Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Substitute adjustment via recovery of latent variables

Jeffrey Adams, Niels Richard Hansen

Abstract

The deconfounder was proposed as a method for estimating causal parameters in a context with multiple causes and unobserved confounding. It is based on recovery of a latent variable from the observed causes. We disentangle the causal interpretation from the statistical estimation problem and show that the deconfounder in general estimates adjusted regression target parameters. It does so by outcome regression adjusted for the recovered latent variable termed the substitute. We refer to the general algorithm, stripped of causal assumptions, as substitute adjustment. We give theoretical results to support that substitute adjustment estimates adjusted regression parameters when the regressors are conditionally independent given the latent variable. We also introduce a variant of our substitute adjustment algorithm that estimates an assumption-lean target parameter with minimal model assumptions. We then give finite sample bounds and asymptotic results supporting substitute adjustment estimation in the case where the latent variable takes values in a finite set. A simulation study illustrates finite sample properties of substitute adjustment. Our results support that when the latent variable model of the regressors hold, substitute adjustment is a viable method for adjusted regression.

Substitute adjustment via recovery of latent variables

Abstract

The deconfounder was proposed as a method for estimating causal parameters in a context with multiple causes and unobserved confounding. It is based on recovery of a latent variable from the observed causes. We disentangle the causal interpretation from the statistical estimation problem and show that the deconfounder in general estimates adjusted regression target parameters. It does so by outcome regression adjusted for the recovered latent variable termed the substitute. We refer to the general algorithm, stripped of causal assumptions, as substitute adjustment. We give theoretical results to support that substitute adjustment estimates adjusted regression parameters when the regressors are conditionally independent given the latent variable. We also introduce a variant of our substitute adjustment algorithm that estimates an assumption-lean target parameter with minimal model assumptions. We then give finite sample bounds and asymptotic results supporting substitute adjustment estimation in the case where the latent variable takes values in a finite set. A simulation study illustrates finite sample properties of substitute adjustment. Our results support that when the latent variable model of the regressors hold, substitute adjustment is a viable method for adjusted regression.
Paper Structure (25 sections, 12 theorems, 125 equations, 4 figures, 3 algorithms)

This paper contains 25 sections, 12 theorems, 125 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Relation to existing literature
  3. Substitute adjustment
  4. The General Model
  5. Substitute Latent Variable Adjustment
  6. Assumption-Lean Substitute Adjustment
  7. Substitute adjustment in a mixture model
  8. The mixture model
  9. Bounding estimation error due to using substitutes
  10. Bounding the mislabeling rate of the substitutes
  11. Asymptotics of the substitute adjustment estimator
  12. Tensor decompositions
  13. Simulation Study
  14. Mixture model simulations and recovery of $Z$
  15. Outcome model simulation and estimation of $\beta_i$
  16. ...and 10 more sections

Key Result

Proposition 1

Fix $i \in \mathbb{N}$ and let $P^{-i}_{z}$ denote a regular conditional distribution of $\mathbf{X}_{-i}$ given $Z = z$. Under Assumptions ass:regpos and ass:variable, the Markov kernel is a regular conditional distribution of $Y$ given $(X_i,Z) = (x,z)$, in which case

Figures (4)

  • Figure 1: Directed Acyclic Graph (DAG) representing the joint distribution of $(X_i, \mathbf{X}_{-i}, Z, Y)$. The variable $Z$ blocks the backdoor from $X_i$ to $Y$.
  • Figure 2: Empirical mislabeling rates as a function of $n = m$ and $p$ and for three different separation scales.
  • Figure 3: Average MSE for substitute adjustment using Algorithm \ref{['alg:leanalgmix']} as a function of sample size $n$ and for two different dimensions, a range of the unobserved confounding levels, and with $\mu_{\mathrm{scale}} = 1$.
  • Figure 4: Average MSE for substitute adjustment using Algorithm \ref{['alg:leanalgmix']} compared to average MSE for the ridge and augmented ridge estimators for two different dimensions, a range of unobserved confounding levels, and with $\mu_{\mathrm{scale}} = 1$.

Theorems & Definitions (38)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Example 1
  • Example 2
  • Definition 2: Regression function
  • Definition 3: Target parameter
  • Proposition 2
  • Remark 2
  • Remark 3
  • ...and 28 more