A Bayesian Perspective on Unmeasured Confounding in Large Administrative Databases

Transcription

1 A Bayesian Perspective on Unmeasured Confounding in Large Administrative Databases Lawrence McCandless Faculty of Health Sciences, Simon Fraser University, Vancouver Canada Summer 2014

2 My Background I work on Bayesian methods for causal inference (epidemiology). Develop Bayesian methods to explore effects of unmeasured confounding. Sensitivity Analysis Application areas: Pharmacoepidemiology Mental health epidemiology Causal inference with large administrative databases (e.g. health records)

3 Today s Talk Causal Mediation Analysis Unmeasured Confounding Bayesian Methods

4 Outline Background: What is causal mediation analysis? Data Example: Mortality in criminal offenders using large administrative databases Partially Missing Confounders: Example of multiple imputation and Bayesian sensitivity analysis

5 What is Mediation Analysis? In health research it is often necessary to disentangle the causal pathways that link exposure to disease. The goals of mediation analyses are to identify the total effect of the exposure on disease, the effect of the exposure that acts through a given set of intermediate variables (indirect effect), and the effect of the exposure unexplained by those same intermediate variables (direct effect). Richiardi et al. Int J Epi (2013)

6 Mediation analysis in epidemiology Mediation analysis concerns intermediate variables on the causal pathway between exposure and outcome Hafeman (2009) Int J Epidemiol

7 Example: Survival Analysis of Time-to-Death in Criminal Offenders Health Criminal Sentences (Log Rate) Gender Mental Illness Age Addiction Death Data Source: Ministry of Justice, Goverment of British Columbia, Canada.

8 How to Estimate Direct & Indirect effects?? The traditional approach to mediation analysis is based on comparing two regression models for the outcome variable, one with and one without adjusting for the intermediate variable. If adjustment for the intermediate variable greatly attenuates the exposure effect, then we conclude that the exposure effect is mediated primarily through the intermediate. This is the Difference in Coefficients Approach described in Baron and Kenny 1980 s.

9 Illustration of Baron & Kenny methods Product of coefficients method There also is a related Product of coefficients approach to mediation analysis. Let T denote time until death or censoring Let X denote a dichotomous exposure variable, Let M denote a continous intermediate variable In a mediation analysis, we write down a model for both the mediator and outcome: P(T, M X) = P(T M, X) }{{} P(M X) }{{} OutcomeModel MediatorModel

10 Illustration of Baron & Kenny method Product of coefficients method Suppose that T follows a proportional hazards model, and M is continuous and normally distributed. Then we could use 1) Weibull outcome model for T : h(t X, M) = exp(β X X + β M M) λt λ 1 2) linear regression model for M: M X = γ 0 + γ X X + ɛ where ɛ N(0, σ 2 ).

11 Illustration of Baron & Kenny method Product of coefficients method The direct effect is β X The indirect effect is γ x β M Indirect Effect M γ X β M X β X T Direct Effect

12 Illustration of Baron & Kenny method The product of coefficients method is criticized because it is invalid for non-linear outcome models, and also invalid if there are interactions between exposure and mediator However, if the disease is rare and there are no interactions, then it approximates the Natural/Controlled Direct and Natural Indirect Effects. Vanderweele (2013) Epidemiol: shows: log HR NDE = β X +... log HR NIE = β M γ X +...

13 Mediation Analysis Results Characteristic Number (%) or Mean n=79088 Outcome Death 1841 (2.3%) Exposure Addiction (14%) Mediator Sentencing rate (sentences/yr) 1 per 2 yrs Covariates Female (20%) Age < (32%) (38%) > (30%) 20+ Other covariates: race/ethnicity; education; mental illness; health services use; hospitalization; disability; type of criminal offense;

14 Mediation Analysis Results Hazard Ratio for Death Direct Effect Indirect Effect Total Effect HR 95% CI HR 95% CI HR 95% CI Addiction 1.20 ( ) 1.40 ( ) 1.68 (1.51, 1.82) Adjusted for 20+ covariates Calculated using method of Vanderweele (2013) + boostrap Conclusion: A large indirect effect. Addiction is associated with mortality that is mediated by high rates of criminal sentencing.

15 Mediation Analysis Results The direct effect is β X = 0.17 The indirect effect is γ x β M = = 0.34 Indirect Effect M γ X =0.23 βm =1.51 X β X =0.17 T Direct Effect

16 The Problem of Confounding Unmeasured confounding can plague causal inferences in administrative databases. The association between mediator and outcome is biased from criminogenic factors. High risk offenders face problems with... Family Peers Poverty Criminal Behavior Mental Illness Cognition

17 The Problem of Confounding This is called Mediator-Outcome confounding Cognition Criminal Sentences (Log Rate) Family Criminal Behavior Mental Illness Peers Poverty Addiction Death

18 Two Important Partially Missing Confounders RNA scores The Risk Need Assement (RNA) score is a validated 21-question instrument that predicts re-offending. % Missing Labels RNA score (Criminal History) 20.4% 1/0 RNA score (Behaviour) 20.4% 1/0 Example: High-risk offenders are more deprived, and consequently more likely to die. Indirect effect is biased away from Null

19 Diagnostics: Analysis of the Complete Data ONLY Hazard Ratio for Death Direct Effect Indirect Effect Total Effect HR 95% CI HR 95% CI HR 95% CI Addiction 1.18 ( ) 1.39 ( ) 1.64 (1.47, 1.81) Addiction 1.17 ( ) 1.27 ( ) 1.48 (1.30, 1.61) Calculated using method of Vanderweele (2013) + boostrap Adjusted for 20+ covariates Adjusted for 20+ covariates and RNA scores Conclusion: When we adjust for RNA scores, we see attenuation of indirect effect.

20 Correlation Among Partially Missing Confounders in the complete data A 2 2 table of the binary missing confounders. RNA score (Behaviour) RNA score (Criminal History) The OR is 2.41 with 95% CI (2.32, 2.49). To adjust for confounding, we require a model for the joint distribution of the 2 partially missing confounders.

21 Bayesian adjustment for partially missing confounders Proposed method: Use Bayesian methods to average over partially missing RNA scores. Similar to multiple imputation. Methodological challenges: We require a joint model for missing confounders (challenging in high dimension) Bayesian MCMC computing is hard in large samples Missing confounders perhaps not missing at random (NMAR) Can be combined with a Bayesian sensitivity analysis for other unmeasured confounders.

22 Bayesian adjustment for partially missing confounders Symbol Description Outcome T Time until death or censoring Exposure variable X Addiction Mediating variable M Rate of criminal sentencing (log) Covariates C Age, Sex, Measures of health status,... Covariates U RNA 1, RNA 2

23 Bayesian adjustment for 2 missing dichotomous confounders We already have P(T X, M, U, C) }{{} Outcome Model P(M X, U, C) }{{} Mediator Model Now we include P(U, C) exp{β U1 U 1 + β U2 U 2 + β U1,U 2 U 1 U } To give a full probability distribution for P(T, M, U, C)

24 Bayesian Computation We assign relatively noninformative prior distributions to model parameters For example, β X, β M, β U1, β U2, β C1,... N(0, 10 6 ) In fact, because MCMC computation is so challenging in large samples, I udpate parameters by sampling from distribution of MLE using standard regression software (e.g. survreg(), lm(), glm())

25 Bayesian Computation Bayesian computation proceeds using MCMC in 2 interative stages: Step 1 Draw Imputations. Sample U from P(U T, X, M, C) P(T X, M, C)P(M X, C)P(U, C) Step 2 Update parameters given imputations Step 1 can be done analytically, but challenging in high dimensional U. Step 2 can approximated using standard regression software.

26 Mediation Analysis Results Hazard Ratio for Death Direct Effect Indirect Effect Total Effect HR 95% CI HR 95% CI HR 95% CI Addiction 1.20 ( ) 1.40 ( ) 1.68 (1.51, 1.82) Addiction 1.20 ( ) 1.29 ( ) 1.55 (1.40, 1.67) Ignoring missing data; Method of Vanderweel (2013) + bootstrap Bayesian adjustment for partially missing confounders

27 Conclusion There are important partially missing confounders that we can control for using Bayesian methods. Note that the complete case analysis produces almost identical answers to the more complex method.

28 Conclusion Additional issues: A quote from from Kropko, Goodrich, Gelman and Hill (2014) Joint vs Conditional Approaches to MI.

29 Conclusion Bayesian approach is useful to explore sensitivity to unmeasured or partially measured confounders. We can model the confounder using a missing data model, and incorporate prior information about the confounder from external data. Very relevant to analysis of large administrative databases, which have large sample sizes. More generally, Bayesian mediation analysis is exciting new area of innovation in biostatistics.

30 References: Thank You! Daniels et al. (2012) Bayesian inference for the causal effect of mediation Biometrics. McCandless LC, Richardson S, Best N. (2012) Adjustment for missing confounders using external validation data and propensity scores. Journal of the American Statistical Association 107: McCandless LC, Gustafson P, Levy AR, Richardson S. (2012) Hierarchical priors for bias parameters in Bayesian sensitivity analysis for unmeasured confounding. Statistics in Medicine 31: McCandless LC, Gustafson P, Levy AR. (2007) Bayesian sensitivity analysis for unmeasured confounding in observational studies. Statistics in Medicine. 26: VanderWeele (2011) Causal mediation analysis with survival data Epidemiology. Lange, Vansteelandt (2012) A simple unified approach to estimating natural direct and indirect effects Am J Epidemiol.