Selection Bias Concepts Hein Stigum Presentation, data and programs at: http://folk.uio.no/heins/talks April 17 H.S.

Questions Given measured appropriate variables: Can you adjust for confounding? Yes Can you adjust for selection bias? Depends on the definition April 17 H.S.

Contents Background Selection bias Examples Size and direction of bias Define bias Selection bias as effect modification (old concept) as collider stratification bias (new concept) DAG structure Examples Size and direction of bias April 17 H.S.

Bias definition Bias Frequency: expected risk ≠ true risk Effect: association ≠ causal effect April 17 H.S.

Selection bias concepts DAG structure Effect responders ≠ Effect non responders Differential response bias Differential loss to follow up Healthy worker bias Berkson’s bias (case control) Effect modification Collider stratification bias April 17 H.S.

Selection bias as effect modification April 17 H.S.

Selection bias: Risk Selection of responders  The prevalence is different among the responders compared to the full population the responders compared to the non responders R0 Non responders Rp Population Rp is the weighted mean of R0 and R1 R1 Responders Rp is the weighted mean of R0 and R1 April 17 H.S.

Effect modification Selection of responders  The effect of E on D is different among the responders compared to the full population the responders compared to the non responders RR0 Non responders RRp Population Smoking intervention No effect among non responders Strong effect among the responders Drug effect Probably same effect among non responders and the responders, same biology RR1 Responders April 17 H.S.

collider stratification bias Problems Is not a bias, RR0 and RR1 are the true effects Is effect modification by selection variable S Leads to the conclusion that: Biolocical effects are protected from bias The bias can not be adjusted for RRp is the average of RR0 and RR1 Not true for collider stratification bias “DAG” structure: S E D April 17 H.S.

Selection bias as collider stratification bias April 17 H.S.

Example with paths Study Milk on bone density Exclude Calcium supplements E milk D bone density S calcium supp.   Path Type Status 1 E®D Causal Open 2 E®[S]D Noncausal E milk D bone density S calcium supp. C family history 2 E®[S][C]®D Noncausal Closed Structure: Collider stratification Lessons learned: Biological effect not protected May adjust for selection bias April 17 H.S.

Examples Differential response Differential loss to follow up alcohol D CHD S respond C education Differential response Survey: Alcohol and CHD Differential loss to follow up Randomized trial: drug and disease Healthy worker effect Cross-section: Melt hall dust and lung disease E drug D disease S loss to follow up C smoking E dust D lung disease S working C health Survey of alcohol on CHD: heavy drinkers do not respond, sex, age and education are common factors for response and disease. Randomized drug trial: side effects of drug causes loss, smoking common to loss and disease Cross-sectional study: dust sensitive change work, health common to work status and (less) lung disease No confounding: for simplicity (1 and 3) or randomization (2) Note: no confounding April 17 H.S.

Selection bias structure April 17 H.S. 13

Paths 1. Causal 2. Confounding 3. Selection bias An open non-causal path without colliders 3. Selection bias A non-causal path that is open due to conditioning on a collider BCVs? C E D B A Confounding C E D B A Selection bias E D B A Causal

Collider stratification bias Selection bias = Collider stratification bias Selection bias, Path definition A non causal path that is open due to conditioning on a collider S E D B A E D S E D S C April 17 H.S.

Selection bias examples April 17 H.S.

Folic acid and cardiac malformation E Folic acid D Card. Mal. C Live born Selection: Study only live born Bias? Yes, E[C]D is open S Grief E Folic acid D Card. Mal. C Live born Selection: Non grieving parents volonteer Bias? Yes, E[C]D is (partially) open Folic acid increases live born by protecting against other malfomations Cardiac malformations decrease live born Not live born increase grief April 17 H.S.

Education and unfaithfulness Study the effect among couples in a relationship (not divorced)? E education D unfaithful S sensation seeking R divorced   Path Type Population Sample 1 E®D Causal Open 2 E®RD Noncausal Closed 3 E®RS®D Selection bias Apr-17 H.S.

Size and Direction of bias April 17 H.S. 19

Example 1, full table (Adjusted) RRs Proportion responding in 1,1 group True and biased RRs Explain: from pop to responders from response to RR true and biased RR RRs in DAG (weighted averages when not equal in both groups)

Example 2 Pattern: Result: Only D influence response CC study: Sample cases with a much higher prob than controls Pattern: Only D influence response Result: RR (and RD) biased, OR unbiased ODS, Case-Control

Example 3 Pattern: Result: Both E and D influence response Works only for true RR=1 ? Pattern: Both E and D influence response Result: Surprise: responders are unbiased Theory: bias in at least one stratum

Example 4 Pattern: Result: Both E and D influence response Can also have both biased downwards Pattern: Both E and D influence response Result: Surprise: both strata biased upwards True RR is not a weighted average

Example 5 Pattern: Result: Both E and D influence response Also an example of both biased downwards For deeper understanding and frustration see: VanderWeele and Robins: Directed Acyclic Graphs, Sufficient Causes, and the Properties of Conditioning on a Common Effect, AJE 2007 Pattern: Both E and D influence response Result: Same DAG, different results The DAG does not fully determine the selection!

Summing up Selection bias as “effect modification”: Is not a bias, should not be called selection bias Has properties different from proper selection bias Selection bias as “collider stratification”: Structure defined in DAG, Distinct from confounding Consistent with Differential response bias Differential loss to follow up Healthy worker bias Berkson’s bias (case control) April 17 H.S.

Litterature Hernan and Robins, Causal Inference April 17 H.S.