Bayesian analysis of longitudinal studies with treatment by indication

Abstract

In a medical setting, observational studies commonly involve patients who initiate a particular treatment (e.g., medication therapy) and others who do not, and the goal is to draw causal inferences about the effect of treatment on a time-to-event outcome. A difficulty with such studies is that the notion of a treatment initiation time is not well-defined for the control group. In this paper, we propose a Bayesian approach to estimate treatment effects in longitudinal observational studies where treatment is given by indication and thereby the exact timing of treatment is only observed for treated units. We present a framework for conceptualizing an underlying randomized experiment in this setting based on separating the time of indication for treatment, which we model using a latent state-space process, from the mechanism that determines assignment to treatment versus control. Next, we develop a two-step inferential approach that uses Markov Chain Monte Carlo (MCMC) posterior sampling to (1) infer the unobserved indication times for units in the control group, and (2) estimate treatment effects based on inferential conclusions from Step 1. This approach allows us to incorporate uncertainty about the unobserved indication times which induces uncertainty in both the selection of the control group and the measurement of time-to-event outcomes for these controls. We demonstrate our approach to study the effects on mortality of inappropriately prescribing phosphodiesterase type 5 inhibitors (PDE5Is), a medication contraindicated for certain types of pulmonary hypertension, using data from the Veterans Affairs (VA) health care system.

Appendix A Posterior sampling

1.1 General sampling scheme

We employ a Gibbs sampler to draw posterior samples from the model described in Sect. 3. In each iteration of the Gibbs sampler, we draw the missing indication times T for units with $M=1$ from the conditional posterior predictive distribution of T given covariates X and the current draw of the parameter $\theta$. The completed indication times can then be used to classify untreated patients into distinct groups of true controls and ineligible controls based on eligibility S, where the true control group consists of patients with $M_i = 1$ and $S_i=1$. For the true controls, we can then calculate values for the potential outcomes Y(0) given the generated values of T. These values are then regarded as the observed potential outcomes, $Y^{obs}=(1-Z)Y(0) + ZY(1)$. We can then update the parameters $\theta _1, \theta _2$ and $\theta _3$ by drawing from the conditional posterior distribution with density function $p(\theta _1,\theta _2,\theta _3|Y^{obs},T,M,X)$.

Posterior inference on the causal effects of interest can be obtained by computing the values of the constructed estimator within each MCMC iteration and summarizing their distribution across the posterior sample. Thus, in each iteration, we can construct a dataset consisting of the observed indication times, the simulated indication times, and all observed potential outcomes, and use these completed data to calculate an estimate of the treatment effect. Alternatively, we could specify a joint distribution for the potential outcomes $Y = (Y(0), Y(1))$ that we could then use to impute the missing potential outcomes $Y^{mis}$ in each iteration by drawing from the conditional distribution with density function $p(Y^{mis}|Y^{obs}, T, X, \theta$). Repeating this process over many such simulated datasets produces the approximate posterior distribution for all causal effects of interest. In the same way, posterior samples of $\theta$ can provide posterior estimates of the parameters that characterize the data-generating process; this is described in greater detail in Sect. 4.

1.2 Full conditionals

For the model described in Sect. 4.1, we employ the Gibbs sampler as the posterior sampling strategy. Using a data augmentation approach, we also let $\Psi _{it}=\theta _{it}+{\varvec{X}}_{it}\varvec{\beta } + \nu _{it}$ where $\nu _{it}\sim N(0,1)$ such that the indication times $T_i$ can be represented as $T_i=\inf \{t\in [0,K]:\Psi _{it} >0 \}$. Note that the latent variables $\Psi _{it}$ are conditionally independent across units i and over t given $\theta _{it}$ and $\varvec{\beta }$.

To begin, we set $j=0$ and draw initial values for the parameters $\Theta =(\rho , \varvec{\beta }, \delta _0, \delta _1)$ and the latent variables $\theta _{i,1:K}$ and $\Psi _{i,1:K}$ for all $i=1,\ldots ,n$. The latent variables $\Psi _{i,1:K}$ are then used to determine the initial values for the missing indication times $T_i$. For each iteration, we then proceed as follows:

• (a) Draw the latent variables $\Psi _{it}$ from the full conditional distribution given below, which is either an unrestricted normal density (when $T>t$) or a truncate normal density on the interval $(-\inf , 0]$ when $T<t$ or $(0,\inf )$ when $T=t$.

  $$\begin{aligned} p(\Psi _{1:K}|\cdot ) = \prod _{i=1}^N \prod _{t=1}^K 1(\Psi _{it}>0)^{T_i=t} 1(\Psi _{it}\le 0)^{T_i < t} \phi (\Psi _{it} - \theta _{it}+{\varvec{X}}_{it}\varvec{\beta }) \end{aligned}$$

  where $\phi (\cdot )$ denotes the probability density function of the standard normal distribution.

• (b) Since Eqs. (5) and (6) define a linear state space model, we sample $\theta _{i,1:K}$ using the Kalman filter. The forward conditional is given by:

  $$\begin{aligned} p(\theta _{1:K}|\cdot ) = \prod _{i=1}^N\prod _{t=1}^K \phi (\theta _{i,t}-\rho \theta _{i,t-1}) \end{aligned}$$

  where $\phi (\cdot )$ denotes the probability density function of the standard normal distribution

• (c) Next, draw the probability of assignment to treatment upon indication from the conditional distribution given by:

  $$\begin{aligned} p(\pi |\cdot ) = \prod _{i=1}^N \prod _{t=1}^K \left( \pi _{it}^{Z_i} (1-\pi _{it})^{1-Z_i}\right) ^{1(T_i=t)} \end{aligned}$$

• (d) Assuming a general multivariate normal prior for $\varvec{\beta }$ of the form $\varvec{\beta }\sim N(\varvec{\beta }_0,\Sigma _0)$, draw $\varvec{\beta }$ from the multivariate normal distribution $\varvec{\beta }|\cdot \sim N_p(\varvec{\beta }_1, \Sigma _1)$ where

  $$\begin{aligned} \varvec{\beta }_1=\Sigma _1\left( \Sigma _0^{-1}\varvec{\beta }_0 + \sum _{i=1}^N\sum _{t=1}^K {\varvec{X}}_{it}(\Psi _{it}-\theta _{it})\right) , \end{aligned}$$

  and

  $$\begin{aligned} \Sigma _1 = \left( \Sigma _0^{-1}+\sum _{i=1}^N \sum _{t=1}^K {\varvec{X}}_{it}{\varvec{X}}_{it}^T\right) ^{-1} \end{aligned}$$

• (e) Draw the autocorrelation parameter $\rho$ from the full conditional distribution given by the truncated normal distribution

  $$\begin{aligned} \rho | \cdot \sim N_{[-1,1]}\left( \frac{\sum _{i=1}^N\sum _{t=1}^K \theta _{i,t}\theta _{i,t-1}}{\sum _{i=1}^N\sum _{t=1}^K \theta _{i,t-1}^2}, \frac{1}{\sum _{i=1}^N\sum _{t=1}^K \theta _{i,t-1}^2}\right) \end{aligned}$$