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Jun 23, 2022 H.S . 1 Selection Bias Concepts Hein Stigum Presentation, data and programs at: http://folk.uio.no/heins/ talks

November 15 H.S.1 Selection Bias Concepts Hein Stigum Presentation, data and programs at:

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  • *H.S.*Selection Bias ConceptsHein StigumPresentation, data and programs at:http://folk.uio.no/heins/talks

    H.S.

  • QuestionsGiven measured appropriate variables:

    Can you adjust for confounding?Yes

    Can you adjust for selection bias?Depends on the definition*H.S.*

    H.S.

  • ContentsBackgroundDefine biasSelection biasas effect modification(old concept)as collider stratification bias(new concept)DAG structureExamplesSize and direction of bias*H.S.*

    H.S.

  • Bias definitionBiasFrequency:expected risk true riskEffect:association causal effect*H.S.*

    H.S.

  • Selection bias conceptsConceptEffect responders Effect non responders

    Differential response biasDifferential loss to follow upHealthy worker biasBerksons bias (case control)

    DAG structure

    Effectmodification

    Colliderstratificationbias*H.S.*

    H.S.

  • Selection bias as effect modification*H.S.*

    H.S.

  • Selection bias: RiskSelection of responders The prevalence is different amongthe responders compared to the full populationthe responders compared to the non responders

    *H.S.*R1R0RpNon respondersRespondersPopulationRp is the weighted mean of R0 and R1

    H.S.

  • Effect modificationSelection of responders The effect of E on D is different amongthe responders compared to the full populationthe responders compared to the non responders

    *H.S.*RR1RR0RRpNon respondersRespondersPopulation

    H.S.

  • ProblemsIs not a bias, RR0 and RR1 are the true effectsIs effect modification by selection variable SLeads to the conclusion that:Biolocical effects are protected from biasThe bias can not be adjusted forRRp is the average of RR0 and RR1*H.S.*EDSDAG structure:Not true for collider stratification bias

    H.S.

  • Selection bias as collider stratification bias*H.S.*

    H.S.

  • Example with pathsStudyMilk on bone densityExclude Calcium supplements

    *H.S.*Lessons learned:Biological effect not protectedMay adjust for selection biasStructure:Collider stratification

    PathTypeStatus1EDCausalOpen2E[S]DNoncausalOpen

    2E[S][C]DNoncausalClosed

    H.S.

  • ExamplesDifferential responseSurvey: Alcohol and CHD

    Differential loss to follow upRandomized trial: drug and disease

    Healthy worker effectCross-section: Melt hall dust and lung disease*H.S.*Note: no confounding

    H.S.

  • Selection bias structure*H.S.*

  • Paths1. Causal2. ConfoundingAn open non-causal path without colliders3. Selection biasA non-causal path that is open due to conditioning on a collider BCVs?

  • Collider stratification biasSelection bias = Collider stratification biasSelection bias, Path definitionA non causal path that is open due to conditioning on a collider

    *H.S.*

    H.S.

  • Selection bias examples*H.S.*

    H.S.

  • Folic acid and cardiac malformationSelection: Study only live bornBias?

    *H.S.*Yes, E[C]D is openSelection: Non grieving parents volonteerBias?

    Yes, E[C]D is (partially) open

    H.S.

  • Education and unfaithfulnessStudy the effect among couples in a relationship (not divorced)?*H.S.*Selection bias

    PathTypePopulationSample1EDCausalOpenOpen2ERDNoncausalClosedOpen3ERSDNoncausalClosedOpen

    H.S.

  • Size and Direction of bias*H.S.*

  • Example 1, full tableProportion responding in 1,1 groupTrue and biased RRs(Adjusted) RRs

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0RR among

    RR200.30.3selected=

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0RR among

    RR200.90.3selected=

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10Ex 3 to 5 are variants of the same pattern

    % disease20%E10.90.92.51.7RR among

    RR200.90.3selected=

    Response=59%10

    PopulationRespondersNon respond2.00.912.3

    DDDWeighted average59%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=03.02.5

    1000591.4285714286408.5714285714D on S|E=11.0

    RR=2.0RR=0.9RR=12.3D on S|E=03.01.6666666667

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.05RD=0.26E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4RR among

    RR200.70.3selected=

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3RR among

    RR200.70.3selected=

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3RR among

    RR200.30.3selected=

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3RR among

    RR200.50.3selected=

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0RR among

    RR100.450.225selected=

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5RR among

    RR200.2250.45selected=

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7RR among

    RR200.90.8selected=

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5RR among

    RR200.4950.33selected=

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5RR among

    RR200.250.1666666667selected=

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7RR among

    RR200.450.45selected=

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    S

    D

    E

    D

    D

    E+D

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0RR among

    RR200.30.3selected=

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0RR among

    RR200.90.3selected=

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10Ex 3 to 5 are variants of the same pattern

    % disease20%E10.90.92.51.7RR among

    RR200.90.3selected=

    Response=59%10

    PopulationRespondersNon respond2.00.912.3

    DDDWeighted average59%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=03.02.5

    1000591.4285714286408.5714285714D on S|E=11.0

    RR=2.0RR=0.9RR=12.3D on S|E=03.01.6666666667

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.05RD=0.26E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4RR among

    RR200.70.3selected=

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3RR among

    RR200.70.3selected=

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3RR among

    RR200.30.3selected=

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3RR among

    RR200.50.3selected=

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0RR among

    RR100.450.225selected=

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5RR among

    RR200.2250.45selected=

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7RR among

    RR200.90.8selected=

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5RR among

    RR200.4950.33selected=

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5RR among

    RR200.250.1666666667selected=

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7RR among

    RR200.450.45selected=

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    S

    D

    E

    D

    D

    E+D

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0RR among

    RR200.30.3selected=

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0RR among

    RR200.90.3selected=

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10Ex 3 to 5 are variants of the same pattern

    % disease20%E10.90.92.51.7RR among

    RR200.90.3selected=

    Response=59%10

    PopulationRespondersNon respond2.00.912.3

    DDDWeighted average59%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=03.02.5

    1000591.4285714286408.5714285714D on S|E=11.0

    RR=2.0RR=0.9RR=12.3D on S|E=03.01.6666666667

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.05RD=0.26E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4RR among

    RR200.70.3selected=

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3RR among

    RR200.70.3selected=

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3RR among

    RR200.30.3selected=

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3RR among

    RR200.50.3selected=

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0RR among

    RR100.450.225selected=

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5RR among

    RR200.2250.45selected=

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7RR among

    RR200.90.8selected=

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5RR among

    RR200.4950.33selected=

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5RR among

    RR200.250.1666666667selected=

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7RR among

    RR200.450.45selected=

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    S

    D

    E

    D

    D

    E+D

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0RR among

    RR200.30.3selected=

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0RR among

    RR200.90.3selected=

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10Ex 3 to 5 are variants of the same pattern

    % disease20%E10.90.92.51.7RR among

    RR200.90.3selected=

    Response=59%10

    PopulationRespondersNon respond2.00.912.3

    DDDWeighted average59%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=03.02.5

    1000591.4285714286408.5714285714D on S|E=11.0

    RR=2.0RR=0.9RR=12.3D on S|E=03.01.6666666667

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.05RD=0.26E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4RR among

    RR200.70.3selected=

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3RR among

    RR200.70.3selected=

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3RR among

    RR200.30.3selected=

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3RR among

    RR200.50.3selected=

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0RR among

    RR100.450.225selected=

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5RR among

    RR200.2250.45selected=

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7RR among

    RR200.90.8selected=

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5RR among

    RR200.4950.33selected=

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5RR among

    RR200.250.1666666667selected=

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7RR among

    RR200.450.45selected=

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    S

    D

    E

    D

    D

    E+D

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    E

    D

    S

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0

    RR200.30.3RR

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0

    RR200.90.3RR

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10Ex 3 to 5 are variants of the same pattern

    % disease20%E10.90.92.51.7

    RR200.90.3RR

    Response=59%10

    PopulationRespondersNon respond2.00.912.3

    DDDWeighted average59%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=03.02.5

    1000591.4285714286408.5714285714D on S|E=11.0

    RR=2.0RR=0.9RR=12.3D on S|E=03.01.6666666667

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.05RD=0.26E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4

    RR200.70.3RR

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3

    RR200.70.3RR

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3

    RR200.30.3RR

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3

    RR200.50.3RR

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0

    RR100.450.225RR

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5

    RR200.2250.45RR

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7

    RR200.90.8RR

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5

    RR200.4950.33RR

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5

    RR200.250.1666666667RR

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7

    RR200.450.45RR

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    R

    D

    E

    D

    D

    E+D

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

  • Example 2Pattern:Only D influence responseResult:RR (and RD) biased, OR unbiasedODS, Case-Control

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0

    RR200.30.3RR

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0

    RR200.90.3RR

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10Ex 3 to 5 are variants of the same pattern

    % disease20%E10.90.51.21.5

    RR200.50.5RR

    Response=55%10

    PopulationRespondersNon respond2.02.90.5

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103143245.71428571430.4E111143154.28571428570.1E on S|D=11.8

    0865146000.10432573000.10432573000.1E on S|D=01.01.2

    1000545.7142857143454.2857142857D on S|E=11.8

    RR=2.0RR=2.9RR=0.5D on S|E=01.01.5333333333

    OR=2.4OR=4.3OR=0.5D=10.250.75

    RD=0.14RD=0.28RD=-0.07E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4

    RR200.70.3RR

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3

    RR200.70.3RR

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3

    RR200.30.3RR

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3

    RR200.50.3RR

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0

    RR100.450.225RR

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5

    RR200.2250.45RR

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7

    RR200.90.8RR

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5

    RR200.4950.33RR

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5

    RR200.250.1666666667RR

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7

    RR200.450.45RR

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    R

    D

    E

    D

    D

    E+D

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0

    RR200.30.3RR

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0

    RR200.90.3RR

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10Ex 3 to 5 are variants of the same pattern

    % disease20%E10.90.51.21.5

    RR200.50.5RR

    Response=55%10

    PopulationRespondersNon respond2.02.90.5

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103143245.71428571430.4E111143154.28571428570.1E on S|D=11.8

    0865146000.10432573000.10432573000.1E on S|D=01.01.2

    1000545.7142857143454.2857142857D on S|E=11.8

    RR=2.0RR=2.9RR=0.5D on S|E=01.01.5333333333

    OR=2.4OR=4.3OR=0.5D=10.250.75

    RD=0.14RD=0.28RD=-0.07E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4

    RR200.70.3RR

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3

    RR200.70.3RR

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3

    RR200.30.3RR

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3

    RR200.50.3RR

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0

    RR100.450.225RR

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5

    RR200.2250.45RR

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7

    RR200.90.8RR

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5

    RR200.4950.33RR

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5

    RR200.250.1666666667RR

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7

    RR200.450.45RR

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    R

    D

    E

    D

    D

    E+D

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

  • Example 3Pattern:Both E and D influence responseResult:Surprise: responders are unbiasedTheory: bias in at least one stratum

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0

    RR200.30.3RR

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0

    RR200.90.3RR

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E-D

    N1000DPositive bias among responders

    % exposed40%10

    % disease20%E10.90.51.21.5

    RR200.50.5RR

    Response=55%10

    PopulationRespondersNon respond2.02.90.5

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103143245.71428571430.4E111143154.28571428570.1E on S|D=11.8

    0865146000.10432573000.10432573000.1E on S|D=01.01.2

    1000545.7142857143454.2857142857D on S|E=11.8

    RR=2.0RR=2.9RR=0.5D on S|E=01.01.5333333333

    OR=2.4OR=4.3OR=0.5D=10.250.75

    RD=0.14RD=0.28RD=-0.07E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4

    RR200.70.3RR

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3

    RR200.70.3RR

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3

    RR200.30.3RR

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3

    RR200.50.3RR

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0

    RR100.450.225RR

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5

    RR200.2250.45RR

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7

    RR200.90.8RR

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5

    RR200.4950.33RR

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5

    RR200.250.1666666667RR

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7

    RR200.450.45RR

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    R

    D

    E

    D

    D

    E+D

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0

    RR200.30.3RR

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0

    RR200.90.3RR

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E-D

    N1000DPositive bias among responders

    % exposed40%10

    % disease20%E10.90.51.21.5

    RR200.50.5RR

    Response=55%10

    PopulationRespondersNon respond2.02.90.5

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103143245.71428571430.4E111143154.28571428570.1E on S|D=11.8

    0865146000.10432573000.10432573000.1E on S|D=01.01.2

    1000545.7142857143454.2857142857D on S|E=11.8

    RR=2.0RR=2.9RR=0.5D on S|E=01.01.5333333333

    OR=2.4OR=4.3OR=0.5D=10.250.75

    RD=0.14RD=0.28RD=-0.07E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4

    RR200.70.3RR

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3

    RR200.70.3RR

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3

    RR200.30.3RR

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3

    RR200.50.3RR

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0

    RR100.450.225RR

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5

    RR200.2250.45RR

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7

    RR200.90.8RR

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142857305.7142857143D on S|E=10.5

    RR=2.0RR=1.1RR=5.4D on S|E=01.1250.7083333333

    OR=2.4OR=1.1OR=8.4D=10.250.75

    RD=0.14RD=0.01RD=0.33E=10.66666666670.33

    Selection:

    Example11ResponseThis and next: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.990.661.52.01.5

    RR200.4950.33RR

    Response=51%210

    PopulationRespondersNon respond2.01.90.1

    DDDWeighted average51%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1113189301.71428571430.4E119798.28571428570.0E on S|D=12.0

    0865146000.1042170212.14285714290.2043345387.85714285710.1E on S|D=02.02

    1000513.8571428571486.1428571429D on S|E=11.5

    RR=2.0RR=1.9RR=0.1D on S|E=01.51.5

    OR=2.4OR=2.4OR=0.1D=10.250.75

    RD=0.14RD=0.18RD=-0.10E=10.66666666670.33

    Selection:

    Example12ResponseThis and previous: same DAG, different results

    N1000D

    % exposed40%10

    % disease20%E10.50.33333333331.52.01.5

    RR200.250.1666666667RR

    Response=26%210

    PopulationRespondersNon respond2.01.91.8

    DDDWeighted average26%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E15795152.3809523810.4E157190247.6190476190.2E on S|D=12.0

    0865146000.102186107.14285714290.2064429492.85714285710.1E on S|D=02.02

    1000259.5238095238740.4761904762D on S|E=11.5

    RR=2.0RR=1.9RR=1.8D on S|E=01.51.5

    OR=2.4OR=2.4OR=2.0D=10.250.75

    RD=0.14RD=0.18RD=0.10E=10.66666666670.33

    Selection:

    Example13Response

    N1000D

    % exposed40%10

    % disease20%E10.90.451.31.7

    RR200.450.45RR

    Response=50%10

    PopulationRespondersNon respond2.03.10.5

    DDDWeighted average50%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103129231.42857142860.4E111157168.57142857140.1E on S|D=12.0

    0865146000.10392312700.10472833300.1E on S|D=01.01.25

    1000501.4285714286498.5714285714D on S|E=12

    RR=2.0RR=3.1RR=0.5D on S|E=011.6666666667

    OR=2.4OR=4.8OR=0.4D=10.250.75

    RD=0.14RD=0.30RD=-0.08E=10.66666666670.33

    ED2

    E

    D

    R

    D

    E

    D

    D

    E+D

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

    E

    D

    R

  • Example 4Pattern:Both E and D influence responseResult:Surprise: both strata biased upwardsTrue RR is not a weighted average

    Sel Bias

    Selection bias examples

    Fill inn gray cells

    Selection:

    Example1ResponseOnly E

    N1000DNo bias

    % exposed40%10

    % disease20%E10.90.93.01.0

    RR200.30.3RR

    Response=54%10

    PopulationRespondersNon respond2.02.02.0

    DDDWeighted average54%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=03.03

    1000540460D on S|E=11.0

    RR=2.0RR=2.0RR=2.0D on S|E=01.01

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.14RD=0.14E=10.66666666670.33

    Selection:

    Example2ResponseOnly D

    N1000DOR not biased

    % exposed40%10

    % disease20%E10.90.31.03.0

    RR200.90.3RR

    Response=42%10

    PopulationRespondersNon respond2.01.62.3

    DDDWeighted average42%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=11.0

    0865146000.1077154231.42857142860.309360368.57142857140.0E on S|D=01.01

    1000420580D on S|E=13.0

    RR=2.0RR=1.6RR=2.3D on S|E=03.03

    OR=2.4OR=2.4OR=2.4D=10.250.75

    RD=0.14RD=0.21RD=0.03E=10.66666666670.33

    Selection:

    Example3ResponseBoth E and D through E-D

    N1000DPositive bias among responders

    % exposed40%10

    % disease20%E10.90.51.21.5

    RR200.50.5RR

    Response=55%10

    PopulationRespondersNon respond2.02.90.5

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103143245.71428571430.4E111143154.28571428570.1E on S|D=11.8

    0865146000.10432573000.10432573000.1E on S|D=01.01.2

    1000545.7142857143454.2857142857D on S|E=11.8

    RR=2.0RR=2.9RR=0.5D on S|E=01.01.5333333333

    OR=2.4OR=4.3OR=0.5D=10.250.75

    RD=0.14RD=0.28RD=-0.07E=10.66666666670.33

    Selection:

    Example4ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.90.92.61.4

    RR200.70.3RR

    Response=57%10

    PopulationRespondersNon respond2.01.04.3

    DDDWeighted average57%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E11032573600.3E11129400.3E on S|D=11.3

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5714285714

    1000574.2857142857425.7142857143D on S|E=11.0

    RR=2.0RR=1.0RR=4.3D on S|E=02.31.4444444444

    OR=2.4OR=1.0OR=5.6D=10.250.75

    RD=0.14RD=0.01RD=0.22E=10.66666666670.33

    Selection:

    Example5ResponseBoth E and D through E or D

    N1000DNegative association among responders

    % exposed40%10

    % disease20%E10.70.92.51.3

    RR200.70.3RR

    Response=55%10

    PopulationRespondersNon respond2.00.88.2

    DDDWeighted average55%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E180257337.14285714290.2E1342962.85714285710.5E on S|D=11.0

    0865146000.1060154214.28571428570.3026360385.71428571430.1E on S|D=03.02.5

    1000551.4285714286448.5714285714D on S|E=10.8

    RR=2.0RR=0.8RR=8.2D on S|E=02.31.2962962963

    OR=2.4OR=0.8OR=16.8D=10.250.75

    RD=0.14RD=-0.04RD=0.48E=10.66666666670.33

    Selection:

    Example6ResponseBoth E and D through E and D

    N1000DPositive association among responders

    % exposed40%10

    % disease20%E10.90.31.52.3

    RR200.30.3RR

    Response=37%10

    PopulationRespondersNon respond2.03.80.4

    DDDWeighted average37%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E110386188.57142857140.5E111200211.42857142860.1E on S|D=13.0

    0865146000.10261541800.10603604200.1E on S|D=01.01.5

    1000368.5714285714631.4285714286D on S|E=13.0

    RR=2.0RR=3.8RR=0.4D on S|E=01.02.3333333333

    OR=2.4OR=7.2OR=0.3D=10.250.75

    RD=0.14RD=0.40RD=-0.09E=10.66666666670.33

    Selection:

    Example7ResponseBoth E and D through E and D

    N1000DBoth strata neg bias

    % exposed40%10

    % disease20%E10.90.82.51.3

    RR200.50.3RR

    Response=53%10

    PopulationRespondersNon respond2.01.41.6

    DDDWeighted average53%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E1103229331.42857142860.3E1115768.57142857140.2E on S|D=11.8

    0865146000.1043154197.14285714290.2043360402.85714285710.1E on S|D=02.72.45

    1000528.5714285714471.4285714286D on S|E=11.1

    RR=2.0RR=1.4RR=1.6D on S|E=01.71.3055555556

    OR=2.4OR=1.6OR=1.7D=10.250.75

    RD=0.14RD=0.09RD=0.06E=10.66666666670.33

    Selection:

    Example8Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.90.452.02.0

    RR100.450.225RR

    Response=38%10

    PopulationRespondersNon respond1.01.00.3

    DDDWeighted average38%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E1803204000.2E1721442160.3E181761840.0E on S|D=12.0

    01204806000.20541081620.30663724380.2E on S|D=02.02

    1000378622D on S|E=12.0

    RR=1.0RR=1.0RR=0.3D on S|E=02.02

    OR=1.0OR=1.0OR=0.3D=10.250.75

    RD=0.00RD=0.00RD=-0.11E=10.66666666670.33

    Selection:

    Example9Response

    N1000DOnly non-responders biased (needs no E-D effect)

    % exposed40%10

    % disease20%E10.450.92.00.5

    RR200.2250.45RR

    Response=56%10

    PopulationRespondersNon respond2.02.23.6

    DDDWeighted average56%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E151257308.57142857140.2E1632991.42857142860.7E on S|D=12.0

    0865146000.1019231250.71428571430.1066283349.28571428570.2E on S|D=02.02

    1000559.2857142857440.7142857143D on S|E=10.5

    RR=2.0RR=2.2RR=3.6D on S|E=00.50.5

    OR=2.4OR=2.4OR=9.4D=10.250.75

    RD=0.14RD=0.09RD=0.50E=10.66666666670.33

    Selection:

    Example10Response

    N1000DStrong opposite effects

    % exposed40%10

    % disease20%E10.30.60.60.7

    RR200.90.8RR

    Response=69%10

    PopulationRespondersNon respond2.01.15.4

    DDDWeighted average69%

    10sumrisk10sumrisk10sumriskRRORRDRRORRD

    E11142864000.3E134171205.71428571430.2E180114194.28571428570.4E on S|D=10.3

    0865146000.1077411488.57142857140.209103111.42857142860.1E on S|D=00.80.6458333333

    1000694.2857142


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