Skip to main content
SpringerLink
Log in

Propensity score weighting for a continuous exposure with multilevel data

  • Published:
Health Services and Outcomes Research Methodology Aims and scope Submit manuscript

Abstract

Propensity score methods (e.g., matching, weighting, subclassification) provide a statistical approach for balancing dissimilar exposure groups on baseline covariates. These methods were developed in the context of data with no hierarchical structure or clustering. Yet in many applications the data have a clustered structure that is of substantive importance, such as when individuals are nested within healthcare providers or within schools. Recent work has extended propensity score methods to a multilevel setting, primarily focusing on binary exposures. In this paper, we focus on propensity score weighting for a continuous, rather than binary, exposure in a multilevel setting. Using simulations, we compare several specifications of the propensity score: a random effects model, a fixed effects model, and a single-level model. Additionally, our simulations compare the performance of marginal versus cluster-mean stabilized propensity score weights. In our results, regression specifications that accounted for the multilevel structure reduced bias, particularly when cluster-level confounders were omitted. Furthermore, cluster mean weights outperformed marginal weights.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Almirall, D., Griffin, B.A., McCaffrey, D.F., Ramchand, R., Yuen, R.A., Murphy, S.A.: Time-varying effect moderation using the structural nested mean model: estimation using inverse-weighted regression with residuals. Stat. Med. 33(20), 3466–3487 (2014)

    Article  PubMed  Google Scholar 

  • Arpino, B., Mealli, F.: The specification of the propensity score in multilevel observational studies. Comput. Stat. Data Anal. 55(4), 1770–1780 (2011)

    Article  Google Scholar 

  • Austin, P.C.: An introduction to propensity score methods for reducing the effects of confounding in observational studies. Multivar. Behav. Res. 46(3), 399–424 (2011)

    Article  Google Scholar 

  • Bates, D., Mächler, M., Bolker, B., Walker, S.: Fitting linear mixed-effects models using lme4. J. Stat. Softw. 67(1), 1–48 (2015)

    Article  Google Scholar 

  • Cohen, J.: Statistical power analysis for the behavioral sciences, 2nd edn. Routledge, Abingdon-on-Thames (1988)

    Google Scholar 

  • Cole, S.R., Hernan, M.A.: Constructing inverse probability weights for marginal structural models. Am. J. Epidemiol. 168(6), 656–664 (2008)

    Article  PubMed  PubMed Central  Google Scholar 

  • Eckardt, P.: Propensity score estimates in multilevel models for causal inference. Nurs. Res. 61(3), 213–223 (2012)

    Article  PubMed  Google Scholar 

  • Fonarow, G.C., Zhao, X., Smith, E.E., Saver, J.L., Reeves, M.J., Bhatt, D.L., Xian, Y., Hernandez, A.F., Peterson, E.D., Schwamm, L.H.: Door-to-needle times for tissue plasminogen activator administration and clinical outcomes in acute ischemic stroke before and after a quality improvement initiative. J. Am. Med. Assoc. 311(16), 1632–1640 (2014)

    Article  CAS  Google Scholar 

  • Fuller, G., Hasler, R.M., Mealing, N., Lawrence, T., Woodford, M., Juni, P., Lecky, F.: The association between admission systolic blood pressure and mortality in significant traumatic brain injury: a multi-centre cohort study. Injury 45(3), 612–617 (2014)

    Article  PubMed  Google Scholar 

  • Greenland, S., Robins, J.M.: Identifiability, exchangeability, and epidemiological confounding. Int. J. Epidemiol. 15(3), 413–419 (1986)

    Article  CAS  PubMed  Google Scholar 

  • Hirano, K., Imbens, G.: Estimation of causal effects using propensity score weighting: an application to data on right heart catheterization. Health Serv. Outcomes Res. Method. 2(3), 259–278 (2001)

    Article  Google Scholar 

  • Hirano, K., Imbens, G.W.: The propensity score with continuous treatments. In: Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives. pp. 73–84 (2004)

  • Imbens, G.W.: The role of the propensity score in estimating dose-response functions. Biometrika 87(3), 706–710 (2000)

    Article  Google Scholar 

  • Kang, J.D.Y., Schafer, J.L.: Demystifying double robustness: a comparison of alternative strategies for estimating a population mean from incomplete data. Stat. Sci. 22(4), 523–539 (2007)

    Article  Google Scholar 

  • Kelcey, B.: Assessing the effects of teachers’ reading knowledge on students’ achievement using multilevel propensity score stratification. Educ. Eval. Policy Anal. 33(4), 458–482 (2011)

    Article  Google Scholar 

  • Kim, J., Seltzer, M.: Causal inference in multilevel settings in which selection processes vary across schools. Center for Study of Evaluation, Los Angeles (2007)

  • Lee, B.K., Lessler, J., Stuart, E.A.: Improving propensity score weighting using machine learning. Stat. Med. 29(3), 337–346 (2009)

    Google Scholar 

  • Lee, B.K., Lessler, J., Stuart, E.A.: Weight trimming and propensity score weighting. PLoS ONE 6(3), e18174 (2011)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Leite, W.L.,  Jimenez, F., Kaya, Y., Stapleton, L.M., MacInnes, J.W., Sandbach, R.: An evaluation of weighting methods based on propensity scores to reduce selection bias in multilevel observational studies. Multivar. Behav. Res. 50(3), 265–284 (2015)

  • Li, F., Zaslavsky, A.M., Landrum, M.B.: Propensity score weighting with multilevel data. Stat. Med. 32(19), 3373–3387 (2013)

    Article  PubMed  PubMed Central  Google Scholar 

  • Lunceford, J.K., Davidian, M.: Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. Stat. Med. 23(19), 2937–2960 (2004)

    Article  PubMed  Google Scholar 

  • McCaffrey, D.F., Ridgeway, G., Morral, A.R.: Propensity score estimation with boosted regression for evaluating causal effects in observational studies. Psychol. Methods 9(4), 403–425 (2004)

    Article  PubMed  Google Scholar 

  • McCormick, M.P., O’Connor, E.E., Cappella, E., McClowry, S.G.: Teacher-child relationships and academic achievement: a multilevel propensity score model approach. J. Sch. Psychol. 51(5), 611–624 (2013)

    Article  PubMed  Google Scholar 

  • Petersen, M.L., Porter, K.E., Gruber, S., Wang, Y., van der Laan, M.J.: Diagnosing and responding to violations in the positivity assumption. Stat. Methods Med. Res. 21(1), 31–54 (2012)

    Article  PubMed  Google Scholar 

  • Pirracchio, R., Petersen, M.L., van der Laan, M.: Improving propensity score estimators’ robustness to model misspecification using super learner. Am. J. Epidemiol. 181(2), 108–119 (2015)

    Article  PubMed  Google Scholar 

  • Potter, F.J.: The effect of weight trimming on nonlinear survey estimates. In: Proceedings of the Section on Survey Research Methods. American Statistical Association (1993)

  • Robins, J., Sued, M., Lei-Gomez, Q., Rotnitzky, A.: Comment: performance of double-robust estimators when “inverse probability” weights are highly variable. Stat. Sci. 22(4), 544–559 (2007)

    Article  Google Scholar 

  • R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/ (2015). Accessed 01 Dec 2015

  • Robins, J.M., Hernan, M.A., Brumback, B.: Marginal structural models and causal inference in epidemiology. Epidemiology 11(5), 550–560 (2000)

    Article  CAS  PubMed  Google Scholar 

  • Rosenbaum, P.R., Rubin, D.B.: The central role of the propensity score in observational studies for causal effects. Biometrika 70(1), 41–55 (1983)

    Article  Google Scholar 

  • Rubin, D.B.: Comment: randomization analysis of experimental data: The Fisher randomization test. J. Am. Stat. Assoc. 75(371), 591–593 (1980)

    Google Scholar 

  • Rubin, D.B.: Statistics and causal inference: comment: Which ifs have causal answers. J. Am. Stat. Assoc. 81(396), 961–962 (1986)

    Google Scholar 

  • Scharfstein, D.O., Rotnitzky, A., Robins, J.M.: Adjusting for non-ignorable drop-out using semiparametric non-response models. J. Am. Stat. Assoc. 94(448), 1096-1120 (1999)

    Article  Google Scholar 

  • Setoguchi, S., Schneeweiss, S., Brookhart, M.A., Glynn, R.J., Cook, E.F.: Evaluating uses of data mining techniques in propensity score estimation: A simulation study. Pharmacoepidemiol Drug Saf. 17(6), 546–555 (2008)

  • Stuart, E.A.: Estimating causal effects using school-level data sets. Educ. Res. 36(4), 187–198 (2007)

    Article  Google Scholar 

  • Stuart, E.A.: Matching methods for causal inference: a review and a look forward. Stat. Sci. 25(1), 1–21 (2010)

    Article  PubMed  PubMed Central  Google Scholar 

  • Su, Y.S., Cortina, J.: What do we gain? Combining propensity score methods and multilevel modeling. Paper presented at the Annual Meeting of the American Political Science Association, Toronto, Canada (2009)

  • Thoemmes, F.J., West, S.G.: The use of propensity scores for nonrandomized designs with clustered data. Multivar. Behav. Res. 46(3), 514–543 (2011)

    Article  Google Scholar 

  • Xiang, Y., Tarasawa, B.: Propensity score stratification using multilevel models to examine charter school achievement effects. J. Sch. Choice 9(2), 179–196 (2015)

    Article  Google Scholar 

  • Zhang, Z., Zhou, J., Cao, W., Zhang, J.: Causal inference with a quantitative exposure. Stat. Methods Med. Res. 25(1), 315–335, (2016)

    Article  PubMed  Google Scholar 

  • Zhu, Y., Coffman, D.L., Ghosh, D.: A boosting algorithm for estimating generalized propensity scores with continuous treatments. J. Causal inference 3(1), 25–40 (2014)

    PubMed  PubMed Central  Google Scholar 

Download references

Acknowledgments

This work was conducted while Megan Schuler was post-doctoral fellow and Wanghuan Chu was a doctoral student at the Pennsylvania State University. This work was funded by awards P50 DA010075, P50 DA039838, and T32 DA017629 from the National Institute on Drug Abuse and K01 ES025437 from the National Institutes of Health Big Data to Knowledge initiative; IGERT award DGE-1144860 from the National Science Foundation. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies.

Author information

Authors and Affiliations

  1. Department of Health Care Policy, Harvard Medical School, 180 Longwood Avenue, Boston, MA, 02215, USA

    Megan S. Schuler

  2. Google, Inc., Mountain View, CA, 94043, USA

    Wanghuan Chu

  3. Department of Epidemiology and Biostatistics, Temple University, Philadelphia, PA, 19122, USA

    Donna Coffman

Authors
  1. Megan S. Schuler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wanghuan Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Donna Coffman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Megan S. Schuler.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 415 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schuler, M.S., Chu, W. & Coffman, D. Propensity score weighting for a continuous exposure with multilevel data. Health Serv Outcomes Res Method 16, 271–292 (2016). https://doi.org/10.1007/s10742-016-0157-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10742-016-0157-5

Keywords

Search

Navigation