Confounding, mediation and colliding

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Confounding, mediation and colliding What types of shared covariates does the sibling comparison design control for? Arvid Sjölander and Johan Zetterqvist

Causal effects and confounding A common aim of epidemiologic research is to estimate the causal effect of an exposure on an outcome In observational studies, confounding is always a concern

Sibling comparison designs A popular way to reduce confounding is to use a sibling comparison design the exposure-outcome association is studied within families instead of between unrelated individuals It is routinly argued that the within-sibling association is controlled for all measured and unmeasured covariates that are shared (constant) within families e.g. socioeconomic status and parental genetic makeup But the sibling comparison design has subtle issues, that are not present in studies of unrelated subjects

Amplification of bias due to unmeasured non-shared confounders Frisell, T., Öberg, S., Kuja-Halkola, R., Sjölander, A. (2012). Sibling comparison designs: bias from non-shared confounders and measurement error. Epidemiology, 23(5), 713-720. U i X i1 X i2 C i1 C i2 Y i1 Y i2

Bias due to carryover effects Sjölander, A., Frisell, T., Kuja-Halkola, R., Öberg, S., Zetterqvist, J. (2016). Carryover effects in sibling comparison designs. Epidemiology, 27(6), 852-858. U i X i1 X i2 C i1 C i2 Y i1 Y i2

Outline Motivating example Main results Correctly specified models Misspecified models Concluding remarks

Outline Motivating example Main results Correctly specified models Misspecified models Concluding remarks

Association between mother s age and ADHD in offspring Attention deficit/hyperactivity disorder (ADHD) is the most common neurodevelopmental disorder in childhood prevalence 5% Studies have shown that young mothers are more likely to get children with ADHD than old mothers But not clear if the association is causal possibly confounded by familial factors, e.g. socioeconomic status and genetics

A sibling comparison study To examine the role of familial confounding, Chang et al. (2014) carried out a sibling comparison study A cohort of 1495543 children born to 896389 mothers, between 1988 and 2003 1.7 children per mother, on average Each child was followed from birth to ADHD diagnosis or 2009.12.31, whichever came first

Analyses Standard analysis: an ordinary Cox proportional hazards model Sibling comparison: a stratified Cox model, with one stratum per family automatically controls for covariates that are shared (constant) in the family model details later

Results Statistically significant association between mother s age and offspring ADHD in the standard analysis No (or inverse) association in the sibling comparison Authors concluded: The association between early maternal age and offspring ADHD is mainly explained by genetic confounding

How about shared mediators? Familial environment (shared) is affected by the mother s age at all her childbirths e.g. having multiple children at an early age may lead to a financially difficult family situation. If the familial environment in turn affects the risk of ADHD, then this covariate is a shared mediator Controlling for this covariate may therefore remove part of the true causal effect and attenuate the association

How about shared colliders? Familial environment is a dynamic condition which may change according to the development of the children e.g. an ADHD diagnosis in one of the siblings may lead to a stressful situation for the whole family If so, then familial environment could also play the role of a shared collider Controlling for this covariate may induce collider-stratification bias, which may also attenuate the association

Confounding, mediation and colliding U i X i1 X i2 M i Y i1 Y i2 W i What types of shared covariates does the sibling comparison design really control for?

Outline Motivating example Main results Correctly specified models Misspecified models Concluding remarks

Outline Motivating example Main results Correctly specified models Misspecified models Concluding remarks

Statistical models that condition on the family Linear model for continuous outcome: E(Y ij family i, X ij ) = α i + βx ij Logistic model for binary outcome: logit{p(y ij = 1 family i, X ij )} = α i + βx ij Cox proportional hazards model for time-to-event outcome: log{λ(y ij family i, X ij )} = α i (y ij ) + βx ij We will focus on the linear model, but all arguments and results hold for the logistic model and the Cox proportional hazards model as well

The role of the intercept E(Y ij family i, X ij ) = α i + βx ij The family-specific intercept α i is intended to absorb those covariates that are shared in the family The exposure effect, β, is controlled for the covariates absorbed into α i Does α i only absorb shared confounders, or shared mediators and colliders as well?

In other words When fitting the conditional (on family) model E(Y ij family i, X ij ) = α i + βx ij, which of the following β s is really estimated? E(Y ij U i, X ij ) = αi 1 + β 1 X ij E(Y ij M i, X ij ) = αi 2 + β 2 X ij E(Y ij W i, X ij ) = αi 3 + β 3 X ij E(Y ij U i, M i, X ij ) = αi 4 + β 4 X ij E(Y ij U i, W i, X ij ) = αi 5 + β 5 X ij E(Y ij M i, W i, X ij ) = αi 6 + β 6 X ij E(Y ij U i, M i, W i, X ij ) = αi 7 + β 7 X ij

Technical note If (U i, X i1, X i2, M i, Y i1, Y i2, W i ) have a multivariate normal distribution, then all the models below are correct E(Y ij U i, X ij ) = αi 1 + β 1 X ij E(Y ij M i, X ij ) = αi 2 + β 2 X ij E(Y ij W i, X ij ) = αi 3 + β 3 X ij E(Y ij U i, M i, X ij ) = αi 4 + β 4 X ij E(Y ij U i, W i, X ij ) = αi 5 + β 5 X ij E(Y ij M i, W i, X ij ) = αi 6 + β 6 X ij E(Y ij U i, M i, W i, X ij ) = αi 7 + β 7 X ij So not a problem of model misspecification per se

Conditional maximum likelihood The answer lies in the way the model is fitted, rather than in the model itself The standard way to fit conditional (on family) models is to use conditional maximum likelihood (CML) leads to conditional logistic regression (for the logistic model) and stratified Cox regression (for the Cox proportional hazards model) Eliminates the family-specific intercept α i by conditioning on a sufficient statistic

Underlying assumptions in CML Let Z i be the set of shared covariates that we control for by conditioning on family i We may thus reformulate the model as E(Y ij family i, X ij ) = α i + βx ij E(Y ij Z i, X ij ) = α i + βx ij The CML estimator of β in this model is consistent if A: Y i1 Y i2 X i1, X i2, Z i B: Y ij X ij X ij, Z i

Comparison with conditional logistic regression Suppose that X ij and Y ij are binary, and that we use the model logit{p(y ij = 1 Z i, X ij )} = α i + βx ij Let D be the exposure-discordant pairs Arbitrary order siblings so that D X i1 = 1 and X i2 = 0 D Y i2 = 0 Y i2 = 1 Y i1 = 0 n 00 n 01 Y i1 = 1 n 10 n 11 exp( ˆβ) = n 10 n 01 p(y i1 = 1, Y i2 = 0 D) p(y i1 = 0, Y i2 = 1 D), Not generally equal to exp(β)

Comparison with conditional logistic regression, cont d However, under we have: A: Y i1 Y i2 X i1, X i2, Z i B: Y ij X ij X ij, Z i p(y i1 = 1, Y i2 = 0 D) p(y i1 = 0, Y i2 = 1 D) = E{p(Y i1 = 1, Y i2 = 0 D, Z i ) D} E{p(Y i1 = 0, Y i2 = 1 D, Z i ) D} A = E{p(Y i1 = 1 D, Z i )p(y i2 = 0 D, Z i ) D} E{p(Y i1 = 0 D, Z i )p(y i2 = 1 D, Z i ) D} B = E{p(Y i1 = 1 X i1 = 1, Z i )p(y i2 = 0 X i2 = 0, Z i ) D} E{p(Y i1 = 0 X i1 = 1, Z i )p(y i2 = 1 X i2 = 0, Z i ) D} { p(yi1 =1 X i1 =1,Z i )p(y i2 =0 X i2 =0,Z i ) p(y p(y i1 =0 X i1 =1,Z i )p(y i2 =1 X i2 =0,Z i ) i1 = 0 X i1 = 1, Z i )p(y i2 = 1 X i2 = 0, Z i ) D E = E{p(Y i1 = 0 X i1 = 1, Z i )p(y i2 = 1 X i2 = 0, Z i ) D} = E {exp(β)p(y i1 = 0 X i1 = 1, Z i )p(y i2 = 1 X i2 = 0, Z i ) D} E{p(Y i1 = 0 X i1 = 1, Z i )p(y i2 = 1 X i2 = 0, Z i ) D} = exp(β) E {p(y i1 = 0 X i1 = 1, Z i )p(y i2 = 1 X i2 = 0, Z i ) D} E{p(Y i1 = 0 X i1 = 1, Z i )p(y i2 = 1 X i2 = 0, Z i ) D} = exp(β) }

Matching by intervention vs nature A: Y i1 Y i2 X i1, X i2, Z i B: Y ij X ij X ij, Z i That the CML estimator requires assumptions A and B is rarely mentioned in standard textbooks Probably since A and B hold by design in studies that have been matched unrelated subjects on Z i by intervention Not guaranteed to hold in studies that have been matched nature, e.g. sibling comparison studies whether A and B hold or not depends on what we consider Z i to be

Implication A: Y i1 Y i2 X i1, X i2, Z i B: Y ij X ij X ij, Z i Since A and B are sufficient for the CML estimator to be consistent we can turn the argument around and use A and B to identify Z i : If there is a Z i such that A and B hold, then the CML estimator of β in model E(Y ij family i, X ij ) = α i + βx ij consistently estimates β in model E(Y ij Z i, X ij ) = α i + βx ij

Causal diagram revisited U i X i1 X i2 M i Y i1 Y i2 W i A: Y i1 Y i2 X i1, X i2, Z i B: Y ij X ij X ij, Z i A and B hold if and only if Z i = (U i, M i )

Conclusion E(Y ij U i, X ij ) = αi 1 + β 1 X ij E(Y ij M i, X ij ) = αi 2 + β 2 X ij E(Y ij W i, X ij ) = αi 3 + β 3 X ij E(Y ij U i, M i, X ij ) = αi 4 + β 4 X ij E(Y ij U i, W i, X ij ) = αi 5 + β 5 X ij E(Y ij M i, W i, X ij ) = αi 6 + β 6 X ij E(Y ij U i, M i, W i, X ij ) = αi 7 + β 7 X ij The CML estimator of β in model E(Y ij family i, X ij ) = α i + βx ij consistently estimates β 4 in model E(Y ij U i, M i, X ij ) = αi 4 + β 4 X ij

Causal interpretation Let Y x,m ij be the potential outcome for a subject with levels X ij = x and M i = m In the absence of non-shared confounders, β is the controlled direct effect E(Y x+1,m ij U i ) E(Y x,m ij U i )

The ADHD study revisited The null finding by Chang et al. (2014) does not imply the absence of a causal effect It may also be explained by a causal effect that is mediated through familial environment because the sibling comparison design implicitly controls for all shared mediators It may not be explained by collider-stratification bias because the sibling comparison design does not make implicit control for any shared colliders

Outline Motivating example Main results Correctly specified models Misspecified models Concluding remarks

Recap The CML estimator of β in model E(Y ij family i, X ij ) = α i + βx ij consistently estimates β 4 in model E(Y ij U i, M i, X ij ) = α 4 i + β 4 X ij...provided that the latter model is correctly specified What if the latter model is incorrect?

General result for linear model Suppose that X ij is binary and define (U i, M i ) = E(Y ij U i, M i, X ij = 1) E(Y ij U i, M i, X ij = 0) The CML estimator of β in model generally converges to E(Y ij family i, X ij ) = α i + βx ij E{ (U i, M i ) D}

General result for logistic model Suppose that X ij is binary and define OR(U i, M i ) = p(y ij = 1 U i, M i, X ij = 1)p(Y ij = 0 U i, M i, X ij = 0) p(y ij = 0 U i, M i, X ij = 1)p(Y ij = 1 U i, M i, X ij = 0) The CML estimator of exp(β) in model logit{p(y ij = 1 family i, X ij )} = α i + βx ij generally converges to E{d(U i, M i )OR(U i, M i ) D}, where d(u i, M i ) is a weight that is proportional to p(y ij = 0 U i, M i, X ij = 1)p(Y ij = 1 U i, M i, X ij = 0)

General result for Cox proportional hazards model Suppose that X ij is binary and define HR(U i, M i ) = λ(y ij U i, M i, X ij = 1)} λ(y ij U i, M i, X ij = 0)}, where we have assumed that the hazard ratio is constant across time y ij, but not across U i and M i The CML estimator of exp(β) in model log{λ(y ij family i, X ij )} = α i (y ij ) + βx ij generally converges to E{b(U i, M i )HR(U i, M i ) D}, where b(u i, M i ) is a weight that is proportional to {1 + HR(U i, M i )} 1

Outline Motivating example Main results Correctly specified models Misspecified models Concluding remarks

No clear cut The sibling comparison design controls for all shared confounders and mediators, but no shared colliders In practice, the distinction between confounders, mediators and colliders may be somewhat artificial e.g. familial environment may act as all three types of covariates Still a useful distinction, since helps to understand the theoretical properties of the design and models

Other estimation alternatives ML, GEE, random effects/frailty model, between-within (BW) model ML gives identical results as CML in linear models, but is inconsistent in non-linear models the incidental parameter problem GEE does not control for any shared confounders since estimates marginal (over families) association Random effects/frailty model is inconsistent if there is shared confounding since the model assumes that the intercept is independent of exposure, thus ruling out shared confounding a priori BW model gives (nearly) identical results as CML in (non-)linear models

References Chang, Z., Lichtenstein, P., D Onofrio, B., Almqvist, C., Kuja-Halkola, R., Sjölander, A., and Larsson, H. (2014). Maternal age at childbirth and risk for adhd in offspring: a population-based cohort study. International Journal of Epidemiology 43, 1815 1824. Sjölander, A, and Zetterqvist, J. (In press). Confounders, mediators or colliders What types of shared covariates does the sibling comparison design control for? Epidemiology.

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