Diagnostics for Confounding of Time-varying and Other Joint Exposures

Abstract

The effects of joint exposures (or exposure regimes) include those of adhering to assigned treatment versus placebo in a randomized controlled trial, duration of exposure in a cohort study, interactions between exposures, and direct effects of exposure, among others. Unlike the setting of a single point exposure (e.g., propensity score matching), there are few tools to describe confounding for joint exposures or how well a method resolves it. Investigators need tools that describe confounding in ways that are conceptually grounded and intuitive for those who read, review, and use applied research to guide policy. We revisit the implications of exchangeability conditions that hold in sequentially randomized trials, and the bias structure that motivates the use of g-methods, such as marginal structural models. From these, we develop covariate balance diagnostics for joint exposures that can (1) describe time-varying confounding, (2) assess whether covariates are predicted by prior exposures given their past, the indication for g-methods, and (3) describe residual confounding after inverse probability weighting. For each diagnostic, we present time-specific metrics that encompass a wide class of joint exposures, including regimes of multivariate time-varying exposures in censored data, with multivariate point exposures as a special case. We outline how to estimate these directly or with regression and how to average them over person-time. Using a simulated example, we show how these metrics can be presented graphically. This conceptually grounded framework can potentially aid the transparent design, analysis, and reporting of studies that examine joint exposures. We provide easy-to-use tools to implement it.

Figure 1

Causal Directed Acyclic Graphs describing unmeasured covariate U, measured covariates C(t), exposures A(t), and outcome Y for a hypothetical trial where exposure is randomized (a) at baseline only (b) at each time within levels of past exposure (c) at baseline only, with continuing data collection among the uncensored at time t i.e. S(t) = 0.

Figure 2

A trellised covariate balance plot for Diagnostic 1 (time varying confounding), indexed by measurement times for exposure (rows) and covariates (columns). Each subplot evaluates the balance of covariates C (i.e. L,M,N,O,P) at time t − k across levels of exposure A = 1 versus A = 0 at time t, for each pattern of exposure history through time t − 1 (dots). When every dot in a horizontal plane aligns at zero, that exposure measurement is not confounded by that covariate. When this holds for all exposure measurements (across all covariates), the exposure is statistically exogenous and there is no measured time-varying confounding. In the simulated data example shown here, imbalance was largest for the most recent covariates and decayed with increasing distance between exposure and covariate times; there was no confounding at t = 0 but the pattern of confounding was the same for t = 1 and t = 2; the least balanced covariates were L (higher among the exposed) and M (higher among the unexposed), and N was the least imbalanced. Note that here, all observed exposure regimes were examined.

Figure 3

Causal Directed Acyclic Graphs describing a hypothetical study with exposures A(t), measured covariates C(t), and unmeasured variables U^* and U, and also outcome Y. Each scenario represents a different structure of bias due to exposure-covariate feedback. In scenario (a) stratifying on C(1) will block the effect of A(0) and Y and open the non-causal path . In scenarios (b) and (c) stratifying on C(1) opens the non-causal pathway . In all scenarios, adjusting for C(1) by stratification leads to bias. Adjusting by a g-method e.g. removing the arrow from C(1) to A(1) does not induce bias. Note that in (c) there is unmeasured confounding by U^* from the path A(0) ← U^* → C(1) → Y and g-methods will not remove it. Because Diagnostic 2 would pick up this unmeasured confounding as an imbalance e.g. A(0) ← U^* → C(1), it is most interpretable when investigators are diagnosing covariate history sufficient enough to support exchangeability assumptions (i.e. no unmeasured confounding).

Figure 4

A trellised covariate balance plot for Diagnostic 2a (exposure-covariate feedback), indexed by measurement times for covariates (rows) and exposure (columns). Each subplot evaluates the balance of covariates C (i.e. L,M,N,O,P) at time t across levels of exposure A = 1 versus A = 0 at time t − k, for each pattern of exposure history through time t − k − 1 (dots). For Diagnostic 2b, the plot would pertain to a specific propensity score stratum and unlike Diagnostic 2a, the dots would not represent a specific exposure history. When all the dots in a horizontal plane align at zero, the covariate measurement is independent of every prior exposure and does not contribute to exposure-covariate feedback. When this holds for every covariate measurement, there is no exposure-covariate feedback. In the simulated data example, covariate measurements for L and M were associated with previous exposures: for L the associations were strong and positive; for M they were weaker and negative. In this simulated data, any causal analysis of exposure regimes would want to consider adjusting for L and M using g-methods. Note that here, all observed exposure regimes were examined.

Figure 5

Demonstration of Diagnostic 3 (residual imbalance in the weighted population) under various sources of residual confounding. Here, we present the balance for covariates and exposures measured at time t = 2. Scenario (a) correct specification of the exposure model. Scenario (b) mis-specification of the exposure model for A(t) by only including recent covariates C(t) and omitting covariate-covariate interactions. Scenarios (c) through (e) have correct specification of the exposure model but with random positivity violations built into the data-generating model for A(t) such that P[A(t)|Ā(t − 1), C̄(t)] = 1/10,000,000 when L(t) = 0 and O(t) = 1. In Scenario (c) the weights are not truncated. In (d) they are truncated to the 1st and 99th percentiles. In (e) they are truncated to the 10th and 90th percentiles. Scenarios (b) through (d) all contained residual confounding which appeared as imbalances for Diagnostic 3.

Figure 6

A trellised plot for Diagnostic 1 among the uncensored, after averaging over exposure history, time, and segments of distance. The panels are indexed by segments of distance between exposure and covariate measurement times (columns). The right panel reports the summary metrics that assess the average balance (adjusting for exposure history) between exposures and proximal covariates at one unit of distance or less: A(1) vs. C(0), A(2) vs. C(1), A(0) vs C(0), A(1) vs C(1), A(2) vs. C(2). The left panel reports the analogous balance metric at two units of distance: A(2) vs. C(0). The average balance across these pools of person-time appear similar, and reflect the same patterns described in Figure 2. Note that here, all observed exposure regimes among the uncensored were examined. Note also that, because censoring is present, this diagnostic should be repeated to assess statistical exogeneity for censoring at time t within levels of uncensored exposure history through time t − 1 (e.g. eFigure 5).