Profile Likelihood Approach Research Articles

A flexible family of transformation cure rate models.

In this paper, we introduce a flexible family of cure rate models, mainly motivated by the biological derivation of the classical promotion time cure rate model and assuming that a metastasis-competent tumor cell produces a detectable-tumor mass only when a specific number of distinct biological factors affect the cell. Special cases of the new model are, among others, the promotion time (proportional hazards), the geometric (proportional odds), and the negative binomial cure rate model. In addition, our model generalizes specific families of transformation cure rate models and some well-studied destructive cure rate models. Exact likelihood inference is carried out by the aid of the expectationŰmaximization algorithm; a profile likelihood approach is exploited for estimating the parameters of the model while model discrimination problem is analyzed by the aid of the likelihood ratio test. A simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data-set. Copyright © 2017 John Wiley & Sons, Ltd.

Parameter estimation for multistage clonal expansion models from cancer incidence data: A practical identifiability analysis.

Many cancers are understood to be the product of multiple somatic mutations or other rate-limiting events. Multistage clonal expansion (MSCE) models are a class of continuous-time Markov chain models that capture the multi-hit initiation–promotion–malignant-conversion hypothesis of carcinogenesis. These models have been used broadly to investigate the epidemiology of many cancers, assess the impact of carcinogen exposures on cancer risk, and evaluate the potential impact of cancer prevention and control strategies on cancer rates. Structural identifiability (the analysis of the maximum parametric information available for a model given perfectly measured data) of certain MSCE models has been previously investigated. However, structural identifiability is a theoretical property and does not address the limitations of real data. In this study, we use pancreatic cancer as a case study to examine the practical identifiability of the two-, three-, and four-stage clonal expansion models given age-specific cancer incidence data using a numerical profile-likelihood approach. We demonstrate that, in the case of the three- and four-stage models, several parameters that are theoretically structurally identifiable, are, in practice, unidentifiable. This result means that key parameters such as the intermediate cell mutation rates are not individually identifiable from the data and that estimation of those parameters, even if structurally identifiable, will not be stable. We also show that products of these practically unidentifiable parameters are practically identifiable, and, based on this, we propose new reparameterizations of the model hazards that resolve the parameter estimation problems. Our results highlight the importance of identifiability to the interpretation of model parameter estimates.

Estimation and Identifiability of Model Parameters in Human Nociceptive Processing Using Yes-No Detection Responses to Electrocutaneous Stimulation.

Healthy or pathological states of nociceptive subsystems determine different stimulus-response relations measured from quantitative sensory testing. In turn, stimulus-response measurements may be used to assess these states. In a recently developed computational model, six model parameters characterize activation of nerve endings and spinal neurons. However, both model nonlinearity and limited information in yes-no detection responses to electrocutaneous stimuli challenge to estimate model parameters. Here, we address the question whether and how one can overcome these difficulties for reliable parameter estimation. First, we fit the computational model to experimental stimulus-response pairs by maximizing the likelihood. To evaluate the balance between model fit and complexity, i.e., the number of model parameters, we evaluate the Bayesian Information Criterion. We find that the computational model is better than a conventional logistic model regarding the balance. Second, our theoretical analysis suggests to vary the pulse width among applied stimuli as a necessary condition to prevent structural non-identifiability. In addition, the numerically implemented profile likelihood approach reveals structural and practical non-identifiability. Our model-based approach with integration of psychophysical measurements can be useful for a reliable assessment of states of the nociceptive system.

An EM type estimation procedure for the destructive exponentially weighted Poisson regression cure model under generalized gamma lifetime

ABSTRACTIn this paper, we assume the number of competing causes to follow an exponentially weighted Poisson distribution. By assuming the initial number of competing causes can undergo destruction and that the population of interest has a cure fraction, we develop the EM algorithm for the determination of the MLEs of the model parameters of such a general cure model. This model is more flexible than the promotion time cure model and also provides an interesting and realistic interpretation of the biological mechanism of the occurrence of an event of interest. Instead of assuming a particular parametric distribution for the lifetime, we assume the lifetime to belong to the wider class of generalized gamma distribution. This allows us to carry out a model discrimination to select a parsimonious lifetime distribution that provides the best fit to the data. Within the EM framework, a two-way profile likelihood approach is proposed to estimate the shape parameters. An extensive Monte Carlo simulation study is carried out to demonstrate the performance of the proposed estimation method. Model discrimination is carried out by means of the likelihood ratio test and information-based methods. Finally, a data on melanoma is analyzed for illustrative purpose.

Improved EDELWEISS-III sensitivity for low-mass WIMPs using a profile likelihood approach

We report on a dark matter search for a Weakly Interacting Massive Particle (WIMP) in the mass range $m_\chi \in [4, 30]\,\mathrm{GeV}/c^2$ with the EDELWEISS-III experiment. A 2D profile likelihood analysis is performed on data from eight selected detectors with the lowest energy thresholds leading to a combined fiducial exposure of 496 kg-days. External backgrounds from $\gamma$- and $\beta$-radiation, recoils from $^{206}$Pb and neutrons as well as detector intrinsic backgrounds were modelled from data outside the region of interest and constrained in the analysis. The basic data selection and most of the background models are the same as those used in a previously published analysis based on Boosted Decision Trees (BDT). For the likelihood approach applied in the analysis presented here, a larger signal efficiency and a subtraction of the expected background lead to a higher sensitivity, especially for the lowest WIMP masses probed. No statistically significant signal was found and upper limits on the spin-independent WIMP-nucleon scattering cross section can be set with a hypothesis test based on the profile likelihood test statistics. The 90% C.L. exclusion limit set for WIMPs with $m_\chi = 4\,\mathrm{GeV/}c^2$ is $1.6 \times 10^{-39}\,\mathrm{cm^2}$, which is an improvement of a factor of seven with respect to the BDT-based analysis. For WIMP masses above $15\,\mathrm{GeV/}c^2$ the exclusion limits found with both analyses are in good agreement.

Non-parametric maximum likelihood estimation of interval-censored failure time data subject to misclassification

The paper considers non-parametric maximum likelihood estimation of the failure time distribution for interval-censored data subject to misclassification. Such data can arise from two types of observation scheme; either where observations continue until the first positive test result or where tests continue regardless of the test results. In the former case, the misclassification probabilities must be known, whereas in the latter case, joint estimation of the event-time distribution and misclassification probabilities is possible. The regions for which the maximum likelihood estimate can only have support are derived. Algorithms for computing the maximum likelihood estimate are investigated and it is shown that algorithms appropriate for computing non-parametric mixing distributions perform better than an iterative convex minorant algorithm in terms of time to absolute convergence. A profile likelihood approach is proposed for joint estimation. The methods are illustrated on a data set relating to the onset of cardiac allograft vasculopathy in post-heart-transplantation patients.

A regularized profile likelihood approach to covariance matrix estimation

Two new orthogonally equivariant estimators of the covariance matrix are proposed. The estimates of the population eigenvalues are isotonized maximum likelihood estimates of the modified profile likelihood obtained from the Wishart distribution, in one case, and of a penalized form of such a likelihood function, in the other, with a penalty that constrains the trace of the sample covariance matrix. Properties of these estimators are studied and numerical risk comparisons with six other well-known estimators are presented to demonstrate the robustness of the proposed estimators for various real and simulated covariance structures.

Joint Modeling of Survival and Longitudinal Ordered Data Using a Semiparametric Approach

SummaryMedical research frequently focuses on the relationship between quality of life (QoL) and survival time of subjects. QoL may be one of the most important factors that could be used to predict survival, making it worth identifying factors that jointly affect survival and QoL. We propose a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. Several popular ordinal models are considered and compared in the item response component, while the Cox proportional hazards model is used in the survival component. We estimate the baseline hazard function and model parameters simultaneously, through a profile likelihood approach. We illustrate the method using an example from a clinical study.

Constructing confidence intervals of extreme rainfall quantiles using Bayesian, bootstrap, and profile likelihood approaches

Hydrological risk is highly dependent on the occurrence of extreme rainfalls. This fact has led to a wide range of studies on the estimation and uncertainty analysis of the extremes. In most cases, confidence intervals (CIs) are constructed to represent the uncertainty of the estimates. Since the accuracy of CIs depends on the asymptotic normality of the data and is questionable with limited observations in practice, a Bayesian highest posterior density (HPD) interval, bootstrap percentile interval, and profile likelihood (PL) interval have been introduced to analyze the uncertainty that does not depend on the normality assumption. However, comparison studies to investigate their performances in terms of the accuracy and uncertainty of the estimates are scarce. In addition, the strengths, weakness, and conditions necessary for performing each method also must be investigated. Accordingly, in this study, test experiments with simulations from varying parent distributions and different sample sizes were conducted. Then, applications to the annual maximum rainfall (AMR) time series data in South Korea were performed. Five districts with 38-year (1973–2010) AMR observations were fitted by the three aforementioned methods in the application. From both the experimental and application results, the Bayesian method is found to provide the lowest uncertainty of the design level while the PL estimates generally have the highest accuracy but also the largest uncertainty. The bootstrap estimates are usually inferior to the other two methods, but can perform adequately when the distribution model is not heavy-tailed and the sample size is large. The distribution tail behavior and the sample size are clearly found to affect the estimation accuracy and uncertainty. This study presents a comparative result, which can help researchers make decisions in the context of assessing extreme rainfall uncertainties.

A corrected profile likelihood method for survival data with covariate measurement error under the Cox model

AbstractIn survival analysis, covariate measurement error has been studied extensively for the Cox model. In this article, we propose a corrected profile likelihood approach, and show that many existing methods can be unified by our approach. Furthermore, we extend our discussion to general measurement error and Berkson models, as opposed to the classical additive error model that has been widely used in the literature. We investigate the impact of model misspecification of the measurement error process and uncover interesting findings. Empirical studies are carried out to illustrate the usage of the proposed methods and to assess their performance. The Canadian Journal of Statistics 43: 454–480; 2015 © 2015 Statistical Society of Canada

Confidence bounds for nonlinear dose-response relationships.

An important aim of drug trials is to characterize the dose-response relationship of a new compound. Such a relationship can often be described by a parametric (nonlinear) function that is monotone in dose. If such a model is fitted, it is useful to know the uncertainty of the fitted curve. It is well known that Wald confidence intervals are based on linear approximations and are often unsatisfactory in nonlinear models. Apart from incorrect coverage rates, they can be unreasonable in the sense that the lower confidence limit of the difference to placebo can be negative, even when an overall test shows a significant positive effect. Bootstrap confidence intervals solve many of the problems of the Wald confidence intervals but are computationally intensive and prone to undercoverage for small sample sizes. In this work, we propose a profile likelihood approach to compute confidence intervals for the dose-response curve. These confidence bounds have better coverage than Wald intervals and are more precise and generally faster than bootstrap methods. Moreover, if monotonicity is assumed, the profile likelihood approach takes this automatically into account. The approach is illustrated using a public dataset and simulations based on the Emax and sigmoid Emax models.

Norovirus Dose-Response: Are Currently Available Data Informative Enough to Determine How Susceptible Humans Are to Infection from a Single Virus?

Two forms of single-hit infection dose-response models have previously been developed to assess available data from human feeding trials and estimate the norovirus dose-response relationship. The mechanistic interpretations of these models include strong assumptions that warrant reconsideration: the first study includes an implicit assumption that there is no immunity to Norwalk virus among the specific study population, while the recent second study includes assumptions that such immunity could exist and that the nonimmune have no defensive barriers to prevent infection from exposure to just one virus. Both models addressed unmeasured virus aggregation in administered doses. In this work, the available data are reanalyzed using a generalization of the first model to explore these previous assumptions. It was hypothesized that concurrent estimation of an unmeasured degree of virus aggregation and important dose-response parameters could lead to structural nonidentifiability of the model (i.e., that a diverse range of alternative mechanistic interpretations yield the same optimal fit), and this is demonstrated using the profile likelihood approach and by algebraic proof. It is also demonstrated that omission of an immunity parameter can artificially inflate the estimated degree of aggregation and falsely suggest high susceptibility among the nonimmune. The currently available data support the assumption of immunity within the specific study population, but provide only weak information about the degree of aggregation and susceptibility among the nonimmune. The probability of infection at low and moderate doses may be much lower than previously asserted, but more data from strategically designed dose-response experiments are needed to provide adequate information.

Profile Hellinger distance estimation

The successful application of the Hellinger distance approach to fully parametric models is well known. The corresponding optimal estimators, known as minimum Hellinger distance (MHD) estimators, are efficient and have excellent robustness properties [Beran R. Minimum Hellinger distance estimators for parametric models. Ann Statist. 1977;5:445–463]. This combination of efficiency and robustness makes MHD estimators appealing in practice. However, their application to semiparametric statistical models, which have a nuisance parameter (typically of infinite dimension), has not been fully studied. In this paper, we investigate a methodology to extend the MHD approach to general semiparametric models. We introduce the profile Hellinger distance and use it to construct a minimum profile Hellinger distance estimator of the finite-dimensional parameter of interest. This approach is analogous in some sense to the profile likelihood approach. We investigate the asymptotic properties such as the asymptotic normality, efficiency, and adaptivity of the proposed estimator. We also investigate its robustness properties. We present its small-sample properties using a Monte Carlo study.

Bivariate log Birnbaum–Saunders distribution

Univariate Birnbaum–Saunders distribution has received a considerable amount of attention in recent years. Rieck and Nedelman (A log-linear model for the Birnbaum–Saunders distribution. Technometrics, 1991;33:51–60) introduced a log Birnbaum–Saunders distribution. The main aim of this paper is to introduce bivariate log Birnbaum–Saunders distribution. The proposed model is symmetric and it has five parameters. It can be obtained using Gaussian copula. Different properties can be obtained using copula structure. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained explicitly. Two-dimensional profile likelihood approach may be adopted to compute the MLEs. We propose some alternative estimators also, which can be obtained quite conveniently. The analysis of one data set is performed for illustrative purposes. Finally, it is observed that this model can be used as a bivariate log-linear model, and its multivariate generalization is also quite straight forward.

Biases on cosmological parameter estimators from galaxy cluster number counts

Sunyaev-Zel'dovich (SZ) surveys are promising probes of cosmology — in particular for Dark Energy (DE) —, given their ability to find distant clusters and provide estimates for their mass. However, current SZ catalogs contain tens to hundreds of objects and maximum likelihood estimators may present biases for such sample sizes. In this work we study estimators from cluster abundance for some cosmological parameters, in particular the DE equation of state parameter w0, the amplitude of density fluctuations σ8, and the Dark Matter density parameter Ωc. We begin by deriving an unbinned likelihood for cluster number counts, showing that it is equivalent to the one commonly used in the literature. We use the Monte Carlo approach to determine the presence of bias using this likelihood and study its behavior with both the area and depth of the survey, and the number of cosmological parameters fitted. Our fiducial models are based on the South Pole Telescope (SPT) SZ survey. Assuming perfect knowledge of mass and redshift some estimators have non-negligible biases. For example, the bias of σ8 corresponds to about 40% of its statistical error bar when fitted together with Ωc and w0. Including a SZ mass-observable relation decreases the relevance of the bias, for the typical sizes of current SZ surveys. Considering a joint likelihood for cluster abundance and the so-called ``distance priors'', we obtain that the biases are negligible compared to the statistical errors. However, we show that the biases from SZ estimators do not go away with increasing sample sizes and they may become the dominant source of error for an all sky survey at the SPT sensitivity. Finally, we compute the confidence regions for the cosmological parameters using Fisher matrix and profile likelihood approaches, showing that they are compatible with the Monte Carlo ones. The results of this work validate the use of the current maximum likelihood methods for present SZ surveys, but highlight the need for further studies for upcoming experiments. To perform the analyses of this work, we developed fast, accurate, and adaptable codes for cluster counts in the framework of the Numerical Cosmology Library.

Constraining supersymmetry using the relic density and the Higgs boson

Recent measurements by Planck, LHC experiments, and Xenon100 have significant impact on supersymmetric models and their parameters. We first illustrate the constraints in the mSUGRA plane and then perform a detailed analysis of the general minimal supersymmetric Standard Model (MSSM) with 13 free parameters. Using SFitter, Bayesian and profile likelihood approaches are applied and their results compared. The allowed structures in the parameter spaces are largely defined by different mechanisms of dark matter annihilation in combination with the light Higgs mass prediction. In mSUGRA the pseudoscalar Higgs funnel and stau coannihilation processes are still avoiding experimental pressure. In the MSSM stau coannihilation, the light Higgs funnel, a mixed bino-Higgsino region including the heavy Higgs funnel, and a large Higgsino region predict the correct relic density. Volume effects and changes in the model parameters impact the extracted mSUGRA and MSSM parameter regions in the Bayesian analysis.

A Semiparametric Approach for Accelerated Failure Time Models with Covariates Subject to Measurement Error

There are relatively few discussions about measurement error in the accelerated failure time (AFT) model, particularly for the semiparametric AFT model. In this article, we propose an adjusted estimation procedure for the semiparametric AFT model with covariates subject to measurement error, based on the profile likelihood approach and simulation and exploration (SIMEX) method. The simulation studies show that the proposed semiparametric SIMEX approach performs well. The proposed approach is applied to a coronary heart disease dataset from the Busselton Health study for illustration.

The Estimation of Threshold Models in Price Transmission Analysis

The threshold vector error correction model is a popular tool for the analysis of spatial price transmission. In the literature, the profile likelihood estimator is the preferred choice for estimating this model. Yet, in many settings this estimator performs poorly. In particular, if the true thresholds are such that one or more regimes contain only a small number of observations, if unknown model parameters are numerous, or if parameters differ little between regimes, the profile likelihood estimator displays large bias and variance. Such settings are likely when studying price transmission. We analyze the weaknesses of the profile likelihood approach and propose an alternative regularized Bayesian estimator, which was developed for simpler but related threshold models. Simulation results show that the regularized Bayesian estimator outperforms profile likelihood in the estimation of threshold vector error correction models. Two empirical applications demonstrate the relevance of this new estimator for spatial price transmission analysis.

Modified profile likelihood approach for certain intraclass correlation coefficients

In this paper we consider the problem of constructing confidence intervals and confidence lower bounds for the intraclass correlation coefficient in an interrater reliability study where the raters are randomly selected from a population of raters. The likelihood function of the interrater reliability is derived and simplified, and the profile likelihood based approach is readily available for computing the confidence intervals of the interrater reliability. Unfortunately, the confidence intervals computed by using the profile likelihood function are in general too narrow to have the desired coverage probabilities. From the practical point of view, a conservative approach, if at least as precise as any existing method, is preferred since it gives the correct results with a probability higher than claimed. Under this rationale, we propose the so-called modified profile likelihood approach in this paper. Simulation study shows that, the proposed method in general has better performance than currently used methods.

Joining forces of Bayesian and frequentist methodology: a study for inference in the presence of non-identifiability

Increasingly complex applications involve large datasets in combination with nonlinear and high-dimensional mathematical models. In this context, statistical inference is a challenging issue that calls for pragmatic approaches that take advantage of both Bayesian and frequentist methods. The elegance of Bayesian methodology is founded in the propagation of information content provided by experimental data and prior assumptions to the posterior probability distribution of model predictions. However, for complex applications, experimental data and prior assumptions potentially constrain the posterior probability distribution insufficiently. In these situations, Bayesian Markov chain Monte Carlo sampling can be infeasible. From a frequentist point of view, insufficient experimental data and prior assumptions can be interpreted as non-identifiability. The profile-likelihood approach offers to detect and to resolve non-identifiability by experimental design iteratively. Therefore, it allows one to better constrain the posterior probability distribution until Markov chain Monte Carlo sampling can be used securely. Using an application from cell biology, we compare both methods and show that a successive application of the two methods facilitates a realistic assessment of uncertainty in model predictions.