Simultaneous Maximum Likelihood Estimation for Piecewise Linear Instrumental Variable Models.

Analysis of instrumental variables is an effective approach to dealing with endogenous variables and unmeasured confounding issue in causal inference. We propose using the piecewise linear model to fit the relationship between the continuous instrumental variable and the continuous explanatory variable, as well as the relationship between the continuous explanatory variable and the outcome variable, which generalizes the traditional linear instrumental variable models. The two-stage least square and limited information maximum likelihood methods are used for the simultaneous estimation of the regression coefficients and the threshold parameters. Furthermore, we study the limiting distribution of the estimators in the correctly specified and misspecified models and provide a robust estimation of the variance-covariance matrix. We illustrate the finite sample properties of the estimation in terms of the Monte Carlo biases, standard errors, and coverage probabilities via the simulated data. Our proposed model is applied to an education-salary data, which investigates the causal effect of children's years of schooling on estimated hourly wage with father's years of schooling as the instrumental variable.

Multiply robust estimation of causal quantile treatment effects.

In causal inference, often the interest lies in the estimation of the average causal effect. Other quantities such as the quantile treatment effect may be of interest as well. In this article, we propose a multiply robust method for estimating the marginal quantiles of potential outcomes by achieving mean balance in (a) the propensity score, and (b) the conditional distributions of potential outcomes. An empirical likelihood or entropy measure approach can be utilized for estimation instead of inverse probability weighting, which is known to be sensitive to the misspecification of the propensity score model. Simulation studies are conducted across different scenarios of correctness in both the propensity score models and the outcome models. Both simulation results and theoretical development indicate that our proposed estimator is consistent if any of the models are correctly specified. In the data analysis, we investigate the quantile treatment effect of mothers' smoking status on infants' birthweight.

Penalized regression procedures for variable selection in the potential outcomes framework.

A recent topic of much interest in causal inference is model selection. In this article, we describe a framework in which to consider penalized regression approaches to variable selection for causal effects. The framework leads to a simple 'impute, then select' class of procedures that is agnostic to the type of imputation algorithm as well as penalized regression used. It also clarifies how model selection involves a multivariate regression model for causal inference problems and that these methods can be applied for identifying subgroups in which treatment effects are homogeneous. Analogies and links with the literature on machine learning methods, missing data, and imputation are drawn. A difference least absolute shrinkage and selection operator algorithm is defined, along with its multiple imputation analogs. The procedures are illustrated using a well-known right-heart catheterization dataset.

Variable selection for causal mediation analysis using LASSO-based methods.

Causal mediation effect estimates can be obtained from marginal structural models using inverse probability weighting with appropriate weights. In order to compute weights, treatment and mediator propensity score models need to be fitted first. If the covariates are high-dimensional, parsimonious propensity score models can be developed by regularization methods including LASSO and its variants. Furthermore, in a mediation setup, more efficient direct or indirect effect estimators can be obtained by using outcome-adaptive LASSO to select variables for propensity score models by incorporating the outcome information. A simulation study is conducted to assess how different regularization methods can affect the performance of estimated natural direct and indirect effect odds ratios. Our simulation results show that regularizing propensity score models by outcome-adaptive LASSO can improve the efficiency of the natural effect estimators and by optimizing balance in the covariates, bias can be reduced in most cases. The regularization methods are then applied to MIMIC-III database, an ICU database developed by MIT.

Using conventional and machine learning propensity score methods to examine the effectiveness of 12-step group involvement following inpatient addiction treatment.

BACKGROUND: Continuing care following inpatient addiction treatment is an important component in the continuum of clinical services. Mutual help, including 12-step groups like Alcoholics Anonymous, is often recommended as a form of continuing care. However, the effectiveness of 12-step groups is difficult to establish using observational studies due to the risks of selection bias (or confounding). OBJECTIVE: To address this limitation, we used both conventional and machine learning-based propensity score (PS) methods to examine the effectiveness of 12-step group involvement following inpatient treatment on substance use over a 12-month period. METHODS: Using data from the Recovery Journey Project - a longitudinal, observational study - we followed an inpatient sample over 12-months post-treatment to assess the effect of 12-step involvement on substance use at 12-months (n = 254). Specifically, PS models were constructed based on 34 unbalanced confounders and four PS-based methods were applied: matching, inverse probability weighting (IPW), doubly robust (DR) with matching, and DR with IPW. RESULTS: Each PS-based method minimized the potential of confounding from unbalanced variables and demonstrated a significant effect (p < 0.001) between high 12-step involvement (i.e., defined as having a home group; having a sponsor; attending at least one meeting per week; and, being involved in service work) and a reduced likelihood of using substances over the 12-month period (odds ratios 0.11 to 0.32). CONCLUSIONS: PS-based methods effectively reduced potential confounding influences and provided robust evidence of a significant effect. Nonetheless, results should be considered in light of the relatively high attrition rate, potentially limiting their generalizability.

A model averaging approach for estimating propensity scores by optimizing balance.

Many approaches, including traditional parametric modeling and machine learning techniques, have been proposed to estimate propensity scores. This paper describes a new model averaging approach to propensity score estimation in which parametric and nonparametric estimates are combined to achieve covariate balance. Simulation studies are conducted across different scenarios varying in the degree of interactions and nonlinearities in the treatment model. The results show that, based on inverse probability weighting, the proposed propensity score estimator produces less bias and smaller standard errors than existing approaches. They also show that a model averaging approach with the objective of minimizing the average Kolmogorov-Smirnov statistic leads to the best performing IPW estimator. The proposed approach is also applied to a real data set in evaluating the causal effect of formula or mixed feeding versus exclusive breastfeeding on a child's body mass index Z-score at age 4. The data analysis shows that formula or mixed feeding is more likely to lead to obesity at age 4, compared to exclusive breastfeeding.

A Kernel-Based Metric for Balance Assessment.

An important goal in causal inference is to achieve balance in the covariates among the treatment groups. In this article, we introduce the concept of distributional balance preserving which requires the distribution of the covariates to be the same in different treatment groups. We also introduce a new balance measure called kernel distance, which is the empirical estimate of the probability metric defined in the reproducing kernel Hilbert spaces. Compared to the traditional balance metrics, the kernel distance measures the difference in the two multivariate distributions instead of the difference in the finite moments of the distributions. Simulation results show that the kernel distance is the best indicator of bias in the estimated casual effect compared to several commonly used balance measures. We then incorporate kernel distance into genetic matching, the state-of-the-art matching procedure and apply the proposed approach to analyze the Early Dieting in Girls study. The study indicates that mothers' overall weight concern increases the likelihood of daughters' early dieting behavior, but the causal effect is not significant.

Estimating controlled direct effects of restrictivefeeding practices in the 'Early dieting in girls' study.

In this article, we examine the causal effect of parental restrictive feeding practices on children's weight status. An important mediator is children's self-regulation status. Recent approaches interpret mediation effects based on the potential outcomes framework. Inverse probability weighting based on propensity scores are used to adjust for confounding and reduce the dimensionality of confounders simultaneously. We show that combining machine learning algorithms and logistic regression to estimate the propensity scores can be more accurate and efficient in estimating the controlled direct effects than using logistic regression alone. A data application shows that the causal effect of mother's restrictive feeding differs according to whether the daughter eats in the absence of hunger.

A Boosting Algorithm for Estimating Generalized Propensity Scores with Continuous Treatments.

In this article, we study the causal inference problem with a continuous treatment variable using propensity score-based methods. For a continuous treatment, the generalized propensity score is defined as the conditional density of the treatment-level given covariates (confounders). The dose-response function is then estimated by inverse probability weighting, where the weights are calculated from the estimated propensity scores. When the dimension of the covariates is large, the traditional nonparametric density estimation suffers from the curse of dimensionality. Some researchers have suggested a two-step estimation procedure by first modeling the mean function. In this study, we suggest a boosting algorithm to estimate the mean function of the treatment given covariates. In boosting, an important tuning parameter is the number of trees to be generated, which essentially determines the trade-off between bias and variance of the causal estimator. We propose a criterion called average absolute correlation coefficient (AACC) to determine the optimal number of trees. Simulation results show that the proposed approach performs better than a simple linear approximation or L2 boosting. The proposed methodology is also illustrated through the Early Dieting in Girls study, which examines the influence of mothers' overall weight concern on daughters' dieting behavior.