Semiparametric Density Ratio Model Research Articles

Joint Statistical Inference for the Area under the ROC Curve and Youden Index under a Density Ratio Model

The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve (AUC) and the Youden index (J) are common summary indices used to evaluate a biomarker’s diagnostic accuracy. Simultaneously examining AUC and J offers a more comprehensive understanding of the ROC curve’s characteristics. In this paper, we utilize a semiparametric density ratio model to link the distributions of a biomarker for healthy and diseased individuals. Under this model, we establish the joint asymptotic normality of the maximum empirical likelihood estimator of (AUC,J) and construct an asymptotically valid confidence region for (AUC,J). Furthermore, we propose a new test to determine whether a biomarker simultaneously exceeds prespecified target values of AUC0 and J0 with the null hypothesis H0:AUC≤AUC0 or J≤J0 against the alternative hypothesis Ha:AUC>AUC0 and J>J0. Simulation studies and a real data example on Duchenne Muscular Dystrophy are used to demonstrate the effectiveness of our proposed method and highlight its advantages over existing methods.

Joint Statistical Inference for the Area under the ROC Curve and Youden Index under a Density Ratio Model

The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve (AUC) and the Youden index (J) are common summary indices used to evaluate a biomarker’s diagnostic accuracy. Simultaneously examining AUC and J offers a more comprehensive understanding of the ROC curve’s characteristics. In this paper, we utilize a semiparametric density ratio model to link the distributions of a biomarker for healthy and diseased individuals. Under this model, we establish the joint asymptotic normality of the maximum empirical likelihood estimator of (AUC,J) and construct an asymptotically valid confidence region for (AUC,J). Furthermore, we propose a new test to determine whether a biomarker simultaneously exceeds prespecified target values of AUC0 and J0 with the null hypothesis H0:AUC≤AUC0 or J≤J0 against the alternative hypothesis Ha:AUC>AUC0 and J>J0. Simulation studies and a real data example on Duchenne Muscular Dystrophy are used to demonstrate the effectiveness of our proposed method and highlight its advantages over existing methods.

Approximate Bayesian computation with semiparametric density ratio model

Approximate Bayesian computation (ABC) is a likelihood-free inference method commonly employed for statistical inference in models with unknown or complex likelihood functions. ABC estimates the posterior distributions of model parameters by comparing summary statistics of the simulated and observed data. However, the selection of informative summary statistics can be challenging in practice. In this study, we propose a summary-statistic-free method called ABC with semiparametric empirical likelihood ratio (ABC-SELR). This method utilises the empirical likelihood within the framework of the semiparametric density ratio model to assess the simulated parameters via testing whether the simulated and observed data conform to the same distribution. By eliminating the need for choosing summary statistics, our approach avoids potential information loss. Furthermore, the proposed method captures the shared characteristics between the simulated and observed data, leading to improved efficiency. We demonstrate the accuracy and efficiency of ABC-SELR by comparing with existing ABC methods through numerical simulations.

Simultaneous selection and incorporation of consistent external aggregate information.

Interest has grown in synthesizing participant level data of a study with relevant external aggregate information. Several efficient and flexible procedures have been developed under the assumption that the internal study and the external sources concern the same population. This homogeneity condition, albeit commonly being imposed, is hard to check due to limitedly available external information in aggregate data forms. Bias may be introduced when the assumption is violated. In this article, we propose a penalized likelihood approach that avoids undesirable bias by simultaneously selecting and synthesizing consistent external aggregate information. The proposed approach provides a general framework which incorporate consistent external information from heterogeneous study populations as long as the conditional distribution of the dependent variable under investigation is same and differences in the independent variable distributions are properly accounted for via a semi-parametric density ratio model. The proposed approach also properly accounts for the sampling errors in the external information. A two-step estimator and an optimization algorithm are proposed for computation. We establish the selection and estimation consistency and the asymptotic normality of the two-step estimator. The proposed approach is illustrated with an analysis of gestational weight gain management studies.

Maximum approximate Bernstein likelihood estimation in a two-sample semiparametric model

Maximum likelihood estimators are proposed for the parameters and the underlying densities in a semiparametric density ratio model in which the nonparametric baseline density is approximated by the Bernstein polynomial model. The EM algorithm is used to obtain the maximum approximate Bernstein likelihood estimates. The proposed method is illustrated by two real data from medical research and is shown by simulation to have better performance than the existing ones. Some asymptotic results are also presented and proved.

Semi-parametric inference on Gini indices of two semi-continuous populations under density ratio models

Summary The Gini index is a popular inequality measure with many applications in social and economic studies. This paper studies semi-parametric inference on the Gini indices of two semi-continuous populations. We characterise the distribution of each semi-continuous population by a mixture of a discrete point mass at zero and a continuous skewed positive component. A semi-parametric density ratio model is then employed to link the positive components of the two distributions. We propose the maximum empirical likelihood estimators of the two Gini indices and their difference, and further investigate the asymptotic properties of the proposed estimators. The asymptotic results enable us to construct confidence intervals, and perform hypothesis tests for the two Gini indices and their difference. The proposed method is also applicable to cases without excessive zero values. The superiority of our proposed method over some existing methods is shown theoretically and numerically. Two real-data applications are presented for illustration.

Semiparametric Density Ratio Model for Survival Data with a Cure Fraction

Semiparametric Density Ratio Model for Survival Data with a Cure Fraction

Joint semiparametric models for case-cohort designs.

Two-phase studies such as case-cohort and nested case-control studies are widely used cost-effective sampling strategies. In the first phase, the observed failure/censoring time and inexpensive exposures are collected. In the second phase, a subgroup of subjects is selected for measurements of expensive exposures based on the information from the first phase. One challenging issue is how to utilize all the available information to conduct efficient regression analyses of the two-phase study data. This paper proposes a joint semiparametric modeling of the survival outcome and the expensive exposures. Specifically, we assume a class of semiparametric transformation models and a semiparametric density ratio model for the survival outcome and the expensive exposures, respectively. The class of semiparametric transformation models includes the proportional hazards model and the proportional odds model as special cases. The density ratio model is flexible in modeling multivariate mixed-type data. We develop efficient likelihood-based estimation and inference procedures and establish the large sample properties of the nonparametric maximum likelihood estimators. Extensive numerical studies reveal that the proposed methods perform well under practical settings. The proposed methods also appear to be reasonably robust under various model mis-specifications. An application to the National Wilms Tumor Study isprovided.

Permutation tests under a rotating sampling plan with clustered data

The distribution of lumber strength of any grade may evolve, for example, due to climate change, forest fire, changes in processing methods, and other factors. So, in North America the forest products industry monitors the evolution of their means, percentiles, or other parameters to ensure the wood products meet the industrial standard. For administrative convenience and informativeness, one may adopt a rotating sampling plan by sampling 36 mills in the initial occasion and having six of them replaced in each successive occasion for the next five occasions. The strength data on a specified number, commonly 10 pieces of lumbers from each sampled mills, are obtained. Under such rotating plans the observations on pieces from the same mill are correlated, and the observations on samples from the same mill taken on different occasions are also correlated. Ignoring these correlations may lead to invalid inference procedures. Yet accommodating a cluster structure in parametric models is difficult and entails a high level of misspecification risk. In this paper we explore symmetry in the clustered data collected via a rotating sampling plan to develop a permutation scheme for testing various hypotheses of interest. We also introduce a semiparametric density ratio model to link the distributions of the response variable over time. The combination retains the validity of the inference methods while extracting maximum information from the sampling plan. A simulation study indicates that the proposed permutation tests firmly control the type I error whether or not the data are clustered. The use of the density ratio model improves the power of the tests. We also apply the proposed tests to data from the motivating application. The proposed permutation tests effectively address many real-world issues with trust worth inference conclusions.

Semiparametric inference on general functionals of two semicontinuous populations

In this paper, we propose new semiparametric procedures for inference on linear functionals in the context of two semicontinuous populations. The distribution of each semicontinuous population is characterized by a mixture of a discrete point mass at zero and a continuous skewed positive component. To utilize the information from both populations, we model the positive components of the two mixture distributions via a semiparametric density ratio model. Under this model setup, we construct the maximum empirical likelihood estimators of the linear functionals. The asymptotic normality of the proposed estimators is established and is used to construct confidence regions and perform hypothesis tests for these functionals. We show that the proposed estimators are more efficient than the fully nonparametric ones. Simulation studies demonstrate the advantages of our method over existing methods. Two real-data examples are provided for illustration.

Synthesizing external aggregated information in the penalized Cox regression under population heterogeneity.

Synthesizing external aggregated information has been proven useful in improving estimation efficiency when conducting statistical analysis using a limited amount of data. In this paper, we develop a unified framework for combining information from high-dimensional individual-level data and potentially low-dimensional external aggregate data under the Cox model. We summarize various forms of external aggregated information by population estimating equations and propose a penalized empirical likelihood approach to borrow information from these estimating equations. The proposed methods possess the flexibility to handle the case where individual-level data and external aggregate data are from heterogeneous populations. Specifically, a penalized empirical likelihood ratio test is developed to check for the potential heterogeneity, and a semiparametric density ratio model is postulated to account for the heterogeneity. Moreover, we study the impact of uncertainty in the auxiliary information on the efficiency gain and propose a modified variance estimator to adjust for the uncertainty. The proposed estimators enjoy the oracle property and are asymptotically more efficient than the penalized partial likelihood estimator that does not exploit the external aggregated information. Simulation studies show improvement in both estimation efficiency and variable selection over the competitors. The proposed approaches are applied to the analysis of a pediatric kidney transplant study for illustration.

New semiparametric regression method with applications in selection‐biased sampling and missing data problems

We propose a new method to estimate a regression function based on the semiparametric density ratio model, which can be viewed as a generalized linear model with a canonical link function and an unspecified baseline distribution function. Under this model, the distribution of the observed data retains the same structure in the presence of selection‐biased sampling or when the predictors are missing at random. In particular, in the latter case, the new method utilizes all the available information and does not need to specify the distribution of the predictors or the probability of observing the predictors. We establish large sample properties of the proposed regression estimators. Simulation studies demonstrate that the proposed estimators perform well in practical situations. Empirical data from the National Health and Nutrition Examination Survey are presented.

Semiparametric inference of the Youden index and the optimal cut‐off point under density ratio models

AbstractThe Youden index is a popular summary statistic for receiver operating characteristic curves. It gives the optimal cut‐off point of a biomarker to distinguish the diseased and healthy individuals. In this article, we model the distributions of a biomarker for individuals in the healthy and diseased groups via a semiparametric density ratio model. Based on this model, we propose using the maximum empirical likelihood method to estimate the Youden index and the optimal cut‐off point. We further establish the asymptotic normality of the proposed estimators and construct valid confidence intervals for the Youden index and the corresponding optimal cut‐off point. The proposed method automatically covers both cases when there is no lower limit of detection (LLOD) and when there is a fixed and finite LLOD for the biomarker. Extensive simulation studies and a real data example are used to illustrate the effectiveness of the proposed method and its advantages over the existing methods.

A Bayesian discriminant analysis method under semiparametric density ratio models

We propose a semiparametric Bayesian discriminant analysis method. Statistical simulation shows that when the hypothesis of normal distribution is satisfied, our method is comparable to and even slightly better than the traditional method; when the hypothesis of normal distribution is not satisfied, our method is obviously better than the traditional one. We also apply our method to analyze a real data set and find that our method is better than the traditional one. Finally, we point out that implementation of our method is easy since the usual polytomous logistic regression procedures in many statistical softwares can be employed.

Small Area Quantile Estimation

SummarySample surveys are widely used to obtain information about totals, means, medians and other parameters of finite populations. In many applications, similar information is desired for subpopulations such as individuals in specific geographic areas and socio‐demographic groups. When the surveys are conducted at national or similarly high levels, a probability sampling can result in just a few sampling units from many unplanned subpopulations at the design stage. Cost considerations may also lead to low sample sizes from individual small areas. Estimating the parameters of these subpopulations with satisfactory precision and evaluating their accuracy are serious challenges for statisticians. To overcome the difficulties, statisticians resort to pooling information across the small areas via suitable model assumptions, administrative archives and census data. In this paper, we develop an array of small area quantile estimators. The novelty is the introduction of a semiparametric density ratio model for the error distribution in the unit‐level nested error regression model. In contrast, the existing methods are usually most effective when the response values are jointly normal. We also propose a resampling procedure for estimating the mean square errors of these estimators. Simulation results indicate that the new methods have superior performance when the population distributions are skewed and remain competitive otherwise.

Understanding Causal Distributional and Subgroup Effects With the Instrumental Propensity Score.

To address issues with measured and unmeasured confounding in observational studies, we developed a unified approach to using an instrumental variable in more flexible ways to evaluate treatment effects. The approach is based on an instrumental propensity score conditional on baseline variables, which can then be incorporated in matching, regression, subclassification, or weighting along with various parametric, semiparametric, or nonparametric methods for the assessment of treatment effects. Therefore, the application of the instrumental propensity score allows different methods for outcome effect evaluations in addition to standard 2-stage least square models while controlling for unmeasured confounders. Several properties of the instrumental propensity score are discussed. The approach is then illustrated using subclassification along with a semiparametric density ratio model and empirical likelihood. This method allows us to evaluate distributional and subgroup treatment effects in addition to the overall average treatment effect. Simulation studies showed that the method works well. We applied our method to a study of the effects of attending a Catholic school versus a public school and found that attending a Catholic school had significant beneficial effects on subsequent wages among a subgroup of subjects.

Bayesian analysis of a density ratio model

AbstractThis work proposes a Bayesian approach for the analysis of a semiparametric density ratio model, a model useful for the integration of data from multiple sources. The proposed Bayesian analysis uses a non‐parametric likelihood and a transformed Gaussian prior for the “non‐parametric part” of the model. The former choice guarantees the validity of the Bayesian analysis in contrast to some semiparametric Bayesian analyses that rely on empirical likelihoods whereas the latter choice allows the representation of an expected smoothness property. We describe a Markov chain Monte Carlo algorithm to fit this model which was found to empirically display good convergence behaviour. The model is illustrated with the analysis of motor vibration data obtained from three different locations on a motor. The Canadian Journal of Statistics 45: 274–289; 2017 © 2017 Statistical Society of Canada

Semiparametric density ratio modeling of survival data from a prevalent cohort.

In this article, we consider methods for assessing covariate effects on survival outcome in the target population when data are collected under prevalent sampling. We investigate a flexible semiparametric density ratio model without the constraints of the constant disease incidence rate and discrete covariates as required in Shen and others 2012. For inference, we introduce two likelihood approaches with distinct computational algorithms. We first develop a full likelihood approach to obtain the most efficient estimators by an iterative algorithm. Under the density ratio model, we exploit the invariance property of uncensored failure times from the prevalent cohort and also propose a computationally convenient estimation procedure that uses a conditional pairwise likelihood. The empirical performance and efficiency of the two approaches are evaluated through simulation studies. The proposed methods are applied to the Surveillance, Epidemiology, and End Results Medicare linked data for women diagnosed with stage IV breast cancer.

Using logistic regression for semiparametric comparison of population means and variances

We propose to compare population means and variances under a semiparametric density ratio model. The proposed method is easy to implement by employing logistic regression procedures in many statistical software, and it often works very well when data are not normal. In this paper, we construct semiparametric estimators of the differences of two population means and variances, and derive their asymptotic distributions. We prove that the proposed semiparametric estimators are asymptotically more efficient than the corresponding non parametric ones. In addition, a simulation study and the analysis of two real data sets are presented. Finally, a short discussion is provided.

Reverse regression: A method for joint analysis of multiple endpoints in randomized clinical trials

In clinical trials, treatment comparisons are often cast in a regression framework that evaluates the dependence of the relevant clinical outcomes on treatment assignment and possibly other baseline characteristics. This article introduces a reverse regression approach to randomized clinical trials, with focus on the dependence of treatment assignment on the clinical outcomes of interest. A reverse regression model is essentially a semiparametric density ratio model for the outcome distributions in the two treatment groups. The resulting inferences can be expected to be more robust than those based on fully parametric models for the outcome distributions and more efficient than nonparametric inferences. In the presence of multiple endpoints, the reverse regression approach leads to a novel procedure for multiplicity adjustment that is readily available in standard logistic regression routines. The proposed approach is evaluated in simulation experiments and illustrated with an example.