Nested case-control and case-cohort studies

Transcription

1 Outne: Nested case-contro and case-cohort studes Ørnuf Borgan Department of Mathematcs Unversty of Oso NORBIS course Unversty of Oso 4-8 December Radaton and breast cancer data Nested case contro studes Case-cohort studes Stratfed case-cohort studes A revew of nested case-contro and case-cohort studes s gven n Chapter 17 by Borgan & Samuesen n «Handbook of survva anayss» (eds Ken et a., Chapman & Ha / CRC Press, Radaton and breast cancer (e.g. Hrubec et a., Cancer Research, 1989 We w for ustraton use data from a cohort of 172 women dscharged from two tubercuoss sanatora n Massachusetts Radaton doses to the breasts due to fuoroscopc examnatons have been estmated for a women he women have been foowed unt end of 198 by whch tme t was 75 breast cancer cases Cohort data and mode Observe events (e.g. occurrences of a dsease n a cohort of n ndvduas (arrows are censored observatons We want to a study the effect of radaton exposure on breast cancer rsk We have data for the fu cohort, but w use t to ustrate nested case-contro and case-cohort studes 3 mes of observed events: 1 2 < < < he ndvdua havng an event at tme (the case s denoted d study tme 4

2 Countng process for ndvdua N ( t = I{ t, = } has ntensty process ( of the Cox regresson form ( λ ( t = Y ( t α ( texp β x at rsk rsk ndcator λ t basene hazard hazard rato (reatve rsk Cohort sampng Cohort studes need nformaton on covarates for a ndvduas at rsk Expensve to coect and check covarate nformaton for a ndvduas n arge cohorts In bomarker studes t w aso mpy a waste of vauabe boogca matera Parta kehood L( β exp ( β x = exp ( β x R Usng a cohort sampng desgn one ony needs to coect covarate nformaton for the cases and a sampe of contros wo types of cohort sampng desgns: Matched desgns: nested case-contro Unmatched desgns: case-cohort We need covarate nformaton for everyone at rsk 5 6 Cassca nested case-contro desgn Seect at random m 1 contros among those at rsk when a case occurs (case excuded,.e. match on study tme Iustraton for two contros per case (m = 3 case at rsk contro 7 We w derve a parta kehood Samped rsk set at tme conssts of the case and ts samped contros Introduce the countng processes N, ( =, =, ɶ r t I t t R = r her ntensty processes are gven by λ, r ( t = λ ( t π ( r t, { } where 1 n( t 1 π ( r t, = m 1 ɶ R ( β x α = Y ( t exp ( t π ( r t, s the probabty of seectng the set r as the samped rsk set f ndvdua fas at tme t

3 Introduce the aggregated processes N ( t = N ( t λ ( t = λ ( t r, r r, r r r he probabty that ndvdua fas gven that a faure occurs at t and gven that the samped rsk set s r : λ, r ( t π ( t, r = λ ( t r = ( β x ( β x Y ( t α ( texp π ( r t, r Y ( t α ( texp π ( r t, he parta kehood s a product of such factors over a faure tmes and samped rsk sets: L ncc λ (, ( = Rɶ β λ ( Rɶ exp ( β x = exp ( β x R ɶ he parta kehood ooks ke the fu cohort parta kehood, but the sums are ony over the samped rsk sets he maxmum parta kehood estmator enoys the «usua propertes» of ML-estmators, and testng may be performed by the kehood rato test Standard software for stratfed Cox regresson may be used by formay specfyng the abe of the samped rsk sets as a stratfcaton varabe (cf. the practca exercses 1 Radaton and breast cancer exampe Seect nested case-contro data wth 2 contros per case (m=3 Consder mode wth og 2 (dose+1 as ony covarate: ˆβ se( ˆ β Z P Cohort: Nested case-contro: Emprca reatve effcency for nested case-contro reatve to fu cohort: 2 Var(cohort.162 reatve effcency = = =.47 2 Var(nested case-contro.237 A note of stratfcaton/matchng Sometmes we woud ke to adopt a stratfed Cox mode where the ntensty process for an ndvdua n stratum c takes the form: ( λ ( t = Y ( t α ( t exp β x c For the radaton and breast cancer exampe, one coud e.g. stratfy on sanatorum For cohort data we then use a modfcaton of the parta kehood where the rsk sets ony contan those under observaton n the same stratum as the case Asymptotc reatve effcency s (m 1/m for a mode wth one covarate wth no effect 11 12

4 Smary, n a nested case-contro study, the contros are seected from the case s stratum (matchng Iustraton for one contro per case Cassca case-cohort desgn Seect by smpe random sampng wthout repacement a subcohort SC consstng of a fracton of the fu cohort m / n Iustraton for n=8 and m=4 subcohort he parta kehood for nested case-contro data remans unchanged Prentce proposed to base estmaton on the pseudo kehood ( exp β x LP ( β = Y ( exp ( β x SC { } Note that a case outsde the subcohort s ony used at ts event tme he pseudo-kehood does not possess kehood propertes Nevertheess, we may prove that s approxmatey mutvarate normay dstrbuted around the true vaue of But the covarance matrx may not be estmated by the observed nformaton and kehood rato tests do not appy However, Wad tests may st be used 15 A sma effcency gan may obtaned by ncudng the cases at a tmes they are at rsk and usng nverse probabty weghted (IPW estmaton We then maxmze the IPW pseudo-kehood ( exp β x LW ( β = Y ( exp ( β x SC Cases Here the weghts are w =1 for a cases and w = n / m for subcohort members who do not experence the event, where n and m are the number of cohort and subcohort members, respectvey, who do not experence the event w 16

5 Radaton and breast cancer data Seect subcohort wth 15 women Mode wth og 2 (dose+1 as ony covarate: ˆβ se( ˆ β Z P Cohort: Nested case-contro: Prentce: IPW: Nested case-contro and case-cohort studes tend to have smar reatve effcences when they are based on (about the same number of ndvduas Here nested case-contro uses data for 21 women and case-cohort uses data for 218 women 17 Stratfed case-cohort sampng From cassca sampng theory we know that we may often mprove our estmates by usng stratfed sampng Often some nformaton, e.g. a surrogate for the exposure of man nterest, may be avaabe for a cohort members In the radaton and breast cancer exampe the number of fuoroscopc examnatons s a surrogate for exposure dose For case-cohort sampng, we may stratfy the cohort nto S sampng strata based on the surrogate nformaton and seect the subcohort by stratfed sampng Stratfed nested case-contro sampng (often denoted «counter-matchng» s not consdered n ths course 18 Stratfed case-cohort Seect the subcohort by stratfed random sampng of a fracton m / n from sampng stratum s s s Iustraton for S = 2, n 1 = n 2 = 4 and m 1 = m 2 = 2 We may here use the IPW pseudo-kehood L W ( exp β x ( β = Y ( exp ( β x SC Cases w Now the weghts are w =1 for a cases and w = n / m for subcohort members from sampng stratum s who do not experence the event, where and are the n s number of cohort and subcohort members from stratum s who do not experence the event m s s s 19 2

6 Radaton and breast cancer data Seect subcohort by stratfed sampng wth 5 women n each of the sampng strata: 1: no fuoroscopc examnatons (698 women 2: fuoroscopc examnatons (765 women 3: 15 fuoroscopc examnatons or more (257 women Mode wth og 2 (dose+1 as ony covarate: ˆβ se( ˆ β Z P Cohort: Nested case-contro: Prentce: IPW: Stratfed case-cohort: Ponts to consder when choosng a cohort sampng desgn Statstca effcency Ease of the statstca anayss Choce of tme scae Pannng the study workfow Mutpe endponts and reuse of contros Need for matchng Stratfed sampng See secton 17.5 n the handbook chapter for a dscusson of these ponts 22