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9 G#"'5($1+$?:A:$ Mortality Rates per 1000 person-years, % Adjusted Mortality Rates using subclasses, % Cigarette Smokers Cigar/Pipe Smokers age subclasses age subclasses age subclasses A"'#.-O$$%".H#&+$FG:$$3H-$-X-.B>-+-(($",$&8l'(@4-+@$",$('<.)&((1q.&B"+$1+$$ #-4">1+2$<1&($1+$"<(-#>&B"+&)$(@'81-(:$$91"4-@#1.($CIJKr$[DO[Isi\C\:$ 9'@$[E$,"'#i.)&(($.">&@-($!$">-#$41))1"+$41))1"+$('<.)&((-($ I$
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12 Study Women Breast Conservation (BC) Estimated Survival Rate for Women Estimated Causal Effect Mastectomy (Mas) BC Mas BC Mas n n % % % US-NCI Milanese French Danish EORTC US-NSABP Single-center trial; Multicenter trial Reference: Rubin DB. Estimated Causal Effects from Large Datasets Using Propensity Scores. Annals of Internal Medicine 1997; 127, 8(II): C[$
13 Propensity Score Subclass Women Breast Conservation (BC) Estimated Survival Rate for Women Estimated Causal Effect Mastectomy (Mas) BC Mas BC Mas n n % % % Averages Across Five Subclasses Reference: Rubin DB. Estimated Causal Effects from Large Datasets Using Propensity Scores. Annals of Internal Medicine 1997; 127, 8(II): C\$
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22 >)D)+1)9" #!"# " F8=/)0"4G" CH+590)-" C$ p$ E$ E$ E$ CCsCD [$ p$ E$ E$ C$ dd \$ U$ C$ E$ E$ [\Ks D$ U$ C$ E$ C$ \D s$ %$ C$ C$ E$ IJJ\ J$ %$ C$ C$ C$ C[ [\JK[ Reference: Sommer and Zeger (1991). On Estimating Efficacy from Clinical Trials. Statistics in Medicine.
23 6+&)7(-($ I)7H49" J*3=27)" C25D85234-" >4B"C4=;20+*4-" " = *33$ ie:ee[j$ \/$D/$s/$a$J$>(:$C$a$[$ " (i@#-&@-8$ ie:eejs$ = s$a$j$>(:$c/$[/$\/$ad$ " Z-#$5#"@".")$ ie:ees[$ = s$a$j$>(:$c$a$[$ Reference: Sommer and Zeger (1991). On Estimating Efficacy from Clinical Trials. Statistics in Medicine.
24 b"b$%6%l$6+&)7(1($ 6%L$W$( ) $ K $U6%L$h$( * $ K $%6%L$ ie:ee[s$w$e:[$ K $U6%L$h$E:K$ K $%6%L$ ie:ee[s$w$e:k$ K $%6%L$!$%6%L$W$iE:EE[soE:K$W$iE:EE\C$
25 Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
26 L(B4&+8$ LM.)'(1"+$ b-&+$ 8->1&B"+$ b-81&+$ $ 5-#.-+B)-$ $ 5-#.-+B)-$ %6%L$ U"$ \:C$ [:s$ \:[$ ie:i$ d:e$ *33 +,-. $ U"$ E:s$ CE:C$ E:[$ icd:c$ Cd:s$ %6%L$ Y-($ \:C$ C:[$ \:C$ C:[$ s:c$ Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
27 Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
28 Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
29 Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
30 /0 1,* y RET0 6 2 RCT0 + 2 y* E/$z0 + 2 y* E/$z0 %0 E:[s0 E$ C$ )RE:C/$E:CJT$ )RE:I/$E:DIT$ -0 E:Ds0 E$ E$ )RC:E/$E:[sT$ )RC:E/$E:[sT$ 90 E:\E0 C$ C$ )RE:E/$E:\JT$ )RE:E/$E:\JT$ Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
31 m+-$a&45)-$ Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
32 4-&+$ 4-81&+$ b-&+$<1&($ b-81&+$ <1&($ (_'&#-8$ -##"#$ b-81&+$ -##"#$ ie:ce$ ie:ed$ E:DK$ E:\E$ ie:ek$ ie:ej$ E:sC$ E:\[$ %">-#&2-$ E:IC$ b-81&+$ C:JC$ bvl$ ie:cd$ ie:c[$ E:sC$ E:\C$ E:dD$ C:CC$ *^L$ E:ss$ E:C\$ [:\C$ E:sD$ E:IC$ [:dk$ Imbens G.W. and Rubin D.B. (1997) Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance. Annals of Statistics 25(1):
33 "+$.Q&$.&+.-#$('#>1>&)$! @#&+(,-#($! \\$
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35 Figure 5.1: Cardia Cancer, Number of People, Subclassified by Propensity Score Small Home Hospital Large Home Hospital Propensity Score Subclasses ;-,-#-+.-$$;'<1+/$0:9:$!"#$m<l-.B>-$%&'(&)$*+,-#-+.-/$0-(12+$3#'45($6+&)7(1(:$$$ 6++&)($",$655)1-8$A@&B(B.(/$[EEK:$ \s$
36 Figure 5.2: Cardia Cancer, Difference in Means for Binary Covariates and Pscore Rural Initial After Pscore Subclassification Male Rural.Male Summer Rural.Summer Male.Summer Pscore Difference in Means ;-,-#-+.-$$;'<1+/$0:9:$!"#$m<l-.B>-$%&'(&)$*+,-#-+.-/$0-(12+$3#'45($6+&)7(1(:$$$ \J$
37 Figure 5.3: Cardia Cancer, t-statistics for Continuous Covariates Age Initial After Pscore Subclassification Age2 AgeL Year Age.Year Age2.Year AgeL.Year t-statistic ;-,-#-+.-$$;'<1+/$0:9:$!"#$m<l-.B>-$%&'(&)$*+,-#-+.-/$0-(12+$3#'45($6+&)7(1(:$$$ \d$
38 5)&'(1<)-$ ITT = " LS ITT LS + " SS ITT SS + " LL ITT LL CACE " ITT LS = 1 N LS % i $ LS (Y i (L)# Y i (S)), ITT = " LS ITT LS, ITT LS = ITT / " LS. \K$
39 Assigned /Randomized Home Hospital Type Treating Hospital Type Approximate N in LS Principal Stratum Subclass h T ! L s S \I$
40 Table 5.3: Cardia Cancer: Observed Counts in Observed Groups and Approximate Counts in Principal Strata Under Monotonicity Assumption Subclass 3 (1) (2) (3) (4) (5) (6) Treating Underlying Approximate Hospital # Principal Proportion Type Strata: in T h= Population in Principal Assigned /Randomized Home Hospital Type (1) (2) h #! 17! s Strata L 17 L L 38% L S 62% S 0 S S 0% Approximate N in LS Principal Stratum 11 (3) (4) s 13 L 5 L L 38% S S 0% S 8 L S 62% 8 ;-,-#-+.-$$;'<1+/$0:9:$!"#$m<l-.B>-$%&'(&)$*+,-#-+.-/$0-(12+$3#'45($6+&)7(1(:$$$ 6++&)($",$655)1-8$A@&B(B.(/$[EEK:$ DE$
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