Comparison of various machine learning algorithms for estimating generalized propensity score

In this paper, we conducted a simulation study to evaluate the performance of four algorithms: multinomial logistic regression (MLR), bagging (BAG), random forest (RF), and gradient boosting (GB), for estimating generalized propensity score (GPS). Similar to the propensity score (PS), the ultimate goal of using GPS is to estimate unbiased average treatment effects (ATEs) in observational studies. We used the GPS estimates computed from these four algorithms with the generalized doubly robust (GDR) estimator to estimate ATEs in observational studies. We evaluated these ATE estimates in terms of bias and mean squared error (MSE). Simulation results show that overall, the GB algorithm produced the best ATE estimates based on these evaluation criteria. Thus, we recommend using the GB algorithm for estimating GPS in practice.     

Comparison of adjusted and unadjusted marginal probabilities for estimating binomial parameters in observational studies

In this study, we demonstrate how generalized propensity score estimators (Imbens’ weighted estimator, the propensity score weighted estimator and the generalized doubly robust estimator) can be used to calculate the adjusted marginal probabilities for estimating the three common binomial parameters: the risk difference (RD), the relative risk (RR), and the odds ratio (OR). We further conduct a simulation study to compare the estimated RD, RR, and OR using the adjusted and the unadjusted marginal probabilities in terms of the bias and mean-squared error (MSE). Although there is no clear winner in terms of the MSE for estimating RD, RR, and OR, simulation results surprisingly show thatthe unadjusted marginal probabilities produce the smallest bias compared with the adjusted marginal probabilities in most of the estimates. Hence, in conclusion, we recommend using the unadjusted marginal probabilities to estimate RD, RR, and OR, in practice.     

Multiple robustness estimation in causal inference

Abstract

Estimation of average treatment effect is crucial in causal inference for evaluation of treatments or interventions in biostatistics, epidemiology, econometrics, sociology. However, existing estimators require either a propensity score model, an outcome vector model, or both is correctly specified, which is difficult to verify in practice. In this paper, we allow multiple models for both the propensity score models and the outcome models, and then construct a weighting estimator based on observed data by using two-sample empirical likelihood. The resulting estimator is consistent if any one of those multiple models is correctly specified, and thus provides multiple protection on consistency. Moreover, the proposed estimator can attain the semiparametric efficiency bound when one propensity score model and one outcome vector model are correctly specified, without requiring knowledge of which models are correct. Simulations are performed to evaluate the finite sample performance of the proposed estimators. As an application, we analyze the data collected from the AIDS Clinical Trials Group Protocol 175.     

Nonparametric likelihood and doubly robust estimating equations for marginal and nested structural models

This article considers Robins's marginal and nested structural models in the cross‐sectional setting and develops likelihood and regression estimators. First, a nonparametric likelihood method is proposed by retaining a finite subset of all inherent and modelling constraints on the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. A profile likelihood is derived by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is intrinsically efficient based on the retained constraints and weakly locally efficient. Second, two regression estimators, named hat and tilde, are derived as first‐order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. The methods are illustrated by data analysis for an observational study on right heart catheterization. The Canadian Journal of Statistics 38: 609–632; 2010 © 2010 Statistical Society of Canada     

Using Inverse Probability Weighting Estimators to Evaluate Various Propensity Scores When Treatment Switching Exists

In this paper, we conduct a Monte Carlo simulation study to evaluate three propensity score (PS) scenarios for estimating an average treatment effect (ATE) in observational studies when treatment switching exists: (a) ignoring treatment switching in subjects (UPS), (b) removing subjects with treatment switching (RPS), and (c) adjusting for treatment switching effect (APS) with two inverse probability weighting estimators, IPW1 and IPW2. We evaluate these six estimators in terms of bias, mean squared error (MSE), empirical standard error (ESE), and coverage probability (CP) under various simulation scenarios. Simulation results show that the IPW2 estimator with RPS has relatively good performance.     

Correlation and efficiency of propensity score-based estimators for average causal effects

Propensity score-based estimators are commonly used to estimate causal effects in evaluation research. To reduce bias in observational studies, researchers might be tempted to include many, perhaps correlated, covariates when estimating the propensity score model. Taking into account that the propensity score is estimated, this study investigates how the efficiency of matching, inverse probability weighting, and doubly robust estimators change under the case of correlated covariates. Propositions regarding the large sample variances under certain assumptions on the data-generating process are given. The propositions are supplemented by several numerical large sample and finite sample results from a wide range of models. The results show that the covariate correlations may increase or decrease the variances of the estimators. There are several factors that influence how correlation affects the variance of the estimators, including the choice of estimator, the strength of the confounding toward outcome and treatment, and whether a constant or non-constant causal effect is present.     

Improved methods for moment restriction models with data combination and an application to two-sample instrumental variable estimation

Combining-100 information from multiple samples is often needed in biomedical and economic studies, but differences between these samples must be appropriately taken into account in the analysis of the combined data. We study the estimation for moment restriction models with data combined from two samples under an ignorability-type assumption while allowing for different marginal distributions of variables common to both samples. Suppose that an outcome regression (OR) model and a propensity score (PS) model are specified. By leveraging semi-parametric efficiency theory, we derive an augmented inverse probability-weighted (AIPW) estimator that is locally efficient and doubly robust with respect to these models. Furthermore, we develop calibrated regression and likelihood estimators that are not only locally efficient and doubly robust but also intrinsically efficient in achieving smaller variances than the AIPW estimator when the PS model is correctly specified but the OR model may be mispecified. As an important application, we study the two-sample instrumental variable problem and derive the corresponding estimators while allowing for incompatible distributions of variables common to the two samples. Finally, we provide a simulation study and an econometric application on public housing projects to demonstrate the superior performance of our improved estimators. The Canadian Journal of Statistics 48: 259–284; 2020 © 2019 Statistical Society of Canada     

Comparison of estimation methods for the finite population mean in simple random sampling: symmetric super-populations

In this paper, a new estimator combined estimator (CE) is proposed for estimating the finite population mean ¯ Y _N in simple random sampling assuming a long-tailed symmetric super-population model. The efficiency and robustness properties of the CE is compared with the widely used and well-known estimators of the finite population mean ¯ Y _N by Monte Carlo simulation. The parameter estimators considered in this study are the classical least squares estimator, trimmed mean, winsorized mean, trimmed L-mean, modified maximum-likelihood estimator, Huber estimator (W24) and the non-parametric Hodges–Lehmann estimator. The mean square error criteria are used to compare the performance of the estimators. We show that the CE is overall more efficient than the other estimators. The CE is also shown to be more robust for estimating the finite population mean ¯ Y _N , since it is insensitive to outliers and to misspecification of the distribution. We give a real life example.     

A Generalized Class of Ratio Type Exponential Estimators of Population Mean Under Linear Transformation of Auxiliary Variable

Recently, Shabbir and Gupta [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] defined a class of ratio type exponential estimators of population mean under a very specific linear transformation of auxiliary variable. In the present article, we propose a generalized class of ratio type exponential estimators of population mean in simple random sampling under a very general linear transformation of auxiliary variable. Shabbir and Gupta's [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] class of estimators is a particular member of our proposed class of estimators. It has been found that the optimal estimator of our proposed generalized class of estimators is always more efficient than almost all the existing estimators defined under the same situations. Moreover, in comparison to a few existing estimators, our proposed estimator becomes more efficient under some simple conditions. Theoretical results obtained in the article have been verified by taking a numerical illustration. Finally, a simulation study has been carried out to see the relative performance of our proposed estimator with respect to some existing estimators which are less efficient under certain conditions as compared to the proposed estimator.     

Doubly robust empirical likelihood inference in covariate-missing data problems

Missing covariate data occurs often in regression analysis. We study methods for estimating the regression coefficients in an assumed conditional mean function when some covariates are completely observed but other covariates are missing for some subjects. We adopt the semiparametric perspective of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866] on regression analyses with missing covariates, in which they pioneered the use of two working models, the working propensity score model and the working conditional score model. A recent approach to missing covariate data analysis is the empirical likelihood method of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503], which effectively combines unbiased estimating equations. In this paper, we consider an alternative likelihood approach based on the full likelihood of the observed data. This full likelihood-based method enables us to generate estimators for the vector of the regression coefficients that are (a) asymptotically equivalent to those of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the working propensity score model is correctly specified, and (b) doubly robust, like the augmented inverse probability weighting (AIPW) estimators of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Am Statist Assoc. 1994;89:846–866]. Thus, the proposed full likelihood-based estimators improve on the efficiency of the AIPW estimators when the working propensity score model is correct but the working conditional score model is possibly incorrect, and also improve on the empirical likelihood estimators of Qin, Zhang and Leung [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the reverse is true, that is, the working conditional score model is correct but the working propensity score model is possibly incorrect. In addition, we consider a regression method for estimation of the regression coefficients when the working conditional score model is correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866]. Finally, we compare the finite-sample performance of various estimators in a simulation study.     

Developing a restricted two-parameter Liu-type estimator: A comparison of restricted estimators in the binary logistic regression model

In the context of estimating regression coefficients of an ill-conditioned binary logistic regression model, we develop a new biased estimator having two parameters for estimating the regression vector parameter β when it is subjected to lie in the linear subspace restriction Hβ = h. The matrix mean squared error and mean squared error (MSE) functions of these newly defined estimators are derived. Moreover, a method to choose the two parameters is proposed. Then, the performance of the proposed estimator is compared to that of the restricted maximum likelihood estimator and some other existing estimators in the sense of MSE via a Monte Carlo simulation study. According to the simulation results, the performance of the estimators depends on the sample size, number of explanatory variables, and degree of correlation. The superiority region of our proposed estimator is identified based on the biasing parameters, numerically. It is concluded that the new estimator is superior to the others in most of the situations considered and it is recommended to the researchers.     

A study on the chain ratio-ratio-type exponential estimator for finite population variance

This paper considers the problem of estimating the population variance S²_y of the study variable y using the auxiliary information in sample surveys. We have suggested the (i) chain ratio-type estimator (on the lines of Kadilar and Cingi (2003)), (ii) chain ratio-ratio-type exponential estimator and their generalized version [on the lines of Singh and Pal (2015)] and studied their properties under large sample approximation. Conditions are obtained under which the proposed estimators are more efficient than usual unbiased estimator s²_y and Isaki (1893) ratio estimator. Improved version of the suggested class of estimators is also given along with its properties. An empirical study is carried out in support of the present study.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

General procedure for estimating the population mean using ranked set sampling

In this paper, we suggest a class of estimators for estimating the population mean ? of the study variable Y using information on X?, the population mean of the auxiliary variable X using ranked set sampling envisaged by McIntyre [A method of unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952), pp. 385–390] and developed by Takahasi and Wakimoto [On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968), pp. 1–31]. The estimator reported by Kadilar et al. [Ratio estimator for the population mean using ranked set sampling, Statist. Papers 50 (2009), pp. 301–309] is identified as a member of the proposed class of estimators. The bias and the mean-squared error (MSE) of the proposed class of estimators are obtained. An asymptotically optimum estimator in the class is identified with its MSE formulae. To judge the merits of the suggested class of estimators over others, an empirical study is carried out.     

Nonparametric estimation with left-truncated and right-censored data when the sample size before truncation is known

In this article, we consider nonparametric estimation of the survival function when the data are subject to left-truncation and right-censoring and the sample size before truncation is known. We propose two estimators. The first estimator is derived based on a self-consistent estimating equation. The second estimator is obtained by using the constrained expectation-maximization algorithm. Simulation results indicate that both estimators are more efficient than the product-limit estimator. When there is no censoring, the performance of the proposed estimators is compared with that of the estimator proposed by Li and Qin [Semiparametric likelihood-based inference for biased and truncated data when total sample size is known, J. R. Stat. Soc. B 60 (1998), pp. 243–254] via simulation study.     

A new method for estimating finite-population quantiles using auxiliary information

A computationally simple method for estimating finite-population quantiles in the presence of auxiliary information is proposed. An algorithm is also found for implementing related approaches for estimating quantiles, including that of Rao et al. (1990), obtained from inverting difference-type estimators of the distribution function. The proposed estimation procedure can be seen as a one-step iteration of the suggested algorithm and is asymptotically equivalent to the limiting estimator. In particular, the proposed method yields a simple and efficient way of approximating Rao et al.'s estimator. Simulation studies based on two real populations show that the approximation can be very satisfactory even for small to moderate samples.     

A comparison of doubly robust estimators of the mean with missing data

We consider data with a continuous outcome that is missing at random and a fully observed set of covariates. We compare by simulation a variety of doubly-robust (DR) estimators for estimating the mean of the outcome. An estimator is DR if it is consistent when either the regression model for the mean function or the propensity to respond is correctly specified. Performance of different methods is compared in terms of root mean squared error of the estimates and width and coverage of confidence intervals or posterior credibility intervals in repeated samples. Overall, the DR methods tended to yield better inference than the incorrect model when either the propensity or mean model is correctly specified, but were less successful for small sample sizes, where the asymptotic DR property is less consequential. Two methods tended to outperform the other DR methods: penalized spline of propensity prediction [Little RJA, An H. Robust likelihood-based analysis of multivariate data with missing values. Statist Sinica. 2004;14:949–968] and the robust method proposed in [Cao W, Tsiatis AA, Davidian M. Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data. Biometrika. 2009;96:723–734].     

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter ln⁡σ of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.     

New types of shrinkage estimators of Poisson means under the normalized squared error loss

In estimating p( ? 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions.     

A restricted Liu estimator for binary regression models and its application to an applied demand system

In this article, we propose a restricted Liu regression estimator (RLRE) for estimating the parameter vector, β, in the presence of multicollinearity, when the dependent variable is binary and it is suspected that β may belong to a linear subspace defined by Rβ?=?r. First, we investigate the mean squared error (MSE) properties of the new estimator and compare them with those of the restricted maximum likelihood estimator (RMLE). Then we suggest some estimators of the shrinkage parameter, and a simulation study is conducted to compare the performance of the different estimators. Finally, we show the benefit of using RLRE instead of RMLE when estimating how changes in price affect consumer demand for a specific product.