Propensity score prediction for electronic healthcare databases using super learner and high-dimensional propensity score methods

ABSTRACT

The optimal learner for prediction modeling varies depending on the underlying data-generating distribution. Super Learner (SL) is a generic ensemble learning algorithm that uses cross-validation to select among a ‘library’ of candidate prediction models. While SL has been widely studied in a number of settings, it has not been thoroughly evaluated in large electronic healthcare databases that are common in pharmacoepidemiology and comparative effectiveness research. In this study, we applied and evaluated the performance of SL in its ability to predict the propensity score (PS), the conditional probability of treatment assignment given baseline covariates, using three electronic healthcare databases. We considered a library of algorithms that consisted of both nonparametric and parametric models. We also proposed a novel strategy for prediction modeling that combines SL with the high-dimensional propensity score (hdPS) variable selection algorithm. Predictive performance was assessed using three metrics: the negative log-likelihood, area under the curve (AUC), and time complexity. Results showed that the best individual algorithm, in terms of predictive performance, varied across datasets. The SL was able to adapt to the given dataset and optimize predictive performance relative to any individual learner. Combining the SL with the hdPS was the most consistent prediction method and may be promising for PS estimation and prediction modeling in electronic healthcare databases.     

A model-based concordance-type index for evaluating the added predictive ability of novel risk factors and markers in the logistic regression models

ABSTRACT

The Concordance statistic (C-statistic) is commonly used to assess the predictive performance (discriminatory ability) of logistic regression model. Although there are several approaches for the C-statistic, their performance in quantifying the subsequent improvement in predictive accuracy due to inclusion of novel risk factors or biomarkers in the model has been extremely criticized in literature. This paper proposed a model-based concordance-type index, C_K, for use with logistic regression model. The C_K and its asymptotic sampling distribution is derived following Gonen and Heller's approach for Cox PH model for survival data but taking necessary modifications for use with binary data. Unlike the existing C-statistics for logistic model, it quantifies the concordance probability by taking the difference in the predicted risks between two subjects in a pair rather than ranking them and hence is able to quantify the equivalent incremental value from the new risk factor or marker. The simulation study revealed that the C_K performed well when the model parameters are correctly estimated for large sample and showed greater improvement in quantifying the additional predictive value from the new risk factor or marker than the existing C-statistics. Furthermore, the illustration using three datasets supports the findings from simulation study.     

On the predictive performance of the almost unbiased Liu estimator

ABSTRACT

Regression models are usually used in forecasting (predicting) unknown values of the response variable y. This article considers the predictive performance of the almost unbiased Liu estimator compared to the ordinary least-squares estimator, principal component regression estimator, and Liu estimator. Finally, we present a numerical example to explain the theoretical results and we obtain a region where the almost unbiased Liu estimator is uniformly superior to the ordinary least-squares estimator, principal component regression estimator, and Liu estimator.     

The p-values for one-sided hypothesis testing in univariate linear calibration

In this article, we focus on the one-sided hypothesis testing for the univariate linear calibration, where a normally distributed response variable and an explanatory variable are involved. The observations of the response variable corresponding to known values of the explanatory variable are used to make inferences on a single unknown value of the explanatory variable. We apply the generalized inference to the calibration problem, and take the generalized p-value as the test statistic to develop a new p-value for one-sided hypothesis testing, which we refer to as the one-sided posterior predictive p-value. The behavior of the one-sided posterior predictive p-value is numerically compared with that of the generalized p-value, and simulations show that the proposed p-value is quite satisfactory in the frequentist performance.     

Simultaneous tests for homogeneity of two zero-inflated (beta) populations

ABSTRACT

Motivated by an example in marine science, we use Fisher’s method to combine independent likelihood ratio tests (LRTs) and asymptotic independent score tests to assess the equivalence of two zero-inflated Beta populations (mixture distributions with three parameters). For each test, test statistics for the three individual parameters are combined into a single statistic to address the overall difference between the two populations. We also develop non parametric and semiparametric permutation-based tests for simultaneously comparing two or three features of unknown populations. Simulations show that the likelihood-based tests perform well for large sample sizes and that the statistics based on combining LRT statistics outperforms the ones based on combining score test statistics. The permutation-based tests have overall better performance in terms of both power and type I error rate. Our methods are easy to implement and computationally efficient, and can be expanded to more than two populations and to other multiple parameter families. The permutation tests are entirely generic and can be useful in various applications dealing with zero (or other) inflation.     

Nonparametric predictive inference for diagnostic test thresholds

Abstract

Measuring the accuracy of diagnostic tests is crucial in many application areas including medicine, machine learning and credit scoring. The receiver operating characteristic (ROC) curve and surface are useful tools to assess the ability of diagnostic tests to discriminate between ordered classes or groups. To define these diagnostic tests, selecting the optimal thresholds that maximize the accuracy of these tests is required. One procedure that is commonly used to find the optimal thresholds is by maximizing what is known as Youden’s index. This article presents nonparametric predictive inference (NPI) for selecting the optimal thresholds of a diagnostic test. NPI is a frequentist statistical method that is explicitly aimed at using few modeling assumptions, enabled through the use of lower and upper probabilities to quantify uncertainty. Based on multiple future observations, the NPI approach is presented for selecting the optimal thresholds for two-group and three-group scenarios. In addition, a pairwise approach has also been presented for the three-group scenario. The article ends with an example to illustrate the proposed methods and a simulation study of the predictive performance of the proposed methods along with some classical methods such as Youden index. The NPI-based methods show some interesting results that overcome some of the issues concerning the predictive performance of Youden’s index.     

ON THE EVALUATION OF CONTROL CHART LIMITS BASED ON PREDICTIVE DISTRIBUTIONS

ABSTRACT

We propose a Bayesian approach to obtaining control charts when there is parameter uncertainty. Our approach consists of two stages, (i) construction of the control chart where we use a predictive distribution based on a Bayesian approach to derive the rejection region, and (ii) evaluation of the control chart where we use a sampling theory approach to examine the performance of the control chart under various hypothetical specifications for the data generation model.     

Matching Using Sufficient Dimension Reduction for Causal Inference

ABSTRACT

To estimate causal treatment effects, we propose a new matching approach based on the reduced covariates obtained from sufficient dimension reduction. Compared with the original covariates and the propensity score, which are commonly used for matching in the literature, the reduced covariates are nonparametrically estimable and are effective in imputing the missing potential outcomes, under a mild assumption on the low-dimensional structure of the data. Under the ignorability assumption, the consistency of the proposed approach requires a weaker common support condition. In addition, researchers are allowed to employ different reduced covariates to find matched subjects for different treatment groups. We develop relevant asymptotic results and conduct simulation studies as well as real data analysis to illustrate the usefulness of the proposed approach.     

TESTING FOR LINEAR DEPENDENCE IN HEAVY-TAILED DATA

ABSTRACT

We use the sample covariation to develop tests for lagged linear dependence in symmetric time series data. We propose tests for both finite and infinite variance processes. The finite sample performance of the tests is investigated using simulated data and compared to tests based on the von Neumann ratio.     

A note on Bayesian estimation of traffic intensity in single-server Markovian queues

ABSTRACT

In queuing theory, a major interest of researchers is studying the behavior and formation process and analyzing the performance characteristics of queues, particularly the traffic intensity, which is defined as the ratio between the arrival rate and the service rate. How these parameters can be estimated using some statistical inferential method is the mathematical problem treated here. This article aims to obtain better Bayesian estimates for the traffic intensity of M/M/1 queues, which, in Kendall notation, stand for Markovian single-server infinity queues. The Jeffreys prior is proposed to obtain the posterior and predictive distributions of some parameters of interest. Samples are obtained through simulation and some performance characteristics are analyzed. It is observed from the Bayes factor that Jeffreys prior is competitive, among informative and non-informative prior distributions, and presents the best performance in many of the cases tested.     

Bayesian Estimation and Model Choice in Item Response Models

Item response models are essential tools for analyzing results from many educational and psychological tests. Such models are used to quantify the probability of correct response as a function of unobserved examinee ability and other parameters explaining the difficulty and the discriminatory power of the questions in the test. Some of these models also incorporate a threshold parameter for the probability of the correct response to account for the effect of guessing the correct answer in multiple choice type tests. In this article we consider fitting of such models using the Gibbs sampler. A data augmentation method to analyze a normal-ogive model incorporating a threshold guessing parameter is introduced and compared with a Metropolis-Hastings sampling method. The proposed method is an order of magnitude more efficient than the existing method. Another objective of this paper is to develop Bayesian model choice techniques for model discrimination. A predictive approach based on a variant of the Bayes factor is used and compared with another decision theoretic method which minimizes an expected loss function on the predictive space. A classical model choice technique based on a modified likelihood ratio test statistic is shown as one component of the second criterion. As a consequence the Bayesian methods proposed in this paper are contrasted with the classical approach based on the likelihood ratio test. Several examples are given to illustrate the methods.     

Predictive Inference and Scientific Reproducibility

ABSTRACT

Most statistical analyses use hypothesis tests or estimation about parameters to form inferential conclusions. I think this is noble, but misguided. The point of view expressed here is that observables are fundamental, and that the goal of statistical modeling should be to predict future observations, given the current data and other relevant information. Further, the prediction of future observables provides multiple advantages to practicing scientists, and to science in general. These include an interpretable numerical summary of a quantity of direct interest to current and future researchers, a calibrated prediction of what’s likely to happen in future experiments, a prediction that can be either “corroborated” or “refuted” through experimentation, and avoidance of inference about parameters; quantities that exists only as convenient indices of hypothetical distributions. Finally, the predictive probability of a future observable can be used as a standard for communicating the reliability of the current work, regardless of whether confirmatory experiments are conducted. Adoption of this paradigm would improve our rigor for scientific accuracy and reproducibility by shifting our focus from “finding differences” among hypothetical parameters to predicting observable events based on our current scientific understanding.     

Regression with fractional polynomials when interactions are erroneously omitted

We propose a new summary tool, so-called average predictive comparison (APC), which summarizes the effect of a particular predictor in a context of regression. Different from the definition in our earlier work (Liu and Gustafson, 2008), the new definition allows a pointwise evaluation of a predictor's effect for any given value of this predictor. We employ this summary tool to examine the consequence of erroneously omitting interactions in regression models. To be able to involve curved relationships between a response variable and predictors, we consider fractional polynomial regression models (Royston and Altman, 1994). We derive the asymptotic properties of the APC estimates under a general setting with p(≥2)$p(\ge 2)$ predictors involved. In particular, when there are only two predictors of interest, we find out that the APC estimator is robust to the model misspecification under some certain conditions. We illustrate the application of the proposed summary tool via a real data example. We also conduct simulation experiments to further check the performance of the APC estimates.     

Identification of drug‐induced toxicity biomarkers for treatment determination

Drug‐induced organ toxicity (DIOT) that leads to the removal of marketed drugs or termination of candidate drugs has been a leading concern for regulatory agencies and pharmaceutical companies. In safety studies, the genomic assays are conducted after the treatment so that drug‐induced adverse effects can occur. Two types of biomarkers are observed: biomarkers of susceptibility and biomarkers of response. This paper presents a statistical model to distinguish two types of biomarkers and procedures to identify susceptible subpopulations. The biomarkers identified are used to develop classification model to identify susceptible subpopulation. Two methods to identify susceptibility biomarkers were evaluated in terms of predictive performance in subpopulation identification, including sensitivity, specificity, and accuracy. Method 1 considered the traditional linear model with a variable‐by‐treatment interaction term, and Method 2 considered fitting a single predictor variable model using only treatment data. Monte Carlo simulation studies were conducted to evaluate the performance of the two methods and impact of the subpopulation prevalence, probability of DIOT, and sample size on the predictive performance. Method 2 appeared to outperform Method 1, which was due to the lack of power for testing the interaction effect. Important statistical issues and challenges regarding identification of preclinical DIOT biomarkers were discussed. In summary, identification of predictive biomarkers for treatment determination highly depends on the subpopulation prevalence. When the proportion of susceptible subpopulation is 1% or less, a very large sample size is needed to ensure observing sufficient number of DIOT responses for biomarker and/or subpopulation identifications. Copyright © 2015 John Wiley & Sons, Ltd.     

PREDICTION INTERVALS FOR SUPPORT VECTOR MACHINE REGRESSION

ABSTRACT

The support vector machine (SVM), first developed by Vapnik and his group at AT&T Bell Laboratories, is being used as a new technique for regression and classification problems. In this paper we present an approach to estimating prediction intervals for SVM regression based on posterior predictive densities. Furthermore, the method is illustrated with a data example.     

Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs

ABSTRACT

This study develops methods for conducting uniform inference on quantile treatment effects for sharp regression discontinuity designs. We develop a score test for the treatment significance hypothesis and Wald-type tests for the hypotheses related to treatment significance, homogeneity, and unambiguity. The bias from the nonparametric estimation is studied in detail. In particular, we show that under some conditions, the asymptotic distribution of the score test is unaffected by the bias, without under-smoothing. For situations where the conditions can be restrictive, we incorporate a bias correction into the Wald tests and account for the estimation uncertainty. We also provide a procedure for constructing uniform confidence bands for quantile treatment effects. As an empirical application, we use the proposed methods to study the effect of cash-on-hand on unemployment duration. The results reveal pronounced treatment heterogeneity and also emphasize the importance of considering the long-term unemployed.     

Bayesian predictive distribution for a Poisson model with a parametric restriction

Abstract

Predictive probability estimation for a Poisson distribution is addressed when the parameter space is restricted. The Bayesian predictive probability against the prior on the restricted space is compared with the non-restricted Bayes predictive probability. It is shown that the former predictive probability dominates the latter under some conditions when the predictive probabilities are evaluated by the risk function relative to the Kullback-Leibler divergence. This result is proved by first showing the corresponding dominance result for estimating the restricted parameter and then translating it into the framework of predictive probability estimation.     

A semiparametric Bayesian approach to simplex regression model with heterogeneous dispersion

ABSTRACT

Simplex regression model is often employed to analyze continuous proportion data in many studies. In this paper, we extend the assumption of a constant dispersion parameter (homogeneity) to varying dispersion parameter (heterogeneity) in Simplex regression model, and present the B-spline to approximate the smoothing unknown function within the Bayesian framework. A hybrid algorithm combining the block Gibbs sampler and the Metropolis-Hastings algorithm is presented for sampling observations from the posterior distribution. The procedures for computing model comparison criteria such as conditional predictive ordinate statistic, deviance information criterion, and averaged mean squared error are presented. Also, we develop a computationally feasible Bayesian case-deletion influence measure based on the Kullback-Leibler divergence. Several simulation studies and a real example are employed to illustrate the proposed methodologies.     

Consistency of non parametric estimators of the area under the ROC curve

ABSTRACT

The area under the receiver operating characteristic (ROC) curve is a popular summary index that measures the accuracy of a continuous-scale diagnostic test to measure its accuracy. Under certain conditions on estimators of distribution functions, we prove a theorem on strong consistency of the non parametric “plugin” estimators of the area under the ROC curve. Based on this theorem, we construct some new “plugin” consistent estimators. The performance of the non parametric estimators considered is illustrated numerically and the estimators are compared in terms of bias, variance, and mean square error.     

A Geometric Approach to Principal Components Analysis

The net promoter score (NPS) is a novel summary statistic used by thousands of companies as a key performance indicator of customer loyalty. While adoption of the statistic has grown rapidly over the last decade, there has been little published on its statistical properties. Common interval estimation techniques are adapted for use with the NPS, and performance assessed on the largest available database of companies’ net promoter scores. Variations on the adjusted Wald, and an iterative score test are found to have superior performance.