Time-varying-propensity-scores-in-SCMMs

Author: Osman Mahic
Date: 15-02-2025

This repository presents a mathematical formulation of the Sequential Conditional Mean Model (SCMM) with Time-Varying Propensity Scores—a Generalized Estimating Equations (GEE)-based approach to modeling time-varying treatment effects, as applied in my study (https://doi.org/10.1093/ndt/gfae069.1616).

This method was first described by:

Keogh, R. H., et al. (2017). Analysis of longitudinal studies with repeated outcome measures: Adjusting for time-dependent confounding using conventional methods. American Journal of Epidemiology, 187(5), 1085–1092. https://doi.org/10.1093/aje/kwx311

Notation

Let $X$ denote the exposure and $Y$ the outcome, both measured repeatedly at times $t = 1, \dots, T$. We define the notation $X_t$ to denote the time-varying exposure at time $t$, and $Y_t$ the outcome at time $t$. Similarly, let $\boldsymbol{L}_t$ denote a vector of (time-varying) covariates at time $t$, and $\bar{\boldsymbol{L}}_{t}$ represent the history up to time $t$.

Estimator for the conditional expectation

We aim to estimate the following conditional expectation at a given time:

$$ \text{E} [Y_t|{\bar{X}}_{t},{\bar{Y}}_{t-1},{\bar{\boldsymbol{L}}}_{t}] = \theta_0 + \theta_1{{X}}_{t} + \theta_2{{X}}_{t-1} + \theta_3{{Y}}_{t-1} + \theta_4^{\top}{\bar{\boldsymbol{L}}}_{t} $$

Because adjustment is made for ${\bar{Y}}_{t-1}$, the parameters of such SCMM can be estimated using a GEE with an identity working correlation matrix:

$$ \text{Corr}(Y_{ij}, Y_{ik}) = \left\lbrace \begin{array}{ll} 1, & \text{if } j = k \\ 0, & \text{if } j \neq k \end{array} \right. $$

Time-varying propensity scores

A SCMM is a doubly-robust estimator when a time-varying propensity score is included as a covariate. Given that $X \in \lbrace 1, \dots, K \rbrace $, we computed the following generalized propensity scores using multinomial logistic regression:

$$ \text{GPS}_t = \text{Pr}[X_t = k | {\bar{X}}_{t-1},{\bar{Y}}_{t-1},{\bar{\boldsymbol{L}}}_{t}] $$

Where the class probabilities were estimated using the softmax function:

$$ \text{Pr}[X_t = k | \boldsymbol{V}] = \frac{1}{\sum_{j=1}^{K} e^{\theta_j' \boldsymbol{V}}} e^{\theta_k' \boldsymbol{V}} $$

With $\boldsymbol{V}$ representing the covariate vector = $({\bar{X}}_{t-1},{\bar{Y}}_{t-1},{\bar{\boldsymbol{L}}}_{t})$. Given that $\sum_{k=1}^{K} \Pr(X_t = k | \boldsymbol{V}) = 1$ holds, it is sufficient to condition on $K-1$ of the $\widehat{\text{GPS}_t}$ in the SCMM.

Software

R version 4.1.3 or higher with the following packages:

• nnet: Feed-Forward Neural Networks and Multinomial Log-Linear Models
• geepack: Generalized Estimating Equation Package