Nonparametric Maximum Likelihood Method Research Articles

Semiparametric regression analysis of length-biased and partly interval-censored data with application to an AIDS cohort study.

Length-biased data occur often in many scientific fields, including clinical trials, epidemiology surveys and genome-wide association studies, and many methods have been proposed for their analysis under various situations. In this article, we consider the situation where one faces length-biased and partly interval-censored failure time data under the proportional hazards model, for which it does not seem to exist an established method. For the estimation, we propose an efficient nonparametric maximum likelihood method by incorporating the distribution information of the observed truncation times. For the implementation of the method, a flexible and stable EM algorithm via two-stage data augmentation is developed. By employing the empirical process theory, we establish the asymptotic properties of the resulting estimators. A simulation study conducted to assess the finite-sample performance of the proposed method suggests that it works well and is more efficient than the conditional likelihood approach. An application to an AIDS cohort study is also provided.

Mixture Modeling of Time-to-Event Data in the Proportional Odds Model

Subgroup analysis with survival data are most essential for detailed assessment of the risks of medical products in heterogeneous population subgroups. In this paper, we developed a semiparametric mixture modeling strategy in the proportional odds model for simultaneous subgroup identification and regression analysis of survival data that flexibly allows the covariate effects to differ among several subgroups. Neither the membership or the subgroup-specific covariate effects are known a priori. The nonparametric maximum likelihood method together with a pair of MM algorithms with monotone ascent property are proposed to carry out the estimation procedures. Then, we conducted two series of simulation studies to examine the finite sample performance of the proposed estimation procedure. An empirical analysis of German breast cancer data is further provided for illustrating the proposed methodology.

Subgroup Identification and Regression Analysis of Clustered and Heterogeneous Interval-Censored Data

Clustered and heterogeneous interval-censored data occur in many fields such as medical studies. For example, in a migraine study with the Netherlands Twin Registry, the information including time to diagnosis of migraine and gender was collected for 3975 monozygotic and dizygotic twins. Since each study subject is observed only at discrete and periodic follow-up time points, the failure times of interest (i.e., the time when the individual first had a migraine) are known only to belong to certain intervals and hence are interval-censored. Furthermore, these twins come from different genetic backgrounds and may be associated with differential risks for developing migraines. For simultaneous subgroup identification and regression analysis of such data, we propose a latent Cox model where the number of subgroups is not assumed a priori but rather data-driven estimated. The nonparametric maximum likelihood method and an EM algorithm with monotone ascent property are also developed for estimating the model parameters. Simulation studies are conducted to assess the finite sample performance of the proposed estimation procedure. We further illustrate the proposed methodologies by an empirical analysis of migraine data.

Semiparametric estimation for nonparametric frailty models using nonparametric maximum likelihood approach.

A consequence of using a parametric frailty model with nonparametric baseline hazard for analyzing clustered time-to-event data is that its regression coefficient estimates could be sensitive to the underlying frailty distribution. Recently, there has been a proposal for specifying both the baseline hazard and the frailty distribution nonparametrically, and estimating the unknown parameters by the maximum penalized likelihood method. Instead, in this paper, we propose the nonparametric maximum likelihood method for a general class of nonparametric frailty models, i.e. models where the frailty distribution is completely unspecified but the baseline hazard can be either parametric or nonparametric. The implementation of the estimation procedure can be based on a combination of either the Broyden-Fletcher-Goldfarb-Shanno or expectation-maximization algorithm and the constrained Newton algorithm with multiple support point inclusion. Simulation studies to investigate the performance of estimation of a regression coefficient by several different model-fitting methods were conducted. The simulation results show that our proposed regression coefficient estimator generally gives a reasonable bias reduction when the number of clusters is increased under various frailty distributions. Our proposed method is also illustrated with two data examples.

Two-Component Mixture Model in the Presence of Covariates

In this article, we study a generalization of the two-groups model in the presence of covariates—a problem that has recently received much attention in the statistical literature due to its applicability in multiple hypotheses testing problems. The model we consider allows for infinite dimensional parameters and offers flexibility in modeling the dependence of the response on the covariates. We discuss the identifiability issues arising in this model and systematically study several estimation strategies. We propose a tuning parameter-free nonparametric maximum likelihood method, implementable via the expectation-maximization algorithm, to estimate the unknown parameters. Further, we derive the rate of convergence of the proposed estimators—in particular we show that the finite sample Hellinger risk for every ‘approximate’ nonparametric maximum likelihood estimator achieves a near-parametric rate (up to logarithmic multiplicative factors). In addition, we propose and theoretically study two ‘marginal’ methods that are more scalable and easily implementable. We demonstrate the efficacy of our procedures through extensive simulation studies and relevant data analyses—one arising from neuroscience and the other from astronomy. We also outline the application of our methods to multiple testing. The companion R package NPMLEmix implements all the procedures proposed in this article.

An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics.

Population pharmacokinetic (PK) modeling has become a cornerstone of drug development and optimal patient dosing. This approach offers great benefits for datasets with sparse sampling, such as in pediatric patients, and can describe between-patient variability. While most current algorithms assume normal or log-normal distributions for PK parameters, we present a mathematically consistent nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions without any assumption about the shape of the distribution. This approach can handle distributions with any shape for all PK parameters. It is shown in convexity theory that the NPML estimator is discrete, meaning that it has finite number of points with nonzero probability. In fact, there are at most N points where N is the number of observed subjects. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. In the simplest case, each point essentially represents the set of PK parameters for one patient. The probability of the points is found by a primal-dual interior-point method; the location of the support points is found by an adaptive grid method. Our method is able to handle high-dimensional and complex multivariate mixture models. An important application is discussed for the problem of population pharmacokinetics and a nontrivial example is treated. Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics. Thereby, this approach presents an important addition to the pharmacometric toolbox for drug development and optimal patient dosing.

The galaxy luminosity function in the LAMOST Complete Spectroscopic Survey of Pointing Area at the Southern Galactic Cap

We present optical luminosity functions (LFs) of galaxies in the 0.1g, 0.1r, 0.1i bands, calculated using data in sky area of the LAMOST Complete Spectroscopic Survey of Pointing Area (LaCoSSPAr) in the Southern Galactic Cap. Redshifts for galaxies brighter than r = 18.1 were obtained mainly with LAMOST. In each band, LFs derived using both parametric and non-parametric maximum likelihood methods agree well with each other. In the 0.1r band, our fitting parameters of the Schechter function are , and , which agree with previous studies. Separate LFs are also derived for emission line galaxies and absorption line galaxies. The LFs of absorption line galaxies show a dip at 0.1r** 18.5 and can be fitted well by a double-Gaussian function, suggesting a bimodality in passive galaxies.

Bias Correction of Multiple MRI Images Based on an Improved Nonparametric Maximum Likelihood Method

With the wide application of nuclear magnetic resonance imaging, the multiplicative bias field in nuclear magnetic resonance images has created great difficulties for doctors in reading diagnostics and for computers in autoprocessing. Most previous methods eliminate the bias field in the image by estimating a single unknown bias field. An improved method that uses the nonparametric maximum likelihood to jointly eliminate bias from multiple magnetic resonance imaging (MRI) images is proposed in this paper. The method uses the statistics from the same location across different patient images, rather than within an image, and builds a “multiresolution” nonparametric tissue model conditioned on image location. We use a separate and nonparametric model to consider the intensity values at each pixel and utilize nonparametric maximum likelihood distance measures to simultaneously eliminate the bias of magnetic resonance (MR) images from different patients. Finally, the performance of the same was tested on a synthetic MRI dataset and a real MRI dataset and is found that the proposed algorithm provides better performance than the method of using entropy minimization across images and the most popular and widely used method, N4.

Mixture regression models for the gap time distributions and illness-death processes.

The aim of this study is to provide an analysis of gap event times under the illness-death model, where some subjects experience "illness" before "death" and others experience only "death." Which event is more likely to occur first and how the duration of the "illness" influences the "death" event are of interest. Because the occurrence of the second event is subject to dependent censoring, it can lead to bias in the estimation of model parameters. In this work, we generalize the semiparametric mixture models for competing risks data to accommodate the subsequent event and use a copula function to model the dependent structure between the successive events. Under the proposed method, the survival function of the censoring time does not need to be estimated when developing the inference procedure. We incorporate the cause-specific hazard functions with the counting process approach and derive a consistent estimation using the nonparametric maximum likelihood method. Simulations are conducted to demonstrate the performance of the proposed analysis, and its application in a clinical study on chronic myeloid leukemia is reported to illustrate its utility.

REBayes: An R Package for Empirical Bayes Mixture Methods

Models of unobserved heterogeneity, or frailty as it is commonly known in survival analysis, can often be formulated as semiparametric mixture models and estimated by maximum likelihood as proposed by Robbins (1950) and elaborated by Kiefer and Wolfowitz (1956). Recent developments in convex optimization, as noted by Koenker and Mizera (2014b), have led to dramatic improvements in computational methods for such models. In this vignette we describe an implementation contained in the R package REBayes with applications to a wide variety of mixture settings: Gaussian location and scale, Poisson and binomial mixtures for discrete data, Weibull and Gompertz models for survival data, and several Gaussian models intended for longitudinal data. While the dimension of the nonparametric heterogeneity of these models is inherently limited by our present gridding strategy, we describe how additional fixed parameters can be relatively easily accommodated via profile likelihood. We also describe some nonparametric maximum likelihood methods for shape and norm constrained density estimation that employ related computational methods.

Comparisons of computational methods for clustered binary data

Clustered binary data are common in medical research and can be fitted to the logistic regression model with random effects which belongs to a wider class of models called the generalized linear mixed model. The likelihood-based estimation of model parameters often has to handle intractable integration which leads to several estimation methods to overcome such difficulty. The penalized quasi-likelihood (PQL) method is the one that is very popular and computationally efficient in most cases. The expectation–maximization (EM) algorithm allows to estimate maximum-likelihood estimates, but requires to compute possibly intractable integration in the E-step. The variants of the EM algorithm to evaluate the E-step are introduced. The Monte Carlo EM (MCEM) method computes the E-step by approximating the expectation using Monte Carlo samples, while the Modified EM (MEM) method computes the E-step by approximating the expectation using the Laplace's method. All these methods involve several steps of approximation so that corresponding estimates of model parameters contain inevitable errors (large or small) induced by approximation. Understanding and quantifying discrepancy theoretically is difficult due to the complexity of approximations in each method, even though the focus is on clustered binary data. As an alternative competing computational method, we consider a non-parametric maximum-likelihood (NPML) method as well. We review and compare the PQL, MCEM, MEM and NPML methods for clustered binary data via simulation study, which will be useful for researchers when choosing an estimation method for their analysis.

Random effects models for assessing diagnostic accuracy of traditional Chinese doctors in absence of a gold standard

Two common problems in assessing the accuracy of traditional Chinese medicine (TCM) doctors in detecting a particular symptom are the unknown true symptom status and the ordinal-scale of the symptom status. Wang et al. (Biostatistics 2011; DOI: 10.1093/biostatistics/kxq075) proposed a nonparametric maximum likelihood method for estimating the accuracy of different TCM doctors without a gold standard when the true symptom status is measured on an ordinal-scale. A key assumption of their work is that the diagnosis results are independent conditional on the gold standard. This assumption can be violated in many practical situations.In this paper, we propose a random effects modeling approach that extends their method to incorporate dependence structure among different tests or doctors. The proposed method is illustrated on a real data set from TCM, which contains the diagnostic results from five doctors for the same patients regarding symptoms related to Chills disease. The same data set was analyzed by Wang et al. under the conditional independence assumption. In addition, we also discuss an ad hoc test for the model fitting and a likelihood ratio test on the random effects.

Evaluation of diagnostic accuracy in detecting ordered symptom statuses without a gold standard

Our research is motivated by 2 methodological problems in assessing diagnostic accuracy of traditional Chinese medicine (TCM) doctors in detecting a particular symptom whose true status has an ordinal scale and is unknown-imperfect gold standard bias and ordinal scale symptom status. In this paper, we proposed a nonparametric maximum likelihood method for estimating and comparing the accuracy of different doctors in detecting a particular symptom without a gold standard when the true symptom status had an ordered multiple class. In addition, we extended the concept of the area under the receiver operating characteristic curve to a hyper-dimensional overall accuracy for diagnostic accuracy and alternative graphs for displaying a visual result. The simulation studies showed that the proposed method had good performance in terms of bias and mean squared error. Finally, we applied our method to our motivating example on assessing the diagnostic abilities of 5 TCM doctors in detecting symptoms related to Chills disease.

Efficient Estimation for the Accelerated Failure Time Model

The accelerated failure time model provides a natural formulation of the effects of covariates on potentially censored response variable. The existing semiparametric estimators are computationally intractable and statistically inefficient. In this article we propose an approximate nonparametric maximum likelihood method for the accelerated failure time model with possibly time-dependent covariates. We estimate the regression parameters by maximizing a kernel-smoothed profile likelihood function. The maximization can be achieved through conventional gradient-based search algorithms. The resulting estimators are consistent and asymptotically normal. The limiting covariance matrix attains the semiparametric efficiency bound and can be consistently estimated. We also provide a consistent estimator for the error distribution. Extensive simulation studies demonstrate that the asymptotic approximations are accurate in practical situations and the new estimators are considerably more efficient than the existing ones. Illustrations with clinical and epidemiologic studies are provided.

A linearization procedure and a VDM/ECM algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models

Suppose independent observations X i , i = 1 , … , n are observed from a mixture model f ( x ; Q ) ≡ ∫ f ( x ; λ ) d Q ( λ ) , where λ is a scalar and Q ( λ ) is a nondegenerate distribution with an unspecified form. We consider to estimate Q ( λ ) by nonparametric maximum likelihood (NPML) method under two scenarios: (1) the likelihood is penalized by a functional g ( Q ) ; and (2) Q is under a constraint g ( Q ) = g 0 . We propose a simple and reliable algorithm termed VDM/ECM for Q-estimation when the likelihood is penalized by a linear functional. We show this algorithm can be applied to a more general situation where the penalty is not linear, but a function of linear functionals by a linearization procedure. The constrained NPMLE can be found by penalizing the quadratic distance | g ( Q ) - g 0 | 2 under a large penalty factor γ > 0 using this algorithm. The algorithm is illustrated with two real data sets.

Copulas, degenerate distributions and quantile tests in competing risk problems

It is well known that, given observable data for a competing risk problem, there is always an independent model consistent with the data. It has been pointed out, however, that this independent model does not necessarily have to be one with proper marginals. One purpose of this paper is to explore the extent to which one might try to use the non-parametric assumption that the marginals are proper in order to test whether or not independence holds. This will lead us naturally to a closely related estimation problem—how we estimate the marginals given that a certain quantile of one variable is reached at the same time as a given quantile of the other variable. The problem will be considered using the copula-based approach of Zheng and Klein. Two different methods are discussed. One is a non-parametric maximum likelihood method. The other is a consistent estimator, called the bilinear adjustment estimator, that can be computed quickly and which thus lends itself more readily to simulation methods such as bootstrapping. The ultimate objective of the work in this paper is to provide an additional analytical tool to understand the effectiveness of preventive maintenance. Preventive maintenance censors component failure data and thus provides an important example of competing risk within the reliability area.

Nonparametric Estimation of ROC Curves in the Absence of a Gold Standard

In the evaluation of diagnostic accuracy of tests, a gold standard on the disease status is required. However, in many complex diseases, it is impossible or unethical to obtain such a gold standard. If an imperfect standard is used, the estimated accuracy of the tests would be biased. This type of bias is called imperfect gold standard bias. In this article we develop a nonparametric maximum likelihood method for estimating ROC curves and their areas of ordinal-scale tests in the absence of a gold standard. Our simulation study shows that the proposed estimators for the ROC curve areas have good finite-sample properties in terms of bias and mean squared error. Further simulation studies show that our nonparametric approach is comparable to the binormal parametric method, and is easier to implement. Finally, we illustrate the application of the proposed method in a real clinical study on assessing the accuracy of seven specific pathologists in detecting carcinoma in situ of the uterine cervix.

Nonparametric Population Pharmacokinetic Analysis of Amikacin in Neonates, Infants, and Children

The therapeutic and toxic effects of amikacin are known to depend on its concentration in plasma, but the pharmacokinetics of this drug in neonates, infants, and children and the influences of clinical and biological variables have been only partially assessed. Therapeutic drug monitoring data collected from 155 patients (49 neonates, 77 infants, and 29 children) receiving amikacin were analyzed by a nonparametric population-based approach, the nonparametric maximum-likelihood method. We assessed the effects of gestational and postnatal age, weight, Apgar score, and plasma creatinine and urea concentrations on pharmacokinetic parameters. There is no specific formulation of amikacin for neonates and infants. We therefore used an error model to account for errors due to dilution during preparation of the infusion. The covariates that reduced the variance of clearance from plasma and the volume of distribution by more than 10% were postnatal age (43 and 28%, respectively) and body weight (30.4 and 17.4%, respectively). The expected reduction of clearance was about 10% for the plasma creatinine concentration. The other covariates studied (Apgar scores, plasma urea concentration, gestational age, sex) were found to have little effect. Simulations showed that a smaller percentage of patients had a maximum concentration in plasma/MIC ratio greater than 8 with a regimen of 7.5 mg/kg of body weight twice daily than with a regimen of 15 mg/kg once a day for MICs of 1 to 8 mg/liter.

Covariate measurement errors in frailty models for clustered survival data

SUMMARY We propose a new class of frailty measurement error models for clustered survival data when covariates are measured with error. We show that the induced hazard function conditional on the observed covariates also follows a frailty model but of a more complicated form. We study the asymptotic bias in regression coefficients and variance components when measurement error is ignored, and the impact of censoring on this asymptotic bias. We show that the naive estimator of the regression coefficient is attenuated and the naive estimator of the variance component is inflated when measurement error is ignored. As the censoring proportion increases, the asymptotic bias in the former becomes larger, but the asymptotic bias in the latter interestingly becomes smaller. We develop a structural approach for parameter estimation using the nonparametric maximum likelihood method, where the baseline hazard is estimated nonparametrically. We prove model identifiability and the existence of the nonparametric maximum likelihood estimators. An EM algorithm is developed for calculating the nonparametric maximum likelihood estimates. The method is applied to the western Kenya parasitaemia data and its performance is evaluated through simulations.

Population pharmacokinetic analysis of mizolastine and validation from sparse data on patients using the nonparametric maximum likelihood method.

A population analysis of the kinetics of mizolastine was performed from concentrations on 449 allergic patients, using the nonparametric maximum likelihood method (NPML). A two-compartment open model with zero-order absorption was used to describe the kinetics of mizolastine after oral administration. A heteroscedastic variance model was assumed for the error. To explain the kinetic variability, eight covariates were introduced in the analysis: gender, pharmaceutical dosage form, age, body weight, serum creatinine concentration, creatinine renal clearance, plasma levels of hepatic transaminases ASAT and ALAT. Their relationships to the kinetic parameters were studied by means of the estimated distribution of each kinetic parameter conditional on different levels of each covariate. An important interindividual kinetic variability was found for all parameters. Moreover, several kinetic parameters among which the duration of absorption were found to be influenced by pharmaceutical dosage form and gender. Body weight and creatinine renal clearance were found to have a little influence on the oral clearance and the smallest disposition rate constant. This population analysis was validated on a separate group of 247 other patients. For each observed concentration of this sample, a predictive distribution was computed using the individual covariates. Predicted concentrations and standardized prediction errors were deduced. The mean and variance of the standardized prediction errors were, respectively, 0.21 and 2.79. Moreover, in the validation sample, the predicted cumulative distribution function of each observed concentration was computed. Empirical distribution of these values was not significantly different from a uniform distribution, as expected under the assumption that the population model estimated by NPML is adequate.