Semiparametric Density Ratio Model Research Articles

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.

Inference on the probability [formula omitted] as a measurement of treatment effect under a density ratio model and random censoring

Let T1 and T2 be the survival times of patients randomized to, respectively, two treatment groups. The probability P(T1<T2) may be used as a measure on the effect of treatments in a randomized clinical trial. With potentially censored data observed and assuming that the density functions of survival times satisfy a semiparametric density ratio model, two estimates of this probability based on, respectively, the conditional and weighted empirical likelihood are proposed. Associated confidence intervals are also derived based on the bootstrap resampling. The assessment on the goodness of fit of the density ratio model is also discussed. The proposed inference procedures are evaluated by Monte Carlo simulations and applied to the analysis of data from a clinical trial on early breast cancer.

A semiparametric hypothesis testing procedure for the ROC curve area under a density ratio model

We propose a semiparametric Wald statistic to test whether a diagnostic test is capable of discriminating between diseased and nondiseased subjects based on the receiver operating characteristic (ROC) curve area under a two-sample semiparametric density ratio model. The proposed Wald test is constructed on the basis of the maximum semiparametric likelihood estimator of the ROC curve area. The proposed test statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. We present some results on a simulation study and on the analysis of two data sets.