Sensitivity analysis for causal inference using inverse probability weighting

Abstract

Evaluation of impact of potential uncontrolled confounding is an important component for causal inference based on observational studies. In this article, we introduce a general framework of sensitivity analysis that is based on inverse probability weighting. We propose a general methodology that allows both non-parametric and parametric analyses, which are driven by two parameters that govern the magnitude of the variation of the multiplicative errors of the propensity score and their correlations with the potential outcomes. We also introduce a specific parametric model that offers a mechanistic view on how the uncontrolled confounding may bias the inference through these parameters. Our method can be readily applied to both binary and continuous outcomes and depends on the covariates only through the propensity score that can be estimated by any parametric or non-parametric method. We illustrate our method with two medical data sets.

Figure 1

Illustration of the feasible regions of E(Y₁) defined by formula (6). f_low, f, and f_up can be viewed as three functions of E(Y₁) that correspond to the three terms in (6): $f_{\text{low}}={\text{ρ}}_1^2{\text{ν}}_1\text{ψ}\left[E\right(Y_1\left)\right(E\left(Y_1\right)-2{\text{μ}}_1)+d]$, $f={({\text{θ}}_1-E(Y_1\left)\right)}^2$, $f_{\text{up}}={\text{ρ}}_1^2{\text{ν}}_1\left[E\right(Y_1\left)\right(1-E\left(Y_1\right))-b]$.The two intervals on X-axis under the gray regions correspond to the feasible regions under ρ₁>0 and ρ₁<0, respectively. Here, θ₁= 0.4, |ρ₁| = 0.5, π = 0.5, ν₁ = 0.5, μ₁ = 0.25, ψ= 1, η₁ = 0.15, b = 0.05, and d = 0.10625.

Figure 2

(A) Estimates of the effect of CD4 testing reminder on CD4 test ordering under different combinations of ρ = ρ₁= −ρ₀ and τ. Dashed lines are the estimates of the average treatment effect (ATE) based on Var_τ(S^*|S) (Eq. 11) and solid lines are the estimates of the ATE based on Var_λ(S^*|S) (Eq. 12). The four dashed lines correspond to (from top to bottom) τ = 0.1, 0.2, 0.3, and 0.4. The four solid lines correspond to (from top to bottom) λ(0.1) = 0.0063, λ(0.2) = 0.0245, λ(0.3) = 0.0526, and λ(0.4) = 0.0881. The horizontal gray line on the top is the estimated ATE assuming no uncontrolled confounding. (B) The lower limit of the one-sided 90% confidence interval of the intervention effect for τ = 0.2 for different values of ρ = ρ₁= −ρ₀.

Figure 3

(A) Estimates of the lower bound of the effect of abciximab on cost under different combinations of ρ = ρ₁=−ρ₀ and τ. Dashed lines are the estimates of the ATE based on Var_τ(S^*|S) (Eq. 11) and solid lines are the ATE based on Var_λ(S^*|S) (Eq. 12). The four dashed lines correspond to (from top to bottom) τ = 0.1, 0.2, 0.3, and 0.4. The four solid lines correspond to (from top to bottom) λ(0.1) = 0.0063, λ(0.2) = 0.0245, λ(0.3) = 0.0526, and λ(0.4)= 0.0881. The horizontal gray line on the top is the estimated ATE assuming no uncontrolled confounding. (B) The lower limit of the one-sided 90% confidence interval of the lower bound of the intervention effect for τ = 0.3 for different values of ρ = ρ₁=− ρ₀.