“HAZARD INCREMENT” ESTIMATEFROM NESTED CASE-CONTROLDATA•fR - case-control set.• wj- More specialized weight than for rateratio estimator.∆bΛeR=1Pj∈eRϕj(bβ) wj1SIMPLE RANDOM SAMPLING OFm − 1 CONTROLSCase-control set estimator of increment:• wj= n/m (n - number in risk set)∆bΛeR(z0) =1Pj∈eRexp[(Zj− z0)bβ + log(n/m)]• Include log(n/m) as an offset in the model.2SIMPLE RANDOM SAMPLING OFm − 1 CONTROLS, MATCHED ON CCase-control set estimator of increment:• wj= n/m (n - number in (unstratified) risk set)∆bΛeR(z0) =1Pj∈eRexp[(Zj− z0)bβ + log(n/m)]• Include log(n/m) as an offset in the model.3COUNTER-MATCHED SAMPLING ONYEARS MINING (<13, 13+ years)Case-control set estimator of increment:• wj= nCj/mCj(nl- number in risk set with C = l)∆bΛeR(z0) =1Pj∈eRexp[(Zj− z0)bβ + log(nCj/mCj)]• Include log(nCj/mCj) as an offset in the model.4PROPERTIES OF THECASE-CONTROL ESTIMATORS• Unbiased for full cohort ∆bΛR(z0) (at β0).• Variance estimated using the weights arevalid.Can use same software for estimation ofrisk as for the full cohort risk sets.5Miners SRS(3) nested case-control analysis dataset (actual data)._setno _rsentry _rstime record _cc timeint offset cr5001 34.4999 34.500 1790 1 1 6.31536 11 34.4999 34.500 2907 0 1 6.31536 11 34.4999 34.500 1379 0 1 6.31536 02 36.4199 36.420 2856 1 1 6.40413 12 36.4199 36.420 2608 0 1 6.40413 02 36.4199 36.420 809 0 1 6.40413 13 38.1699 38.170 1338 1 1 6.47028 13 38.1699 38.170 2682 0 1 6.47028 03 38.1699 38.170 324 0 1 6.47028 04 39.9199 39.920 185 1 1 6.53233 14 39.9199 39.920 1731 0 1 6.53233 04 39.9199 39.920 1174 0 1 6.53233 15 39.9999 40.000 258 1 1 6.53233 15 39.9999 40.000 3231 0 1 6.53233 05 39.9999 40.000 521 0 1 6.53233 16 40.0799 40.080 3242 1 1 6.53766 06 40.0799 40.080 3152 0 1 6.53766 16 40.0799 40.080 2650 0 1 6.53766 1offset = log(weight) = log(n/m)6CM(2:2) DATA FOR RISK ESTIMATION_setno _rsentry _rstime record _cc mine13(C_i) m(C_i) n(C_i) _offset cr500(Z)1 34.4999 34.500 316 0 0 2 1498 6.61874 01 34.4999 34.500 2114 0 0 2 1498 6.61874 11 34.4999 34.500 1790 1 1 2 161 4.38826 11 34.4999 34.500 2127 0 1 2 161 4.38826 12 36.4199 36.420 2445 0 0 2 1612 6.69208 02 36.4199 36.420 2715 0 0 2 1612 6.69208 12 36.4199 36.420 2856 1 1 2 201 4.61016 12 36.4199 36.420 1618 0 1 2 201 4.61016 13 38.1699 38.170 1283 0 0 2 1679 6.73281 13 38.1699 38.170 761 0 0 2 1679 6.73281 03 38.1699 38.170 1338 1 1 2 258 4.85981 13 38.1699 38.170 1256 0 1 2 258 4.85981 14 39.9199 39.920 1240 0 0 2 1767 6.78389 14 39.9199 39.920 640 0 0 2 1767 6.78389 04 39.9199 39.920 185 1 1 2 294 4.99043 14 39.9199 39.920 1700 0 1 2 294 4.99043 15 39.9999 40.000 258 1 0 2 1768 6.78446 15 39.9999 40.000 1330 0 0 2 1768 6.78446 15 39.9999 40.000 2160 0 1 2 293 4.98703 05 39.9999 40.000 2991 0 1 2 293 4.98703 1offset= nCi/mCi.7RISK ESTIMATION FOR <500 WLMAND 500 WLM+ MINERS: COXREGRESSION MODELWith z the 500 WLM+ variable:λ(t, z; β) = λ0(t) exp(zβ)=(λ0(t) for <500 WLMλ0(t) exp(β) for 500 WLM+• Standard Cox regression software to estimate risk:– Entry time = rsentry– Exit time = rstime– Failure indicator = cc– Covariate = cr500 (risk for <500 WLM) andcr500-1 (risk for 500 WLM+).– offset = log sampling weight.– (For time interval risk estimates) Strata =timeint.Ask for baseline cumulative hazard.8SAS CODE FOR MINERS NESTEDCASE-CONTROL DATANON-PARAMETRICtitle4 ’Non-parametric’;data covs;input _offset;datalines;0;run;proc phreg data=cm.cm2_2 nosummary;model _rstime*_cc(0)= / entry=_rsentry offset=_offset rl;baseline out=ch_rs covariates=covs cumhaz=cumhazlowercumhaz=lci uppercumhaz=uci/nomean;strata cr500 _rstime (50,70);run;Only difference compared to full cohort is inclusion of thelog weight as an offset in the model and in cov file.9Estimated risk of lung cancer death betweenages 50-69 from case-control dataProportional hazard modelWLM Full cohort SRS(3)<500 6.2% (4.7-8.2) 6.3% (4.7-8.4)≥500 26.3% (22.1-31.2) 28.1% (22.8-34.7)Non-parametric (Nelson-Aalen)WLM Full cohort SRS(3)<500 7.3% (5.3-10.2) 6.8% (4.6-10.1)≥500 24.9% (20.8-29.9) 22.7% (18.6-27.7)10Estimated risk of lung cancer death betweenages 50-69 from case-control data, continuedProportional hazard modelWLM SRS(3) Matched CM(2:2)<500 6.4% (4.8-8.5) 5.8% (4.3-7.7)≥500 27.4% (22.0-34.2) 29.8% (24.3-36.6)Non-parametric (Nelson-Aalen)WLM SRS(3) Matched CM(2:2)<500 6.9 (4.8-9.9) 7.0% (4.9-10.1)≥500 22.0% (18.0-26.8) 27.8% (22.4-35.4)11RISK ESTIMATION FROM NESTEDCASE-CONTROL DATA• Analysis is completely analogous to fullcohort, just need to include sampling weights.• Need to include weights to connect case-control set size to risk set size.• Might need “different weights” than neededfor rate ratio estimation, but rate ratioestimate will not