(confounding)
Confounding Exposure? Outcome Confounders adjustment (confounders)
Not Confounding? 2 exposure 3rd factor 3rd factor
NEJM 2000; 342: 1930-6. (adjustment) univariate or crude relative risk (univariate or crude OR)adjusted relative risk (or adjusted OR) multivariate adjustment adjusted value 10
(OR = 14.3; 95%CI: 2.9 70.7) 13,10011.600 13,0003.9
Overestimation
(RR=0.3; 95%CI: 0.2 0.8) (<10%)
Overestimation
14.317.3,
Underestimation
(adjustment) (restriction)
Exposure Outcome Randomization Matching restriction Confounder Adjustment (confounders)
Confounder vs. Effect modification Confounder exposure outcome ( Effect modification: Effect modifier exposure, outcome ()
Crude data vs. Mantel-Haenszel adjustment crude data adjusted data exposure factor disease risk ratio (RR), odds ratio (OR) RRcrude = (a/n 1 ) / (c/n 0 ) ORcrude = ad/bc 203070 adjustment Mantel-Haenszel adjustment(mh) RR MH = [(a i )(N 0i )/T i ]/[(c i )(N 1i )/T i ] = [(w i )(RR i )]/[(w i )] w i = (c i )(N 1i )/T i = [T i ][(N 1i N 0i )/(T i T i )][c i /N 0i ] OR MH = [(a i )(d i )/T i ]/[ (b i )(c i )/T i ] = [(w i )(OR i )]/[ w i ] w i = (b i )(c i )/T i RR crude = RR MH, OR crud e = OR MH RR crude RR MH, OR crude OR MH
i closed cohort study :
? 120 80 200 80 320 400 210 290 500 90 410 500 300 700 1000 RR crude = (210/500) / (90/500) = 2.33 200 400 600 RR boy = = (120/200) / (80/400) = 3.00 90 210 300 10 90 100 100 300 400 RR girl (90/300) / (10/100) = = 3.00
Mantel Haenszel RRcrude RRadjusted confounder RRboy = RRgirl effect modifier
adjustment boys 120 80 200 80 320 400 200 400 600 girls 90 210 300 10 90 100 100 300 400 (120 x 400)/600 + (90 x 100)/400 = 80 + 22.5 = 102.5 (80 x 200)/600 + (10 x 300)/400 = 26.7 + 7.5 = 34.2 102.5 / 34.2 = 3.00 = RR adjusted
confounder effect modifier
? 210 290 500 70 430 500 280 720 1000 RR crude = (210/500) / (70/500) = 3.00 120 80 200 40 160 200 160 240 400 RR boy = (120/200) / (40/200) = 3.00 90 210 300 30 270 300 120 480 600 RR girl (90/300) / (30/300) = = 3.00
RRcrude = RRadjusted confounder RRboy = RRgirl effect modifier
adjustment boys 120 80 200 40 160 200 200 400 400 girls 90 210 300 30 270 300 120 480 600 (120 x 200)/400 + (90 x 300)/600 = 60 + 45 = 105 (40 x 200)/400 + (30 x 300)/600 = 20 + 15 = 35 105 / 35 = 3.00 = RRadjusted
confounder effect modifier
? 210 290 500 70 430 500 280 720 1000 RR crude = (210/500) / (70/500) = 3.00 160 40 200 40 160 200 160 240 400 RR boy = (160/200) / (40/200) = 4.00 50 250 300 30 270 300 80 520 600 RR girl (50/300) / (30/300) = = 1.67
RRcrude = RRadjusted confounder RRboy RRgirl effect modifier crude 10effect modification crude homogeneity test effect modification effect modification effect modification effect modification
adjustment boys 160 40 200 40 160 200 200 200 400 girls 50 250 300 30 270 300 80 520 600 (160 x 200)/400 + (50 x 300)/600 = 80 + 25 = 105 (40 x 200)/400 + (30 x 300)/600 = 20 + 15 = 35 105 / 35 = 3.00 = RRadjusted
Stratification open cohort study i = 1,,,,, I strata stratum i,
i opened cohort study :
British Doctors Study revised 1.72
Stratification (open cohort study) 35-44 45-54 55-64 65-74 75-84 s ns s ns s ns s ns s ns 32 2 104 12 206 28 186 28 102 31 PY 52,407 18,790 43,248 10,673 28,612 5,710 12,663 2,585 5,317 1,462 British Doctors Study S: smoker, ns: non-smokerpy: person-years British Doctors Study revised
1.72
Just simply summarized s ns 630 101 PY 142,247 39,220 IR 44.29 25.79 IRR 1.72 incidence rate ratio (IRR)1.72
IRR effect modifier confounder chi 2 = (O E) 2 / var O: observed, E: expected, var:variance Open cohort study chi 2 = [X E(X/H 0 ) ] 2 / var(x/h 0 ) X = = a E(X/H 0 ) = confounder = M1(N1/T) Var(X/H 0 ) = M 1 (N 1 /T)(1- N 1 /T)=M 1 N 1 N 0 /T 2 (crude table) = [630 (142247 731/181467)] 2 / (731 x 142247 x 39220/1814672) = 26.24
crude table chi2= [Xi Ei (X i /H0 i) ]2 / vari (X i /H0 i) I Xi = = ai Ei(Xi/Hi0) = confounder = M1i(N1i/Ti) Vari(Xi/H0i) = M1i(N1i/Ti)(1- N1i/Ti)=M1iN1iN0i/Ti2 H0: HA: Xi 630
Effect Modification in Stratified Table IR: incidence rate, IRR: incidence rate ratio
Ei(Xi/Hi0) Vari(Xi/H0i) Ei(Xi/Hi0) = 595.2 Vari(Xi/H0i) = 110.1 chi2= [Xi Ei (X i /H0 i) ]2 / vari (X i /H0 i)=(630 595.2)2 / 110.1 = 11.02 Pr (chi2> 11.02) = 0.001 H0 control (control 101 :residual confounder)10 (residual confounder)( ),
c 226.23( )c211.02 crude data adjust Effect modification (restriction) PY IRR(weight) PY(weight) Inverse variance weights {wi = 1/var[ln(RRi)] = 1/(1/ai + 1/bi)}a or b 0weight 0 strata Mantel- Haenszel weights
Adjusting to exclude confounding by age PY IR IRR E i (X i /H i0 ) Var i (X i /H 0i ) E(X/H 0 ) = V(X/H 0 ) = (32+2) x 52407 = 25.03 52407 + 18790 (32+2) x 52407 x 18790 = 6.61 (52407 + 18790) 2 variance
wi = bin1i/ti effect modification weight RRMH = wi RRi / wi =bin1i/ti(ai/n1i)/(bi/n0) /(bin1i/ti) =(ain1i/ti)/(bin1i/ti) ain1i/ti=116.835 bin1i/ti=82.09 RRMH = (ain1i/ti)/(bin1i/ti)=1.42 IRR1.721.42, 42
value 95% confidence interval (CI) lnirrmh 1.96var(lnIRRMH) var(lnirrmh)= A/BC A = M1iN1iN0I / Ti2 B= ain0i /Ti C= bin1i/ti var(lnirrmh)= A/BC=0.01149 ln(1.42) 1.960.01149 = (0.141, 0.560) e(0.141, 0.560)= (1.15, 1.76) adjust 1576 95 IRR1.72positive confounding
Calculate at each stratified table
rate difference (RD) RD = wi RDi / wi Var (RD) = a/n12 + b/n02 = (an02 + bn12)/n12n02 w i = 1/vari(RDi) = N1i2N0i2 /(N0i2ai + N1i2b1) RD = wi RDi / wi =0.00061, 10,000PY6.1 RD 1.96var(RD) Var(RD) = 1 / wi 0.00061 1.960.0000000154 =(3.7/104 PY, 8.5/104 PY) RDcrude = 1.85/1,000PY (12.4/104 PY, 24.6/104 PY) RDcrudeRDadjusted RDconfounder
Calculate at each stratified table