ], to b e S ca y [ D | E a 1 ] ≡ Z A P y ( D | E a 1 , A ) dP ∗ ( A ) . (6) s ca y [ D | e ∗ 1 ] denotes the SCA standard ized rate giv en E a 1 = e ∗ 1 . In the Annual R ep orts standardized rates are pro d uced using th e S CA op erator: A is age, and th e s upp ort p oints are the fi v e year in terv als into whic h age is categorized. The wei gh ting d istribution, P ∗ ( E ), is the co v ariate distribution in yea r 2000, P 2000 ( E ). Thus, P ∗ ( A ) is th e age frequency in yea r 2000. F or example, let D b e col on cancer, E a 1 = (gender), and E a 2 = (race, ethnicit y , place). T h e S E ER S CA estimate of the standardized rate of colon cancer conditional on gender b eing male is s ca y [colon cancer | male] = N X j =1 P y (colon cancer | male, age = a j ) × P 2000 (age = a j ) . A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 7 Here P y (colon cancer | male, age = a j ) is the frequency of colon cancer in y ear y f or males in age category a j , and P 2000 (age = a j ) is the frequency of age group a j in y ear 2000. W e defin e the SCC op erator , S cc y [ D | E 1 ], to b e the functional of P y ( D , E ), P ∗ ( E ), and E 1 , giv en by the int egral in ( 7 ): S cc y [ D | E 1 ] ≡ Z E 2 P y ( D | E 1 , E 2 ) dP ∗ ( E 2 | E 1 ) . (7) Using the s ame factorizatio n of E , and the w eigh ting distribution sp ecified b y ( 7 ), the SCC estimate of the standardized colon cancer rate conditional on gender equals male is s cc y [colon cancer | male] = X E 2 P y (colon cancer | male, age, race, ethnicit y , place) × dP 2000 (age, race, ethnicit y , place | male) . Here P y (colon cancer | male, age, race, ethnicit y , place) is th e frequency of colon cancer in y ear y for males within strata d efined b y age, race, ethnicit y , and p lace; dP 2000 (age, race, ethnicit y , place | m ale) is the fr equency among males for a giv en (age, race, ethnicit y , place). F or completeness w e note that we ha ve explicitly pr esen ted the SCA and SCC op erators in terms of th e rand om v ariables ( D , E ) and the empirical distributions P y ( D , E ). If we restrict the data set to some D ‡ ⊂ D , and/or E ‡ ⊂ E , the op erators are defi n ed in terms of the empirical distribution P y ( D ‡ , E ‡ ). In practice, giv en the strong correlat ion of cancer t yp e with age, suc h analyses are common. In Section 4 we present standardized r ates for colon cancer and br east cancer. These rates were made fr om data limited to su b jects 40 y ears of age or older. Similarly , when W ard et al. rep ort standardized rates for childhoo d cancers, they restrict sub jects to those age 19 or less. 3. Con trasting the p rop erties of the SCC an d SCA op erators. Insp ec- tion of the form u las f or the SCA ( 6 ) and S CC ( 7 ) op erators rev eals t wo imp ortant differences. In the S CA op erator the crude-cancer rate v aries as a function of E 1 , bu t the w eigh t do es not. In the SCC op erator the crude rate is alw ays the finest-crude-cancer rate, and the we igh t dep end s on E 1 . In this section we examine the consequen ces of these differences on the prop erties of the standardized rates. W e b egin by formalizing the goals of standardization discussed at the end of Section 2.1 . 8 S. D. MARK 3.1. Standar dization op er ators and the c ontr ol of c onfounding by me a- sur e d risk factors. The Annual R ep orts compare y ear-to-y ear differences in cancer rates conditional on some s u bset E 1 of E ( 1 ). The n eed for standard - ization arises b ecause of the concern that d ifferences in the crude-cancer rates ma y reflect y ear-to-y ear differences in the distribution of E 2 . F r om ( 2 ), w e see that the distribution of E 2 that affects the crude-cancer rate is P y ( E 2 | E 1 ). If for years y † and y †† , P y † ( E 2 | E 1 ) = P y †† ( E 2 | E 1 ) , (8) w e say there is n o E 2 confounding of the E 1 crude rate d ifferences , P y † ( D | E 1 ) − P y †† ( D | E 1 ) . (9) When ( 8 ) is true, con trasts of the crude rates pr o vide the b est rates of th e y ear-to-y ear differences. F rom ( 8 ) we kn o w that the differences in the crude- cancer r ates cannot b e d ue to differences in the E 2 distribution; trivially , the cru de rate differences ac hieve the desired goal of ha vin g the standardized con trasts pr eserve the observ ed magnitude of the d ifferences in crude r ates. Assume n o w th at ( 8 ) is true for all y ∈ Y , and that th e weigh ting dis- tribution used is P 2000 ( E ). By insp ection of ( 7 ), w e see that for all y , and an y factorization of E , standardized r ates pro d uced by the SCC op erator equal the crude-cancer rates. Thus, if there is no E 2 confounding of the E 1 crude rate differences, and one us es the SC C op erator, con trasts of the SCC standardized rates are contrasts of the un confounded cru de rates. T o see that this is not the case for the SCA op erator, we r e-express ( 6 ) in terms of the fi n est-crude-cancer rates: S ca y [ D | E a 1 ] = Z A  Z E 2 P y ( D | E a 1 , E a 2 , a ) dP y ( E a 2 | E a 1 , a )  dP ∗ ( A = a ) . (10) Supp ose in ( 10 ) th at y = 2000. Ev en were ( 8 ) true, and P ∗ ( A ) = P 2000 ( A ), the SCA op erator do es not retur n the cru de-cancer rate. Thus, cont rary to the stated justification for the c hoice of weig h ts, the SCA standardized rates in y ear 2000 conditional on E a 1 do not equal the observe d cancer r ates in y ear 2000, ev en though the w eights used are the age d istribution of y ear 2000. This will b e graphically demonstrated in Figure 1 . Consisten t with our defin ition of no confounding of crude r ate differences ( 8 ), w e defi ne standardized r ate differences, s ∗ y † [ D | e ∗ 1 ] − s ∗ y †† [ D | e ∗ 1 ] , (11) to b e unconfoun d ed by E 2 , if the standardized rates are pro duced b y a stan- dardization op erator, S ∗ y [ D | E 1 ], that can b e expr essed as an in tegral of the finest-crude-cancer rates with resp ect to a measure that dep ends on ly on A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 9 the factorization of E [see ( A.1 ) for f ormal definition]. W e refer to such op erators as standardization op erators w ith no E 2 confounding (SONC op erators) . The SCC op erator is one suc h op erator. In fact, if w e c hose P ∗ ( E ) = P 2000 ( E ), it is th e uniqu e standardization op erator that pr o duces the standardized cance r rates that “w ould ha v e b een seen in yea r y had the co v ariate distr ib ution in y b een identica l to the co v ariate d istribution in y ear 2000.” Note th at SONC op erators with no E 2 confounding p ro du ce standardized rate differences that are n ot confounded by E 2 regardless of whether ( 8 ) is true. It is clear from ( 10 ) that the SCA op er ator do es not, in general, pro d u ce standardized rate differences that are un confounded. In fact, for a giv en factorizat ion of E and for s p ecific y † , y †† ∈ Y , the SCA op erator p ro duces unconfound ed standard ized r ate d ifferences if and only if P y † ( E a 2 | E a 1 , a ) = P y †† ( E a 2 | E a 1 , a ) . (12) Since ( 8 ) do es not imply ( 12 ), the SCA op erator can pro duce s tandardized rate differences that are confounded by E 2 ev en when ( 8 ) is true and th e crude rate differences are not confound ed. F or S CA to pro du ce unconfound ed s tandardized r ate d ifferen ces for all factorizat ions of E requ ires ( 13 ): P y † ( E a | a ) = P y †† ( E a | a ) . (13) When ( 13 ) is true for all p ossible com b ination of y ears, ( y † , y †† ), then con- ditional on age, the distribu tion of the other r isk factors are id en tical for all y ears. Th u s, for the SCA op erator, which “standardizes only for age,” to pro du ce u nconfound ed standard ized rate d ifferences requires that the P y ( E ) distributions are id en tical except p ossibly f or the marginal distr ibution of age. The equ alit y in ( 13 ) do es not exist for the analysis of the SEER data w e p r esen t in Section 4 . If the P ∗ ( E ) and P y ( E ) distribu tions are such that P ∗ ( E a | A ) = P y ( E a | A ) and P y ( E a | A ) = P y ( E a ) , (14) then the rates pro duced b y the SCA and SCC op erator are identical . Th us, if ( 8 ) is true, the SEER S CA op erator alw ays return s the crud e-cancer rates iff the E a distributions are identica l for all yea r s and age is indep end en t of E a . It is ins tructiv e to consid er the p rop erties of the SCA and SCC op era- tors with regard to confounding by measured risk factors in terms of the familiar regression mo del approac h to con trol for suc h confounding. In this con text, we regard the P y ( D , E ) as r andom samples fr om larger p opu la- tions. F or concreteness, we conceptualize that the P y ( D | E ) ha ve log-linear P oisson distributions. Giv en that age is the pr imary determinant of cancer 10 S. D. MARK rates, we prop ose Po isson mo d els stratified on age cat egory , and obtain es- timates of P y ( D | E ) from w eigh ted sums of th e pred icted p r obabilities in eac h age stratum. Our goal is to fin d the most parsimonious P oisson mo d- els from w hic h to m ak e inferences r egarding the effect of sex and race on within- and b et ween-y ear cancer r ates. W e b egin with the within-age stra- tum mo dels saturated in (sex, race, ethnicit y , place). T he pr edicted proba- bilities from these saturated mod els are iden tical to the probabilities giv en b y the finest-crude-cancer rates, P y ( D | E ). W e would mak e in ferences from the m ore p arsimonious mo dels saturated only in (age, sex, race) if: (1) the regression coefficients f or all co v ariates that are a fun ction of either ethnicit y or place w ere equal to zero; or (2) w ithin eac h age stratum , (sex, race) were (in the d ata) statistically indep enden t of (ethnicit y , p lace). Th ese criteria corresp ond to the usu al rubric that co v ariates E 2 are not confoun ders of the effect of E 1 pro v id ed either E 2 are not risk factors for d isease, or E 2 are uncorrelated with E 1 . The cond itions required for a standardization op erator to b e in the class of SONC op erators are r elated to, but more stringen t than, those giv en ab o ve. SONC op erators are in tegrals of the finest-crude-cancer rates with resp ect to a measure that do es not d ep end on y ear ( A.1 ). If the first regression criteria for non-confoundin g we r e true, then the in tegrand in the SC A op erator ( 6 ) w ould b e equiv alen t to the fi nest-crude-cancer rates. Ho wev er, the second criterion sp ecifies only that P y ( E a 2 | E a 1 , a ) = P y ( E a 2 | a ), not that P y ( E a 2 | a ) is in v ariant to y ear. The latt er is clearly necessary for SCA to b e a SONC op erator ( 10 ). 4. Using the SCA and SC C op erators to analyze S E ER data. In this section we present results from our analyses of the S E ER data fr om 13 r eg- istries, yea r s 1992–20 03 [SEER 13 Regs Limited Use ( 2005b ); henceforth called SEER 13]. W e consider the sub set of SEER 13 where age is greater than or equal to 40, race is limited to blac k or white, and ethnicit y is limited to either Hispanic or non-Hispanic. Because of the restriction we p lace on race, w e exclude Alask a and consider only 12 of th e 13 cancer registries. In our analyses w e limit the co v ariates to the E d efi ned in ( 1 ). Th e inten t of this section is to demonstrate the existence of differences in the stan- dardized r ates pro duced by the SCA and S CC op erators, particularly those differences discussed in Section 3 . W e mak e no commen ts ab out the statis- tical significance of these find ings and provide no formal estimates of trends (see Discussion in S ection 7 ). All rates giv en are p er 100,0 00 p ersons. Figure 1 is a graph of the crude and standard ized (SCA, and SCC) race-and-gender sp ecific colon cancer incidence for eac h y ear fr om 1992 to 2003. F or the SCA and SCC op erators all rates are pro duced with P ∗ ( E ) = P 2000 ( E ). A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 11 The SCA (dashed line) and SCC (solid line) rates differ for all groups and all years. Thus, ( 14 ) is false. F or the SCA op erator to pro du ce r ate d ifferen ces that are not confoun d ed, ( 12 ) m u st b e true f or this factorization of E . In SEER 13 ( 12 ) is false: there exists v ariation in the year-to -year P y (Hispanic, place | age, gender, race = white). During the time p erio d 1992 to 2003 the frequency of Hispanic ethnicit y in creased in eve ry place, for b oth genders, and (with rare exception) for eve ry age group (data not sho wn ). Note, h o w- ev er, th at in Figure 1 th e slop e of eac h segment of the SCA and SCC plots for white males, and b oth white and b lac k females, are virtually iden tical. Th is indicates that though S CA r ate differences ma y b e confounded ( 11 ), infer- ences ab out the existence of trend s ma y b e iden tical. The slop e of the SC A curv e is determined by b etw een-y ear differences in th e v alue of the integral of the fin est-crude-cancer rates with resp ect to the measur e P y ( E a 2 | E a 1 , a ) ( 10 ); confounding of SCA differences requires only yea r-to-y ear v ariation of P y ( E a 2 | E a 1 , a ). The principal motiv atio n f or us ing the weigh ts sp ecified by d ir ect stan- dardization is the d esire to pr o duce stand ardized rates that reflect the tr ue Fig. 1. Comp aring sex and age sp e cific crude-c anc er r ates with SCA and SCC stan- dar dize d r ates for the ye ars 1992–2003. The dotte d line is the crude c olon c anc er r ate. The dashe d line i s the SC A standar dize d c ol on c anc er r ate. The solid line is the SCC standar di ze d c ol on c anc er r ate. 12 S. D. MARK Fig. 2. The absolute value of the p er c ent di ffer enc e b etwe en the SCA standar dize d c olon c anc er r ate and the crude c olon c anc er r ate by sex and r ac e for the ye ars 1992–2003. The plotte d values for e ach ye ar wer e pr o duc e d by the fol lowi ng formula: | SCA standar di ze d c olon c anc er r ate − Crude c olon c anc er r ate | SCA standar di ze d c olon c anc er r ate × 100 . The dashe d lines ar e the absolut e value of the p er c ent di ffer enc es for whites. The solid lines ar e the absolute value of the p er c ent differ enc e for blacks. absolute v alues of the crude-cancer rates in the E 1 group. In Figure 1 w e see that standard ized rates from the SCC op erator more closely trac k the crude-cancer rates th an those p ro du ced by the SCA op erator. In fact, as indicated earlier, the SCA standard ized rate in the y ear 2000 do es not equal the cru d e-cancer rate that w ou ld hav e b een seen if the age distribution were iden tical to th at of the yea r 200 0. Figure 2 is a graph of the magnitude (the absolute v alue) of the p ercen t difference in the SCA and cru de-cancer rates. F or b oth males and females the magnitude of these differences is greatest for blac ks (sol id line). This phenomenon is due to the fact that the P 2000 (age) distribu tion is m uch closer to th e age d istribution for whites than f or b lacks. In addition, the ye ar- to-y ear v ariation in the p ercent deviation app ears to b e greater for blac ks. Th us, graph ically , b lac ks alw ays app ear to ha ve larger y ear-to-y ear c hanges in cancer inciden ce than do whites. Th ese d ifferences are more pr ominen t when cancer rates are compared within groups d efined b y ethn icity (data not sho wn ). Em pirically we fi n d that the lo w er the p opu lation frequency of a group, the great er the d eviation of S CA standard ized rates from the actual crude-cancer rates. One pr eviously unm en tioned limitat ion of the S EER SCA op er ator is that it cannot pro duce age-sp ecific cancer rates that cont rol for differences in the distribution of other risk factors. Since age is b y far the largest risk factor for cancer, comparing within-age-strata rates may r ev eal trends that are other- wise not visible. Figure 3 is a graph of the SCC standardized breast cancer A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 13 rates for white females for eac h y ear from 1992 thr ough 2003. This graph indicates an o ve rall increase in breast cancer rates from 199 2 to 1998, and a decrease f rom 1998 to 2003. Figure 4 con tains a plot of the SCC b reast cancer rates for white females in the yea r s 1992 (dotted line), 1997 (solid line), and 2003 (dashed line), within eac h of the fi v e-ye ar ag e categories 40 y ears old or greater. Th e shap e of the graphs of standard ized rates are s im i- lar for all three yea r s. Consisten t with Figure 3 , we see that the lo west rates are for year 2003, and the h ighest for 1997. What cannot b e d iscerned fr om Figure 3 is that the differences in cancer rates for the y ears 200 3 and 1997 are greatest for females older than 60; and that the d ifferen ces b et we en 2003 and 1992 rates are almost entirely due to rate differences in females ov er 60. The inf orm ation in Figure 4 suggests that w hen considering p ossible causes of the calendar trend s sho wn in Figure 3 , one should fo cu s on change s that w ere more prominent in f emales age 60 and older. Giv en th at the usual ap- proac h in epidemiology is to mak e inferences ab out cancer rates w ithin age groups, and n ot from m arginal rates obtained b y inte grating out age, w e an ticipate that age-sp ecific standardized rates will p r o vide imp ortan t add i- tional information ab out cancer trends and b et ween su bgroup d ifferences in cancer rates. Note that if one were to employ a “stratification str ategy” and use the SCA op erator to calculate separate standardized rates for eac h age group, the age standardized rates pr o duced w ould in fact b e th e crude breast cancer rates, P y (breast cancer | white, female, age = a j ). Fig. 3. The SCC standar di ze d br e ast c anc er r ates for white femal es age forty and older for the ye ars 1992–2003. 14 S. D. MARK Fig. 4. The SCC standar dize d br e ast c anc er r ates by five-ye ar interval for white females age f orty and older. The dotte d line is the SCC standar dize d br e ast c anc er r ates for 1992. The solid line is the SCC standar dize d br e ast c anc er r ates for 1997. The dashe d line is the SCC standar dize d br e ast c anc er r ates for 2003. 5. Making inferences from observ ed rate differences of SCC standard ized rates. In S ection 3 w e established that the SCC op erator is in the class of op erators in w hic h differences in the P y ( E 2 | E 1 ) do not affect the standard - ized rates pro du ced by th ose op erators. Thus, if the SCC stand ardized rates for year y † and y †† differ, we can conclude that the finest-crude-cancer r ates differ for those y ears. Ho wev er, desp ite the nomenclature, w e do not kno w whether for fixed E 2 the finest-crude-cancer rates differ as a fu n ction of E 1 , for fi xed E 1 they differ as a f u nction of E 2 , or whether the differences in finest-crude-cancer rates dep end on b oth E 1 and E 2 . T o m ak e th e inf erences of in terest to the SE ER inv estigators [W ard et al. ( 2006 )], w e require that w e consider d isease rates as a fu nction of b oth mea- sured and unmeasured risk factors. T o incorp orate the effect of unmeasur ed risk factors on in ference, we use the fundamenta l disease pr obabilit y (FDP) paradigm for inf erence p r op osed by Mark ( 2004 , 2005 , 2006 , 2008 ). T he re- sults we presen t in this sectio n dep end only on the definitions presen ted in this sectio n , and require no kno w ledge of, or r esults from, an y of the other material con tained in Mark ( 2004 , 2005 , 2006 , 2008 ). W e defi n e the fun damen tal disease p robabilit y for yea r y to b e the pr oba- bilit y of disease conditional on all risk factors. W e denote this b y P y ( D | E , U ). Here E are the measur ed risk factors; U is a set of u nmeasured risk factors that, along with E , completely determine the probabilit y of d isease. Th e A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 15 relationship b et w een the FDP and th e finest-crude-cancer r ates, P y ( D | E ), is P y ( D | E ) = Z U P y ( D | E , U ) dP y ( U | E ) . (15) Using th e FDP paradigm for inf er en ce, w e are able to falsify a subset of assum p tions ab out the u nmeasured risk factors based on contrasts of SCC standardized rates. W e d efine t wo assu mptions: the iden tical disease probabilit y (IDP) assum ption, P y † ( D | E , U ) = P y †† ( D | E , U ) , (16) and the comparable-confounding assumption, P y † ( U | E ) = P y †† ( U | E ) . (17) Without loss of generalit y , w e u se as example the inferen ces that can b e made from cont rasts of the ov erall (marginal) S CC standardized cancer rates in yea r s y † and y †† . T h ese are the standardized p opulation cancer rates not conditional on any risk f actors ( E 1 = ∅ ). In terms of the FDP form ulation this standardized rate is s cc y [ D ] = Z E  Z U P y ( D | E , U ) P y ( U | E )  dP ∗ ( E ) . (18) If IDP ( 16 ) is tru e, and s cc y † [ D ] 6 = s cc y †† [ D ], then w e can conclud e that the assumption of comparable-confounding ( 17 ) is false. F or instance, dietary factors suc h as folate in take are susp ected of b eing risk f actors for colon can- cer [Gio v annucci ( 2002 )]. SEER conta ins no measurement of folate intak e. If w ithin lev els of E , the in tak e of folate has c hanged ov er time, then ( 17 ) is false. In fact, in the United States a p opulation-wide folate sup plemen tation program b egan in 1998 ; it is known th at folate int ak e in the p opulation has increased considerably since then [Qu inliv an an d Gregory ( 2003 )]. Is th e IDP assumption reasonable? If w e b eliev e that the determinan ts of a disease, and th e impact of those determinants on the probability of disease, are inheren t to the biology of humans and do n ot v ary with y ear, then IDP is true. Suc h b elief is consisten t with our cur ren t conceptualization of biologica l pr o cesses. Ho w ev er, hidden in the IDP assumption is th e assertion that the classification of disease and exp osures is iden tical in year y † and y †† . I f diagnostic criteria for colon cancer ha ve c h anged, or, if diagnostic pro cedur es for detecting colon cancer h a ve c hanged (for instance, an increase in pro cedures th at lead to early detection of colon cancer), then D y † and D y †† ma y not in fact represen t the s ame b iologica l outcome. S imilarly , if the measuremen t to ols for ascertaining ethnicit y and race h a ve changed, then E † and E †† ma y not measure the same attributes. In either case, we would exp ect IDP ( 16 ) to b e false. 16 S. D. MARK Iden tical reasoning prov es th at if comparable confoun ding is true ( 17 ), then IDP ( 16 ) is f alse. In summary , if the observed SC C s tand ardized rate d ifferences are nonzero, w e can conclude that either IDP ( 16 ) and/or comparable confounding ( 17 ) are false. There is a direct corresp ond ence b et we en falsification of the abov e as- sumptions and the conclusions made in the Annual R ep orts to the Nation . W ard et al. b egin their p ap er, Interpr eting Canc er T r ends , with the f ollo w- ing sen tence: “T emp oral trends in th e incidence of particular typ es of cance r ma y reflect c hanges in exp osu re to underlying etiologic factors, changes in classification, or the introdu ction of n ew screening or diagnostic tests.” Th e “c hanges in underlyin g etiologic factors” corresp onds to comparable con- founding b eing false ( 17 ); the “c hanges in classification, or the introd uction of new screening or d iagnostic tests,” corresp ond s to IDP ( 16 ) b eing false. The FDP inferences giv en ab ov e apply to an y SONC op er ator. They do not apply to the SCA standardization operator used in W ard et al. ( 2006 ) or an y of the Annual R e p orts to the N ation . 6. Nested standardized rates and within -y ear mo d el b u ilding. Th e An- nual R e p orts provide and interpret tr en ds in cancer rates o v er time for v ari- ous demographic subgroup s. The ma jority of the rep ort examines con trasts in o v erall cancer rates, cont rasts cond itional on gender and r ace, and con- trasts conditional only on gender or only on race. Ho wev er, u n lik e in th e usual regression analysis, no attempt is m ad e to construct a “parsimonious mo del.” W ere the analyses in th e Annual R ep orts u sed only to d escrib e within-group trends o ve r time, su c h mo del develo pment migh t b e of no in- terest. When used to make the t yp e of inferences describ ed in th e first p ara- graph of this pap er, the abilit y to test nested mo dels assumes imp ortance. Whether standardized r ates conditional on race and gender are identi cal to standardized rates conditional on race alone has implications for the allo ca- tion of health care resources, the constr u ction of preve n tive pr ograms, and the fo cus of f u ture etiolog ic researc h. Regarding standardized rates as the output of stand ardization op erators allo ws us to ev aluate th e r elationship b et we en standardized rates pro d uced b y the same op erator on the same data. W e defin e the standardized rate s ∗ y [ D | E †† 1 ] to b e nested in s ∗ y [ D | E † 1 ] provided the follo wing three conditions are true: (1) b oth are pro du ced b y the same standardization op erator, S ∗ y [ D | E 1 ], (2) the argumen ts of the op erator, ( D y , E y ) and P ∗ ( E ) , are iden tical, (3) E †† 1 is a p r op er subset of E † 1 . The relationship of n ested standardized rates pr o duced b y the SCC op- erator has familiar prop erties. The SCC op erator is recursive in the sense A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 17 that s cc y [ D | E †† 1 ] = S cc y [[ s cc y [ D | E † 1 ]] | E †† 1 ] . (19) The righ t-hand side of ( 19 ) is defined to b e S cc y [[ s cc y [ D | E † 1 ]] | E †† 1 ] ≡ Z E † 1 \ E †† 1 s cc y [ D | E † 1 ] dP ∗ ( E † 1 \ E †† 1 | E †† 1 ) . (20) The SCC op erator do es not “discard information.” The standardized rate obtained from equation ( 7 ) when E 1 = E †† 1 is the same estimate obtained b y replacing the fin est-crud e-cancer rate in (7) with s cc y [ D | E † 1 ]. Thus, nested rates pro duced b y the SCC op erator ha v e the same p rop erties as nested estimates in regression mo d els of conditional exp ectations. W e are cur ren tly dev eloping inferential pr o cedures analogous to those that exist for regression. Though the iden tity in ( 19 ) can easily b e verified b y su bstitution, a more instructiv e pro of is based on th e functional form of the S CC operator. W e define the probab ility measure P y ∗ ( D , E ) ≡ P y ( D | E ) P ∗ ( E ). The SCC op- erator can b e regarded as the conditional exp ectation of the fin est-crude- cancer r ates, P y ( D | E 1 , E 2 ), with resp ect to P y ∗ ( E 2 | E 1 ). Th u s, s cc y [ D | E †† 1 ] is the p ro jection (conditional exp ectation) of the finest-crude-cancer rates on the subs pace (su bsigma algebra) defined by E †† . Pro jections (conditional ex- p ectations) are en tirely determined (a.e. u nique) by the sub space (subsigma algebra) on wh ic h they are defined (measur ab le) [Dudley ( 1989 )]. Recursion cannot b e defin ed for SCA. The P y ( D | E a 1 , A ) in the integral in ( 6 ) is alw ays a function of A ; s ca y [ D | E 1 ] is n ev er a function of A . Mimic king the form of ( 20 ) one migh t defin e r ecursion for S CA to b e Z E † 1 s ca y [ D | E † 1 ] P ∗ ( E † 1 | E †† 1 ) . (21) The integ r al in ( 21 ) do es not equal the s ca y [ D | E †† 1 ] obtained from ( 6 ). Th e SCA op erator is not a pro jection op erator. 7. Discussion. In order to mak e infer en ce from observ ational data, re- searc hers attempt to separate the effect of the exp osures of interest from the effect of other disease determinants that co v ary with the exp osures of in terest. W e ha ve divided these other determinant s into tw o m u tually ex- clusiv e sets: d eterminan ts that are m easured and determinants that are u n- measured. W e refer to pro cedures that atte mpt to separate the asso ciation of the co v ariates of in terest, E 1 , from the association of the other measured disease determinan ts, E 2 , as p r o cedures that control for confoun d ing. Standardization is one su c h pr o cedure. It is the m ost common pro cedur e used to con trol for confounding in the analyses of p opulation-studies or d ata 18 S. D. MARK from disease registries. In this pap er w e examined the abilit y of v arious stan- dardization pr o cedures to con trol for measured risk factors, and the inter- pretabilit y of differen ces in standardized r ates in the presence of u nmeasured risk factors. Our motiv ation for conducting th is research w as to ev aluate the prop erties of the “age adjustment usin g the direct metho d” standardization pro cedur e (SCA standardization) u sed in the analysis of SEER cancer reg- istry d ata and , if n eeded, to deve lop standardization pro cedures with b etter prop erties. W e define a general class of standardization operators, and regard stan- dardized rates to b e the output from a standardization op erator. T h e general class of op erators are an y fun ctionals that are inte grals of a cru de-cancer r ate with resp ect to a user-d efined “weig h ting” distrib ution ( 5 ). Since all cru de- cancer rates are themselves in tegrals of the finest-crude-cancer rates ( 2 ), P y ( D | E ), stand ardization op erators differ only with resp ect to the measure used to in tegrate P y ( D | E ). Based on this formulatio n , w e defined b et we en- y ear differences in crude-cancer rates conditional on E 1 as b eing u ncon- founded b y E 2 , p ro vid ed the d istribution of E 2 conditional on E 1 is the same in b oth yea rs ( 8 ). By extension, we defin ed a sub class of standard - ization op erators with no E 2 confounding (SONC op erators). This sub class consists of standardization op erators in whic h the finest-crude-cancer rates for eac h y are int egrated with resp ect to a distr ib ution that is the same for all y ( A.1 ). The SCA op erator is n ot a SO NC op erator ( 10 ). If the differences in crude-cancer rates are not confounded, the SCA op erator can int ro duce con- founding and pro d uce b et wee n-y ear differences in s tandardized r ates that are confou n ded. In Section 3.1 we s ho wed that the S C A op erator will in - tro duce confoundin g u nless the P y ( E ) distr ib utions d iffer only in terms of the marginal distribution of age. This criterion was not met in the SEER 13 data that w e analyzed. Figure 1 from that analysis provides a graphic represent ation of the f act that the standard ized rates pro d u ced by th e SCA op erator for y ear y are n ot the “cancer rates one w ould ha v e seen” had the age distribution in y ear y b een iden tical to the age distribution that generated the w eigh ts. It is clear that one should alwa y s choose a stand ardization op erator fr om the sub class of op erators with no E 2 confounding. W e prop osed and exam- ined the pr op erties and p erformance of one su c h op erator: the standardiza- tion con tr olling for cov ariates op erator (SCC). A desirable c h aracteristic of the SC C op er ator is that the y ear-to-y ear differences in standard ized cancer rates p reserv e the magnitud e of the differences of the crud e-cancer rates. When the cr u de-cancer rates are not confounded, and thus n o standard - ization pr o cedure is required, the SCC op erator is the only op erator for whic h th e standardized rates equal the crude-cancer rates. Figure 1 pro- vides a grap h ic illustration of these pr op erties. Figure 2 sho ws that the A GENERAL FORMULA TION F O R ST A NDARDIZA TION OF RA TES 19 largest d ifferences b etw een standard ized rates f rom the S CA op erator and crude-cancer rates o ccurs in m inorit y p opulations. If standard ized rates pro duced by standardize op erators with no E 2 con- founding are different in y ear y † and y †† , then differen ces m ust exist in the finest-crude-cancer rates. Ho we v er, to m ak e the inferences of in terest to the authors of the Annual R ep orts [W ard et al. ( 2006 )], one must consid er the impact of unmeasured cancer r isk factors on the y ear-to-y ear differences in standardized rates. In Section 5 w e used th e fu ndamenta l d isease p robabilit y paradigm for inference p rop osed by Mark ( 2004 , 2005 , 2006 , 2008 ) to pro v e that nonzero con trasts of standardized rates falsify assumptions ab out the conditional probabilities of disease giv en all the risk factors [the iden tical disease probab ilit y assumption , ( 16 )], and /or the conditional prob ab ilities of unmeasured risk factors giv en the measured risk factors [the comparable- confounding assumption , ( 17 )]. W e describ e the one-to -one corresp ond ence that exists b etw een the inferences made in the SEER A nnual R ep orts [W ard et al. ( 2006 )], and the falsification of these assumptions. The analyses in the Annual R ep orts only examine b et we en-y ear d ifferences in cance r rates. In Section 6 w e argue that, giv en the t yp e of inferences made from these rep orts, it would b e desirable to examine within-y ear con trasts. W e defined the concept of nested standardized r ates. W e prov ed that, like regression mo dels for cond itional exp ectations, the SC C op erator is a pro jec- tion op erator. In our current r esearc h w e are dev eloping metho ds for testing nested mo dels analogous to those used in regression. Though w e h a ve discussed the SCC op erator in terms of estimating the conditional exp ectatio n of a binary v ariable, these op erators extend in an ob vious manner to the estimation of conditional exp ectations in general. The particular op erators w e hav e defined are nonparametric estimators. O n e could constru ct parametric or semiparametric op erators b y , for instance, replacing P y ( D | E ) with a parametric or s emip arametric mo d el, P y ( D | E ; θ ). The n ext step in our research is to deriv e SCC estimators for the statis- tics currently used for inf erence in the A nnual R ep ort . Our eve n tual goal is to program theseestimators in the R -language [R Dev elopment Core T eam ( 2007 )] and pro duce f reely a v ailable user-friendly soft w are comparable to the SEER*Stat f r eew are [SEE R *S tat.6.3.6. ( 2007 )] a v ailable for S CA analyses of SEER d ata. APPENDIX: DEFINING S T AND ARDIZA TION OPE RA TORS WITH NO E 2 CONF O UNDING F or all factorizat ions E = ( E 1 , E 2 ), let F ∗ ( E 1 , E 2 ) b e a family of proba- bilit y measures indexed by e ∗ 1 , with the prop erty that R E 2 dF ∗ ( e ∗ 1 , E 2 ) = 1 for all e ∗ 1 ∈ E 1 . 20 S. D. MARK S ∗ y [ D | E 1 ] is a standard ization op erator unconfound ed b y E 2 if it can b e expressed in the form S ∗ y [ D | E 1 ] = Z E 2 P y ( D | E 1 , E 2 ) dF ∗ ( E 1 , E 2 ) . (A.1) Note that for the S CC op erator ( 7 ), the P ∗ ( E ) sp ecifies the F ∗ ( E 1 , E 2 ) for all factorization of E , and all realizatio ns e ∗ ∈ E . In general, this n eed n ot b e the case. SUPPLEMENT AR Y MA TERIAL F undamenta l disease probabilit y inference: A new p aradigm for causal inference in th e biologi cal sciences (DOI: 10.1214/08 -AO AS170SUPP ; .p df ). REFERENCES Anderson, R. N. and Rosenber g, H. M. ( 1998). Age standardization of death rates: Implementation of the yea r 2000 standard. National Vital Statistics R ep orts 47 . Dudley, R. M . (1989). R e al An alysis and Pr ob abili ty , 1st ed. W adswo rth, Paci fic Grov e, CA. MR0982264 Edw ards, B. K., Ho we, H. L. , Ries, L. A. G. et al. (2002). 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Dep ar tment of Biost a tistics and Biometrics University of Colorado School of Public Heal th 4200 East Ninth A venue, RM1615 Denver, Colorado 80262 USA E-mail: stev en.mark@uchsc.edu