Nonparametric Maximum Likelihood Approach Research Articles

Survival analysis with a random change-point.

Contemporary works in change-point survival models mainly focus on an unknown universal change-point shared by the whole study population. However, in some situations, the change-point is plausibly individual-specific, such as when it corresponds to the telomere length or menopausal age. Also, maximum-likelihood-based inference for the fixed change-point parameter is notoriously complicated. The asymptotic distribution of the maximum-likelihood estimator is non-standard, and computationally intensive bootstrap techniques are commonly used to retrieve its sampling distribution. This article is motivated by a breast cancer study, where the disease-free survival time of the patients is postulated to be regulated by the menopausal age, which is unobserved. As menopausal age varies across patients, a fixed change-point survival model may be inadequate. Therefore, we propose a novel proportional hazards model with a random change-point. We develop a nonparametric maximum-likelihood estimation approach and devise a stable expectation-maximization algorithm to compute the estimators. Because the model is regular, we employ conventional likelihood theory for inference based on the asymptotic normality of the Euclidean parameter estimators, and the variance of the asymptotic distribution can be consistently estimated by a profile-likelihood approach. A simulation study demonstrates the satisfactory finite-sample performance of the proposed methods, which yield small bias and proper coverage probabilities. The methods are applied to the motivating breast cancer study.

Efficient Estimation and Inference in the Proportional Odds Model for Survival Data

In modeling time-to-event data with long-term survivors, the proportional hazards model is widely used for its easy and direct interpretation as well as the flexibility to accommodate the past information and allow time-varying predictors. This becomes most relevant when the mortality of individuals converges with time, and the estimation and inference based upon the proportional odds model can often yield more accurate and reasonable results than the classical Cox’s proportional hazards model. Along with the fast development of the data science technologies, computational challenges for survival data with increasing sample size and diverging parameter dimension exist. Currently, existing methods for analyzing such data are computationally inconvenient. In this paper, we propose efficient computational methods for analyzing survival data in the proportional odds model, where the nonparametric maximum likelihood approach is combined with the minorization-maximization (MM) algorithm and the regularization scheme to yield fast and accurate estimation and inferential procedures. The illustration of the methodology using extensive simulation studies and then the application to the Veterans’ Administration lung cancer data is also given.

Identification in ascending auctions, with an application to digital rights management

This study provides new identification and estimation results for ascending (traditional English or online) auctions with unobserved auction‐level heterogeneity and an unknown number of bidders. When the seller's reserve price and two order statistics of bids are observed, we derive conditions under which the distributions of buyer valuations, unobserved heterogeneity, and number of participants are point identified. We also derive conditions for point identification in cases where reserve prices are binding and present general conditions for partial identification. We propose a nonparametric maximum likelihood approach for estimation and inference. We apply our approach to the online market for used iPhones and analyze the effects of recent regulatory changes banning consumers from circumventing digital rights management technologies used to lock phones to service providers. We find that buyer valuations for unlocked phones dropped by 39% on average after the unlocking ban took effect, from $231.30 to $141.50.

Predictive accuracy of markers or risk scores for interval censored survival data.

Methods for the evaluation of the predictive accuracy of biomarkers with respect to survival outcomes subject to right censoring have been discussed extensively in the literature. In cancer and other diseases, survival outcomes are commonly subject to interval censoring by design or due to the follow up schema. In this article, we present an estimator for the area under the time-dependent receiver operating characteristic (ROC) curve for interval censored data based on a nonparametric sieve maximum likelihood approach. We establish the asymptotic properties of the proposed estimator and illustrate its finite-sample properties using a simulation study. The application of our method is illustrated using data from a cancer clinical study. An open-source R package to implement the proposed method is available on Comprehensive R Archive Network.

Functional estimation and hypothesis testing in nonparametric boundary models

Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int\Phi(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Hölder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $\Phi$ which are satisfied for $\Phi(u)=|u|^{p}$, $p\ge1$. As an application, we consider the problem of estimating the $L^{p}$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.

An extended semiparametric model for short-term and long-term hazard ratios

ABSTRACTIn survival analysis, one way to deal with non-proportional hazards is to model short-term and long-term hazard ratios. The existing model of this nature has no control over how fast the hazard ratio is changing over time. We add a parameter to the existing model to allow the hazard ratio to change over time at different speed. A nonparametric maximum likelihood approach is used to estimate the model parameters. The existing model is a special case of the extended model when the speed parameter is 0, which leads naturally to a way of testing the adequacy of the existing model. Simulation results show that there can be substantial bias in the estimation of the short-term and long-term hazard ratio if the speed parameter is fixed incorrectly at 0 rather than estimated. The extended model is fitted to three real data sets to shed new insights, including the observation that converging hazards does not necessarily imply the odds are proportional.

Gumbel regression models for a monotone increasing continuous biomarker subject to measurement error

Right- or left-skewed continuous biomarker measurements are often confronted in practice. The extreme values located in the long tail of skewed distributions indicate abnormal health status, and characterizing the features of a subgroup with abnormally high(or low) values is important for risk management. Furthermore, a biomarker is usually subject to measurement error. In the presence of these two challenging issues, existing methods for analyzing skewed biomarker data are no longer applicable. In this paper, we propose a semiparametric method based on Gumbel distributions while accounting for measurement error in a biomarker and adjusting for a monotone age effect on biomarker levels. We estimate the model parameters using the nonparametric maximum likelihood approach and implement computation via the EM algorithm. We establish the asymptotic properties of the proposed estimators and summarize simulation results to assess the numerical performance of the proposed method. The method is illustrated through an application to data from a diabetes ancillary study to the Atherosclerosis Risk in Communities (ARIC) Study.

SEMIPARAMETRIC TRANSFORMATION MODELS WITH MULTILEVEL RANDOM EFFECTS FOR CORRELATED DISEASE ONSET IN FAMILIES

Large cohort studies are commonly launched to study risk of genetic variants or other risk factors on age at onset (AAO) of a chronic disorder. In these studies, family history data including AAO of disease in family members are collected to provide additional information and can be used to improve efficiency. Statistical analysis of these data is challenging due to missing genotypes in family members and the heterogeneous dependence attributed to both shared genetic back-ground and shared environmental factors (e.g., life style). In this paper, we propose a class of semiparametric transformation models with multilevel random effects to tackle these challenges. The proposed models include both proportional hazards model and proportional odds model as special cases. The multilevel random effects contain individual-specific random effects including kinship correlation structure dependent on the family pedigree, and a shared random effect to account for unobserved environment exposure. We use nonparametric maximum likelihood approach for inference and propose an expectation-maximization algorithm for computation in the presence of missing genotypes among family members. The obtained estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient. Simulation studies demonstrate that the proposed method performs well with finite sample sizes. Finally, the proposed method is applied to study genetic risks in an Alzheimer's disease study.

A nonparametric maximum likelihood approach for survival data with observed cured subjects, left truncation and right-censoring.

We consider observational studies in pregnancy where the outcome of interest is spontaneous abortion (SAB). This at first sight is a binary 'yes' or 'no' variable, albeit there is left truncation as well as right-censoring in the data. Women who do not experience SAB by gestational week 20 are 'cured' from SAB by definition, that is, they are no longer at risk. Our data is different from the common cure data in the literature, where the cured subjects are always right-censored and not actually observed to be cured. We consider a commonly used cure rate model, with the likelihood function tailored specifically to our data. We develop a conditional nonparametric maximum likelihood approach. To tackle the computational challenge we adopt an EM algorithm making use of "ghost copies" of the data, and a closed form variance estimator is derived. Under suitable assumptions, we prove the consistency of the resulting estimator which involves an unbounded cumulative baseline hazard function, as well as the asymptotic normality. Simulation results are carried out to evaluate the finite sample performance. We present the analysis of the motivating SAB study to illustrate the advantages of our model addressing both occurrence and timing of SAB, as compared to existing approaches in practice.

Efficient estimation of functionals in nonparametric boundary models

For nonparametric regression with one-sided errors and a boundary curve model for Poisson point processes, we consider the problem of efficient estimation for linear functionals. The minimax optimal rate is obtained by an unbiased estimation method which nevertheless depends on a Holder condition or monotonicity assumption for the underlying regression or boundary function. We first construct a simple blockwise estimator and then build up a nonparametric maximum-likelihood approach for exponential noise variables and the point process model. In that approach also non-asymptotic efficiency is obtained (UMVU: uniformly minimum variance among all unbiased estimators). The proofs rely essentially on martingale stopping arguments for counting processes and the point process geometry. The estimators are easily computable and a small simulation study confirms their applicability.

Estimating the Proportion of True Null Hypotheses in Multiple Testing Problems

The problem of estimating the proportion, π0, of the true null hypotheses in a multiple testing problem is important in cases where large scale parallel hypotheses tests are performed independently. While the problem is a quantity of interest in its own right in applications, the estimate of π0 can be used for assessing or controlling an overall false discovery rate. In this article, we develop an innovative nonparametric maximum likelihood approach to estimate π0. The nonparametric likelihood is proposed to be restricted to multinomial models and an EM algorithm is also developed to approximate the estimate of π0. Simulation studies show that the proposed method outperforms other existing methods. Using experimental microarray datasets, we demonstrate that the new method provides satisfactory estimate in practice.

Efficient semiparametric mixture inferences on cure rate models for competing risks

AbstractCancer patients may die from causes other than the diagnosed cancer. In a study of patients treated for soft tissue sarcoma, the patients may die from the disease or die without experiencing disease recurrence. In addition, a substantial proportion of the patients will remain cancer‐free after surgical resection of the tumour, and therefore will not be at increased risk of any type of failure. Our goal is to describe the effect of adjuvant chemotherapy simultaneously on the probabilities of long‐term survival, death from cancer, or death from other causes. To this end, we propose a semiparametric mixture model to determine the effects of factors on the probability of occurrence, allowing the surviving fraction, and the hazard rate conditional on each of the failure types. These quantities are combined in the mixture approach using a multinomial logistic model and a class of semiparametric transformation models. Estimation of the regression and nonparametric parameters is achieved with a novel nonparametric maximum likelihood approach. Statistical inferences can be conveniently made from the inverse of the observed information matrix. Simulation studies show that the procedures work well in practical settings. The methodology is illustrated with data from the soft tissue sarcoma study. The Canadian Journal of Statistics 43: 420–435; 2015 © 2015 Statistical Society of Canada

Nonparametric maximum likelihood approach to multiple change-point problems

In multiple change-point problems, different data segments often follow different distributions, for which the changes may occur in the mean, scale or the entire distribution from one segment to another. Without the need to know the number of change-points in advance, we propose a nonparametric maximum likelihood approach to detecting multiple change-points. Our method does not impose any parametric assumption on the underlying distributions of the data sequence, which is thus suitable for detection of any changes in the distributions. The number of change-points is determined by the Bayesian information criterion and the locations of the change-points can be estimated via the dynamic programming algorithm and the use of the intrinsic order structure of the likelihood function. Under some mild conditions, we show that the new method provides consistent estimation with an optimal rate. We also suggest a prescreening procedure to exclude most of the irrelevant points prior to the implementation of the nonparametric likelihood method. Simulation studies show that the proposed method has satisfactory performance of identifying multiple change-points in terms of estimation accuracy and computation time.

A mixed non‐homogeneous hidden Markov model for categorical data, with application to alcohol consumption

Hidden Markov models (HMMs) are frequently used to analyse longitudinal data, where the same set of subjects is repeatedly observed over time. In this context, several sources of heterogeneity may arise at individual and/or time level, which affect the hidden process, that is, the transition probabilities between the hidden states. In this paper, we propose the use of a finite mixture of non-homogeneous HMMs (NH-HMMs) to face the heterogeneity problem. The non-homogeneity of the model allows us to take into account observed sources of heterogeneity by means of a proper set of covariates, time and/or individual dependent, explaining the variations in the transition probabilities. Moreover, we handle the unobserved sources of heterogeneity at the individual level, due to, for example, omitted covariates, by introducing a random term with a discrete distribution. The resulting model is a finite mixture of NH-HMM that can be used to classify individuals according to their dynamic behaviour or to estimate a mixed NH-HMM without any assumption regarding the distribution of the random term following the non-parametric maximum likelihood approach. We test the effectiveness of the proposal through a simulation study and an application to real data on alcohol abuse.

The robust estimation method for a finite mixture of Poisson mixed-effect models

When analyzing clustered count data derived from several latent subpopulations, the finite mixture of the Poisson mixed-effect model is an immediate strategy to model the underlying heterogeneity. Within the generalized linear mixed model framework, parameters in such a model are often estimated through the residual maximum likelihood estimation approach. However, the method is vulnerable to outliers. To develop robust estimators, the minimum Hellinger distance (MHD) estimation approach has been proposed by Xiang et al. (Xiang, L., Yau, K.K.W., Lee, A.H., Hui, Y.V., 2008. Minimum Hellinger distance estimation for k-component Poisson mixture with random effects. Biometrics 64, 508–518) with the random effects following a normal distribution. In some circumstances, there is little information available on the joint distributional form of the random effects. Without prescribing a parametric form for the random effects distribution, we consider embedding the non-parametric maximum likelihood (NPML) approach within the MHD estimation to extend the robust estimation method for a finite mixture of Poisson mixed-effect models. The NPML estimation not only avoids the problem of numerical integration in deriving the MHD estimating equations, but also enhances the robustness characteristic because of its resistance to possible misspecification of the parametric distribution for the random effects. The performance of the new method is evaluated and compared with that of the existing MHD estimation using simulations. Application to analyze a real data set of recurrent urinary tract infections is illustrated.

How individual characteristics affect university students drop-out: a semiparametric mixed-effects model for an Italian case study

University drop-out is a topic of increasing concern in Italy as well as in other countries. In empirical analysis, university drop-out is generally measured by means of a binary variable indicating the drop-out versus retention. In this paper, we argue that the withdrawal decision is one of the possible outcomes of a set of four alternatives: retention in the same faculty, drop out, change of faculty within the same university, and change of institution. We examine individual-level data collected by the administrative offices of “Sapienza” University of Rome, which cover 117 072 students enrolling full-time for a 3-year degree in the academic years from 2001/2002 to 2006/2007. Relying on a non-parametric maximum likelihood approach in a finite mixture context, we introduce a multinomial latent effects model with endogeneity that accounts for both heterogeneity and omitted covariates. Our estimation results show that the decisions to change faculty or university have their own peculiarities, thus we suggest that caution should be used in interpreting results obtained without modeling all the relevant alternatives that students face.

Maximum likelihood analysis of semicompeting risks data with semiparametric regression models

The "semicompeting risks" include a terminal event and a non-terminal event. The terminal event may censor the non-terminal event but not vice versa. Because times to the two events are usually correlated, the non-terminal event is subject to dependent/informative censoring by the terminal event. We seek to conduct marginal regressions and joint association analyses for the two event times under semicompeting risks. The proposed method is based on the modeling setup where the semiparametric transformation models are assumed for marginal regressions, and a copula model is assumed for the joint distribution. We propose a nonparametric maximum likelihood approach for inferences, which provides a martingale representation for the score function and an analytical expression for the information matrix. Direct theoretical developments and computational implementation are allowed for the proposed approach. Simulations and a real data application demonstrate the utility of the proposed methodology.

Semiparametric Transformation Models with Random Effects for Joint Analysis of Recurrent and Terminal Events

We propose a broad class of semiparametric transformation models with random effects for the joint analysis of recurrent events and a terminal event. The transformation models include proportional hazards/intensity and proportional odds models. We estimate the model parameters by the nonparametric maximum likelihood approach. The estimators are shown to be consistent, asymptotically normal, and asymptotically efficient. Simple and stable numerical algorithms are provided to calculate the parameter estimators and to estimate their variances. Extensive simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. Applications to two HIV/AIDS studies are presented.

New multi-sample nonparametric tests for panel count data

This paper considers the problem of multi-sample nonparametric comparison of counting processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For the problem considered, we construct two new classes of nonparametric test statistics based on the accumulated weighted differences between the rates of increase of the estimated mean functions of the counting processes over observation times, wherein the nonparametric maximum likelihood approach is used to estimate the mean function instead of the nonparametric maximum pseudo-likelihood. The asymptotic distributions of the proposed statistics are derived and their finite-sample properties are examined through Monte Carlo simulations. The simulation results show that the proposed methods work quite well and are more powerful than the existing test procedures. Two real data sets are analyzed and presented as illustrative examples.

A semiparametric approach to hidden Markov models under longitudinal observations

We propose a hidden Markov model for longitudinal count data where sources of unobserved heterogeneity arise, making data overdispersed. The observed process, conditionally on the hidden states, is assumed to follow an inhomogeneous Poisson kernel, where the unobserved heterogeneity is modeled in a generalized linear model (GLM) framework by adding individual-specific random effects in the link function. Due to the complexity of the likelihood within the GLM framework, model parameters may be estimated by numerical maximization of the log-likelihood function or by simulation methods; we propose a more flexible approach based on the Expectation Maximization (EM) algorithm. Parameter estimation is carried out using a non-parametric maximum likelihood (NPML) approach in a finite mixture context. Simulation results and two empirical examples are provided.