Best, N

Geneletti, S

Richardson, S

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Adjusting for selection bias in case control studiesS.Geneletti, S.Richardson, N.BestDepartment of Epidemiology and Public Health, Imperial College01/07/2008OUTLINE1. DAGs and conditional independence2. Examples of selection bias in case-controls studies3. DAG expression4. Odds ratios5. Bias breaking model6. Application7. Simulation8. Further workDAGsDAGs are directed acyclic graphsI All arrows have directionI No cycles A→ B → AI Arrows are not causal unless extra assumptions made -time ordering, interventionSESSMKBMIMICONDITIONAL INDEPENDENCEDAGs are used to encode conditional independencestatementsI A⊥⊥C|B [1] means p(A,C|B) = p(A|B)p(C|B)I In words if we know about C, knowing about A gives us noextra clues about B (and vice-versa)A B CA BACB CI Causal interpretation from observational data is difficultI Need to make additional explicit assumptionsI Not all DAGs have others that are Markov EquivalentSIMPLE EXAMPLE - INHERITANCEM F1. Male and female are independent2. Then they meet and have a child.SIMPLE EXAMPLE - INHERITANCEM FC1. Male and female are independent2. Then they meet and have a child.SIMPLE EXAMPLE - INHERITANCEM FC1. Male and female are independent2. Then they meet and have a child3. Now they are dependent through childSIMPLE EXAMPLE - INHERITANCEM FCI In terms of conditional independence we have thatI Initially M⊥⊥FI Later M 6⊥⊥F |SEXAMPLES of SELECTION BIASCase selection biasI Study in 70’s found oestrogen use associated withendometrial cancer [2]I Selecting cases mainly amongst women with vaginalbleeding (associated to oestrogen use)I induces a false association between endometrial cancerand oestrogen use.Control selection biasI Recent studies find a weak association between exposureto magnetic fields (EMF) and childhood leukaemia [3]I Eligible controls with lower SES are less likely to allow EMFmeasurements in their homes,I this induces a false association between leukaemia andEMF when only “full” controls included.SELECTION BIAS DAGBasic premiseSelection bias comes about by conditioning on a common childwhere we don’t know distribution of child given parentsW Y W YS SI Y is the outcome of interest, W the exposure, S theselection indicator.I Left: conditioning induces relationshipI Right: conditioning distorts relationshipI Both share v-structureProblem - we don’t know p(S|Y )CONDITIONAL INDEPENDENCE in SBDAGs in previous slide represent the following conditional(in)dependences :I Left: Y⊥⊥WI Right: None (and ME to Y →W )However, both share the same v-structureW YSwhich “charcterises” the selection bias problem.ODDS RATIOTrue Odds ratioψ =p(Y = 1|W = 1)p(Y = 0|W = 0)p(Y = 0|W = 1)p(Y = 1|W = 0)=p(W = 1|Y = 1)p(W = 0|Y = 0)p(W = 0|Y = 1)p(W = 1|Y = 0)(1)Observed Odds ratioψo =p(Y = 1,W = 1|S = 1)p(Y = 0,W = 0|S = 1)p(Y = 0,W = 1|S = 1)p(Y = 1,W = 0|S = 1)(2)BIAS BREAKING MODEL1. The problem can be addressed if we can find a biasbreaking variable B s.t. we can somehow separateexposure W from selection S for example we canassume A1 thatW⊥⊥S|(Y , B) (3)2. It also necessary to find additional data3. s.t. we can obtain an unbiased estimate of thedistribution of p(B|Y ) - see why below.IDEA OF “SEPARATION”The conditional independence A1 W⊥⊥S|(Y ,B) allows us toW Y1. separate the exposure disease mechanism of inferentialinterest2. from the niusance selection bias mechanism3. by using B to separate these mechanismsIDEA OF “SEPARATION”The conditional independence A1 W⊥⊥S|(Y ,B) allows us toW YS1. separate the exposure disease mechanism of inferentialinterest2. from the niusance selection bias mechanism3. by using B to separate these mechanismsIDEA OF “SEPARATION”The conditional independence A1 W⊥⊥S|(Y ,B) allows us toW YB S1. separate the exposure disease mechanism of inferentialinterest2. from the niusance selection bias mechanism3. by using B to separate these mechanismsEXAMPLE DAGsW YSBW YSBBB MODELEssential assumptions:A1 Have B such that W⊥⊥S|(Y ,B) holdsA2 Case and control selection are independentThis is usually plausible as case and control recruitmentprocesses are essentially differentSome assumptions for simplicity:S1 There is no selection bias in the cases i.e.p(W = 1|Y = 1,S = 1) = p(W = 1|Y = 1).S2 Stratify B if it is not discreteBB MODELNow we can estimate p(W = 1|Y = 0) asp(W |Y = 0,S = 1,B) = p(W |Y = 0,B) by A1 and∑Bp(W |Y = 0,B)p(B|Y = 0) = p(W |Y = 0)I Focus is on finding estimates of p(B|Y ) as p(W |Y ,B) isestimated by stratum specific proportion of exposedcases/controlsI similar argument can be applied to case selection biasESTIMATES OF p(B|Y )There are various options depending on the source ofadditional dataData sources1. Partial study data OR2. External (eg census) data.... and also on the type of estimate:Type of estimate1. Conditional estimate - based on p(B|Y ) OR2. Marginal estimate - based on p(B) -3. Marginal estimate valid to adjust for control selection biaswhen p(B|Y = 0) ≈ p(B).HYPOSPADIAS CASE CONTROL STUDYStoryI Hypospadias is a congenital malformation of newborn boysI Is it associated to gestational age or smoking? [4, 5]I Concern that controls have a higher SES than cases-selection bias?I SES measured using the Carstairs score - an area (ward)level index of deprivation ([6])HYPOSPADIAS CASE CONTROL STUDYDataDue to data collection process we hadI Carstairs score of people who participated - fullparticipants (indexed by f)I Carstairs score of many people who were asked toparticipate but declined as their ward was known - partialparticipants (indexed by p)I Finally, Carstairs score of people who lived in the regionthe study was conducted from censusBoxplotIs there also case selection bias? partial participant cases (pcs)have low SES (high Carstairs)HYPOSPADIAS CASE CONTROL STUDYTo adjust for selection bias we need additional data to get an“unbiased” estimate of p(B|Y )I Pooling partial and full participant data and assuming thisis a representative sample of the target population gives usan internal adjustment can estimate:I p(B|Y ) conditionalI as well as p(B) marginal - this is assuming that p(B|Y = 0)can be approximated by p(B)I Using data from the census means that we can do anexternal adjustment based on just p(B) marginal ext, againassuming p(B|Y = 0) ≈ p(B)RESULTSHYPOSPADIAS CASE CONTROL STUDYConclusionsI There appears to be no selection bias mediated by SESI Naive and adjusted are all very similarI Do not read too much into small differencesI Validates the study resultsSIMULATIONSSet-upI True OR = 1, 2, 2.41 (only show 2 and 2.41)I When OR=2.41, B is also a confounderI B has 3 levels - imagine this is SESI Introduce bias by changing the probability of beingselected into study if in 3rd level (p(S = 1|B = 3))I for different probabilities of being in 3rd level. (p(B = 3))I Have two simulation studies, one emulates theHypospadias case-control study with full and partialparticipantsI The second emulates the Hypospadias case-control studywith full participants and census informationSIMULATIONSMonitor1. No bias estimate (Logistic regression coefficient with B ascovariate in data that is not biased)2. Naive estimate3. Logistic regression coefficient with B as covariate4. Marginal estimator based on all data on B5. Marginal estimator based only on external data on Bwe compare our estimators to logistic regression coefficients asthese are standard approaches in EpidemiologyRESULTSFINAL COMMENTSConclusions1. Our methods adjust well for selection bias2. Marginal estimators in particular as they use more datathan others3. The estimators do not introduce bias when it is not present4. Can be used for sensitivity analysis and validation5. Similar to post-stratification [7]6. Comes out in next issue of BiostatisticsFurther work1. Have developed Baysian version2. Are applying it to EMF data from the US [8]BIBLIOGRAPY[1] A. P. Dawid. Conditional Independence in Statistical Theory. Journal of the Royal Statistical Society, Series B(Statisical Methodology), 41(1):1–31, 1979.[2] RI. Horwitz and AR. Feinstein. Alternative analytic methods for case-control studies of estrogens andendometrial cancer. New England Journal of Medicine, 299(20):1089–1094, 1978.[3] G. Mezei and L. Kheifets. Selection bias and its implications for case-control studies: a case study of magneticfield exposure and childhood leukaemia. International Journal of Epidemiology, 35:397–406, 2006.[4] G. Ormond, M.J. Nieuwenhuijsen, P. Nelson, N. Izatt, S. Geneletti, M. Toledano, and P. Elliott. Folatesupplementation, endocrine disruptors and hypospadias: case-control study. under review in BMJ, 2008.[5] M. Nieuwenhuijsen, P. Nelson, and P. Elliott. Occupational exposure of pregnant women in the south east ofEngland. Epidemiology, 15(4):S165, 2004.[6] V. Carstairs and R. Morris. Deprivation and Health in Scotland. Aberdeen University Press, Aberdeen, 1991.[7] A. Gelman. Struggles with survey weighting and regression modelling. Statistical Science, 22:153–164, 2007.[8] E.E. Hatch, R.A. Kleinerman, M.S. Linet, R.E. Tarone, W.T. Kaune, A. Anssi, B. Dasul, L.L. Robison, andS. Wacholder. Do confounding or selection factors of residential wire codings and magnetic fields distortfindings of electromagnetic field studies? Epidemiology, (11):189–198, 2000.

Adjusting for selection bias in case control studies

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Adjusting for selection bias in case control studies

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