Profile Likelihood Approach Research Articles

A novel finite mixture model based on generalized scale mixtures of generalized normal distributions with application to stock dataset

This paper introduces a novel family of distributions known as generalized scale mixtures of generalized normal distributions (GSMGN). These distributions incorporate two additional shape parameters that serve to regulate the shape and tails of the distribution. A finite mixture model based on this family is presented to address clustering heterogeneous data in the presence of leptokurtic and heavy-tailed outcomes. The estimation of the parameters of this model are obtained by developing an ECM-PLA ensemble algorithm which combine the profile likelihood approach (PLA) and the classical Expectation Conditional Maximization (ECM) algorithm, and the observed information matrix is obtained. The applicability of this new family and the numerical performance of the proposed methodology is discussed through simulated and real data examples.

A novel skewed generalized normal distribution: properties, statistical inference, and its applications

In this paper, a novel skewed generalized normal distribution (NSGN), which encompasses several well-known distributions, is a two-component mixture model based on the generalized normal distribution. The NSGN distribution offers flexibility in modeling skewed data characterized by high kurtosis and strong skewness, making it highly applicable in finance, medicine, and engineering. The primary characteristics and properties of this new distribution are introduced, and the explicit expressions for the moments of order statistics are provided using recursive relations. The maximum likelihood estimation based on a profile likelihood approach, L-moments estimation, and a two-step estimation method combining Bayesian posterior maximum estimation with moment estimates are presented to estimate the four parameters of NSGN distribution. A simulation study compares the performance of these methods across different sample sizes and different parameters. The appropriateness of the proposed distribution has been tested by comparing it with several skewed distributions. Additionally, applications to real datasets showcase the beneficial properties of the NSGN for applied statistical research.

An SIR‐based Bayesian framework for COVID‐19 infection estimation

AbstractEstimating the COVID‐19 infection fatality rate, inferring the latent incidence and predicting the future epidemic evolution are critical to public health surveillance, but often challenging due to limited data availability or quality. Recently, a Bayesian framework combining time series deconvolution of deaths with a parametric Susceptible–Infectious–Recovered (SIR) model was proposed by Irons and Raftery, 2021. We assess the parameter identifiability of the model using the profile likelihood approach and simulations, when only the time series of deaths and seroprevalence survey data are available. The robustness of the model to the more complex but also more realistic Susceptible–Exposed–Infectious–Recovered (SEIR)‐based epidemics is evaluated through simulations; the influence of potential biases in the serosurveys on the inference is also investigated. We use a stationary first‐order autoregressive prior to account for the variability of transmission rate over time. The results suggest that the model is relatively robust to SEIR‐based epidemics, especially when the reproductive number is low, given sufficient information from serosurveys or priors. However, the lack of parameter identifiability under limited data availability cannot be neglected. We apply the model to infer the COVID‐19 infections in Ontario and Quebec, Canada during the Omicron era.

Cox proportional hazards regression in small studies of predictive biomarkers

Predictive biomarkers are essential for personalized medicine since they select the best treatment for a specific patient. However, of all biomarkers that are evaluated, only few are eventually used in clinical practice. Many promising biomarkers may be erroneously abandoned because they are investigated in small studies using standard statistical techniques which can cause small sample bias or lack of power. The standard technique for failure time endpoints is Cox proportional hazards regression with a multiplicative interaction term between binary variables of biomarker and treatment. Properties of this model in small studies have not been evaluated so far, therefore we performed a simulation study to understand its small sample behavior. As a remedy, we applied a Firth correction to the score function of the Cox model and obtained confidence intervals (CI) using a profile likelihood (PL) approach. These methods are generally recommended for small studies of different design. Our results show that a Cox model estimates the biomarker-treatment interaction term and the treatment effect in one of the biomarker subgroups with bias, and overestimates their standard errors. Bias is however reduced and power is increased with Firth correction and PL CIs. Hence, the modified Cox model and PL CI should be used instead of a standard Cox model with Wald based CI in small studies of predictive biomarkers.

On an asymmetric functional-coefficient ARCH-M model

This paper proposes an asymmetric functional-coefficient autoregressive conditional heteroscedasticity in mean (ARCH-M) model, which allows for asymmetry in the volatility. The profile likelihood approach is applied to estimate the parametric and nonparametric components. Under some regularity assumptions, we derive asymptotic behavior of the proposed estimator. To avoid model misspecification, the Wald, quasi-likelihood ratio test statistic and generalized likelihood ratio test statistic are put forward to detect ARCH effect, asymmetric effect and goodness-of-fit, respectively. Moreover, their asymptotic distributions are established under both null and alternative hypotheses. Some Monte Carlo simulations are conducted to evaluate the finite sample performance of the proposed estimation methodology and testing procedure. Also, real data sets are analyzed to demonstrate the applications of the proposed model.

Parameter identifiability and model selection for partial differential equation models of cell invasion.

When employing mechanistic models to study biological phenomena, practical parameter identifiability is important for making accurate predictions across wide ranges of unseen scenarios, as well as for understanding the underlying mechanisms. In this work, we use a profile-likelihood approach to investigate parameter identifiability for four extensions of the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) model, given experimental data from a cell invasion assay. We show that more complicated models tend to be less identifiable, with parameter estimates being more sensitive to subtle differences in experimental procedures, and that they require more data to be practically identifiable. As a result, we suggest that parameter identifiability should be considered alongside goodness-of-fit and model complexity as criteria for model selection.

Generalized Extreme Value Distribution for Modeling Earthquake Risk in Makran Subduction Zone Using Extreme Value Theory

The long-term pattern of severe incidents is one of the most crucial and fascinating topics of seismic events. This work aims to analyze the maximum annual earthquake magnitude in the Makran subduction zone using extreme value theory by implementing the block maxima method. The seismic data utilized for the current study was collected from the International Seismological Center (ISC) ranging from 1934 to 2022. The extreme parameters have fitted utilizing the generalized extreme value distribution. Numerous plots of the generalized extreme value distribution approach gave the accuracy of the used model when fitted to seismic data of the Makran subduction zone. Using the profile likelihood approach, the shape parameters (?) calculated is 0.29. According to the model fit, the Fréchet distribution is the best model for predicting the annual maximum earthquake magnitude in the Makran subduction zone. The estimated return levels for different return periods 10, 20, 50, and 100 are 6.35, 6.81, 7.58, and 8.31, respectively, indicating that an earthquake's maximum magnitude is increasing across the future 100 years. We also computed the profile likelihood to achieve a precise confidence interval. Thus, the 1945 earthquake of the Makran region with magnitude 8.1(Mw) was one of the most significant events in this area and occurred once every 100 years. The significance of this research is to inform decision-makers to implement suitable risk-mitigation methods.

Blood‐derived mitochondrial DNA copy number is associated with Alzheimer’s Disease through the regulation of plasma lipid and amino acid metabolism

AbstractBackgroundBlood‐derived mitochondrial DNA copy number (mtDNA‐CN) is the proxy measurement of mitochondrial function in the peripheral and central systems. Abnormal mtDNA‐CN not only indicates impaired mtDNA replication and transcription machinery but also dysregulated biological processes such as energy and lipid metabolism. However, the relationship between mtDNA‐CN and Alzheimer’s Disease (AD) and the underlying mechanisms of mitochondrial dysfunction in the pathogenesis of AD are still unclear.MethodTo investigate the causal relationship between mtDNA‐CN and AD, we performed two‐sample Mendelian randomization (MR) using publicly available summary statistics from GWAS for mtDNA‐CN and AD. Further, we estimated mtDNA‐CN using whole‐genome sequence data from blood samples of 1,521 ADNI participants and used a Cox proportional hazard model, adjusted for age, sex, and study phase, to assess the association between mtDNA‐CN and AD risk. The association of AD biomarkers measured in brain and plasma metabolites with mtDNA‐CN was evaluated using linear regression. We conducted causal mediation analysis to test the natural indirect effects of mtDNA‐CN change on AD risk through the significantly associated biomarkers and metabolites.ResultMR analysis using a profile likelihood approach suggested a causal relationship between decreased blood‐derived mtDNA‐CN and increased risk of AD (OR = 0.71; P = 0.023). Survival analysis showed that decreased mtDNA‐CN was significantly associated with increased risk of conversion from mild cognitive impairment to AD (HR = 0.80; P = 0.0017). We also identified significant associations between mtDNA‐CN and brain glucose metabolism (β = 0.050; P = 0.047) and amyloid‐β (Aβ) (β = 0.069; P = 0.015) and CSF Aβ (β = ‐0.051; P = 0.048). Plasma Aβ was, however, not associated. Several lipid species, amino acids, biogenic amines, and inflammatory proteins in plasma were also significantly associated with mtDNA‐CN. Finally, causal mediation analyses showed that the effect of change in mtDNA‐CN on AD risk is mediated by plasma neurofilament light (β = ‐0.055; P = 0.010), glycine (β = 0.016; P = 0.044), and hexadecenoylcarnitine (β = ‐0.017; P = 0.049).ConclusionOur study indicates that mtDNA‐CN measured in blood is predictive of AD risk. Associations of mtDNA‐CN with Aβ in CSF and brain, but not plasma, suggest cross‐tissue regulation of mitochondria. Decreased mtDNA‐CN might increase AD risk through lipid and amino acid metabolism. These results also suggest potential development of mtDNA‐CN as a blood‐based biomarker.

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.

Parameter estimation and identifiability analysis for a bivalent analyte model of monoclonal antibody-antigen binding

Surface plasmon resonance (SPR) is an extensively used technique to characterize antigen-antibody interactions. Affinity measurements by SPR typically involve testing the binding of antigen in solution to monoclonal antibodies (mAbs) immobilized on a chip and fitting the kinetics data using 1:1 Langmuir binding model to derive rate constants. However, when it is necessary to immobilize antigens instead of the mAbs, a bivalent analyte (1:2) binding model is required for kinetics analysis. This model is lacking in data analysis packages associated with high throughput SPR instruments and the packages containing this model do not explore multiple local minima and parameter identifiability issues that are common in non-linear optimization. Therefore, we developed a method to use a system of ordinary differential equations for analyzing 1:2 binding kinetics data. Salient features of this method include a grid search on parameter initialization and a profile likelihood approach to determine parameter identifiability. Using this method we found a non-identifiable parameter in data set collected under the standard experimental design. A simulation-guided improved experimental design led to reliable estimation of all rate constants. The method and approach developed here for analyzing 1:2 binding kinetics data will be valuable for expeditious therapeutic antibody discovery research.

Structure Identification in Varying-Coefficient Partially Linear Accelerated Failure Time Models via the Group MCP

The accelerated failure time (AFT) model is commonly used for the analysis of survival data in the presence of right censored due to the interpretability. In some practical cases, especially when some covariates are time-related, it is not realistic to assume the linear predictors in the AFT model. We propose a varying-coefficient partially linear AFT model for right censored data, allowing the nonlinear effects of covariates. To tackle challenges in estimation, we propose a penalized profile likelihood approach which utilizes a group minimax concave penalty to determine the nonlinear effects of covariates. Under the suitable conditions, we show that the proposed method can correctly select linear components and nonlinear components with high probability. Simulation results demonstrate the satisfactory performance of the proposed method in finite sample cases.

Destructive cure models with proportional hazards lifetimes and associated likelihood inference

In survival analysis, cure models have gained much importance due to rapid advancements in medical sciences. More recently, a subset of cure models, called destructive cure models, have been studied extensively under competing risks scenario wherein initial competing risks undergo a destructive process. In this article, we study destructive cure models by assuming a flexible weighted Poisson distribution (exponentially weighted Poisson, length biased Poisson and negative binomial distributions) for the initial number of competing causes and lifetimes of the susceptible individuals being defined by proportional hazards. The expectation-maximization (EM) algorithm and profile likelihood approach are made use of to estimate the model parameters. An extensive simulation study is carried out under various parameter settings to examine the properties of the models, and accuracy and the robustness of the proposed estimation technique. Effects of model mis-specification on the parameter estimates are also discussed in detail. For further illustration of the proposed methodology, a real-life cutaneous melanoma data set is analyzed.

Dark biportals at direct detection

We present a study of the constraining power of low-threshold, high-resolution direct-detection experiments for a biportal simplified dark matter model. In the scenario we consider here, dark matter and Standard Model particles interact through two dark vector mediators---one light and one heavy with respect to the momentum transfer in the experiment. Interference effects at the level of scattering amplitudes can lead to novel marked shape features in the differential recoil spectra, which are best exploited by high-resolution, low-threshold experiments. We identify the region in parameter space for our model where such effects are dominant and show that composite-target experiments with large atomic mass differences are ideal to explore these scenarios. We develop a profile likelihood approach to analyze presently available and future data. Using published results by the CRESST-III experiment and projections of future sensitivities for the COSINUS experiment, we constrain the parameter space in our model, thereby showing the potential of such an analysis for a class of dark matter models that exhibit nonstandard features in the recoil spectra.

New constraints on axion-gauge field dynamics during inflation from Planck and BICEP/Keck data sets

We present new constraints on spectator axion-U(1) gauge field interactions during inflation using the latest Planck (PR4) and BICEP/Keck 2018 data releases. This model can source tensor perturbations from amplified gauge field fluctuations, driven by an axion rolling for a few e-folds during inflation. The gravitational waves sourced in this way have a strongly scale-dependent (and chiral) spectrum, with potentially visible contributions to large/intermediate scale B-modes of the CMB. We first derive theoretical bounds on the model imposing validity of the perturbative regime and negligible backreaction of the gauge field on the background dynamics. Then, we determine bounds from current CMB observations, adopting a frequentist profile likelihood approach. We study the behaviour of constraints for typical choices of the model's parameters, analyzing the impact of different dataset combinations. We find that observational bounds are competitive with theoretical ones and together they exclude a significant portion of the model's parameter space. We argue that the parameter space still remains large and interesting for future CMB experiments targeting large/intermediate scales B-modes.

Optimal Experimental Design Based on Two-Dimensional Likelihood Profiles.

Dynamic behavior of biological systems is commonly represented by non-linear models such as ordinary differential equations. A frequently encountered task in such systems is the estimation of model parameters based on measurement of biochemical compounds. Non-linear models require special techniques to estimate the uncertainty of the obtained model parameters and predictions, e.g. by exploiting the concept of the profile likelihood. Model parameters with significant uncertainty associated with their estimates hinder the interpretation of model results. Informing these model parameters by optimal experimental design minimizes the additional amount of data and therefore resources required in experiments. However, existing techniques of experimental design either require prior parameter distributions in Bayesian approaches or do not adequately deal with the non-linearity of the system in frequentist approaches. For identification of optimal experimental designs, we propose a two-dimensional profile likelihood approach, providing a design criterion which meaningfully represents the expected parameter uncertainty after measuring data for a specified experimental condition. The described approach is implemented into the open source toolbox Data2Dynamics in Matlab. The applicability of the method is demonstrated on an established systems biology model. For this demonstration, available data has been censored to simulate a setting in which parameters are not yet well determined. After determining the optimal experimental condition from the censored ones, a realistic evaluation was possible by re-introducing the censored data point corresponding to the optimal experimental condition. This provided a validation that our method is feasible in real-world applications. The approach applies to, but is not limited to, models in systems biology.

Parameter identifiability and model selection for sigmoid population growth models

Sigmoid growth models, such as the logistic, Gompertz and Richards’ models, are widely used to study population dynamics ranging from microscopic populations of cancer cells, to continental-scale human populations. Fundamental questions about model selection and parameter estimation are critical if these models are to be used to make practical inferences. However, the question of parameter identifiability – whether a data set contains sufficient information to give unique or sufficiently precise parameter estimates – is often overlooked. We use a profile-likelihood approach to explore practical parameter identifiability using data describing the re-growth of hard coral. With this approach, we explore the relationship between parameter identifiability and model misspecification, finding that the logistic growth model does not suffer identifiability issues for the type of data we consider whereas the Gompertz and Richards’ models encounter practical non-identifiability issues. This analysis of parameter identifiability and model selection is important because different growth models are in biological modelling without necessarily considering whether parameters are identifiable. Standard practices that do not consider parameter identifiability can lead to unreliable or imprecise parameter estimates and potentially misleading mechanistic interpretations. For example, using the Gompertz model, the estimate of the time scale of coral re-growth is 625 days when we estimate the initial density from the data, whereas it is 1429 days using a more standard approach where variability in the initial density is ignored. While tools developed here focus on three standard sigmoid growth models only, our theoretical developments are applicable to any sigmoid growth model and any appropriate data set. MATLAB implementations of all software are available on GitHub.

Inference on covariance-mean regression

In this article, we introduce a covariance-mean regression model with heterogeneous similarity matrices. It not only links the covariance of responses to heterogeneous similarity matrices induced by auxiliary information, but also establishes the relationship between the mean of responses and covariates. Under this new model setting, however, two statistical inference challenges are encountered. The first challenge is that the consistency of the covariance estimator based on the standard profile likelihood approach breaks down. Hence, we propose an adjustment and develop the Z-estimation and unconstrained/constrained ordinary least squares estimation methods. We demonstrate that the resulting estimators are consistent and asymptotically normal. The second challenge is testing the adequacy of the covariance-mean regression model comprising both the multivariate mean regression and the heterogeneous covariance matrices. Correspondingly, we introduce two diagnostic test statistics and then obtain their theoretical properties. The proposed estimators and tests are illustrated via extensive simulations and an empirical example study of the stock return comovement in the US stock market.

On the estimation of population size under dependent dual-record system: an adjusted profile-likelihood approach

Motivated by various applications in epidemiology, population studies, criminology, etc., the problem of estimating size of a homogeneous human population based on two-sample capture–recapture experiment is considered in this article. The Lincoln–Petersen estimate, assuming independence between the samples, is widely used though often its relevance is unanimously criticized. Time and behavioural response variation model (denoted by ) is the most suitable here. Effect of model mis-specification on the Lincoln–Petersen estimate is studied in terms of efficiency and robustness. We study the accuracy and efficiency of this estimator under existence of dependence between the samples. This article shows that profile-likelihood and its existing modifications fail to make inference from the model . Therefore, an adjustment over profile likelihood is newly proposed and evaluated through an extensive simulation study. Finally, two real data sets with different characteristics are analyzed as practical illustrations of the proposed method.

A smooth dynamic network model for patent collaboration data

The development and application of models, which take the evolution of network dynamics into account, are receiving increasing attention. We contribute to this field and focus on a profile likelihood approach to model time-stamped event data for a large-scale dynamic network. We investigate the collaboration of inventors using EU patent data. As event we consider the submission of a joint patent and we explore the driving forces for collaboration between inventors. We propose a flexible semiparametric model, which includes external and internal covariates, where the latter are built from the network history.

On the Problem of Comparing the Means and Medians of Two Independent Lognormal Distributions

Inferences about the ratio of two lognormal means δ can depend on plausible values of ρ, the ratio of the normal standard deviations associated to these distributions. This aspect is not usually considered in most of the analyses carried out in some applied sciences. In this paper we propose a profile likelihood approach that allows the comparison of two independent lognormal data sets in a more exhaustive way. Inferences about δ, ρ and (δ, ρ) are jointly analyzed through a simple closed-form expression obtained for the profile likelihood function of the parameter vector (δ, ρ). A similar analysis is done for ψ and ρ, where ψ is the ratio of two lognormal medians, obtaining also a simple closed-form expression for the profile likelihood function of these parameters. These expressions allow us to construct likelihood contour plots that capture most of the information provided by the samples and become valuable to identify if a trade-off between the parameters under study occurs; in case of that, individual inferences should be analyzed carefully. A detailed series of Monte Carlo simulations are included; they illustrate the performance of profile likelihood and parametric bootstrap approaches, for different sample sizes and parameter values.