Evaluations of Bayesian and maximum likelihood methods in PK models with below‐quantification‐limit data

Pharmacokinetic (PK) data often contain concentration measurements below the quantification limit (BQL). While specific values cannot be assigned to these observations, nevertheless these observed BQL data are informative and generally known to be lower than the lower limit of quantification (LLQ). Setting BQLs as missing data violates the usual missing at random (MAR) assumption applied to the statistical methods, and therefore leads to biased or less precise parameter estimation. By definition, these data lie within the interval [0, LLQ], and can be considered as censored observations. Statistical methods that handle censored data, such as maximum likelihood and Bayesian methods, are thus useful in the modelling of such data sets. The main aim of this work was to investigate the impact of the amount of BQL observations on the bias and precision of parameter estimates in population PK models (non‐linear mixed effects models in general) under maximum likelihood method as implemented in SAS and NONMEM, and a Bayesian approach using Markov chain Monte Carlo (MCMC) as applied in WinBUGS. A second aim was to compare these different methods in dealing with BQL or censored data in a practical situation. The evaluation was illustrated by simulation based on a simple PK model, where a number of data sets were simulated from a one‐compartment first‐order elimination PK model. Several quantification limits were applied to each of the simulated data to generate data sets with certain amounts of BQL data. The average percentage of BQL ranged from 25% to 75%. Their influence on the bias and precision of all population PK model parameters such as clearance and volume distribution under each estimation approach was explored and compared. Copyright © 2009 John Wiley & Sons, Ltd.     

Influence diagnostics for Student-t censored linear regression models

In this paper, we extend the censored linear regression model with normal errors to Student-t errors. A simple EM-type algorithm for iteratively computing maximum-likelihood estimates of the parameters is presented. To examine the performance of the proposed model, case-deletion and local influence techniques are developed to show its robust aspect against outlying and influential observations. This is done by the analysis of the sensitivity of the EM estimates under some usual perturbation schemes in the model or data and by inspecting some proposed diagnostic graphics. The efficacy of the method is verified through the analysis of simulated data sets and modelling a real data set first analysed under normal errors. The proposed algorithm and methods are implemented in the R package CensRegMod.     

Local influence on bandwidth estimation for kernel smoothing

This paper examines the problem of assessing local influence on the optimal bandwidth estimation in kernel smoothing based on cross validation. The bandwidth for kernel smoothing plays an important role in the model fitting and is often estimated using the cross-validation criterion. Following the argument of the second-order approach to local influence suggested by Wu and Luo (1993), we develop a new diagnostic statistic to examine the local influence of the observations on the estimation of the optimal bandwidth, where the perturbation may belong to one of three schemes. These are the response perturbation, the perturbation in the explanatory variable, and the case-weight

perturbation. The proposed diagnostic is nonparametric and is capable of identifying influential observations with strong influence on the bandwidth estimation. An example is presented to illustrate the application of the proposed diagnostic, and the usefulness of the nonparametric approach is illustrated in comparison with some other approaches to the assessment of local influence     

A Graphical Diagnostic for Identifying Influential Model Choices in Bayesian Hierarchical Models

Abstract. Real‐world phenomena are frequently modelled by Bayesian hierarchical models. The building‐blocks in such models are the distribution of each variable conditional on parent and/or neighbour variables in the graph. The specifications of centre and spread of these conditional distributions may be well motivated, whereas the tail specifications are often left to convenience. However, the posterior distribution of a parameter may depend strongly on such arbitrary tail specifications. This is not easily detected in complex models. In this article, we propose a graphical diagnostic, the Local critique plot, which detects such influential statistical modelling choices at the node level. It identifies the properties of the information coming from the parents and neighbours (the local prior) and from the children and co‐parents (the lifted likelihood) that are influential on the posterior distribution, and examines local conflict between these distinct information sources. The Local critique plot can be derived for all parameters in a chain graph model.     

Influential observations in GARCH models

This paper examines local influence assessment in generalized autoregressive conditional heteroscesdasticity models with Gaussian and Student-t errors, where influence is examined via the likelihood displacement. The analysis of local influence is discussed under three perturbation schemes: data perturbation, innovative model perturbation and additive model perturbation. For each case, expressions for slope and curvature diagnostics are derived. Monte Carlo experiments are presented to determine the threshold values for locating influential observations. The empirical study of daily returns of the New York Stock Exchange composite index shows that local influence analysis is a useful technique for detecting influential observations; most of the observations detected as influential are associated with historical shocks in the market. Finally, based on this empirical study and the analysis of simulated data, some advice is given on how to use the discussed methodology.     

INFLUENCE DIAGNOSTICS FOR THE NORMAL LINEAR MODEL WITH CENSORED DATA

Methods of detecting influential observations for the normal model for censored data are proposed. These methods include one-step deletion methods, deletion of observations and the empirical influence function. Emphasis is placed on assessing the impact that a single observation has on the estimation of coefficients of the model. Functions of the coefficients such as the median lifetime are also considered. Results are compared when applied to two sets of data.     

Censored time series analysis with autoregressive moving average models

The authors consider time series observations with data irregularities such as censoring due to a detection limit. Practitioners commonly disregard censored data cases which often result in biased estimates. The authors present an attractive remedy for handling autocorrelated censored data based on a class of autoregressive and moving average (ARMA) models. In particular, they introduce an imputation method well suited for fitting ARMA models in the presence of censored data. They demonstrate the effectiveness of their technique in terms of bias, efficiency, and information loss. They also describe its adaptation to a specific context of meteorological time series data on cloud ceiling height, which are measured subject to the detection limit of the recording device.     

On the asymptotic distribution of Cook's distance in logistic regression models

It sometimes occurs that one or more components of the data exert a disproportionate influence on the model estimation. We need a reliable tool for identifying such troublesome cases in order to decide either eliminate from the sample, when the data collect was badly realized, or otherwise take care on the use of the model because the results could be affected by such components. Since a measure for detecting influential cases in linear regression setting was proposed by Cook [Detection of influential observations in linear regression, Technometrics 19 (1977), pp. 15–18.], apart from the same measure for other models, several new measures have been suggested as single-case diagnostics. For most of them some cutoff values have been recommended (see [D.A. Belsley, E. Kuh, and R.E. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, 2nd ed., John Wiley & Sons, New York, Chichester, Brisban, (2004).], for instance), however the lack of a quantile type cutoff for Cook's statistics has induced the analyst to deal only with index plots as worthy diagnostic tools. Focussed on logistic regression, the aim of this paper is to provide the asymptotic distribution of Cook's distance in order to look for a meaningful cutoff point for detecting influential and leverage observations.     

Bayesian influence diagnostics in radiocarbon dating

Linear models constitute the primary statistical technique for any experimental science. A major topic in this area is the detection of influential subsets of data, that is, of observations that are influential in terms of their effect on the estimation of parameters in linear regression or of the total population parameters. Numerous studies exist on radiocarbon dating which propose a value consensus and remove possible outliers after the corresponding testing. An influence analysis for the value consensus from a Bayesian perspective is developed in this article.     

A Class of Weighted Estimating Equations for Semiparametric Transformation Models with Missing Covariates

In survival analysis, covariate measurements often contain missing observations; ignoring this feature can lead to invalid inference. We propose a class of weighted estimating equations for right‐censored data with missing covariates under semiparametric transformation models. Time‐specific and subject‐specific weights are accommodated in the formulation of the weighted estimating equations. We establish unified results for estimating missingness probabilities that cover both parametric and non‐parametric modelling schemes. To improve estimation efficiency, the weighted estimating equations are augmented by a new set of unbiased estimating equations. The resultant estimator has the so‐called ‘double robustness’ property and is optimal within a class of consistent estimators.     

Heteroscedastic log-exponentiated Weibull regression model

We introduce a new class of heteroscedastic log-exponentiated Weibull (LEW) regression models. The class of regression models can be applied to censored data and be used more effectively in survival analysis. Maximum likelihood estimation of the model parameters with censored data as well as influence diagnostics for the new regression model is investigated. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the heteroscedastic LEW regression model. The normal curvatures for studying local influence are derived under various perturbation schemes. An empirical application to a real data set is provided to illustrate the usefulness of the new class of heteroscedastic regression models.     

Asymmetric autoregressive models: statistical aspects and a financial application under COVID-19 pandemic

In the present study, we provide a motivating example with a financial application under COVID-19 pandemic to investigate autoregressive (AR) modeling and its diagnostics based on asymmetric distributions. The objectives of this work are: (i) to formulate asymmetric AR models and their estimation and diagnostics; (ii) to assess the performance of the parameters estimators and of the local influence technique for these models; and (iii) to provide a tool to show how data following an asymmetric distribution under an AR structure should be analyzed. We take the advantages of the stochastic representation of the skew-normal distribution to estimate the parameters of the corresponding AR model efficiently with the expectation-maximization algorithm. Diagnostic analytics are conducted by using the local influence technique with four perturbation schemes. By employing Monte Carlo simulations, we evaluate the statistical behavior of the corresponding estimators and of the local influence technique. An illustration with financial data updated until 2020, analyzed using the methodology introduced in the present work, is presented as an example of effective applications, from where it is possible to explain atypical cases from the COVID-19 pandemic.     

Log-Weibull extended regression model: Estimation,sensitivity and residual analysis

The failure rate function commonly has a bathtub shape in practice. In this paper we discuss a regression model considering new Weibull extended distribution developed by Xie et al. (2002) that can be used to model this type of failure rate function. Assuming censored data, we discuss parameter estimation: maximum likelihood method and a Bayesian approach where Gibbs algorithms along with Metropolis steps are used to obtain the posterior summaries of interest. We derive the appropriate matrices for assessing the local influence on the parameter estimates under different perturbation schemes, and we also present some ways to perform global influence. Also, some discussions on case deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback–Leibler divergence. Besides, for different parameter settings, sample sizes and censoring percentages, are performed various simulations and display and compare the empirical distribution of the Martingale-type residual with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to the martingale-type residual in log-Weibull extended models with censored data. Finally, we analyze a real data set under a log-Weibull extended regression model. We perform diagnostic analysis and model check based on the martingale-type residual to select an appropriate model.     

Stepwise local influence in generalized autoregressive conditional heteroskedasticity models

Detection of outliers or influential observations is an important work in statistical modeling, especially for the correlated time series data. In this paper we propose a new procedure to detect patch of influential observations in the generalized autoregressive conditional heteroskedasticity (GARCH) model. Firstly we compare the performance of innovative perturbation scheme, additive perturbation scheme and data perturbation scheme in local influence analysis. We find that the innovative perturbation scheme give better result than other two schemes although this perturbation scheme may suffer from masking effects. Then we use the stepwise local influence method under innovative perturbation scheme to detect patch of influential observations and uncover the masking effects. The simulated studies show that the new technique can successfully detect a patch of influential observations or outliers under innovative perturbation scheme. The analysis based on simulation studies and two real data sets show that the stepwise local influence method under innovative perturbation scheme is efficient for detecting multiple influential observations and dealing with masking effects in the GARCH model.     

Diagnostic tools in generalized Weibull linear regression models

We propose some statistical tools for diagnosing the class of generalized Weibull linear regression models [A.A. Prudente and G.M. Cordeiro, Generalized Weibull linear models, Comm. Statist. Theory Methods 39 (2010), pp. 3739–3755]. This class of models is an alternative means of analysing positive, continuous and skewed data and, due to its statistical properties, is very competitive with gamma regression models. First, we show that the Weibull model induces ma-ximum likelihood estimators asymptotically more efficient than the gamma model. Standardized residuals are defined, and their statistical properties are examined empirically. Some measures are derived based on the case-deletion model, including the generalized Cook's distance and measures for identifying influential observations on partial F-tests. The results of a simulation study conducted to assess behaviour of the global influence approach are also presented. Further, we perform a local influence analysis under the case-weights, response and explanatory variables perturbation schemes. The Weibull, gamma and other Weibull-type regression models are fitted into three data sets to illustrate the proposed diagnostic tools. Statistical analyses indicate that the Weibull model fitted into these data yields better fits than other common alternative models.     

Bayesian local influence for survival models

The aim of this paper is to develop a Bayesian local influence method (Zhu et al. 2009, submitted) for assessing minor perturbations to the prior, the sampling distribution, and individual observations in survival analysis. We introduce a perturbation model to characterize simultaneous (or individual) perturbations to the data, the prior distribution, and the sampling distribution. We construct a Bayesian perturbation manifold to the perturbation model and calculate its associated geometric quantities including the metric tensor to characterize the intrinsic structure of the perturbation model (or perturbation scheme). We develop local influence measures based on several objective functions to quantify the degree of various perturbations to statistical models. We carry out several simulation studies and analyze two real data sets to illustrate our Bayesian local influence method in detecting influential observations, and for characterizing the sensitivity to the prior distribution and hazard function.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

A semi-parametric estimation of a survival function from incomplete and doubly censored data

The purpose of this paper is to present a semi-parametric estimation of a survival function when analyzing incomplete and doubly censored data. Under the assumption that the chance of censoring is not related to the individual's survivorship, we propose a consistent estimation of survival. The derived estimator treats the uncensored observations nonparametrically and uses parametric models for both right and left censored data. Some asymptotic properties and simulation studies are also presented in order to analyze the behavior of the proposed estimator.     

A Study of Interval Censoring in Parametric Regression Models

Parametric models for interval censored data can now easily be fitted with minimal programming in certain standard statistical software packages. Regression equations can be introduced, both for the location and for the dispersion parameters. Finite mixture models can also be fitted, with a point mass on right (or left) censored observations, to allow for individuals who cannot have the event (or already have it). This mixing probability can also be allowed to follow a regression equation.Here, models based on nine different distributions are compared for three examples of heavily censored data as well as a set of simulated data. We find that, for parametric models, interval censoring can often be ignored and that the density, at centres of intervals, can be used instead in the likelihood function, although the approximation is not always reliable. In the context of heavily interval censored data, the conclusions from parametric models are remarkably robust with changing distributional assumptions and generally more informative than the corresponding non-parametric models.     

Robust methods for generalized linear models with nonignorable missing covariates

The EM algorithm is often used for finding the maximum likelihood estimates in generalized linear models with incomplete data. In this article, the author presents a robust method in the framework of the maximum likelihood estimation for fitting generalized linear models when nonignorable covariates are missing. His robust approach is useful for downweighting any influential observations when estimating the model parameters. To avoid computational problems involving irreducibly high‐dimensional integrals, he adopts a Metropolis‐Hastings algorithm based on a Markov chain sampling method. He carries out simulations to investigate the behaviour of the robust estimates in the presence of outliers and missing covariates; furthermore, he compares these estimates to the classical maximum likelihood estimates. Finally, he illustrates his approach using data on the occurrence of delirium in patients operated on for abdominal aortic aneurysm.