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Assessing Time-Varying Causal Effect Moderation in Mobile Health

Audrey Boruvka, Daniel Almirall, Katie Witkiewitz, Susan A. Murphy

arXiv:1601.00237v3stat.ME

TL;DR

Mobile health data create a need for causal effect moderation methods accommodating repeated treatment, time-varying moderators, and treatment-influenced histories. The paper defines such effects using potential outcomes and develops centered, weighted estimation; it reports that common unweighted GEEs and non-independence working correlations can be biased, while the proposed approach targets consistent estimation under its assumptions.

  • Problem

    Time-varying treatment can make common GEE and random-effects approaches unable to consistently estimate causal moderated effects in intensive mobile health data.

  • Method

    The paper defines proximal and lagged moderated effects with potential outcomes and estimates them using centered, weighted least squares.

  • Results

    Unweighted GEEs and non-independence working correlations are not guaranteed to consistently estimate β∗ and can generally introduce bias.

  • Takeaways & Limitations

    Moderated mobile-health effects should be defined and estimated with methods that account for treatment-influenced histories and the treatment distribution.

Abstract

from arXiv · show

In mobile health interventions aimed at behavior change and maintenance, treatments are provided in real time to manage current or impending high risk situations or promote healthy behaviors in near real time. Currently there is great scientific interest in developing data analysis approaches to guide the development of mobile interventions. In particular data from mobile health studies might be used to examine effect moderators-i.e., individual characteristics, time-varying context or past treatment response that moderate the effect of current treatment on a subsequent response. This paper introduces a formal definition for moderated effects in terms of potential outcomes, a definition that is particularly suited to mobile interventions, where treatment occasions are numerous, individuals are not always available for treatment, and potential moderators might be influenced by past treatment. Methods for estimating moderated effects are developed and compared. The proposed approach is illustrated using BASICS-Mobile, a smartphone-based intervention designed to curb heavy drinking and smoking among college students.

1 Introduction

Mobile health studies offer intensive, time-varying data for examining when treatments work better or worse. The paper defines moderated treatment effects for this setting and develops a centered, weighted estimator.

  • mHealth interventions use mobile devices to deliver treatment and sense individuals’ current context.
  • Intensive longitudinal data can support analyses of individual, contextual, and treatment-response factors that strengthen or weaken current treatment effects.
  • Moderators can guide treatment delivery toward settings where interventions are most efficacious or suggest alternative strategies when benefit is limited.
  • Treatment effects may change over an intervention, making functions of time potential moderators.
  • The paper defines effects suited to numerous treatment occasions and moderators influenced by past treatment, then develops centered and weighted least squares estimation.
  • Time-varying treatment can make common GEE and random-effects approaches unable to consistently estimate causal treatment effects, whereas the proposed method provides unbiased estimation.

2 Proximal and Other Lagged Treatment Effects

The paper represents repeated treatments, responses, histories, and treatment-influenced moderators using potential outcomes. It defines proximal and lagged effects conditional on selected history summaries and a data-based treatment distribution.

  • Motivating Example: BASICS-Mobile delivers mindfulness or general health-information modules after afternoon and evening self-reports and examines subsequent smoking responses.
  • Notation and Data: Each treatment occasion is represented by A_t, the subsequent proximal response by Y_t+1, and time-varying context by X_t.
  • Notation and Data: Potential outcomes allow past treatment to influence later responses, measurements, and treatments, so histories include treatment-dependent quantities.
  • Moderated Treatment Effects: The proximal effect contrasts treatment versus no treatment on the next response, conditional on selected moderator summaries rather than the full history.
  • Moderated Treatment Effects: Lagged effects capture the impact of one additional treatment on responses k treatment occasions later, including treatment-dependent future actions.
  • Moderated Treatment Effects: The reference treatment regime assigns treatment with probabilities between zero and one corresponding to the treatment distribution in the data.

3 Estimation

The proposed estimator models lagged treatment effects with centered treatment indicators and probability-based weights. Under stated assumptions, it supports consistent estimation while remaining robust to misspecification of a nuisance model.

  • The paper models each lagged treatment effect as a function of selected history summaries and time, with separate models allowed for different lags.
  • Under sequential ignorability, randomization, invertibility, and moment conditions, the estimating equation yields an asymptotically normal estimator with consistently estimable variance.
  • The estimator centers treatment indicators and weights estimating functions so effects can be marginalized over history variables omitted from the treatment-effect model.
  • The estimator remains consistent when the working model for E[W_tY_t+k | H_t] is misspecified, providing robustness in high-dimensional histories.
  • Weights equal one when randomization probabilities depend only on selected summaries, and constant randomization simplifies estimation to unweighted regression with centered treatment.
  • Bias can result when the weight numerator depends on variables outside the selected summaries or when non-independence working correlations are used.

4 Availability

Because participants may be unavailable, treatment effects are defined through decision rules that prevent treatment delivery when unavailable. The resulting estimator conditions on availability and modifies the weighting accordingly.

  • Availability matters because treatment may be unreasonable, counterproductive, or unethical when participants cannot engage with the intervention.
  • BASICS-Mobile treated participants as available after fully completing a self-report, while availability in other interventions can be detected through sensors or device interaction.
  • Decision rules set delivered treatment to zero when unavailable and make potential responses depend on treatment-induced availability histories.
  • Repeated treatment may reduce later engagement and therefore lower subsequent availability.
  • Under limited availability, effects are defined among individuals available at occasion t, whose composition can change across occasions.
  • Estimation conditions treatment and working models on availability and replaces W_t with I_tW_t.

5 Implementation

The implementation uses GEE-compatible weighting and centering, with variance adjustments for estimated treatment probabilities and small samples.

  • The weighting and centering method can be implemented with standard GEE software using ItWt as prior weights and an independence working correlation matrix.
  • When n ≤50, the method applies a small-sample correction to the sandwich variance estimator.
  • Inference uses t or Hotelling’s T-squared critical values for univariate or multivariate tests.
  • Sandwich variance estimation must account for sampling error when treatment probabilities are estimated.
  • R code is provided for calculating standard errors.

6 Simulation Study

The simulations evaluate the proposed weighted and centered estimator against GEE alternatives under moderation, weighting, and correlation-structure scenarios. Results show that omitted moderation and inappropriate weighting choices can bias GEE or weighted estimates, whereas the proposed method can retain nominal coverage under its stated conditions.

  • The simulation study evaluates the proposed centering and weighting method using repeated samples and compares it with GEE estimators.Across the main experiments, results include point estimates, standard deviations, RMSE, and 95% confidence-interval coverage.
  • When an important moderator is omitted, GEE conditional-mean models are misspecified and their estimated marginal treatment effects are expected to be biased.The proposed method does not require a conditional-mean model for consistency under the stated assumptions.
  • The proposed weighted and centered estimator is unbiased across moderation strengths, while GEE generally fails to achieve nominal 95% coverage.GEE has RMSE at least as large as the proposed method in the reported comparisons.
  • 89% coverage occurred when the weighting numerator depended on the moderator even though the target effect was marginal.That numerator choice also exhibited bias, illustrating the limited ability to stabilize weights when relevant variables are omitted from S_kt.
  • 65% coverage occurred for the moderator-dependent numerator under the alternative working-correlation simulation.The reported result is associated with bias under the evaluated correlation structure.

7 Application

In BASICS-Mobile, the mindfulness message reduced next-reported smoking only for users with stable or decreased self-regulatory need. No overall lag-2 treatment effect was supported.

  • BASICS-Mobile was a pilot study with n = 28 and T = 28, and participants were considered available after completing the preceding self-report.
  • Treatment delivery followed a complex decision rule based mainly on preceding urge-to-smoke reports and early treatment occasions.The analysis modeled treatment probabilities using current smoking, urge, and an indicator for the first three occasions.
  • The proximal moderator was whether the user reported an increase in need to self-regulate over the preceding two self-reports.
  • The mindfulness message reduced average next-reported smoking when self-regulatory need was stable or decreased, with a 95% CI of −5.45 to −0.15 cigarettes per day.
  • No proximal treatment effect was apparent when self-regulatory need increased, and no overall lag-2 effect was supported.The lag-2 estimate had a 95% CI of −1.74 to 0.76 cigarettes per day.

8 Discussion

The paper defines treatment effects for intensive longitudinal mHealth data and proposes centered, weighted least squares estimation because common GEE and random-effects approaches can be biased. It also identifies scope boundaries and several directions for future work.

  • Contributions: The paper defines treatment effects suited to frequent measurements and treatment delivery when moderators may be influenced by past treatment.The definition is atypical in causal inference and targets mobile interventions.
  • Estimation: GEE and random-effects approaches can cause bias when estimating causal effects with time-varying treatment.Their conditional mean functions can coincide, and likelihood-based methods using induced correlation may generally be biased.
  • Estimation: The proposed centered and weighted least squares method provides unbiased estimation, whereas non-independence working correlations can introduce bias despite improving precision.The discussion specifically contrasts unbiased estimation with exchangeable or AR(1)-type working structures.
  • Future work: Future work should examine incorporating random effects, alternative lagged-effect definitions, penalized working models, and time-series or Markovian approaches.The paper considered continuous responses, binary treatment decisions, and longitudinal-style analyses.

Supplementary Material

The supplement connects the paper’s moderated effects to structural nested mean model treatment blips. It replaces the usual fixed reference regime with a stochastic regime matching the observed treatment distribution and derives lagged effects from blips.

  • Connection to SNMM: The supplement connects the paper’s lagged moderated effects to treatment blips from the structural nested mean model framework.It shows how treatment blips are additive on the conditional mean of the potential proximal response.
  • Reference regime: The reference treatment regime is stochastic and matches the conditional treatment distribution given observed history.For each occasion, the potential treatment follows the observed probability conditional on history.
  • Treatment blips: The treatment blip contrasts a fixed treatment with the stochastic treatment on the proximal response.The general blip is extended across treatment occasions and histories.
  • Derivation: Under consistency and sequential ignorability, the lag-k treatment effect can be expressed as an expected contrast of treatment blips.The derivation also invokes the stated causal-identification conditions and nuisance-function constraints.

A.2 Identification from Data

The supplement derives the lag-k treatment effect from potential outcomes under consistency, positivity, and sequential ignorability. The derivation uses observed treatment histories and conditional treatment indicators to connect potential and observed-data expressions.

  • Identification assumptions: The lag-k treatment effect is derived under consistency, positivity, and sequential ignorability.These conditions support the identification argument.
  • Identification steps: The derivation first uses consistency to replace potential histories with observed histories and moderators.It then applies sequential ignorability to treatment indicators conditional on history.
  • Identification steps: Sequential ignorability links potential outcomes under treatment to observed outcomes among individuals receiving treatment a.The argument applies this relation across future treatment occasions and lagged responses.

B Model Specification

The supplement explains why lag-specific treatment-effect models can be specified separately, while conditional-mean models across lags may impose restrictive compatibility requirements. Binary response predictors make parsimony and correctness difficult to achieve simultaneously.

  • Lag-specific specification: Treatment effects at different lags can be specified separately without constraining one another.This follows from the treatment-blip representation when the conditional mean of the potential response is not restricted to certain values.
  • Cross-lag compatibility: A lag-1 conditional-mean model can constrain the form of a lag-2 treatment effect when predictors are influenced by prior treatment.The supplement uses a binary predictor example to demonstrate this incompatibility.
  • Cross-lag compatibility: The induced lag-2 effect can be nonlinear in prior history, so the lag-1 response model and lag-2 effect model cannot both be correct in general.The nonlinearity arises because the conditional probability of the binary predictor is bounded between 0 and 1.
  • Model restrictions: Parsimony and correctness are difficult to achieve with binary or other non-continuous response predictors.Coherence across lags requires special settings such as multivariate normal predictors or conditional-mean centering, which impose strong restrictions.

C Large Sample Properties

The paper develops large-sample theory for a centered and weighted estimator of time-varying treatment effects, allowing observational treatments, estimated weights, and intermittent availability. Under stated modeling and ignorability assumptions, the estimator has asymptotically normal behavior, while misspecification changes the estimand through the numerator weights.

  • The supplement allows individuals to be unavailable and provides a general estimating-function framework for observational treatments under sequential ignorability.
  • The treatment and numerator-weight models require correct specification, regular estimating-equation behavior, finite moments, and treatment probabilities bounded away from zero and one.
  • The proposed estimator is obtained by solving the centered and weighted estimating equation with estimated or prespecified treatment and numerator-weight parameters.
  • Under these assumptions, the estimator has an asymptotically normal distribution with a variance-covariance matrix that can be consistently estimated.
  • When the modeling assumption is incorrect, the analyst’s choice of numerator weights determines the estimand, yielding a weighted projection rather than necessarily the intended effect.
  • With no moderators, the scalar estimand is the availability-weighted average of proximal treatment effects.

D Additional simulation results

Additional simulations vary sample sizes, time points, and working correlation structures. Across these settings, the weighted and centered estimator remained unbiased with standard errors close to Monte Carlo variability, while competing GEE procedures showed bias or worsening coverage in moderator-dependent settings.

  • The extended simulations vary sample sizes n = 30, 60 and time points T = 30, 50 across three experiments.
  • Across scenarios, average standard errors for the weighted and centered estimator closely approximated the Monte Carlo standard deviation.
  • When an important moderator exists, the weighted and centered estimator was unbiased across values of β∗11, whereas GEE-IND and GEE-AR(1) bias increased with moderator magnitude.
  • Coverage probabilities for GEE-IND and GEE-AR(1) generally worsened for larger n and T in the moderator simulation.
  • The weight-stabilization simulation produced results similar to those previously reported, with standard errors close to the Monte Carlo standard deviation.
  • Under a non-independent working correlation, coverage probability worsened for larger n, while independent working correlation retained close standard-error agreement with Monte Carlo variability.
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