Linear mixed models with endogenous covariates: modeling sequential treatment effects with application to a mobile health study

Mobile health is a rapidly developing field in which behavioral treatments are delivered to individuals via wearables or smartphones to facilitate health-related behavior change. Micro-randomized trials (MRT) are an experimental design for developing mobile health interventions. In an MRT the treatments are randomized numerous times for each individual over course of the trial. Along with assessing treatment effects, behavioral scientists aim to understand between-person heterogeneity in the treatment effect. A natural approach is the familiar linear mixed model. However, directly applying linear mixed models is problematic because potential moderators of the treatment effect are frequently endogenous---that is, may depend on prior treatment. We discuss model interpretation and biases that arise in the absence of additional assumptions when endogenous covariates are included in a linear mixed model. In particular, when there are endogenous covariates, the coefficients no longer have the customary marginal interpretation. However, these coefficients still have a conditional-on-the-random-effect interpretation. We provide an additional assumption that, if true, allows scientists to use standard software to fit linear mixed model with endogenous covariates, and person-specific predictions of effects can be provided. As an illustration, we assess the effect of activity suggestion in the HeartSteps MRT and analyze the between-person treatment effect heterogeneity

Performance guarantees for individualized treatment rules

Because many illnesses show heterogeneous response to treatment, there is increasing interest in individualizing treatment to patients [Arch. Gen. Psychiatry 66 (2009) 128--133]. An individualized treatment rule is a decision rule that recommends treatment according to patient characteristics. We consider the use of clinical trial data in the construction of an individualized treatment rule leading to highest mean response. This is a difficult computational problem because the objective function is the expectation of a weighted indicator function that is nonconcave in the parameters. Furthermore, there are frequently many pretreatment variables that may or may not be useful in constructing an optimal individualized treatment rule, yet cost and interpretability considerations imply that only a few variables should be used by the individualized treatment rule. To address these challenges, we consider estimation based on $l_1$-penalized least squares. This approach is justified via a finite sample upper bound on the difference between the mean response due to the estimated individualized treatment rule and the mean response due to the optimal individualized treatment rule.Comment: Published in at http://dx.doi.org/10.1214/10-AOS864 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Observed Information in Semiparametric Models

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90945/1/observed_information_semi-parametric_models.pdf6512512