Multivariate Probit Analysis of Binary Time Series Data with Missing Responses

In the present paper a mixed approach is proposed for the simultaneously estimation of regression and correlation structure parameters in multivariate probit models using generalized estimating equations for the former and pseudo-score equations for the latter. The finite sample properties of the corresponding estimators are compared to estimators proposed by Qu, Williams, Beck and Medendorp (1992) and Qu, Piedmonte and Williams (1994) using generalized estimating equations for both sets of parameters via a Monte Carlo experiment. As a `reference' estimator for an equicorrelation model, the maximum likelihood (ML) estimator of the random effects probit model is calculated. The results show the mixed approach to be the most robust approach in the sense that the number of datasets for which the corresponding estimates converged was largest relative to the other two approaches. Furthermore, the mixed approach led to the most efficient non-ML estimators and to very efficient estimators for regression and correlation structure parameters relative to the ML estimator if individual covariance matrices were used.

Analyses of multivariate ordinal probit models typically use data augmentation to link the observed (discrete) data to latent (continuous) data via a censoring mechanism defined by a collection of “cutpoints.” Most standard models, for which effective Markov chain Monte Carlo (MCMC) sampling algorithms have been developed, use a separate (and independent) set of cutpoints for each element of the multivariate response. Motivated by the analysis of ratings data, we describe a particular class of multivariate ordinal probit models where it is desirable to use a common set of cutpoints. While this approach is attractive from a data-analytic perspective, we show that the existing efficient MCMC algorithms can no longer be accurately applied. Moreover, we show that attempts to implement these algorithms by numerically approximating required multivariate normal integrals over high-dimensional rectangular regions can result in severely degraded estimates of the posterior distribution. We propose a new data augmentation that is based on a covariance decomposition and that admits a simple and accurate MCMC algorithm. Our data augmentation requires only that univariate normal integrals be evaluated, which can be done quickly and with high accuracy. We provide theoretical results that suggest optimal decompositions within this class of data augmentations, and, based on the theory, recommend default decompositions that we demonstrate work well in practice. This article has supplementary material online.

The present paper deals with the comparison of the performance of different estimation methods for regression models with correlated binary responses. Throughout, we consider probit models where an underlying latent continous random variable crosses a threshold. The error variables in the unobservable latent model are assumed to be normally distributed. The estimation procedures considered are (1) marginal maximum likelihood estimation using Gauss-Hermite quadrature, (2) generalized estimation equations (GEE) techniques with an extension to estimate tetrachoric correlations in a second step, and, (3) the MECOSA approach proposed by Schepers, Arminger and Kusters (1991) using hierarchical mean and covariance structure models. We present the results of a simulation study designed to evaluate the small sample properties of the different estimators and to make some comparisons with respect to technical aspects of the estimation procedures and to bias and mean squared error of the estimators. The results show that the calculation of the ML estimator requires the most computing time, followed by the MECOSA estimator. For small and moderate sample sizes the calculation of the MECOSA estimator is problematic because of problems of convergence as well as a tendency of underestimating the variances. In large samples with moderate or high correlations of the errors in the latent model, the MECOSA estimators are not as efficient as ML or GEE estimators. The higher the `true' value of an equicorrelation structure in the latent model and the larger the sample sizes are, the more is the efficiency gain of the ML estimator compared to the GEE and MECOSA estimators. Using the GEE approach, the ML estimates of tetrachoric correlations calculated in a second step are biased to a smaller extent than using the MECOSA approach.

In this paper, we offer generalized exponentiated gamma distribution. We consider the maximum likelihood and Bayesian estimators of unknown parameters and propose Markov chain Monte Carlo (MCMC) method to generate samples from the posterior distribution. The mean square error of estimations is computed and comparisons are made using Monte Carlo simulation. Key words: Maximum likelihood estimator, Bayesian estimator, squared error loss function, generalized exponentiated gamma distribution.

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.

Building an accurate and reasonable time-series model of wind power is of great significance for the power system operation and planning. This paper proposes a Markov chain Monte Carlo (MCMC) method to simulate the time series of the wind power. Taking into account the nonstationarity and stochastics of the wind power, this model constructs Markov chain for time series of the wind power to keep the stochastics and utilize the Gibbs sampling to realize the probability transition matrix. A numerical case study with one-year real measurement of the wind power from one large wind farm in Gansu province of China is applied to illustrate the validity the proposed model: the simulated wind-power time series of arbitrary length that accurately reproduce the statistical behavior of the original series.

The occurrence patterns of important words found in six texts (one historical pamphlet and five renowned academic books) are analyzed using both univariate and multivariate Hawkes processes. By treating the occurrence patterns as binary time-series data along the texts, we investigate how effectively univariate and multivariate Hawkes processes capture the characteristics of these word occurrence signals. Through maximum likelihood estimation and subsequent simulations, we found that the multivariate Hawkes process clearly outperforms the univariate Hawkes process in modeling word occurrence signals. Moreover, we found that the multivariate Hawkes process can provide a Hawkes graph, which serves as an intuitive representation of the relationships between concepts appearing in the analyzed text. Furthermore, our study demonstrates that the importance of concepts within a given text can be quantitatively estimated based on the optimized parameter values of the multivariate Hawkes process.

Introduction: There has been little study regarding prediction of acute exacerbation. Aims and Objective: This study aimed to develop prediction model of COPD acute exacerbation with big data by machine learning methods. Methods: We investigated data from 594 COPD patients who were enrolled in the KOCOSS cohort. Smoking status, lung function, body mass index, and COPD assessment test (CAT) score were collected from cohort data. We merged patients’ information with the Korean Health Insurance database. Comorbidity, health care utilization, moderate to severe exacerbation, and COPD medications between 2008 and 2012 were collected. We also collected daily air pollution level, temperature, humidity, and wind velocity. Data on the activities of respiratory viruses were collected. Prediction model was developed by deep learning methods and also statistical method. Deep neural network (DNN), random forest, and generalized estimating equation (GEE) were utilized. Results: Area under the curve (AUC) value for prediction of acute exacerbation was highest by random forest method (0.8722) followed by GEE (0.8617) and DNN (0.8493). Sensitivity was highest by random forest (0.8613) followed by DNN (0.8000) and GEE (0.7613). In GEE analysis, female, CAT score, FEV1 (%), number of exacerbations in previous one year, use of bronchodilator, history of asthma, influenza, and human coronavirus were significantly associated with acute exacerbation. Conclusions: Prediction model for acute exacerbation of COPD was developed with big data by machine learning methods. AUC and sensitivity were higher in model by machine learning methods compared with GEE.

저수분석(low flow analysis)은 수자원공학에서 중요한 분야 중 하나이며, 특히 저수량 빈도분석(low flow frequency analysis)의 결과는 저수(貯水)용량의 설계, 물 수급계획, 오염원의 배치 및 관개와 생태계의 보존을 위한 수량과 수질의 관리에 중요하게 사용된다. 그러므로 본 연구에서는 저수량 빈도분석을 위한 점 빈도분석을 수행하였으며, 특히 빈도분석에 있어서의 불확실성을 탐색하기 위하여 Bayesian 방법을 적용하고 그 결과를 기존에 사용되던 불확실성 탐색방법과 비교하였다. 본 논문의Ⅰ편에서는 Bayesian 방법 중 사전분포(prior distribution)와 우도함수(likelihood function)의 복잡성에 상관없이 계산이 가능한 Bayesian MCMC(Bayesian Markov Chain Monte Carlo) 방법과 Metropolis-Hastings 알고리즘을 사용하기 위한 여러 과정의 이론적 배경과 Bayesian 방법에서 가장 중요한 요소인 사전분포를 구축하고 이를 비교 및 평가하였다. 고려된 사전분포는 자료에 기반하지 않은 사전분포와 자료에 기반한 사전분포로써 두 사전분포를 이용하여 Metropolis-Hastings 알고리즘을 수행하고 그 결과를 비교하여 저수량 빈도분석에 합리적인 사전분포를 선정하였다. 또한 알고리즘의 수행과정에서 필요한 제안분포(proposal distribution)를 적용하여 그에 따른 알고리즘의 효율성을 채택률(acceptance rate)을 산정하여 검증해 보았다. 사전분포의 분석 결과, 자료에 기반한 사전분포가 자료에 기반하지 않은 사전분포보다 정확성 및 불확실성의 표현에 있어서 우수한 결과를 제시하는 것을 확인할 수 있었고, 채택률을 이용한 알고리즘의 효용성 역시 기존 연구자들이 제시하였던 만족스러운 범위를 가지는 것을 알 수 있었다. 최종적으로 선정된 사전분포는 본 연구의 II편에서 Bayesian MCMC방법의 사전분포로 이용되었으며, 그 결과를 기존 불확실성의 추정방법의 하나인 2차 근사식을 이용한 최우추정(maximum likelihood estimation)방법의 결과와 비교하였다. The low flow analysis is an important part in water resources engineering. Also, the results of low flow frequency analysis can be used for design of reservoir storage, water supply planning and design, waste-load allocation, and maintenance of quantity and quality of water for irrigation and wild life conservation. Especially, for identification of the uncertainty in frequency analysis, the Bayesian approach is applied and compared with conventional methodologies in at-site low flow frequency analysis. In the first manuscript, the theoretical background for the Bayesian MCMC (Bayesian Markov Chain Monte Carlo) method and Metropolis-Hasting algorithm are studied. Two types of the prior distribution, a non-data- based and a data-based prior distributions are developed and compared to perform the Bayesian MCMC method. It can be suggested that the results of a data-based prior distribution is more effective than those of a non-data-based prior distribution. The acceptance rate of the algorithm is computed to assess the effectiveness of the developed algorithm. In the second manuscript, the Bayesian MCMC method using a data-based prior distribution and MLE(Maximum Likelihood Estimation) using a quadratic approximation are performed for the at-site low flow frequency analysis.

In this paper, we discuss the problem of constant-stress accelerated life test when the failure data are progressive type-I interval censored. Both classical and Bayesian inferential approaches of the distribution parameters and reliability characteristics are discussed. In the classical scenario, the maximum likelihood estimates are approximated using the EM algorithm and the mid-point approximation method. Furthermore, the model's parameters are estimated by method of moments. Next in the Bayesian framework, the point estimates of unknown parameters are obtained using Tierney-Kadane's technique and Markov Chain Monte Carlo (MCMC) method. In addition, both approximate and credible confidence intervals (CIs) of the estimators are constructed. For illustration purpose, a Monte Carlo simulation is conducted to investigate the performance of the proposed estimators and a real data set is analysed.

In repeated measures data, the observations tend to be correlated within each subject, and such data are often analyzed using Generalized Estimating Equations (GEE), which are robust to assumptions that many methods hold. The main limitation of GEE is that its method of estimation is quasi-likelihood. The recent framework of the copula is very popular for handling repeated data. The maximum likelihood-based analysis for repeated data can be obtained using Gaussian copula regression. The purpose of this study is to show the handling and analysis of the repeated data using the Gaussian copula regression approach and compare the findings with GEE. The prospective, double-blinded, randomized controlled trial data for this study was obtained from the Department of Anesthesia, Christian Medical College, and Vellore. ASA I and II patients were randomized into three groups. Hemodynamic parameters were obtained for 88 patients at thirteen-time points. The outcome of interest was mean arterial pressure. Both GEE and Gaussian copula regression were compared assuming four different correlation structures. The optimal correlation structures were selected with the Akaike Information Criterion (AIC) and Correlation Information Criterion (CIC) goodness of fit criteria according to the method of estimation of Gaussian copula regression and GEE, respectively. The correlation structures, unstructured and autoregressive, were found to be optimal for Gaussian copula regression and GEE based on AIC and CIC criteria values respectively. A comparison between the estimated values of the selected models showed no major differences. Gaussian copula regression found that intrathecal morphine has a significant reduction in MAP over time, this significance is considered important as the study uses randomized controlled trial data. Both methods have almost similar results, but Gaussian copula regression provides better results by identifying significant findings associated with the outcome using maximum likelihood estimation that GEE fails to identify using quasi-likelihood estimation.

Classical and Bayesian inferential approaches using Lomax model under progressively type-I hybrid censoring

We describe a Bayesian approach to incorporate between-individual heterogeneity associated with parameters of complicated biological models. We emphasize the use of the Markov chain Monte Carlo (MCMC) method in this context and demonstrate the implementation and use of MCMC by analysis of simulated overdispersed Poisson counts and by analysis of an experimental data set on preneoplastic liver lesions (their number and sizes) in the presence of heterogeneity. These examples show that MCMC-based estimates, derived from the posterior distribution with uniform priors, may agree well with maximum likelihood estimates (if available). However, with heterogeneous parameters, maximum likelihood estimates can be difficult to obtain, involving many integrations. In this case, the MCMC method offers substantial computational advantages.

This study demonstrated the equivalence between the Rasch testlet model and the three‐level one‐parameter testlet model and explored the Markov Chain Monte Carlo (MCMC) method for model parameter estimation in WINBUGS. The estimation accuracy from the MCMC method was compared with those from the marginalized maximum likelihood estimation (MMLE) with the expectation‐maximization algorithm in ConQuest and the sixth‐order Laplace approximation estimation in HLM6. The results indicated that the estimation methods had significant effects on the bias of the testlet variance and ability variance estimation, the random error in the ability parameter estimation, and the bias in the item difficulty parameter estimation. The Laplace method best recovered the testlet variance while the MMLE best recovered the ability variance. The Laplace method resulted in the smallest random error in the ability parameter estimation while the MCMC method produced the smallest bias in item parameter estimates. Analyses of three real tests generally supported the findings from the simulation and indicated that the estimates for item difficulty and ability parameters were highly correlated across estimation methods.

In this article, the estimation of the parameters and the reliability and hazard functions for Weibull inverted exponential (WIE) distribution is considered based on progressive first-failure censoring (PFFC) data. For non-Bayesian inference, maximum likelihood (ML) estimators are acquired; meanwhile, their existence is verified. Via asymptotic normality of ML estimators and delta method, the corresponding confidence intervals (CIs) of the parameters and the reliability and hazard functions are constructed. For Bayesian inference, Lindley’s approximation and Markov chain Monte Carlo (MCMC) techniques are proposed to obain the Bayes estimators and the corresponding credible intervals (CRIs). To this end, both symmetric and asymmetric loss functions are used. A large number of Monte Carlo simulations are implemented to evaluate the efficiency of the developed methods. Eventually, a numerical example is analyzed for illustrative purposes.