Two-sample Semiparametric Model Research Articles

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.

Minimum Profile Hellinger Distance Estimation for A Two-Sample Location-Shifted Model

Minimum Hellinger distance estimation (MHDE) for parametric model is obtained by minimizing the Hellinger distance between an assumed parametric model and a nonparametric estimation of the model. MHDE receives increasing attention for its efficiency and robustness. Recently, it has been extended from parametric models to semiparametric models. This manuscript considers a two-sample semiparametric location-shifted model where two independent samples are generated from two identical symmetric distributions with different location parameters. We propose to use profiling technique in order to utilize the information from both samples to estimate unknown symmetric function. With the profiled estimation of the function, we propose a minimum profile Hellinger distance estimation (MPHDE) for the two unknown location parameters. This MPHDE is similar to but dif- ferent from the one introduced in Wu and Karunamuni (2015), and thus the results presented in this work is not a trivial application of their method. The difference is due to the two-sample nature of the model and thus we use different approaches to study its asymptotic properties such as consistency and asymptotic normality. The efficiency and robustness properties of the proposed MPHDE are evaluated empirically though simulation studies. A real data from a breast cancer study is analyzed to illustrate the use of the proposed method.

Efficient estimation and computation of parameters and nonparametric functions in generalized semi/non-parametric regression models

The efficiency of estimation for the parameters in semiparametric models has been widely studied in the literature (Bickel et al., 1993). In this paper, we study efficient estimators for both parameters and nonparametric functions in a class of generalized semi/non-parametric regression models, which cover commonly used semiparametric models such as partially linear models, partially linear single index models, and two-sample semiparametric models. We propose a maximum likelihood principle combined with the local linear technique for estimating the parameters and nonparametric functions. The proposed estimators of the parameters and a linear functional of the nonparametric functions are consistent and asymptotically normal and are further shown to be semiparametrically efficient. An efficient computational algorithm to achieve the maximization is proposed. Extensive simulation experiments show the superiority of the proposed methods. Three real data examples are analyzed and presented as an illustration.

Bayesian superposition of pure-birth destructive cure processes for tumor latency

In this paper, a new Bayesian stochastic cure rate model, based on the individual frailty as the total number of descendent cells of each damaged cell (known as the volume of the tumor) up to specific time and the repair mechanism (known as the destructive mechanism), is formulated as a superposition of pure-birth stochastic processes (Yule processes). The proposed model deals with combined modeling of long-term effects (cure rate) and short-term effects (tumor growth) which can be regarded as a Bayesian alternative to the Cox regression and promotion cure rate models when the proportional hazards assumption is violated, such as in the case of crossing survival functions. A simulation study and an application to a melanoma data, using the Bayesian RStan package, are provided to illustrate the usefulness of the proposed model and also to evaluate the impact of cancer treatment over the long-term effect and the progression stage of the tumor (tumor growth). Besides this interesting new biological interpretation of the tumor growth, the model can be viewed as an extended Bayesian version of the destructive cure rate model of Rodrigues et al. and an alternative to the two-sample semiparametric model of Yang and Prentice.

Minimum Hellinger distance estimation in a two-sample semiparametric model

We investigate the estimation problem of parameters in a two-sample semiparametric model. Specifically, let X 1 , … , X n be a sample from a population with distribution function G and density function g . Independent of the X i ’s, let Z 1 , … , Z m be another random sample with distribution function H and density function h ( x ) = exp [ α + r ( x ) β ] g ( x ) , where α and β are unknown parameters of interest and g is an unknown density. This model has wide applications in logistic discriminant analysis, case-control studies, and analysis of receiver operating characteristic curves. Furthermore, it can be considered as a biased sampling model with weight function depending on unknown parameters. In this paper, we construct minimum Hellinger distance estimators of α and β . The proposed estimators are chosen to minimize the Hellinger distance between a semiparametric model and a nonparametric density estimator. Theoretical properties such as the existence, strong consistency and asymptotic normality are investigated. Robustness of proposed estimators is also examined using a Monte Carlo study.

Constrained Maximum Likelihood Estimation under Logistic Regression Models Based on Case-Control Data

Abstract We study constrained maximum likelihood estimation and tests under the logistic regression model based on case-control data. Our approach is based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the semiparametric likelihood ratio statistic, the Lagrangian multiplier statistic, and the Wald statistic are asymptotically equivalent and that they have an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we demonstrate that the three test statistics and their asymptotic distributions may be obtained by fitting the prospective logistic regression model to case-control data. We present some results on simulation and on the analysis of two real data sets.

Comparing correlated ROC curves for continuous diagnostic tests under density ratio models

A family of nonparametric statistics to comparing ROC curves for continuous diagnostic tests was proposed by Wieand et al. [Wieand, S., Gail, M.H., James, B.R., James, K.L., 1989. A family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data. Biometrika 76, 585–592]. In this paper, we study the semiparametric counterpart. We propose a two-sample semiparametric bivariate density ratio model, under which new ROC curve estimators are constructed and a family of semiparametric statistics for comparing ROC curves are proposed. We derive the asymptotic results on the newly proposed ROC curve estimators and show that they are more efficient than the nonparametric counterparts. We also show the proposed method for comparing ROC curves is more efficient than the nonparametric counterpart. A simulation study and the analysis of two real examples are also presented.

Weighted empirical likelihood in some two-sample semiparametric models with various types of censored data

In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator (θ̃n, F̃n) for the underlying parameter θ0 and distribution F0 is derived, and the strong consistency of (θ̃n, F̃n) and the asymptotic normality of θ̃n are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that $\sqrt{n}(\tilde{F}_{n}-F_{0})$ weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models.

Testing treatment effect by combining weighted log-rank tests and using empirical likelihood

For testing treatment effect with time to event data, combinations of several tests are often desired when the hazard functions of the two groups are nonproportional. Yang and Prentice [2005. Semiparametric analysis of short term and long term hazard ratios with two sample survival data. Biometrika 92, 1–17.] defined a new two-sample semiparametric model that accommodates nonproportional hazard functions and contains the Cox model and the proportional odds model as two submodels. They also obtained a χ 2 test on the parameter that reduces to the χ 2 test based on weighted log-rank tests for testing the null hypothesis of no treatment effect. In this paper, we consider a new χ 2 test using the empirical likelihood method. Extensive simulation studies were conducted to compare the performance of the test with other related ones, for a variety of combinations of the short-term and long-term treatment effects.

Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.

Density estimation under a two-sample semiparametric model

We consider estimating a density function under a two-sample semiparametric model in which the log ratio of two density functions is a quadratic function of data. This two-sample semiparametric model, arising naturally from case–control studies and logistic discriminant analysis, can be regarded as a biased sampling model. Under this model, the difference between the two samples is quantified. A kernel-based density estimator is constructed by smoothing the increments of the maximum semiparametric likelihood estimator of the underlying distribution function. The required computation for our method can be accomplished by using the standard statistical software packages for categorical data analysis. We establish some asymptotic results on the proposed kernel density estimator. In addition, we present some results on a simulation study and on the analysis of two data sets to demonstrate the utility of the proposed density estimator.

Bias-corrected maximum semiparametric likelihood estimation under logistic regression models based on case–control data

We consider the corrective approach (Theoretical Statistics, Chapman & Hall, London, 1974, p. 310) and preventive approach (Biometrica 80 (1993) 27) to bias reduction of maximum likelihood estimators under the logistic regression model based on case–control data. The proposed bias-corrected maximum likelihood estimators are based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the prospective and retrospective analyses on the basis of the corrective approach to bias reduction produce identical bias-corrected maximum likelihood estimators of the odds ratio parameter, but this does not hold when using the preventive approach unless the case and control sample sizes are identical. We present some results on simulation and on the analysis of two real data sets.

Quantile Estimation under a Two-Sample Semi-Parametric Model

We consider quantile estimation under a two-sample semi-parametric model in which the log ratio of two unknown density functions has a known parametric form. This two-sample semi-parametric model, arising naturally from case-control studies and logistic discriminant analysis, can be regarded as a biased sampling model. A new quantile estimator is constructed on the basis of the maximum semi-parametric likelihood estimator of the underlying distribution function. It is shown that the proposed quantile estimator is asymptotically normally distributed with smaller asymptotic variance than that of the standard quantile estimator. Also presented are some results on simulation and on analysis of a real data set.

A goodness-of-fit test for logistic regression models based on case-control data

SUMMARY We test the logistic regression assumption under a case-control sampling plan. After reparameterisation, the assumed logistic regression model is equivalent to a two-sample semiparametric model in which the log ratio of two density functions is linear in data. By identifying this model with a biased sampling model, we propose a Kolmogorov-Smirnovtype statistic to test the validity of the logistic link function. Moreover, we point out that this test statistic can also be used in mixture sampling. We present a bootstrap procedure along with some results on simulation and on analysis of two real datasets.

Efficient estimation in the two-sample semiparametric location-scale models

Efficient estimation in the two-sample semiparametric location-scale models

Rank estimates in a class of semiparametric two-sample models

We consider a two-sample semiparametric model involving a real parameter θ and a nuisance parameter F which is a distribution function. This model includes the proportional hazard, proportional odds, linear transformation and Harrington-Fleming models (1982, Biometrika, 69, 533–546). We propose two types of estimates based on ranks. The first is a rank approximation to Huber's M-estimates (1981, Robust Statistics, Wiley) and the second is a Hodges-Lehmann type rank inversion estimate (1963, Ann. Math. Statist., 34, 598–611). We obtain asymptotic normality and efficiency results. The estimates are consistent and asymptotically normal generally but fully efficient only for special cases.

Inefficiency of inferences with the partial likelihood

In two-sample semiparametric survival models other than the Cox proportional-hazards regression model, it is shown that partial-likelihood inference of structural parameters in the presence of fully nonpararnetric nuisance-hazards typically has relative efficiency zero compared with fuii-Iikelihood infer -ence. The practical Interpretation of efficiencies in the pres-ence of infinite-dimensional nuisance-parameters is discussed, with reference to two important examples, namely a recent sur-vival regression-model of Clayton and Cuzick and a class of additive excess-risk models. Under the excess-risk models, a formula is derived for the large-sample information [which here is the same as the limiting Fisher information when the nuisance-parameter dimension gets large] for estimating the parameter of difference between two samples, as the nuisance function becomes fully nonpararnetric.