An alternative empirical likelihood method in missing response problems and causal inference

Missing responses are common problems in medical, social, and economic studies. When responses are missing at random, a complete case data analysis may result in biases. A popular debias method is inverse probability weighting proposed by Horvitz and Thompson. To improve efficiency, Robins et al. proposed an augmented inverse probability weighting method. The augmented inverse probability weighting estimator has a double‐robustness property and achieves the semiparametric efficiency lower bound when the regression model and propensity score model are both correctly specified. In this paper, we introduce an empirical likelihood‐based estimator as an alternative to Qin and Zhang (2007). Our proposed estimator is also doubly robust and locally efficient. Simulation results show that the proposed estimator has better performance when the propensity score is correctly modeled. Moreover, the proposed method can be applied in the estimation of average treatment effect in observational causal inferences. Finally, we apply our method to an observational study of smoking, using data from the Cardiovascular Outcomes in Renal Atherosclerotic Lesions clinical trial. Copyright © 2016 John Wiley & Sons, Ltd.     

Doubly robust estimation and causal inference for recurrent event data

Many longitudinal databases record the occurrence of recurrent events over time. In this article, we propose a new method to estimate the average causal effect of a binary treatment for recurrent event data in the presence of confounders. We propose a doubly robust semiparametric estimator based on a weighted version of the Nelson-Aalen estimator and a conditional regression estimator under an assumed semiparametric multiplicative rate model for recurrent event data. We show that the proposed doubly robust estimator is consistent and asymptotically normal. In addition, a model diagnostic plot of residuals is presented to assess the adequacy of our proposed semiparametric model. We then evaluate the finite sample behavior of the proposed estimators under a number of simulation scenarios. Finally, we illustrate the proposed methodology via a database of circus artist injuries.     

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.     

Causal inference for community‐based multi‐layered intervention study

Estimating causal treatment effect for randomized controlled trials under post‐treatment confounding, that is, noncompliance and informative dropouts, is becoming an important problem in intervention/prevention studies when the treatment exposures are not completely controlled. When confounding is present in a study, the traditional intention‐to‐treat approach could underestimate the treatment effect because of insufficient exposure of treatment. In the recent two decades, many papers have been published to address such confounders to investigate the causal relationship between treatment and outcome of interest based on different modeling strategies. Most of the existing approaches, however, are suitable only for standard experiments. In this paper, we propose a new class of structural functional response model to address post‐treatment confounding in complex multi‐layered intervention studies within a longitudinal data setting. The new approach offers robust inference and is readily implemented. We illustrate and assess the performance of the proposed structural functional response model using both real and simulated data. Copyright © 2014 John Wiley & Sons, Ltd.     

Doubly robust estimation of the weighted average treatment effect for a target population

The weighted average treatment effect is a causal measure for the comparison of interventions in a specific target population, which may be different from the population where data are sampled from. For instance, when the goal is to introduce a new treatment to a target population, the question is what efficacy (or effectiveness) can be gained by switching patients from a standard of care (control) to this new treatment, for which the average treatment effect for the control estimand can be applied. In this paper, we propose two estimators based on augmented inverse probability weighting to estimate the weighted average treatment effect for a well-defined target population (ie, there exists a predefined target function of covariates that characterizes the population of interest, for example, a function of age to focus on elderly diabetic patients using samples from the US population). The first proposed estimator is doubly robust if the target function is known or can be correctly specified. The second proposed estimator is doubly robust if the target function has a linear dependence on the propensity score, which can be used to estimate the average treatment effect for the treated and the average treatment effect for the control. We demonstrate the properties of the proposed estimators through theoretical proof and simulation studies. We also apply our proposed methods in a comparison of glucagon-like peptide-1 receptor agonists therapy and insulin therapy among patients with type 2 diabetes, using the UK Clinical Practice Research Datalink data.     

Bootstrap confidence intervals for medical costs with censored observations

Medical costs data with administratively censored observations often arise in cost-effectiveness studies of treatments for life-threatening diseases. Mean of medical costs incurred from the start of a treatment until death or a certain time point after the implementation of treatment is frequently of interest. In many situations, due to the skewed nature of the cost distribution and non-uniform rate of cost accumulation over time, the currently available normal approximation confidence interval has poor coverage accuracy. In this paper, we propose a bootstrap confidence interval for the mean of medical costs with censored observations. In simulation studies, we show that the proposed bootstrap confidence interval had much better coverage accuracy than the normal approximation one when medical costs had a skewed distribution. When there is light censoring on medical costs (< or =25 per cent), we found that the bootstrap confidence interval based on the simple weighted estimator is preferred due to its simplicity and good coverage accuracy. For heavily censored cost data (censoring rate > or =30 per cent) with larger sample sizes (n > or =200), the bootstrap confidence intervals based on the partitioned estimator has superior performance in terms of both efficiency and coverage accuracy. We also illustrate the use of our methods in a real example.     

A new estimation with minimum trace of asymptotic covariance matrix for incomplete longitudinal data with a surrogate process

Missing data is a very common problem in medical and social studies, especially when data are collected longitudinally. It is a challenging problem to utilize observed data effectively. Many papers on missing data problems can be found in statistical literature. It is well known that the inverse weighted estimation is neither efficient nor robust. On the other hand, the doubly robust (DR) method can improve the efficiency and robustness. As is known, the DR estimation requires a missing data model (i.e., a model for the probability that data are observed) and a working regression model (i.e., a model for the outcome variable given covariates and surrogate variables). Because the DR estimating function has mean zero for any parameters in the working regression model when the missing data model is correctly specified, in this paper, we derive a formula for the estimator of the parameters of the working regression model that yields the optimally efficient estimator of the marginal mean model (the parameters of interest) when the missing data model is correctly specified. Furthermore, the proposed method also inherits the DR property. Simulation studies demonstrate the greater efficiency of the proposed method compared with the standard DR method. A longitudinal dementia data set is used for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimating treatment effects on healthcare costs under exogeneity: is there a ��magic bullet��?

Methods for estimating average treatment effects, under the assumption of no unmeasured confounders, include regression models; propensity score adjustments using stratification, weighting, or matching; and doubly robust estimators (a combination of both). Researchers continue to debate about the best estimator for outcomes such as health care cost data, as they are usually characterized by an asymmetric distribution and heterogeneous treatment effects,. Challenges in finding the right specifications for regression models are well documented in the literature. Propensity score estimators are proposed as alternatives to overcoming these challenges. Using simulations, we find that in moderate size samples (n= 5000), balancing on propensity scores that are estimated from saturated specifications can balance the covariate means across treatment arms but fails to balance higher-order moments and covariances amongst covariates. Therefore, unlike regression model, even if a formal model for outcomes is not required, propensity score estimators can be inefficient at best and biased at worst for health care cost data. Our simulation study, designed to take a 'proof by contradiction' approach, proves that no one estimator can be considered the best under all data generating processes for outcomes such as costs. The inverse-propensity weighted estimator is most likely to be unbiased under alternate data generating processes but is prone to bias under misspecification of the propensity score model and is inefficient compared to an unbiased regression estimator. Our results show that there are no 'magic bullets' when it comes to estimating treatment effects in health care costs. Care should be taken before naively applying any one estimator to estimate average treatment effects in these data. We illustrate the performance of alternative methods in a cost dataset on breast cancer treatment.     

Semi-parametric accelerated failure time regression analysis with application to interval-censored HIV/AIDS data

This paper demonstrates a way to investigate a potentially non-linear relationship between an interval-censored response variable and a continuously distributed explanatory variable. A potentially non-linear effect of a continuous explanatory variable on the response is incorporated into an accelerated failure time model, forming a partial linear model. A sieve maximum likelihood estimator (MLE) is suggested to simultaneously estimate all the parameters. The sieve MLE is shown to be asymptotically efficient and normally distributed. Simulation studies show that the proposed estimators for the scale and regression parameters are robust and efficient, and the estimator for the non-linear function is able to capture the shape of a variety of smooth non-linear functions. The model is applied to observational HIV data, where the response variable is the time to suppression of HIV viral load after initiation of antiretroviral therapy, and baseline viral load is investigated as a potentially non-linear effect.     

Interpretable inference on the mixed effect model with the Box–Cox transformation

We derived results for inference on parameters of the marginal model of the mixed effect model with the Box–Cox transformation based on the asymptotic theory approach. We also provided a robust variance estimator of the maximum likelihood estimator of the parameters of this model in consideration of the model misspecifications. Using these results, we developed an inference procedure for the difference of the model median between treatment groups at the specified occasion in the context of mixed effects models for repeated measures analysis for randomized clinical trials, which provided interpretable estimates of the treatment effect. From simulation studies, it was shown that our proposed method controlled type I error of the statistical test for the model median difference in almost all the situations and had moderate or high performance for power compared with the existing methods. We illustrated our method with cluster of differentiation 4 (CD4) data in an AIDS clinical trial, where the interpretability of the analysis results based on our proposed method is demonstrated. Copyright © 2017 John Wiley & Sons, Ltd.     

Model misspecification and robustness in causal inference: comparing matching with doubly robust estimation

In this paper, we compare the robustness properties of a matching estimator with a doubly robust estimator. We describe the robustness properties of matching and subclassification estimators by showing how misspecification of the propensity score model can result in the consistent estimation of an average causal effect. The propensity scores are covariate scores, which are a class of functions that removes bias due to all observed covariates. When matching on a parametric model (e.g., a propensity or a prognostic score), the matching estimator is robust to model misspecifications if the misspecified model belongs to the class of covariate scores. The implication is that there are multiple possibilities for the matching estimator in contrast to the doubly robust estimator in which the researcher has two chances to make reliable inference. In simulations, we compare the finite sample properties of the matching estimator with a simple inverse probability weighting estimator and a doubly robust estimator. For the misspecifications in our study, the mean square error of the matching estimator is smaller than the mean square error of both the simple inverse probability weighting estimator and the doubly robust estimators.     

On the conditional logistic estimator in two‐arm experimental studies with non‐compliance and before–after binary outcomes

The behavior of the conditional logistic estimator is analyzed under a causal model for two‐arm experimental studies with possible non‐compliance in which the effect of the treatment is measured by a binary response variable. We show that, when non‐compliance may only be observed in the treatment arm, the effect (measured on the logit scale) of the treatment on compliers and that of the control on non‐compliers can be identified and consistently estimated under mild conditions. The same does not happen for the effect of the control on compliers. A simple correction of the conditional logistic estimator is then proposed, which allows us to considerably reduce the bias in estimating this quantity and the causal effect of the treatment over control on compliers. A two‐step estimator results on the basis of which we can also set up a Wald test for the hypothesis of absence of a causal effect of the treatment. The asymptotic properties of the estimator are studied by exploiting the general theory on maximum likelihood estimation of misspecified models. Finite‐sample properties of the estimator and of the related Wald test are studied by simulation. The extension of the approach to the case of missing responses is also outlined. The approach is illustrated by an application to a dataset deriving from a study on the efficacy of a training course on the breast self examination practice. Copyright © 2010 John Wiley & Sons, Ltd.     

A doubly robust estimator for the attributable benefit of a treatment regime

The identification and study of treatment regimes (algorithms or policies for dictating treatments to patients) are a growing area of study in the statistical sciences. Many methods have been put forth to identify the ‘best’ or optimal treatment regime from observed data. Once the optimal treatment regime is identified, a secondary question of interest is to determine the public health impact of that health policy. Simply put, what is the benefit that can be attributed to using such a regime in practice? The attributable benefit of a treatment regime is a measure of the reduction in poor outcomes that would have been observed had the regime of interest been utilized. Methods for identifying the optimal treatment regime can use statistical modeling techniques which are susceptible to model misspecification in the identification of both the optimal treatment regime and its attributable benefit. Using notions from causal inference and building upon previous work, this paper identifies an estimator for attributable benefit that offers a second layer of protection in cases where an outcome regression model may be misspecified. The estimator is dubbed doubly robust in that it is unbiased for the true benefit if either a model for the outcome or a propensity model for treatment is correctly specified. Large sample properties are explored, and two sets of confidence intervals proposed. Simulation studies compare the proposed estimator with previous work, with a focus on model misspecification. The estimator is applied to real data to examine its utility in practice. Copyright © 2014 John Wiley & Sons, Ltd.     

A method for the analysis of repeated binary outcomes in randomized clinical trials with non-compliance

When analysing repeated binary data from randomized trials, the model-based approaches, such as generalized estimating equations, are frequently used. Such methods ignore compliance information and give the model-based intention-to-treat estimate of treatment effect. In this paper, the design-based (randomization-based) semi-parametric estimation procedure is given in the estimation of causal risk difference. The resulting risk difference estimator is interpreted as an extension of the instrumental variables estimator for a binary outcome which has the causal interpretation. Extension of the proposed method to stratified analysis is given for data from stratified randomization or meta-analysis. It yields a Mantel-Haenszel type risk difference estimator. As a special case of stratified analysis, the pattern mixture model which stratifies the data by pattern of missing data is performed. Application of the proposed method to a trial in which endpoints were the occurrences of fever over three courses is provided. The same ideas are applied to the causal risk ratio estimation.     

Estimation of treatment effect under non‐proportional hazards and conditionally independent censoring

In clinical trials with time‐to‐event outcomes, it is common to estimate the marginal hazard ratio from the proportional hazards model, even when the proportional hazards assumption is not valid. This is unavoidable from the perspective that the estimator must be specified a priori if probability statements about treatment effect estimates are desired. Marginal hazard ratio estimates under non‐proportional hazards are still useful, as they can be considered to be average treatment effect estimates over the support of the data. However, as many have shown, under non‐proportional hazard, the ‘usual’ unweighted marginal hazard ratio estimate is a function of the censoring distribution, which is not normally considered to be scientifically relevant when describing the treatment effect. In addition, in many practical settings, the censoring distribution is only conditionally independent (e.g., differing across treatment arms), which further complicates the interpretation. In this paper, we investigate an estimator of the hazard ratio that removes the influence of censoring and propose a consistent robust variance estimator. We compare the coverage probability of the estimator to both the usual Cox model estimator and an estimator proposed by Xu and O'Quigley (2000) when censoring is independent of the covariate. The new estimator should be used for inference that does not depend on the censoring distribution. It is particularly relevant to adaptive clinical trials where, by design, censoring distributions differ across treatment arms. Copyright © 2012 John Wiley & Sons, Ltd.     

Doubly robust inference for targeted minimum loss–based estimation in randomized trials with missing outcome data

Missing outcome data is a crucial threat to the validity of treatment effect estimates from randomized trials. The outcome distributions of participants with missing and observed data are often different, which increases bias. Causal inference methods may aid in reducing the bias and improving efficiency by incorporating baseline variables into the analysis. In particular, doubly robust estimators incorporate 2 nuisance parameters: the outcome regression and the missingness mechanism (ie, the probability of missingness conditional on treatment assignment and baseline variables), to adjust for differences in the observed and unobserved groups that can be explained by observed covariates. To consistently estimate the treatment effect, one of these nuisance parameters must be consistently estimated. Traditionally, nuisance parameters are estimated using parametric models, which often precludes consistency, particularly in moderate to high dimensions. Recent research on missing data has focused on data‐adaptive estimation to help achieve consistency, but the large sample properties of such methods are poorly understood. In this article, we discuss a doubly robust estimator that is consistent and asymptotically normal under data‐adaptive estimation of the nuisance parameters. We provide a formula for an asymptotically exact confidence interval under minimal assumptions. We show that our proposed estimator has smaller finite‐sample bias compared to standard doubly robust estimators. We present a simulation study demonstrating the enhanced performance of our estimators in terms of bias, efficiency, and coverage of the confidence intervals. We present the results of an illustrative example: a randomized, double‐blind phase 2/3 trial of antiretroviral therapy in HIV‐infected persons.     

A data-adaptive strategy for inverse weighted estimation of causal effects

In most nonrandomized observational studies, differences between treatment groups may arise not only due to the treatment but also because of the effect of confounders. Therefore, causal inference regarding the treatment effect is not as straightforward as in a randomized trial. To adjust for confounding due to measured covariates, the average treatment effect is often estimated by using propensity scores. Typically, propensity scores are estimated by logistic regression. More recent suggestions have been to employ nonparametric classification algorithms from machine learning. In this article, we propose a weighted estimator combining parametric and nonparametric models. Some theoretical results regarding consistency of the procedure are given. Simulation studies are used to assess the performance of the newly proposed methods relative to existing methods, and a data analysis example from the Surveillance, Epidemiology and End Results database is presented.     

Links between analysis of surrogate endpoints and endogeneity

There has been substantive interest in the assessment of surrogate endpoints in medical research. These are measures that could potentially replace ‘true’ endpoints in clinical trials and lead to studies that require less follow‐up. Recent research in the area has focused on assessments using causal inference frameworks. Beginning with a simple model for associating the surrogate and true endpoints in the population, we approach the problem as one of endogenous covariates. An instrumental variables estimator and general two‐stage algorithm are proposed. Existing surrogacy frameworks are then evaluated in the context of the model. In addition, we define an extended relative effect estimator as well as a sensitivity analysis for assessing what we term the treatment instrumentality assumption. A numerical example is used to illustrate the methodology. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimating marginal causal effects in a secondary analysis of case‐control data

When an initial case‐control study is performed, data can be used in a secondary analysis to evaluate the effect of the case‐defining event on later outcomes. In this paper, we study the example in which the role of the event is changed from a response variable to a treatment of interest. If the aim is to estimate marginal effects, such as average effects in the population, the sampling scheme needs to be adjusted for. We study estimators of the average effect of the treatment in a secondary analysis of matched and unmatched case‐control data where the probability of being a case is known. For a general class of estimators, we show the components of the bias resulting from ignoring the sampling scheme and demonstrate a design‐weighted matching estimator of the average causal effect. In simulations, the finite sample properties of the design‐weighted matching estimator are studied. Using a Swedish diabetes incidence register with a matched case‐control design, we study the effect of childhood onset diabetes on the use of antidepressant medication as an adult. Copyright © 2017 John Wiley & Sons, Ltd.     

Variance estimation for clustered recurrent event data with a small number of clusters

Often in biomedical studies, the event of interest is recurrent and within-subject events cannot usually be assumed independent. In semi-parametric estimation of the proportional rates model, a working independence assumption leads to an estimating equation for the regression parameter vector, with within-subject correlation accounted for through a robust (sandwich) variance estimator; these methods have been extended to the case of clustered subjects. We consider variance estimation in the setting where subjects are clustered and the study consists of a small number of moderate-to-large-sized clusters. We demonstrate through simulation that the robust estimator is quite inaccurate in this setting. We propose a corrected version of the robust variance estimator, as well as jackknife and bootstrap estimators. Simulation studies reveal that the corrected variance is considerably more accurate than the robust estimator, and slightly more accurate than the jackknife and bootstrap variance. The proposed methods are used to compare hospitalization rates between Canada and the U.S. in a multi-centre dialysis study. Copyright (c) 2005 John Wiley & Sons, Ltd.