Selection bias

Classic definition

• The magnitude of the association is different for participants and non-participants

  Structural definition

• Occurs when conditioning on the common child or its descendants of two variables

  Encompasses various biases

8.1 The structure of selection bias

DAG:

• A: exposure (folic acid supplements)
• Y: outcome, binary (cardiac malformation)
• C: common effect (death before birth) observed only among live-born children (C=0)

Under the null

• The biasing path A ➡️ [C] ⬅️ Y

• The associational risk ratio does not equal causal risk ratio

  • $\frac{Pr[Y=1|A=1,C=0]}{Pr[Y=1|A=0,C=0]}$ is not $\frac{Pr\left[Y^{a=1}\right]}{Pr\left[Y^{a=0}\right]}$

Selection bias

DAG:

• A: exposure (folic acid supplements)
• Y: outcome, binary (cardiac malformation)
• C: common effect (death before birth) observed only among live-born children (C=0)
• S: parental grief (restricted to S=0)

Under null we would still see an association between A and Y due to the collider stratification bias

• The biasing path A ➡️ [C] ⬅️ Y

Selection bias

• DAG 8.3:
  • A: anti-retroviral treatment
  • Y: 3-year risk of death
  • U: unmeasured, immunosuppression (U=1 are under greater risk of death)
  • C: censored (C=1)
  • L: unmeasured, immunosuppression symptoms

Biasing path:

• A ➡️ [C] ⬅️ L ⬅️ U ➡️ Y

Selection bias

• DAG 8.4:
  • A: anti-retroviral treatment
  • Y: 3-year risk of death
  • U: unmeasured, immunosuppression (U=1 are under greater risk of death)
  • C: censored (C=1)
  • L: unmeasured, immunosuppression symptoms

Biasing path:

• A ➡️ [L] ⬅️ U ➡️ Y

Selection bias

• DAG 8.5 (M-bias):
  • A: anti-retroviral treatment
  • Y: 3-year risk of death
  • U: unmeasured, immunosuppression (U=1 are under greater risk of death)
  • C: censored (C=1)
  • L: unmeasured, immunosuppression symptoms
  • W: unmeasured, lifestyle

Selection bias

• DAG 8.6 (M-bias):
  • A: anti-retroviral treatment
  • Y: 3-year risk of death
  • U: unmeasured, immunosuppression (U=1 are under greater risk of death)
  • C: censored (C=1)
  • L: unmeasured, immunosuppression symptoms
  • W: unmeasured, lifestyle

8.2 Examples of selection bias

• Differential loss to follow-up/informative censoring (8.3-8.6)
• Missing data bias, non-response bias
  • Missing data on the outcome for any reason (8.3-8.6)
• Healthy worker bias
  • U: unmeasured, underlying health
  • C: at work or not (C=0 being at work)
• Self-selection bias, volunteer bias
• Selection affected by treatment received before study entry

Selection bias can occur in any follow-up study (observational or RCT)

• Those who stayed in the study (uncensored, C=0) are not exchangeable with those who did not stay in the study (censored, C=1)
  • Can only compute observational $\frac{Pr[Y=1|A=1,C=0]}{Pr[Y=1|A=0,C=0]}$, but not counterfactual $\frac{Pr\left[Y^{A=1,C=c}\right]}{Pr\left[Y^{A=0,C=c}\right]}$ under both levels of C={0, 1}

8.3 Selection bias and confounding

Sources of lack of exchangeability:

• Confounding
• Selection bias

DAG 8.7:

• A: physical activity
• Y: heart disease
• L: family SES
• C: becoming a firefighter (restricted to C=1)
• U: unmeasured, personal preference for physical activity professions

There is no confounding between A and Y $\frac{Pr[Y=1|A=1,C=0]}{Pr[Y=1|A=0,C=0]}$ is $\frac{Pr\left[Y^{a=1}\right]}{Pr\left[Y^{a=0}\right]}$

8.4 Selection bias and censoring

• Censoring as "treatment"
• Identifiability conditions of exchangeability, positivity, and consistency hold for both A and C
• Analytical methods

8.5 How to adjust for selection bias

• IP weighting (or standardization)
  • Assigning a weight to each selected individual (C = 0) that accounts for the individuals with same A and L, but C=1
  • Weights constructed from the probability of selection model: $Pr[C=0|L,A]$
• Effect measure in the population had no one been censored

8.6 Selection without bias

• A: surgery
• Y: death
• E: haplotype

• "Collider stratification is not always a source of selection bias"

References

Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC (v. 31jan21)