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Comments on ‘A weighting analogue to pair matching in propensity score analysis’ by L. Li and T. Greene

  • Shunichiro Orihara ORCID logo EMAIL logo , Taishi Kawamura and Masataka Taguri
Published/Copyright: March 24, 2022
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The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 19 Issue 1

Abstract

Li and Greene (A weighting analogue to pair matching in propensity score analysis. Int J Biostat 2013;9:215–34) propose that estimates derived by the matching weight (MW) estimator are similar to those derived by the one-to-one propensity score matching estimator. The MW estimator has some useful properties, however, some regularity conditions need to be confirmed to derive an asymptotic distribution since the MW has a non-differentiable point. In this letter, we confirm the asymptotic distribution of the MW estimator and the sufficient conditions to achieve it.

Keywords: asymptotic normality; causal inference; matching weight
Corresponding author: Shunichiro Orihara, Graduate School of Data Science, Yokohama City University, Yokohama, Kanagawa, Japan, E-mail:

Acknowledgement

We would like to express our thanks to editors for their useful comments.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix: Regularity conditions and assumptions

  1. Moment conditions

    1. For all θ ∈ Θ,

      E Z ( Y − μ 1 ) Z e ( β ) + ( 1 − Z ) ( 1 − e ( β ) ) 1 < ∞ , E ( 1 − Z ) ( Y − μ 0 ) Z e ( β ) + ( 1 − Z ) ( 1 − e ( β ) ) 1 < ∞

    1. For all θ in a neighborhood N of θ 0 and δ ∈ (0, 0.5),

      E ∂ ∂ β e ( β ) Y 1 − μ 1 e ( β ) I 3 ( δ ) 1 < ∞ , E ∂ ∂ β e ( β ) Y 0 − μ 0 1 − e ( β ) I 1 ( δ ) 1 < ∞ ,

    1. For all θ in a neighborhood N of θ 0 and δ ∈ (0, 0.5),

      E ∂ ∂ β e ( β ) Y − μ 1 6 I 2 ( δ ) < ∞ , E ∂ ∂ β e ( β ) Y − μ 0 6 I 2 ( δ ) < ∞

    1. For all θ in a neighborhood N of θ 0 and δ ∈ (0, 0.5), for each β j ,

      sup ∂ ∂ β j e ( β ) Y − μ 1 I 2 ( δ ) < ∞ , sup ∂ ∂ β j e ( β ) Y − μ 0 I 2 ( δ ) < ∞

  1. Distribution of the propensity score

    1. A distribution function of the propensity score F e (e) is twice differentiable around e = 1 2 .

    2. A distribution function of the propensity score F e (e) is twice differentiable around e = 1 2 . In addition, the density function of the propensity score f e (e) has

      f e 1 2 = 0 .

  1. Asymptotic normality

    1. θ ̂ → P θ 0

    2. θ 0 is in the interior of Θ, where Θ is compact.

    3. E ϕ ( θ 0 ) ϕ ( θ 0 ) ⊤ < ∞

  1. Norm

    For all x ∈ X ⊂ R p ,

    ‖ x ‖ q ≔ ∑ j = 1 p | x j | q 1 q , q ≥ 1

References

1. Li, L, Greene, T. A weighting analogue to pair matching in propensity score analysis. Int J Biostat 2013;9:215–34. https://doi.org/10.1515/ijb-2012-0030.Search in Google Scholar PubMed

2. Yoshida, K, Hernández-Díaz, S, Solomon, DH, Jackson, JW, Gagne, JJ, Glynn, RJ, et al.. Matching weights to simultaneously compare three treatment groups: comparison to three-way matching. Epidemiology 2017;28:387–95. https://doi.org/10.1097/ede.0000000000000627.Search in Google Scholar

3. Newey, WK, McFadden, D. Large sample estimation and hypothesis testing. Handb Econom 1994;4:2111–245. https://doi.org/10.1016/s1573-4412(05)80005-4.Search in Google Scholar

4. Rosenbaum, PR, Rubin, DB. The central role of the propensity score in observational studies for causal effects. Biometrika 1983;70:41–55. https://doi.org/10.1093/biomet/70.1.41.Search in Google Scholar

5. Li, F, Morgan, KL, Zaslavsky, AM. Balancing covariates via propensity score weighting. J Am Stat Assoc 2018;113:390–400. https://doi.org/10.1080/01621459.2016.1260466.Search in Google Scholar

6. Van der Vaart, AW. Asymptotic statistics, vol. 3. Cambridge University Press; 2000.Search in Google Scholar
Received: 2021-08-20
Revised: 2022-03-07
Accepted: 2022-03-07
Published Online: 2022-03-24

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijb-2021-0090
Keywords for this article
asymptotic normality; causal inference; matching weight

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Two-sample t α -test for testing hypotheses in small-sample experiments
  4. Estimating risk and rate ratio in rare events meta-analysis with the Mantel–Haenszel estimator and assessing heterogeneity
  5. Estimating population-averaged hazard ratios in the presence of unmeasured confounding
  6. Commentary
  7. Comments on ‘A weighting analogue to pair matching in propensity score analysis’ by L. Li and T. Greene
  8. Research Articles
  9. Variable selection for bivariate interval-censored failure time data under linear transformation models
  10. A quantile regression estimator for interval-censored data
  11. Modeling sign concordance of quantile regression residuals with multiple outcomes
  12. Robust statistical boosting with quantile-based adaptive loss functions
  13. A varying-coefficient partially linear transformation model for length-biased data with an application to HIV vaccine studies
  14. Application of the patient-reported outcomes continual reassessment method to a phase I study of radiotherapy in endometrial cancer
  15. Borrowing historical information for non-inferiority trials on Covid-19 vaccines
  16. Multivariate small area modelling of undernutrition prevalence among under-five children in Bangladesh
  17. The optimal dynamic treatment rule superlearner: considerations, performance, and application to criminal justice interventions
  18. Estimators for the value of the optimal dynamic treatment rule with application to criminal justice interventions
  19. Efficient estimation of pathwise differentiable target parameters with the undersmoothed highly adaptive lasso
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