Skip to main content

Estimation of Causal Effect Measures in the Presence of Measurement Error in Confounders

Abstract

The odds ratio, risk ratio, and the risk difference are important measures for assessing comparative effectiveness of available treatment plans in epidemiological studies. Estimation of these measures, however, is often challenged by the presence of error-contaminated confounders. In this article, by adapting two correction methods for measurement error effects applicable to the noncausal context, we propose valid methods which consistently estimate the causal odds ratio, causal risk ratio, and the causal risk difference for settings with error-prone confounders. Furthermore, we develop a bootstrap-based procedure to construct estimators with improved asymptotic efficiency. Numerical studies are conducted to assess the performance of the proposed methods.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

    Babanezhad M, Vansteelandt S, Goetghebeur E (2010) Comparison of causal effect estimators under exposure misclassification. J Stat Plan Inference 140:1306–1319

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Baiocchi M, Small DS, Lorch S, Rosenbaum PR (2010) Building a stronger instrument in an observational study of perinatal care for premature infants. J Am Stat Assoc 105:1285–1296

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Blakely T, McKenzie S, Carter K (2013) Misclassification of the mediator matters when estimating indirect effects. J Epidemiol Commun Health 67:458–466

    Article  Google Scholar 

  4. 4.

    Buonaccorsi JP (2010) Measurement error: models, methods, and applications. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  5. 5.

    Carroll RJ, Spiegelman CH, Lan KG, Bailey KT, Abbott RD (1984) On errors-in-variables for binary regression models. Biometrika 71:19–25

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Carroll RJ, Ruppert D, Stefanski LA, Crainiceanu CM (2006) Measurement error in nonlinear models: a modern perspective. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  7. 7.

    Cornfield J (1962) Joint dependence of risk of coronary heart disease on serum cholesterol and systolic blood pressure: a discriminant function analysis. Fed Proc 21:59–61

    Google Scholar 

  8. 8.

    Edwards JK, Cole SR, Westreich D (2015) All your data are always missing: incorporating bias due to measurement error into the potential outcomes framework. Int J Epidemiol 44:1452–1459

    Article  Google Scholar 

  9. 9.

    Efron B (1982) The jackknife, the bootstrap and other resampling plans, vol 38. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  10. 10.

    Fuller WA (1987) Measurement error models, vol 305. Wiley, New York

    Book  MATH  Google Scholar 

  11. 11.

    Gustafson P (2003) Measurement error and misclassification in statistics and epidemiology: impacts and Bayesian adjustments. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  12. 12.

    Hernán MA, Cole SR (2009) Invited commentary: causal diagrams and measurement bias. Am J Epidemiol 170:959–962

    Article  Google Scholar 

  13. 13.

    Hernán MA, Robins JM (2016) Causal inference. Chapman & Hall/CRC, Boca Raton forthcoming

    Google Scholar 

  14. 14.

    Huang Y, Wang C (2001) Consistent functional methods for logistic regression with errors in covariates. J Am Stat Assoc 96:1469–1482

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Imai K, Yamamoto T (2010) Causal inference with differential measurement error: nonparametric identification and sensitivity analysis. Am J Polit Sci 54:543–560

    Article  Google Scholar 

  16. 16.

    Kyle RP, Moodie EE, Klein MB, Abrahamowicz M (2016) Correcting for measurement error in time-varying covariates in marginal structural models. Am J Epidemiol 184:249–258

    Article  Google Scholar 

  17. 17.

    Lockwood J, McCaffrey DF (2016) Matching and weighting with functions of error-prone covariates for causal inference. J Am Stat Assoc 111:1831–1839

    MathSciNet  Article  Google Scholar 

  18. 18.

    Lunceford JK, Davidian M (2004) Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. Stat Med 23:2937–2960

    Article  Google Scholar 

  19. 19.

    McCaffrey DF, Lockwood J, Setodji CM (2013) Inverse probability weighting with error-prone covariates. Biometrika 100:671–680

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Ogburn EL, VanderWeele TJ (2012) Analytic results on the bias due to nondifferential misclassification of a binary mediator. Am J Epidemiol 176:555–561

    Article  Google Scholar 

  21. 21.

    Pearl J (2009) On measurement bias in causal inference. Technical Report R-357, Department of Computer Science, University of California, Los Angeles

  22. 22.

    Regier MD, Moodie EE, Platt RW (2014) The effect of error-in-confounders on the estimation of the causal parameter when using marginal structural models and inverse probability-of-treatment weights: a simulation study. Int J Biostat 10:1–15

    MathSciNet  Article  Google Scholar 

  23. 23.

    Robins JM (1999) Marginal structural models versus structural nested models as tools for causal inference. In Statistical models in epidemiology: the environment and clinical trials, pp 95–134. Springer, New York

  24. 24.

    Robins JM, Hernán MA, Brumback B (2000) Marginal structural models and causal inference in epidemiology. Epidemiology 11:550–560

    Article  Google Scholar 

  25. 25.

    Rosenbaum PR (1987) Model-based direct adjustment. J Am Stat Assoc 82:387–394

    Article  MATH  Google Scholar 

  26. 26.

    Rosenbaum PR (1998) Propensity score. In: Armitage P, Colton T (eds) Encyclopedia of Biostatistics, vol 5. Wiley, Chichester, pp 3551–3555

    Google Scholar 

  27. 27.

    Rosenbaum PR, Rubin DB (1983) The central role of the propensity score in observational studies for causal effects. Biometrika 70:41–55

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Rosenbaum PR, Rubin DB (1984) Reducing bias in observational studies using subclassification on the propensity score. J Am Stat Assoc 79:516–524

    Article  Google Scholar 

  29. 29.

    Rothman KJ, Greenland S, Lash TL (2008) Modern Epidemiology. Lippincott Williams & Wilkins, Philadelphia

    Google Scholar 

  30. 30.

    Small DS, Rosenbaum PR (2008) War and wages: the strength of instrumental variables and their sensitivity to unobserved biases. J Am Stat Assoc 103:924–933

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Stefanski LA, Carroll RJ (1987) Conditional scores and optimal scores for generalized linear measurement-error models. Biometrika 74:703–716

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Yi GY (2017) Statistical Analysis with Measurement Error or Misclassification: Strategy, Method and Application. Springer, New York

    Book  MATH  Google Scholar 

  33. 33.

    Yi GY, He W (2006) Methods for bivariate survival data with mismeasured covariates under an accelerated failure time model. Commun Stat 35:1539–1554

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers for their comments on the initial version. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and partially supported by a Collaborative Research Team Project of the Canadian Statistical Sciences Institute (CANSSI).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Grace Y. Yi.

Appendix: Proof of Theorem

Appendix: Proof of Theorem

Let \(\Delta _{i}(k)= X^*_{ik}+\{A_i(k)-1/2\} {\varvec{\Sigma }}_{\epsilon k}{{\varvec{\gamma }}}_{\mathrm{X}k}\) and \(G_i(k)= 1+\exp [\{-\gamma _{0k}-{{\varvec{\gamma }}}_{\mathrm{A}k}^T\bar{A}_i(k-1)-{{\varvec{\gamma }}}_{\mathrm{Z}k}^TZ_i(k)-{{\varvec{\gamma }}}_{\mathrm{X}k}^T\Delta _i(k)\}\{2A_i(k)-1\}]\). We first show \( E\left\{ YI(\bar{A}=\bar{a})\prod _{k=0}^K G_k \right\} =E(Y_{\bar{a}}) \).

$$\begin{aligned}&E\left\{ YI(\bar{A}=\bar{a})\prod _{k=0}^K G_k\right\} \\&\quad =P({\bar{A}}={\bar{a}})\iiiint y_{\bar{a}}\prod _{k=0}^K g_kf(y_{\bar{a}},{\bar{z}}, {\bar{x}},{\bar{x}}^*|{\bar{A}}={\bar{a}})d{\bar{x}}^*d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}} \left\{ \int \prod _{k=0}^K g_kf({\bar{x}}^*|{\bar{z}}, {\bar{x}},y_{\bar{a}},{\bar{A}}={\bar{a}})d{\bar{x}}^*\right\} f({\bar{z}}, {\bar{x}}, y_{\bar{a}}|{\bar{A}}={\bar{a}}) d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}} \left\{ \prod _{k=0}^K \int g_k f\{x^*_k| x(k)\}d x^*_k \right\} f({\bar{z}}, {\bar{x}}, y_{\bar{a}}|{\bar{A}}={\bar{a}}) d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}}\prod _{k=0}^K\frac{1}{P\{a(k)|,\bar{a}(k-1), z(k), x(k)\}} f({\bar{z}}, {\bar{x}}, y_{\bar{a}}|{\bar{A}}={\bar{a}})d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}}\prod _{k=0}^K\frac{1}{P\{a(k)|,\bar{a}(k-1), z(k), x(k)\}} \dfrac{P({\bar{A}}={\bar{a}}|{\bar{z}}, {\bar{x}}, y_{\bar{a}})f({\bar{z}}, {\bar{x}}, y_{\bar{a}})}{P({\bar{A}}={\bar{a}})}d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =\iiint y_{\bar{a}}\dfrac{1}{P({\bar{A}}={\bar{a}}|{\bar{z}}, {\bar{x}})} {P({\bar{A}}={\bar{a}}|{\bar{z}}, {\bar{x}})f({\bar{z}}, {\bar{x}}, y_{\bar{a}})}d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =E(Y_{{\bar{a}}}). \end{aligned}$$

Using similar arguments, it can then be shown that \( E\left\{ I(\bar{A}=\bar{a})\prod _{k=0}^K G_k \right\} =1. \) Therefore, the causal mean \(E(Y_{\bar{a}})\) can be consistently estimated by

$$\begin{aligned} \dfrac{\sum _{i=1}^n {\hat{w}}_iY_iI(\bar{A}_i=\bar{a})}{n}\bigg /\dfrac{\sum _{i=1}^n {\hat{w}}_iI(\bar{A}_i=\bar{a})}{n}= \dfrac{\sum _{i=1}^n {\hat{w}}_iY_iI(\bar{A}_i=\bar{a})}{\sum _{i=1}^n {\hat{w}}_iI(\bar{A}_i=\bar{a})}. \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shu, D., Yi, G.Y. Estimation of Causal Effect Measures in the Presence of Measurement Error in Confounders. Stat Biosci 10, 233–254 (2018). https://doi.org/10.1007/s12561-018-9213-8

Download citation

Keywords

  • Causal effect measures
  • Causal inference
  • Comparative effectiveness
  • Confounding
  • Measurement error