Epidemiologists often wish to estimate quantities that are easy to communicate and correspond to the results of realistic public health interventions. Methods from causal inference can answer these questions. We adopt the language of potential outcomes under Rubin’s original Bayesian framework and show that the parametric g-formula is easily amenable to a Bayesian approach. We show that the frequentist properties of the Bayesian g-formula suggest it improves the accuracy of estimates of causal effects in small samples or when data are sparse. We demonstrate an approach to estimate the effect of environmental tobacco smoke on body mass index among children aged 4–9 years who were enrolled in a longitudinal birth cohort in New York, USA. We provide an algorithm and supply SAS and Stan code that can be adopted to implement this computational approach more generally.
GlassTAGoodmanSNHernánMAet al.Causal inference in public health. Annu Rev Public Health2013; 34: 61–75.
2.
RobinsJM. A new approach to causal inference in mortality studies with a sustained exposure period—Application to control of the healthy worker survivor effect. Math Mod1986; 7: 1393–1512.
3.
KeilAPEdwardsJKRichardsonDBet al.The parametric g-formula for time-to-event data: intuition and a worked example. Epidemiology2014; 25: 889–897.
ThomasDCWitteJSGreenlandS. Dissecting effects of complex mixtures: who’s afraid of informative priors?Epidemiology2007; 18: 186–190.
6.
EfronBMorrisCN. Stein’s paradox in statistics. Scientific Am1977; 236: 119–127.
7.
Stein C, et al. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In: Proceedings of the third Berkeley symposium on mathematical statistics and probability. Berkeley, Calif: University of California Press, 1956, vol. 1, pp.197–206.
8.
KeilAPDanielsJLHertz-PicciottoI. Autism spectrum disorder, flea and tick medication, and adjustments for exposure misclassification: the CHARGE (CHildhood Autism Risks from Genetics and Environment) case–control study. Environ Health2014; 13: 889–897.
WangWScharfsteinDWangCet al.Estimating the causal effect of low tidal volume ventilation on survival in patients with acute lung injury. J R Stat Soc Ser C Appl Stat2011; 60: 475–496.
14.
TaubmanSLRobinsJMMittlemanMAet al.Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. Int J Epidemiol2009; 38: 1599–1611.
15.
AhernJHubbardAGaleaS. Estimating the effects of potential public health interventions on population disease burden: a step-by-step illustration of causal inference methods. Am J Epidemiol2009; 169: 1140–1147.
16.
WestreichDColeSRYoungJGet al.The parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident aids or death. Stat Med2012; 31: 2000–2009.
17.
KeilAPRichardsonDB. Reassessing the link between airborne arsenic exposure among anaconda copper smelter workers and multiple causes of death using the parametric g-formula. Environ Health Perspect2017; 125: 608–614.
18.
ParkTCasellaG. The Bayesian lasso. J Am Stat Assoc2008; 103: 681–686.
19.
ColeSRHudgensMGBrookhartMAet al.Risk. Am J Epidemiol2015; 181: 246–250.
20.
PearlJ. On the consistency rule in causal inference: axiom, definition, assumption, or theorem?Epidemiology2010; 21: 872–875.
21.
RosenbaumP. The consequences of adjustment for a concomitant variable that has been affected by the treatment. J R Stat Soc Ser A-G1984; 147: 656–666.
22.
RubinD. Bayesian inference for causal effects: the role of randomization. Ann Stat1978; 6: 34–58.
23.
EdwardsJKColeSRWestreichD. All your data are always missing: incorporating bias due to measurement error into the potential outcomes framework. Int J Epidemiol2015; 44: 1452–1459.
GreenlandSMansourniaMA. Penalization, bias reduction, and default priors in logistic and related categorical and survival regressions. Stat Med2015; 34: 3133–3143.
26.
RobinsJMHernánMAEstimation of the causal effects of time-varying exposures. In: FitzmauriceGDavidianMVerbekeGet al.(eds). Longitudinal Data Analysis, boca raton: Chapman & Hall/CRC, 2009; Vol. 26: 553–600.
27.
EngelSMBerkowitzGSBarrDBet al.Prenatal organophosphate metabolite and organochlorine levels and performance on the brazelton neonatal behavioral assessment scale in a multiethnic pregnancy cohort. Am J Epidemiol2007; 165: 1397–1404.
28.
BuckleyJPEngelSMMendezMAet al.Prenatal phthalate exposures and childhood fat mass in a New York City cohort. Environ Health Perspect2015; 124: 507–513.
29.
WhiteH. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econom J Econom Soc1980; 48: 817–838.
30.
ColeSRFrangakisCE. The consistency statement in causal inference: a definition or an assumption?Epidemiology2009; 20: 3–5.
31.
McConnellRShenEGillilandFDet al.A longitudinal cohort study of body mass index and childhood exposure to secondhand tobacco smoke and air pollution: the Southern California children’s health study. Environ Health Perspect2015; 123: 360–366.
32.
MøllerSEAjslevTAAndersenCSet al.Risk of childhood overweight after exposure to tobacco smoking in prenatal and early postnatal life. PLoS One2014; 9: e109184–e109184.
33.
RobinsonOMartínezDAurrekoetxeaJJet al.The association between passive and active tobacco smoke exposure and child weight status among Spanish children. Obesity (Silver Spring)2016; 24: 1767–1777.
34.
YangSDeckerAKramerMS. Exposure to parental smoking and child growth and development: a cohort study. BMC Pediatr2013; 13: 104–104.
35.
BerzuiniCDawidPBernardinellL. Causality: statistical perspectives and applications, West Sussex, UK: John Wiley & Sons, 2012.
36.
ArjasESaarelaO. Optimal dynamic regimes: presenting a case for predictive inference. Int J Biostat2010; 6: 1–21.
37.
SaarelaOArjasEStephensDAet al.Predictive Bayesian inference and dynamic treatment regimes. Biom J2015.
38.
RobinsJMGreenlandS. Causal inference without counterfactuals: comment. J Am Stat Assoc2000; 95: 431–435.
39.
De Los CamposGNayaHGianolaDet al.Predicting quantitative traits with regression models for dense molecular markers and pedigree. Genetics2009; 182: 375–385.
40.
van der LaanMJRubinD. Targeted maximum likelihood learning. Int J Biostat2006; 2: 1–1.
Madigan D, Raftery AE, Volinsky C, et al. Bayesian model averaging. In: Proceedings of the AAAI workshop on integrating multiple learned models, Portland, OR: AAAI Press, 1996, pp.77–83.
45.
BuckleyJPKeilAPMcGrathLJet al.Evolving methods for inference in the presence of healthy worker survivor bias. Epidemiology2015; 26: 204–212.
46.
GreenlandS. Principles of multilevel modelling. Int J Epidemiol2000; 29: 158–167.
47.
RobinsJM. General methodological considerations. J Econom2003; 112: 89–106.
48.
RobinsJMCausal inference from complex longitudinal data. In: BerkaneM (ed). Latent variable modeling applications to causality, New York: Springer, 1997; Vol. 120: 69–117.
49.
YoungJGCainLERobinsJMet al.Comparative effectiveness of dynamic treatment regimes: an application of the parametric g-formula. Stat Biosci2011; 3: 119–143.
