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Causal inference under over-simplified longitudinal causal models

  • Lola Étiévant ORCID logo EMAIL logo and Vivian Viallon
Published/Copyright: November 1, 2021
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The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 18 Issue 2

Abstract

Many causal models of interest in epidemiology involve longitudinal exposures, confounders and mediators. However, repeated measurements are not always available or used in practice, leading analysts to overlook the time-varying nature of exposures and work under over-simplified causal models. Our objective is to assess whether – and how – causal effects identified under such misspecified causal models relates to true causal effects of interest. We derive sufficient conditions ensuring that the quantities estimated in practice under over-simplified causal models can be expressed as weighted averages of longitudinal causal effects of interest. Unsurprisingly, these sufficient conditions are very restrictive, and our results state that the quantities estimated in practice should be interpreted with caution in general, as they usually do not relate to any longitudinal causal effect of interest. Our simulations further illustrate that the bias between the quantities estimated in practice and the weighted averages of longitudinal causal effects of interest can be substantial. Overall, our results confirm the need for repeated measurements to conduct proper analyses and/or the development of sensitivity analyses when they are not available.

Keywords: causal inference; identifiability; longitudinal model; structural causal model
Corresponding author: Lola Étiévant, Institut Camille Jordan, Villeurbanne 69622, France, E-mail:

Acknowledgments

The authors are grateful to Stijn Vansteelandt for insightful comments on preliminary versions of this article, and to the reviewers of the International Journal of Biostatistics for valuable comments and suggestions.

  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.

  4. Disclaimers: Where authors are identified as personnel of the International Agency for Research on Cancer/World Health Organization, the authors alone are responsible for the views expressed in this article and they do not necessarily represent the decisions, policy or views of the International Agency for Research on Cancer/World Health Organization.

Appendix A. Proof of Theorem 1

Consider a longitudinal model (L) and assume that the only available data regarding the exposure of interest consists in X , which is a deterministic function of X ̲ t 1 t 2 , with 1 ≤ t 1 ≤ t 2 ≤ T. Let x and x * , x ≠ x * , be two given possible values of X , and assume that there exists some W ⊂ Z , taking its values in some space Ω W , such that 0 < P ( X = x ∣ W = w ) < 1 and 0 < P ( X = x * ∣ W = w ) < 1 , for all w such that P ( W = w ) > 0 . Now, consider an over-simplified model (S) and assume that ( Y X = x X ∣ W ) S . Then, following usual arguments of causal inference [8, 10, 11], the causal effect of interest, A T E S x ; x * ≔ E S Y X = x − Y X = x * , would be estimated under this over-simplified model (S) as

A T E ̃ S x ; x * = ∑ w ∈ Ω W E Y ∣ W = w , X = x − E Y ∣ W = w , X = x * × P ( W = w ) .

Now assume that there exists some t 0 ∈ 1 , t 1 , some t 3 ∈ t 2 , T , such that ( Y X ̲ t 0 t 3 = x ̲ t 0 t 3 X ̲ t 0 t 3 ∣ W ) L . Let w be any given possible value of W such that P ( W = w ) > 0 , and x ̲ t 0 t 3 and x ̲ t 0 t 3 * in 0,1 t 3 − t 0 + 1 be any two possible profiles of X ̲ t 0 t 3 leading to X = x and X = x * , respectively, and such that P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ X = x , W = w ) > 0 and P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 * ∣ X = x * , W = w ) > 0 . It follows that P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w ) > 0 and P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 * ∣ W = w ) > 0 . Next, usual arguments of causal inference [8, 10, 11] yield

A T E L ∣ W = w x ̲ t 0 t 3 ; x ̲ t 0 t 3 * = ∑ w ∈ Ω W E L Y X ̲ t 0 t 3 = x ̲ t 0 t 3 − Y X ̲ t 0 t 3 = x ̲ t 0 t 3 * ∣ W = w , = ∑ w ∈ Ω W E L Y X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w , X ̲ t 0 t 3 = x ̲ t 0 t 3 − E L Y X ̲ t 0 t 3 = x ̲ t 0 t 3 * ∣ W = w , X ̲ t 0 t 3 = x ̲ t 0 t 3 * , = ∑ w ∈ Ω W E Y ∣ W = w , X ̲ t 0 t 3 = x ̲ t 0 t 3 − E Y ∣ W = w , X ̲ t 0 t 3 = x ̲ t 0 t 3 * .

Because X ̲ t 0 t 3 d-separates X and W under model (L) [9, 44], we have, for any w in Ω W and any x ̲ t 0 t 3 in { 0,1 } t 3 − t 0 + 1 such that P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w ) > 0 ,

E Y ∣ W = w , X ̲ t 0 t 3 = x ̲ t 0 t 3 = E Y ∣ W = w , X = x , X ̲ t 0 t 3 = x ̲ t 0 t 3 ,

with x corresponding to the value taken by X when X ̲ t 0 t 3 = x ̲ t 0 t 3 . In other respect, we have

E Y ∣ W = w , X = x = ∑ x ̲ t 0 t 3 ∈ 0,1 t 3 − t 0 + 1 E Y ∣ W = w , X = x , X ̲ t 0 t 3 = x ̲ t 0 t 3 × P X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w , X = x , = ∑ x ̲ t 0 t 3 ∈ 0,1 t 3 − t 0 + 1 E Y ∣ W = w , X ̲ t 0 t 3 = x ̲ t 0 t 3 × P X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w , X = x , = ∑ x ̲ t 0 t 3 ∈ 0,1 t 3 − t 0 + 1 E L Y X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w × P X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w , X = x ,

where the second equality comes from the fact that X ̲ t 0 t 3 = x ̲ t 0 t 3 ⇒ X = x , for any x ̲ t 0 t 3 such that P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w , X = x ) ≠ 0 . This finally yields

A T E ̃ S ( x ; x * ) = ∑ w ∈ Ω W ∑ x ̲ t 0 t 3 , x ̲ t 0 t 3 * ∈ 0,1 t 3 − t 0 + 1 A T E L ∣ W = w x ̲ t 0 t 3 ; x ̲ t 0 t 3 * × P X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ X = x , W = w × P X ̲ t 0 t 3 = x ̲ t 0 t 3 * ∣ X = x * , W = w × P ( W = w ) ,

where the sums are over all x ̄ t 0 t 3 and x ̄ t 0 t 3 * in { 0,1 } t 3 − t 0 + 1 such that P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 ∣ W = w , X = x ) and P ( X ̲ t 0 t 3 = x ̲ t 0 t 3 * ∣ W = w , X = x * ) , respectively, are not null.

The proof of the result under condition (T.Uncond) follows from similar, but simpler, arguments and is therefore omitted.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/ijb-2020-0081).

Received: 2020-06-02
Revised: 2021-10-13
Accepted: 2021-10-14
Published Online: 2021-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijb-2020-0081
Keywords for this article
causal inference; identifiability; longitudinal model; structural causal model

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Doubly robust adaptive LASSO for effect modifier discovery
  4. Increasing the efficiency of randomized trial estimates via linear adjustment for a prognostic score
  5. Review
  6. Review and comparison of treatment effect estimators using propensity and prognostic scores
  7. Research Articles
  8. Error rate control for classification rules in multiclass mixture models
  9. Regression trees and ensembles for cumulative incidence functions
  10. Causal inference under over-simplified longitudinal causal models
  11. Causal inference under interference with prognostic scores for dynamic group therapy studies
  12. Bayesian multi-response nonlinear mixed-effect model: application of two recent HIV infection biomarkers
  13. A Bayesian semiparametric accelerate failure time mixture cure model
  14. Quantifying the extent of visit irregularity in longitudinal data
  15. An improved method for analysis of interrupted time series (ITS) data: accounting for patient heterogeneity using weighted analysis
  16. A robust hazard ratio for general modeling of survival-times
  17. Penalized likelihood estimation of the proportional hazards model for survival data with interval censoring
  18. A parametric approach to relaxing the independence assumption in relative survival analysis
  19. The number of response categories in ordered response models
  20. A comparison of joint dichotomization and single dichotomization of interacting variables to discriminate a disease outcome
  21. Spike detection for calcium activity
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