Estimation of time-varying treatment effects using marginal structural models dependent on partial treatment history

TL;DR

The study tackles estimating time-varying treatment effects under time-varying confounding by improving IP-weighted MSMs. It introduces history-dependent IP-weights (PSW) and a closed testing procedure to select the relevant partial treatment history, linking MSMs and HRMSMs through a unifying estimand framework. Theoretical results establish asymptotic behavior and test validity, while simulations and a hemodialysis data application show that PSW-based approaches improve efficiency and bias control, particularly when past treatment effects and time-point correlations are strong. Practically, the methods enable more reliable inference about the impact of continuing treatment across time, with guidance on weight choice and history selection, and potential extensions to double robust or covariate-balancing strategies. Overall, the proposed framework enhances estimation of causal time-varying effects under realistic data settings with partial history considerations.

Abstract

Inverse probability (IP) weighting of marginal structural models (MSMs) can provide consistent estimators of time-varying treatment effects under correct model specifications and identifiability assumptions, even in the presence of time-varying confounding. However, this method has two problems: (i) inefficiency due to IP-weights cumulating all time points and (ii) bias and inefficiency due to the MSM misspecification. To address these problems, we propose (i) new IP-weights for estimating parameters of the MSM that depends on partial treatment history and (ii) closed testing procedures for selecting partial treatment history (how far back in time the MSM depends on past treatments). We derive the theoretical properties of our proposed methods under known IP-weights and discuss their extension to estimated IP-weights. Although some of our theoretical results are derived under additional assumptions beyond standard identifiability assumptions, some of which can be checked empirically from the data. In simulation studies, our proposed methods outperformed existing methods both in terms of performance in estimating time-varying treatment effects and in selecting partial treatment history. Our proposed methods have also been applied to real data of hemodialysis patients with reasonable results.

Figures (13)

• Figure 1: Plots of the selection probability of $m\in\{1,2,3,4\}$ corresponding to the main effect model over 1000 simulation runs based on the data generation process described in Section \ref{['subsec: sim set']} with $(\alpha_0,\alpha_1, \alpha_2, \pi_1, \delta_0, \delta_1, \delta_2, \delta_3)=(0,0,1,\pi_1,0,\delta_1,2,0)$, (a) setting $\pi_1=2.5$ and changing $\delta_1\in\{0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00\}$ and (b) setting $\delta_1=1.5$ and changing $\pi_1\in\{0.5, 1.0, 1.5, 2.0, 2.5, 3.0,3.5,4.0\}$. In (a), the x-axis represents $\delta_1$ multiplied by 100, whose change is corresponding to the change of the effect of past treatment $\delta_1\alpha_2$. In (b), the x-axis represents $\pi_1$ multiplied by 10, whose change is corresponding to the change of the association between time-varying treatments. The first row is existing selection methods, where QICw is $\tilde{m}_{{\text{QICw}}}$ and cQICw is $\tilde{m}_{{\text{cQICw}}}$. The bottom two rows are proposed selection methods, where ztest05, ztest20, pztest05, pztest20 is $\tilde{m}_{0.05}$, $\tilde{m}_{0.20}$, $\hat{m}_{0.05}$, $\hat{m}_{0.20}$, respectively. True is $m^*=2$.
• Figure 2: Box-plots of estimates of $\theta^{(K)}$ over 1000 runs of the first simulation with $(\alpha_0,\alpha_1, \alpha_2, \pi_1, \delta_0, \delta_1, \delta_2, \delta_3)=(0,0,1,4,0,1,2,1)$. The horizontal line is drawn at true value $\theta^{(K)}=4$. Twenty-two methods for estimating $\theta^{(K)}$ with combinations of selection methods and IP-weights are compared. Six gray blocks represent selection methods, where QICw, cQICw, ztest05, ztest20, pztest05, pztest20 is $\tilde{m}_{{\text{QICw}}}$, $\tilde{m}_{{\text{cQICw}}}$, $\tilde{m}_{0.05}$, $\tilde{m}_{0.20}$, $\hat{m}_{0.05}$, $\hat{m}_{0.20}$, respectively. For $m\in \{\tilde{m}_{\text{QICw}}, \tilde{m}_{\text{cQICw}}, \tilde{m}_{0.05}, \tilde{m}_{0.20}, \hat{m}_{0.05}, \hat{m}_{0.20}\}$, SW, RSW, PSW is $\hat{\theta}_{sw}^{(m)}$, $\hat{\theta}_{rsw}^{(m)}$, $\hat{\theta}_{psw}^{(m)}$, respectively. For $m\in \{\tilde{m}_{0.05}, \tilde{m}_{0.20}\}$, PSW_SW, PSW_RSW is $\hat{\theta}_{sw/psw}^{(m)}$, $\hat{\theta}_{rsw/psw}^{(m)}$, respectively.
• Figure D.1: Box-plots of estimates of $\theta^{(K)}$ over 1000 runs of the second simulation with $(\alpha_0,\alpha_1, \alpha_2, \pi_1, \delta_0, \delta_1, \delta_2, \delta_3)=(0,0,1,4,0,1,2,0)$. The horizontal line is drawn at true value $\theta^{(K)}=3$. Twenty-two methods for estimating $\theta^{(K)}$ with combinations of selection methods and IP-weights are compared. Six gray blocks represent selection methods, where QICw, cQICw, ztest05, ztest20, pztest05, pztest20 is $\tilde{m}_{\text{QICw}}$, $\tilde{m}_{\text{cQICw}}$, $\tilde{m}_{0.05}$, $\tilde{m}_{0.20}$, $\hat{m}_{0.05}$, $\hat{m}_{0.20}$, respectively. For $m\in \{\tilde{m}_\text{QICw}, \tilde{m}_\text{cQICw}, \tilde{m}_{0.05}, \tilde{m}_{0.20}, \hat{m}_{0.05}, \hat{m}_{0.20}\}$, SW, RSW, PSW is $\hat{\theta}_{sw,main}^{(m)}$, $\hat{\theta}_{rsw,main}^{(m)}$, $\hat{\theta}_{psw,main}^{(m)}$, respectively. For $m\in \{\tilde{m}_{0.05}, \tilde{m}_{0.20}\}$, PSW_SW, PSW_RSW is $\hat{\theta}_{sw/psw,main}^{(m)}$, $\hat{\theta}_{rsw/psw,main}^{(m)}$, respectively.
• Figure D.2: Box-plots of estimates of $\theta^{(K)}$ over 1000 runs of the third simulation with $(\alpha_0,\alpha_1, \alpha_2, \pi_1, \delta_0, \delta_1, \delta_2, \delta_3)=(0.5,0,1,4,0.5,1,2,0)$. The horizontal line is drawn at true value $\theta^{(K)}=3$. Twenty-two methods for estimating $\theta^{(K)}$ with combinations of selection methods and IP-weights are compared. Six gray blocks represent selection methods, where QICw, cQICw, ztest05, ztest20, pztest05, pztest20 is $\tilde{m}_{\text{QICw}}$, $\tilde{m}_{\text{cQICw}}$, $\tilde{m}_{0.05}$, $\tilde{m}_{0.20}$, $\hat{m}_{0.05}$, $\hat{m}_{0.20}$, respectively. For $m\in \{\tilde{m}_\text{QICw}, \tilde{m}_\text{cQICw}, \tilde{m}_{0.05}, \tilde{m}_{0.20}, \hat{m}_{0.05}, \hat{m}_{0.20}\}$, SW, RSW, PSW is $\hat{\theta}_{sw,main}^{(m)}$, $\hat{\theta}_{rsw,main}^{(m)}$, $\hat{\theta}_{psw,main}^{(m)}$, respectively. For $m\in \{\tilde{m}_{0.05}, \tilde{m}_{0.20}\}$, PSW_SW, PSW_RSW is $\hat{\theta}_{sw/psw,main}^{(m)}$, $\hat{\theta}_{rsw/psw,main}^{(m)}$, respectively.
• Figure D.3: Box-plots of estimates of $\theta^{(K)}$ over 1000 runs of the third simulation with $(\alpha_0,\alpha_1, \alpha_2, \pi_1, \delta_0, \delta_1, \delta_2, \delta_3)=(0.5,0,1,4,0.5,1,2,0)$. The horizontal line is drawn at true value $\theta^{(K)}=3$. Twelve methods for estimating $\theta^{(K)}$ with combinations of selection methods and IP-weights are compared. Four gray blocks represent selection methods, where QICw, cQICw, ztest05, ztest20 is $\tilde{m}_{\text{QICw}}$, $\tilde{m}_{\text{cQICw}}$, $\tilde{m}_{0.05}$, $\tilde{m}_{0.20}$, respectively. For $m\in \{\tilde{m}_\text{QICw}, \tilde{m}_\text{cQICw}, \tilde{m}_{0.05}, \tilde{m}_{0.20}\}$, SW, RSW, PSW is $\hat{\theta}_{sw,main}^{(m)}$, $\hat{\theta}_{rsw,main}^{(m)}$, $\hat{\theta}_{psw,main}^{(m)}$, respectively.

Theorems & Definitions (14)

• Lemma 1
• Lemma 2
• Corollary 1
• Corollary 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• proof