Density Ratio Model Research Articles

Local Empirical Likelihood Inference for Varying-Coefficient Density-Ratio Models Based on Case-Control Data

In this article, we develop a varying-coefficient density-ratio model for case-control studies. The case and control samples come from two different distributions. Under the model assumption, the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function. A special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates. We propose a local empirical likelihood (EL) approach to estimate the nonparametric coefficient functions. Under some regularity assumptions, the proposed estimators are shown to be consistent and asymptotically normally distributed. The sieve empirical likelihood ratio (SELR) test statistic for detecting whether the varying-coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi-squared distribution. We introduce a modified bootstrap procedure to estimate the null distribution of the SELR when sample size is small. We also examine the performance of proposed method for finite sample sizes through simulation studies and illustrate it with a real dataset. Supplementary materials for this article are available online.

Semiparametric empirical likelihood confidence intervals for the difference of areas under two correlated ROC curves under density ratio model

The area under the receiver operating characteristic (ROC) curve, AUC, is commonly used to assess the ability of a diagnostic test to correctly classify individuals into diseased and nondiseased populations. When there are two diagnostic tests available, it is of interest to evaluate and compare their performances. Based on the difference of two placement values, we propose a two-sample empirical likelihood method for comparing AUCs of two ROC curves. The proposed empirical likelihood ratio statistic converges in distribution to a scaled chi-squared random variable. Simulation results show that the proposed empirical likelihood method has better finite-sample performance than other competitors.

Likelihood approaches for proportional likelihood ratio model with right‐censored data

Regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the Cox regression model and the proportional odds model. However, their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases. As an alternative, in this paper, the proportional likelihood ratio model introduced by Luo and Tsai is extended to flexibly model the relationship between survival outcome and covariates. This model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems. Compared with the semiparametric transformation model, the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation. We present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions. Based on a conditional likelihood approach using uncensored failure times, a numerically simple estimation procedure is developed by maximizing a pairwise pseudo-likelihood. We also develop a full likelihood approach, and the most efficient maximum likelihood estimator is obtained by a profile likelihood. Simulation studies are conducted to assess the finite-sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches. An application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non-proportionality. The relative merits of these methods are discussed in concluding remarks.

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.

Local Empirical Likelihood Diagnosis of Varying Coefficient Density-Ratio Models Based on Case-Control Data

In this paper, a varying-coefficient density-ratio model for case-control studies is developed. We investigate the local empirical likelihood diagnosis of varying coefficient density-ratio model for case-control data. The local empirical log-likelihood ratios for the nonparametric coefficient functions are introduced. First, the estimation equations based on empirical likelihood method are established. Then, a few of diagnostic statistics are proposed. At last, we also examine the performance of proposed method for finite sample sizes through simulation studies.

Semiparametric ROC surface estimation for continuous diagnostic tests via polytomous logistic regression procedures

In this paper, we propose a semiparametric method of estimating receiver operating characteristic (ROC) surfaces for continuous diagnostic tests under density ratio models. Implementation of our method is easy since the usual polytomous logistic regression procedures in many statistical software packages can be employed. A simulated example is provided to facilitate the implementation of our method. Simulation results show that the proposed semiparametric ROC surface estimator is more efficient than the nonparametric counterpart and the parametric counterpart whether the normality assumption of data holds or not. Moreover, some simulation results on the underlying semiparametric distribution function estimators are also reported. In addition, some discussions on the proposed method as well as analysis of a real data set are provided.

Semiparametric Inference for the Two‐way Layout Under Order Restrictions

ABSTRACTThere are many situations in which a researcher would like to analyse data from a two‐way layout. Often, the assumptions of linearity and normality may not hold. To address such situations, we introduce a semiparametric model. The model extends the well‐known density ratio model from the one‐way to the two‐way layout and provides a useful framework for semiparametric analysis of variance type problems under order restrictions. In particular, the likelihood ratio order is emphasized. The model enables highly efficient inference without resorting to fully parametric assumptions or the use of transformations. Estimation and testing procedures under order restrictions are developed and investigated in detail. It is shown that the model is robust to misspecification, and several simulations suggest that it performs well in practice. The methodology is illustrated using two data examples; in the first, the response variable is discrete, whereas in the second, it is continuous.

Semiparametric empirical likelihood confidence intervals for AUC under a density ratio model

Inferences on the area under a receiver operating characteristic curve (AUC) are usually based on a fully parametric approach or a fully nonparametric approach. A semiparametric empirical likelihood method is proposed to construct confidence intervals for AUC by assuming a density ratio model for the diseased and non-diseased population densities. The limiting distribution of the semiparametric empirical log likelihood ratio statistic for AUC has a scaled chi-square distribution. The proposed semiparametric empirical likelihood approach is shown, via a simulation study, to be more robust than a fully parametric approach and is more accurate than a fully nonparametric approach. Some results on simulation and an analysis of two real examples are presented.

Quantile and quantile-function estimations under density ratio model

Population quantiles and their functions are important parameters in many applications. For example, the lower quantiles often serve as crucial quality indices for forestry products. Given several independent samples from populations satisfying the density ratio model, we investigate the properties of empirical likelihood (EL) based inferences. The induced EL quantile estimators are shown to admit a Bahadur representation that leads to asymptotically valid confidence intervals for functions of quantiles. We rigorously prove that EL quantiles based on all the samples are more efficient than empirical quantiles based on individual samples. A simulation study shows that the EL quantiles and their functions have superior performance when the density ratio model assumption is satisfied and when it is mildly violated. An example is used to demonstrate the new method and the potential cost savings.

Survival analysis without survival data: connecting length-biased and case-control data

We show that relative mean survival parameters of a semiparametric log-linear model can be estimated using covariate data from an incident sample and a prevalent sample, even when there is no prospective follow-up to collect any survival data. Estimation is based on an induced semiparametric density ratio model for covariates from the two samples, and it shares the same structure as for a logistic regression model for case-control data. Likelihood inference coincides with well-established methods for case-control data. We show two further related results. First, estimation of interaction parameters in a survival model can be performed using covariate information only from a prevalent sample, analogous to a case-only analysis. Furthermore, propensity score and conditional exposure effect parameters on survival can be estimated using only covariate data collected from incident and prevalent samples.

Mark‐Specific Hazard Ratio Model with Multivariate Continuous Marks: An Application to Vaccine Efficacy

In randomized placebo-controlled preventive HIV vaccine efficacy trials, an objective is to evaluate the relationship between vaccine efficacy to prevent infection and genetic distances of the exposing HIV strains to the multiple HIV sequences included in the vaccine construct, where the set of genetic distances is considered as the continuous multivariate "mark" observed in infected subjects only. This research develops a multivariate mark-specific hazard ratio model in the competing risks failure time analysis framework for the assessment of mark-specific vaccine efficacy. It allows improved efficiency of estimation by employing the semiparametric method of maximum profile likelihood estimation in the vaccine-to-placebo mark density ratio model. The model also enables the use of a more efficient estimation method for the overall log hazard ratio in the Cox model. In addition, we propose testing procedures to evaluate two relevant hypotheses concerning mark-specific vaccine efficacy. The asymptotic properties and finite-sample performance of the inferential procedures are investigated. Finally, we apply the proposed methods to data collected in the Thai RV144 HIV vaccine efficacy trial.

Semiparametric regression in testicular germ cell data

It is possible to approach regression analysis with random covariates from a semiparametric perspective where information is combined from multiple multivariate sources. The approach assumes a semiparametric density ratio model where multivariate distributions are "regressed" on a reference distribution. A kernel density estimator can be constructed from many data sources in conjunction with the semiparametric model. The estimator is shown to be more efficient than the traditional single-sample kernel density estimator, and its optimal bandwidth is discussed in some detail. Each multivariate distribution and the corresponding conditional expectation (regression) of interest are estimated from the combined data using all sources. Graphical and quantitative diagnostic tools are suggested to assess model validity. The method is applied in quantifying the effect of height and age on weight of germ cell testicular cancer patients. Comparisons are made with multiple regression, generalized additive models (GAM) and nonparametric kernel regression.

A semiparametric hypothesis testing method on the difference of two population means

In the literature, some semiparametric statistical methods were recently developed under density ratio models, and these methods are usually more robust than parametric methods and more effective than nonparametric methods. In this paper, we propose a semiparametric hypothesis testing method on the difference of two population means under a semiparametric density ratio model. This method is essentially built upon semiparametric estimation of the difference of two population means. Some theoretical results as well as simulation results are provided, which suggest that our proposed method is slightly superior to some commonly used parametric and nonparametric methods when data are normal, and is significantly better than them when data are not normal. We also apply our proposed method to analysis to two real examples.

Likelihood approaches for the invariant density ratio model with biased-sampling data.

The full likelihood approach in statistical analysis is regarded as the most efficient means for estimation and inference. For complex length-biased failure time data, computational algorithms and theoretical properties are not readily available, especially when a likelihood function involves infinite-dimensional parameters. Relying on the invariance property of length-biased failure time data under the semiparametric density ratio model, we present two likelihood approaches for the estimation and assessment of the difference between two survival distributions. The most efficient maximum likelihood estimators are obtained by the em algorithm and profile likelihood. We also provide a simple numerical method for estimation and inference based on conditional likelihood, which can be generalized to k-arm settings. Unlike conventional survival data, the mean of the population failure times can be consistently estimated given right-censored length-biased data under mild regularity conditions. To check the semiparametric density ratio model assumption, we use a test statistic based on the area between two survival distributions. Simulation studies confirm that the full likelihood estimators are more efficient than the conditional likelihood estimators. We analyse an epidemiological study to illustrate the proposed methods.

Maximum Likelihood Estimation for Semiparametric Density Ratio Model

In the statistical literature, the conditional density model specification is commonly used to study regression effects. One attractive model is the semiparametric density ratio model, under which the conditional density function is the product of an unknown baseline density function and a known parametric function containing the covariate information. This model has a natural connection with generalized linear models and is closely related to biased sampling problems. Despite the attractive features and importance of this model, most existing methods are too restrictive since they are based on multi-sample data or conditional likelihood functions. The conditional likelihood approach can eliminate the unknown baseline density but cannot estimate it. We propose efficient estimation procedures based on the nonparametric likelihood. The nonparametric likelihood approach allows for general forms of covariates and estimates the regression parameters and the baseline density simultaneously. Therefore, the nonparametric likelihood approach is more versatile than the conditional likelihood approach especially when estimation of the conditional mean or other quantities of the outcome is of interest. We show that the nonparametric maximum likelihood estimators are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in practical settings. A real example is used for illustration.

J056026 格子ボルツマン法を用いた固体高分子形燃料電池ガス拡散層内での気液二相流解析

For practical use of a polymer electrolyte fuel cell (PEFC), a flooding phenomenon is one of the problems to be solved. The flooding is caused by liquid water accumulated in the gas diffusion layer (GDL) and the gas flow channels and deteriorates the performance of the PEFC. Therefore, it is necessary to clarify the gas and liquid flow in the PEFC and to control the liquid water behavior appropriately. Our research group applied the lattice Boltzmann method (LBM) for the two phase flow with high density ratio, considering the effect of wettability, and has carried out the analyses of gas and liquid water to propose optimal structures of the PEFC. Here, the LBM is a method to reproduce the macroscopic flow by considering fluid as an aggregate of a virtual particle and calculating those collisions and translation sequentially, and suitable for the two-phase flow simulation with complex boundary geometries. In this study, we simulated liquid water behavior inside of the GDL applying a low density ratio LBM with shorter computational time instead of the high density ratio model. It was confirmed that the effect of large density difference on the liquid water behavior is negligible because of the weak gas flow inside of the GDL. Using the LBM with low density ratio, we tried efficient large-scale analyses in the GDL to elucidate the effect of the GDL structure and wettability on the liquid water behavior inside of the GDL.

A proportional likelihood ratio model

We propose a semiparametric proportional likelihood ratio model which is particularly suitable for modelling a nonlinear monotonic relationship between the outcome variable and a covariate. This model extends the generalized linear model by leaving the distribution unspecified, and has a strong connection with semiparametric models such as the selection bias model (Gilbert et al., 1999), the density ratio model (Qin, 1998; Fokianos & Kaimi, 2006), the single-index model (Ichimura, 1993) and the exponential tilt regression model (Rathouz & Gao, 2009). A maximum likelihood estimator is obtained for the new model and its asymptotic properties are derived. An example and simulation study illustrate the use of the model. Copyright 2012, Oxford University Press.

Density-ratio matching under the Bregman divergence: a unified framework of density-ratio estimation

Estimation of the ratio of probability densities has attracted a great deal of attention since it can be used for addressing various statistical paradigms. A naive approach to density-ratio approximation is to first estimate numerator and denominator densities separately and then take their ratio. However, this two-step approach does not perform well in practice, and methods for directly estimating density ratios without density estimation have been explored. In this paper, we first give a comprehensive review of existing density-ratio estimation methods and discuss their pros and cons. Then we propose a new framework of density-ratio estimation in which a density-ratio model is fitted to the true density-ratio under the Bregman divergence. Our new framework includes existing approaches as special cases, and is substantially more general. Finally, we develop a robust density-ratio estimation method under the power divergence, which is a novel instance in our framework.

Lattice Boltzmann simulations of incompressible liquid–gas systems on partial wetting surfaces

A three-dimensional Lattice Boltzmann two-phase model capable of dealing with large liquid and gas density ratios and with a partial wetting surface is introduced. This is based on a high density ratio model combined with a partial wetting boundary method. The predicted three-dimensional droplets at different partial wetting conditions at equilibrium are in good agreement with analytical solutions. Despite the large density ratio, the spurious velocity around the interface is not substantial, and is rather insensitive to the examined liquid and gas density and viscosity ratios. The influence of the gravitational force on the droplet shape is also examined through the variations of the Bond number, where the droplet shape migrates from spherical to flattened interface in tandem with the increase of the Bond number. The predicted interfaces under constant Bond number are also validated against measurements with good agreements.