Bounds on Average and Quantile Treatment Effects on Duration Outcomes Under Censoring,Selection, and Noncompliance

Abstract

We consider the problem of assessing the effects of a treatment on duration outcomes using data from a randomized evaluation with noncompliance. For such settings, we derive nonparametric sharp bounds for average and quantile treatment effects addressing three pervasive problems simultaneously: self-selection into the spell of interest, endogenous censoring of the duration outcome, and noncompliance with the assigned treatment. Ignoring any of these issues could yield biased estimates of the effects. Notably, the proposed bounds do not impose the independent censoring assumption—which is commonly used to address censoring but is likely to fail in important settings—or exclusion restrictions to address endogeneity of censoring and selection. Instead, they employ monotonicity and stochastic dominance assumptions. To illustrate the use of these bounds we assess the effects of the Job Corps (JC) training program on its participants’ last complete employment spell duration. Our estimated bounds suggest that JC participation may increase the average duration of the last complete employment spell before week 208 after randomization by at least 5.6 log points (5.8%) for individuals who comply with their treatment assignment and experience a complete employment spell whether or not they enrolled in JC. The estimated quantile treatment effects suggest the impacts may be heterogeneous, and strengthen our conclusions based on the estimated average effects.     

The Minimum Wage

Do moderate increases in the minimum wage reduce employment? If not, do they nevertheless raise wages? To examine these questions, we apply techniques of time series analysis and systems estimation that are commonly used in macroeconomics and finance to five panels of data that contain between 11 and 34 low-wage industries. Our answers are “No” and “Yes,” respectively. We find that increases in the federal minimum wage between 1947 and 1997 have raised average wages in many of these industries, especially the lowest wage ones. The effect on employment, however, is mixed and typically nonsignificant, even when average wages have risen.     

Testing the Unconfoundedness Assumption via Inverse Probability Weighted Estimators of (L)ATT

We propose inverse probability weighted estimators for the local average treatment effect (LATE) and the local average treatment effect for the treated (LATT) under instrumental variable assumptions with covariates. We show that these estimators are asymptotically normal and efficient. When the (binary) instrument satisfies one-sided noncompliance, we propose a Durbin–Wu–Hausman-type test of whether treatment assignment is unconfounded conditional on some observables. The test is based on the fact that under one-sided noncompliance LATT coincides with the average treatment effect for the treated (ATT). We conduct Monte Carlo simulations to demonstrate, among other things, that part of the theoretical efficiency gain afforded by unconfoundedness in estimating ATT survives pretesting. We illustrate the implementation of the test on data from training programs administered under the Job Training Partnership Act in the United States. This article has online supplementary material.     

Controlled Direct and Mediated Effects: Definition,Identification and Bounds

Abstract. Results are given which provide bounds for controlled direct effects when nounmeasured confounding assumptions required for the identification of these effects do not hold. Previous results concerning bounds for controlled direct effects rely on monotonicity relationships between the treatment, mediator and the outcome themselves; the results presented in this article instead assume that monotonicity relationships hold between the unmeasured confounding variable or variables and the treatment, mediator and outcome. Whereas prior results give bounds that contain the null hypothesis of no direct effect, the results presented here will in many instances yield bounds that do not contain the null hypothesis of no direct effect. For contexts in which a set of variables intercepts all paths between a treatment and an outcome, it is possible to provide a definition for a controlled mediated effect. We discuss the identification of these controlled mediated effects; the bounds for controlled direct effects are applicable also to controlled mediated effects. An example is given to illustrate how the results in the article can be used to draw inferences about direct and mediated effects in the presence of unmeasured confounding variables.     

Using information on realized effects to determine prospective causal effects

The potential outcomes approach to causal inference postulates that each individual has a number of possibly latent outcomes, each of which would be observed under a different treatment. For any individual, some of these outcomes will be unobservable or counterfactual. Information about post-treatment characteristics sometimes allows statements about what would have happened if an individual or group with these characteristics had received a different treatment. These are statements about the realized effects of the treatment. Determining the likely effect of an intervention before making a decision involves inference about effects in populations defined only by characteristics observed before decisions about treatment are made. Information on realized effects can tighten bounds on these prospectively defined measures of the intervention effect. We derive formulae for the bounds and their sampling variances and illustrate these points with data from a hypothetical study of the efficacy of screening mammography.     

An Approach for Estimating Causal Effects in Randomized Trials with Noncompliance

We developed methods for estimating the causal risk difference and causal risk ratio in randomized trials with noncompliance. The developed estimator is unbiased under the assumption that biases due to noncompliance are identical between both treatment arms. The biases are defined as the difference or ratio between the expectations of potential outcomes for a group that received the test treatment and that for the control group in each randomly assigned group. Although the instrumental variable estimator yields an unbiased estimate under a sharp null hypothesis but may yield a biased estimate under a non-null hypothesis, the bias of the developed estimator does not depend on whether this hypothesis holds. Then the estimate of the causal effect from the developed estimator may have a smaller bias than that from the instrumental variable estimator when the treatment effect exists. There is not yet a standard method for coping with noncompliance, and thus it is important to evaluate estimates under different assumptions. The developed estimator can serve this purpose. Its application to a field trial for coronary heart disease is provided.     

Semiparametric estimation and inference for distributional and general treatment effects

Summary.  There is a large literature on methods of analysis for randomized trials with noncompliance which focuses on the effect of treatment on the average outcome. The paper considers evaluating the effect of treatment on the entire distribution and general functions of this effect. For distributional treatment effects, fully non-parametric and fully parametric approaches have been proposed. The fully non-parametric approach could be inefficient but the fully parametric approach is not robust to the violation of distribution assumptions. We develop a semiparametric instrumental variable method based on the empirical likelihood approach. Our method can be applied to general outcomes and general functions of outcome distributions and allows us to predict a subject's latent compliance class on the basis of an observed outcome value in observed assignment and treatment received groups. Asymptotic results for the estimators and likelihood ratio statistic are derived. A simulation study shows that our estimators of various treatment effects are substantially more efficient than the currently used fully non-parametric estimators. The method is illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices.     

Regression Discontinuity Designs With Sample Selection

This article extends the standard regression discontinuity (RD) design to allow for sample selection or missing outcomes. We deal with both treatment endogeneity and sample selection. Identification in this article does not require any exclusion restrictions in the selection equation, nor does it require specifying any selection mechanism. The results can therefore be applied broadly, regardless of how sample selection is incurred. Identification instead relies on smoothness conditions. Smoothness conditions are empirically plausible, have readily testable implications, and are typically assumed even in the standard RD design. We first provide identification of the “extensive margin” and “intensive margin” effects. Then based on these identification results and principle stratification, sharp bounds are constructed for the treatment effects among the group of individuals that may be of particular policy interest, that is, those always participating compliers. These results are applied to evaluate the impacts of academic probation on college completion and final GPAs. Our analysis reveals striking gender differences at the extensive versus the intensive margin in response to this negative signal on performance.     

Treatment Effects With Unobserved Heterogeneity: A Set Identification Approach

We propose the sharp identifiable bounds of the potential outcome distributions using panel data. We allow for the possibility that statistical randomization of treatment assignments is not achieved until unobserved heterogeneity is properly controlled for. We use certain stationarity assumptions to obtain the sharp bounds. Our approach allows for dynamic treatment decisions, where the current treatment decisions may depend on the past treatments or the past observed outcomes. As an empirical illustration, we study the effect of smoking during pregnancy on infant birthweight. We find that for the group of switchers the infant birthweight of a smoking mother is first-order stochastically dominated by that of a nonsmoking mother.     

基于面板数据的公务员工资溢出效应研究

为了考察中国公务员工资的增长是否会对本地其它行业工资产生溢出效应,以及这种溢出效应的大小在不同的行业类别和区域之间是否存在差异,利用中国31个省份在2003-2014年的有关面板数据,使用工具变量的方法,尽力消除公共部门工资和私人部门工资之间的逆向因果关系和影响,估计了中国公务员工资增长对本地其它非公共部门行业工资增长的溢出效应。研究发现,公务员工资每增加1%,会带来其它行业工资约0.741%的上升,其中对劳动力流动性较大以及与公共部门关联较密切行业的工资溢出效应较大。此外,中国公务员工资的溢出效应在不同的省份之间存在区域差异,东部省份公务员工资的溢出效应大于西部省份。     

On the Predictive Distributions of Outcome Gains in the Presence of an Unidentified Parameter

In this article we describe methods for obtaining the predictive distributions of outcome gains in the framework of a standard latent variable selection model. Although most previous work has focused on estimation of mean treatment parameters as the method for characterizing outcome gains from program participation, we show how the entire distributions associated with these gains can be obtained in certain situations. Although the out-of-sample outcome gain distributions depend on an unidentified parameter, we use the results of Koop and Poirier to show that learning can take place about this parameter through information contained in the identified parameters via a positive definiteness restriction on the covariance matrix. In cases where this type of learning is not highly informative, the spread of the predictive distributions depends more critically on the prior. We show both theoretically and in extensive generated data experiments how learning occurs, and delineate the sensitivity of our results to the prior specifications. We relate our analysis to three treatment parameters widely used in the evaluation literature—the average treatment effect, the effect of treatment on the treated, and the local average treatment effect—and show how one might approach estimation of the predictive distributions associated with these outcome gains rather than simply the estimation of mean effects. We apply these techniques to predict the effect of literacy on the weekly wages of a sample of New Jersey child laborers in 1903.     

A note on confidence bounds after fixed-sequence multiple tests

We are concerned with the problem of estimating the treatment effects at the effective doses in a dose-finding study. Under monotone dose-response, the effective doses can be identified through the estimation of the minimum effective dose, for which there is an extensive set of statistical tools. In particular, when a fixed-sequence multiple testing procedure is used to estimate the minimum effective dose, Hsu and Berger (1999) show that the confidence lower bounds for the treatment effects can be constructed without the need to adjust for multiplicity. Their method, called the dose-response method, is simple to use, but does not account for the magnitude of the observed treatment effects. As a result, the dose-response method will estimate the treatment effects at effective doses with confidence bounds invariably identical to the hypothesized value. In this paper, we propose an error-splitting method as a variant of the dose-response method to construct confidence bounds at the identified effective doses after a fixed-sequence multiple testing procedure. Our proposed method has the virtue of simplicity as in the dose-response method, preserves the nominal coverage probability, and provides sharper bounds than the dose-response method in most cases.     

Effect of heteroscedasticity between treatment groups on mixed‐effects models for repeated measures

Mixed‐effects models for repeated measures (MMRM) analyses using the Kenward‐Roger method for adjusting standard errors and degrees of freedom in an “unstructured” (UN) covariance structure are increasingly becoming common in primary analyses for group comparisons in longitudinal clinical trials. We evaluate the performance of an MMRM‐UN analysis using the Kenward‐Roger method when the variance of outcome between treatment groups is unequal. In addition, we provide alternative approaches for valid inferences in the MMRM analysis framework. Two simulations are conducted in cases with (1) unequal variance but equal correlation between the treatment groups and (2) unequal variance and unequal correlation between the groups. Our results in the first simulation indicate that MMRM‐UN analysis using the Kenward‐Roger method based on a common covariance matrix for the groups yields notably poor coverage probability (CP) with confidence intervals for the treatment effect when both the variance and the sample size between the groups are disparate. In addition, even when the randomization ratio is 1:1, the CP will fall seriously below the nominal confidence level if a treatment group with a large dropout proportion has a larger variance. Mixed‐effects models for repeated measures analysis with the Mancl and DeRouen covariance estimator shows relatively better performance than the traditional MMRM‐UN analysis method. In the second simulation, the traditional MMRM‐UN analysis leads to bias of the treatment effect and yields notably poor CP. Mixed‐effects models for repeated measures analysis fitting separate UN covariance structures for each group provides an unbiased estimate of the treatment effect and an acceptable CP. We do not recommend MMRM‐UN analysis using the Kenward‐Roger method based on a common covariance matrix for treatment groups, although it is frequently seen in applications, when heteroscedasticity between the groups is apparent in incomplete longitudinal data.     

The time series concept of invertibility

Box and JENKINS introduced the concept of invertibility for reasons which are argued to be largely irrelevant. However, the concept has some value since the boundary between invertible and ”strongly“ non-invertible moving average paramter sets, gives rise to bounds on the autocorrelations. As well as being of academic interest, these bounds may be useful for identifying processes.     

Likelihood methods in randomized trials with noncompliance and subsequent nonresponse

This article considers likelihood methods for estimating the causal effect of treatment assignment for a two-armed randomized trial assuming all-or-none treatment noncompliance and allowing for subsequent nonresponse. We first derive the observed data likelihood function as a closed form expression of the parameter given the observed data where both response and compliance state are treated as variables with missing values. Then we describe an iterative procedure which maximizes the observed data likelihood function directly to compute a maximum likelihood estimator (MLE) of the causal effect of treatment assignment. Closed form expressions at each iterative step are provided. Finally we compare the MLE with an alternative estimator where the probability distribution of the compliance state is estimated independent of the response and its missingness mechanism. Our work indicates that direct maximum likelihood inference is straightforward for this problem. Extensive simulation studies are provided to examine the finite sample performance of the proposed methods.     

A Note on the Role of the Propensity Score for Estimating Average Treatment Effects

Hahn [Hahn, J. (1998). On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica 66:315-331] derived the semiparametric efficiency bounds for estimating the average treatment effect (ATE) and the average treatment effect on the treated (ATET). The variance of ATET depends on whether the propensity score is known or unknown. Hahn attributes this to “dimension reduction.” In this paper, an alternative explanation is given: Knowledge of the propensity score improves upon the estimation of the distribution of the confounding variables.     

The effects of exchange rate regimes on economic growth: evidence from propensity score matching estimates

This is the first study that employs the propensity score matching framework to examine the average treatment effect of exchange rate regimes on economic growth. Previous studies examining the effects of different exchange regimes on growth often apply time series or panel data techniques and provide mixed results. This study employs a variety of non-parametric matching methods to address the self-selection problem, which potentially causes a bias in the traditional linear regressions. We evaluate the average treatment effect of the floating exchange rate regime on economic growth in 164 countries. Time period of the quasi experiment starts in 1970, capturing the collapse of the Bretton Woods fixed exchange rate commitment system. Results show that the average treatment effect of floating exchange rate regimes on economic growth is statistically insignificant. Verifying the results with the Rosenbaum's bounds, our findings are strong and robust. The research states that there is no evidence that employing a floating exchange rate regime compared to a fixed one leads to a higher economic growth for the countries that use this particular policy.     

Estimating the causal effects in randomized trials for survival data with a cure fraction and non compliance

We consider causal inference in randomized studies for survival data with a cure fraction and all-or-none treatment non compliance. To describe the causal effects, we consider the complier average causal effect (CACE) and the complier effect on survival probability beyond time t (CESP), where CACE and CESP are defined as the difference of cure rate and non cured subjects’ survival probability between treatment and control groups within the complier class. These estimands depend on the distributions of survival times in treatment and control groups. Given covariates and latent compliance type, we model these distributions with transformation promotion time cure model whose parameters are estimated by maximum likelihood. Both the infinite dimensional parameter in the model and the mixture structure of the problem create some computational difficulties which are overcome by an expectation-maximization (EM) algorithm. We show the estimators are consistent and asymptotically normal. Some simulation studies are conducted to assess the finite-sample performance of the proposed approach. We also illustrate our method by analyzing a real data from the Healthy Insurance Plan of Greater New York.     

Estimation of risk ratio in a noncompliance randomized clinical trial with trichotomous dose levels

In randomized clinical trials (RCTs), we may come across the situation in which some patients do not fully comply with their assigned treatment. For an experimental treatment with trichotomous levels, we derive the maximum likelihood estimator (MLE) of the risk ratio (RR) per level of dose increase in a RCT with noncompliance. We further develop three asymptotic interval estimators for the RR. To evaluate and compare the finite sample performance of these interval estimators, we employ Monte Carlo simulation. When the number of patients per treatment is large, we find that all interval estimators derived in this paper can perform well. When the number of patients is not large, we find that the interval estimator using Wald’s statistic can be liberal, while the interval estimator using the logarithmic transformation of the MLE can lose precision. We note that use of a bootstrap variance estimate in this case may alleviate these concerns. We further note that an interval estimator combining interval estimators using Wald’s statistic and the logarithmic transformation can generally perform well with respect to the coverage probability, and be generally more efficient than interval estimators using bootstrap variance estimates when RR>1. Finally, we use the data taken from a study of vitamin A supplementation to reduce mortality in preschool children to illustrate the use of these estimators.