Estimating the effect of a treatment on a given outcome, conditioned on a vector of covariates, is central in many applications. However, learning the impact of a treatment on a continuous temporal response, when the covariates suffer extensively from measurement error and even the timing of the treatments is uncertain, has not been addressed. We introduce a novel data-driven method that can estimate treatment-response trajectories in this challenging scenario. We model personalized treatment-response curves as a combination of parametric response functions, hierarchically sharing information across individuals, and a sparse Gaussian process for the baseline trend. Importantly, our model considers measurement error not only in treatment covariates, but also in treatment times, a problem which arises in practice for example when treatment information is based on self-reporting. In a challenging and timely problem of estimating the impact of diet on continuous blood glucose measurements, our model leads to significant improvements in estimation accuracy and prediction.
: In contrast to the vast body of research concerning data-driven methods for the prediction of a treatment response, very little is known about their performance when measurement error is present. Authors of @cite_27 study a method for inferring causal directions using EIV models, but they do not focus on the response conditioned on specific treatments. Regression on various student covariates, incorporating EIV, has been used to predict standardized test scores @cite_25 . In @cite_28 , the effect of measurement error on a binary treatment response is analyzed, underlining the devastating impact of ignoring such errors. A model for measurement errors has been used to quantify uncertainty in order to increase the confidence in detecting genuine treatment changes for liver metastases @cite_10 .
@cite_28 @cite_27 @cite_10 @cite_25
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Researchers in economics and other disciplines are often interested in the causal effect of a binary treatment on outcomes. Econometric methods used to estimate such effects are divided into one of two strands depending on whether they require the conditional independence assumption (i.e., independence of potential outcomes and treatment assignment conditional on a set of observable covariates). When this assumption holds, researchers now have a wide array of estimation techniques from which to choose. However, very little is known about their performance – both in absolute and relative terms – when measurement error is present. In this study, the performance of several estimators that require the conditional independence assumption, as well as some that do not, are evaluated in a Monte Carlo study. In all cases, the data-generating process is such that conditional independence holds with the 'real' data. However, measurement error is then introduced. Specifically, three types of measurement error are considered: (i) errors in treatment assignment, (ii) errors in the outcome, and (iii) errors in the vector of covariates. Recommendations for researchers are provided. A common strategy for estimating treatment effects in observational studies using individual student-level data is analysis of covariance (ANCOVA) or hierarchical variants of it, in which outcomes (often standardized test scores) are regressed on pretreatment test scores, other student characteristics, and treatment group indicators. Measurement error in the prior test scores, which typically is both large and heteroscedastic, is regularly overlooked in empirical analyses and may erode the ability of regression models to adjust for student factors and may result in biased treatment effect estimates. We develop extensions of method-of-moments, Simulation-Extrapolation, and latent regression approaches to correcting for measurement error using the conditional standard errors of measure of test scores, and demonstrate their effectiveness relative to simpler alternatives using both simulation and a case study of teacher value-added effect estimation using longitudinal data from a large suburban school district. In this paper we propose methods for estimating heterogeneity in causal effects in experimental and observational studies and for conducting hypothesis tests about the magnitude of differences in treatment effects across subsets of the population. We provide a data-driven approach to partition the data into subpopulations that differ in the magnitude of their treatment effects. The approach enables the construction of valid confidence intervals for treatment effects, even with many covariates relative to the sample size, and without “sparsity” assumptions. We propose an “honest” approach to estimation, whereby one sample is used to construct the partition and another to estimate treatment effects for each subpopulation. Our approach builds on regression tree methods, modified to optimize for goodness of fit in treatment effects and to account for honest estimation. Our model selection criterion anticipates that bias will be eliminated by honest estimation and also accounts for the effect of making additional splits on the variance of treatment effect estimates within each subpopulation. We address the challenge that the “ground truth” for a causal effect is not observed for any individual unit, so that standard approaches to cross-validation must be modified. Through a simulation study, we show that for our preferred method honest estimation results in nominal coverage for 90 confidence intervals, whereas coverage ranges between 74 and 84 for nonhonest approaches. Honest estimation requires estimating the model with a smaller sample size; the cost in terms of mean squared error of treatment effects for our preferred method ranges between 7–22 . Treatment effects can be estimated from observational data as the difference in potential outcomes. In this paper, we address the challenge of estimating the potential outcome when treatment-dose levels can vary continuously over time. Further, the outcome variable may not be measured at a regular frequency. Our proposed solution represents the treatment response curves using linear time-invariant dynamical systems---this provides a flexible means for modeling response over time to highly variable dose curves. Moreover, for multivariate data, the proposed method: uncovers shared structure in treatment response and the baseline across multiple markers; and, flexibly models challenging correlation structure both across and within signals over time. For this, we build upon the framework of multiple-output Gaussian Processes. On simulated and a challenging clinical dataset, we show significant gains in accuracy over state-of-the-art models.