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This work presents a new model and estimation procedure for the illness–death survival data where the hazard functions follow accelerated failure time (AFT) models. A shared frailty variate induces positive dependence among failure times of a subject for handling the unobserved dependency between the nonterminal and the terminal failure times given the observed covariates. The motivation behind the proposed modeling approach is to leverage the well‐known interpretability advantage of AFT models with respect to the observed covariates, while also benefiting from the simple and intuitive interpretation of the hazard functions. A semiparametric maximum likelihood estimation procedure is developed via a kernel smoothed‐aided expectation‐maximization algorithm, and variances are estimated by weighted bootstrap. We consider existing frailty‐based illness–death models and place particular emphasis on highlighting the contribution of our current research. The breast cancer data of the Rotterdam tumor bank are analyzed using the proposed as well as existing illness–death models. The results are contrasted and evaluated based on a new graphical goodness‐of‐fit procedure. Simulation results and data analysis nicely demonstrate the practical utility of the shared frailty variate with the AFT regression model under the illness–death framework. [ABSTRACT FROM AUTHOR]

Copula is a popular method for modeling the dependence among marginal distributions in multivariate censored data. As many copula models are available, it is essential to check if the chosen copula model fits the data well for analysis. Existing approaches to testing the fitness of copula models are mainly for complete or right‐censored data. No formal goodness‐of‐fit (GOF) test exists for interval‐censored or recurrent events data. We develop a general GOF test for copula‐based survival models using the information ratio (IR) to address this research gap. It can be applied to any copula family with a parametric form, such as the frequently used Archimedean, Gaussian, and D‐vine families. The test statistic is easy to calculate, and the test procedure is straightforward to implement. We establish the asymptotic properties of the test statistic. The simulation results show that the proposed test controls the type‐I error well and achieves adequate power when the dependence strength is moderate to high. Finally, we apply our method to test various copula models in analyzing multiple real datasets. Our method consistently separates different copula models for all these datasets in terms of model fitness. [ABSTRACT FROM AUTHOR]

Evaluating the goodness of fit of logistic regression models is crucial to ensure the accuracy of the estimated probabilities. Unfortunately, such evaluation is problematic in large samples. Because the power of traditional goodness of fit tests increases with the sample size, practically irrelevant discrepancies between estimated and true probabilities are increasingly likely to cause the rejection of the hypothesis of perfect fit in larger and larger samples. This phenomenon has been widely documented for popular goodness of fit tests, such as the Hosmer‐Lemeshow test. To address this limitation, we propose a modification of the Hosmer‐Lemeshow approach. By standardizing the noncentrality parameter that characterizes the alternative distribution of the Hosmer‐Lemeshow statistic, we introduce a parameter that measures the goodness of fit of a model but does not depend on the sample size. We provide the methodology to estimate this parameter and construct confidence intervals for it. Finally, we propose a formal statistical test to rigorously assess whether the fit of a model, albeit not perfect, is acceptable for practical purposes. The proposed method is compared in a simulation study with a competing modification of the Hosmer‐Lemeshow test, based on repeated subsampling. We provide a step‐by‐step illustration of our method using a model for postneonatal mortality developed in a large cohort of more than 300 000 observations. [ABSTRACT FROM AUTHOR]

Summary: Assessing a binary regression model based on ungrouped data is a commonly encountered but very challenging problem. Although tests, such as Hosmer–Lemeshow test and le Cessie–van Houwelingen test, have been devised and widely used in applications, they often have low power in detecting lack of fit and not much theoretical justification has been made on when they can work well. In this article, we propose a new approach based on a cross‐validation voting system to address the problem. In addition to a theoretical guarantee that the probabilities of type I and II errors both converge to zero as the sample size increases for the new method under proper conditions, our simulation results demonstrate that it performs very well. [ABSTRACT FROM AUTHOR]

In statistical inference, one has to make sure that the underlying regression model is correctly specified otherwise the resulting estimation may be biased. Model checking is an important method to detect any departure of the regression model from the true one. Missing data are a ubiquitous problem in social and medical studies. If the underlying regression model is correctly specified, recent researches show great popularity of the doubly robust (DR) estimates method for handling missing data because of its robustness to the misspecification of either the missing data model or the conditional mean model, that is, the model for the conditional expectation of true regression model conditioning on the observed quantities. However, little work has been devoted to the goodness of fit test for DR estimates method. In this article, we propose a testing method to assess the reliability of the estimator derived from the DR estimating equation with possibly missing response and always observed auxiliary variables. Numerical studies demonstrate that the proposed test can control type I errors well. Furthermore the proposed method can detect departures from model assumptions in the marginal mean model of interest powerfully. A real dementia data set is used to illustrate the method for the diagnosis of model misspecification in the problem of missing response with an always observed auxiliary variable for cross-sectional data. [ABSTRACT FROM AUTHOR]

This article describes applications of extensions of bivariate rank sum statistics to the crossover design with four sequence groups for two treatments. A randomized clinical trial in ophthalmology provides motivating background for the discussion. The bilateral design for this study has four sequence groups T:T, T:P, P:T, and P:P, respectively, for T as test treatment or P as placebo in the corresponding order for the left and right eyes. This article describes how to use the average of the separate Wilcoxon rank sum statistics for the left and right eyes for the overall comparison between T and P with the correlation between the two eyes taken into account. An extension of this criterion with better sensitivity to potential differences between T and P through reduction of the applicable variance has discussion in terms of a conceptual model with constraints for within-side homogeneity of groups with the same treatment and between-side homogeneity of the differences between T and P. Goodness of fit for this model can have assessment with test statistics for its corresponding constraints. Simulation studies for the conceptual model confirm better power for the extended test statistic with its full invocation than other criteria without this property. The methods summarized here are illustrated for the motivating clinical trial in ophthalmology, but they are applicable to other situations with the crossover design with four sequence groups for either two locations for two treatments at the same time for a patient or two successive periods for the assigned treatments for a recurrent disorder. This article also notes that the methods based on its conceptual model can have unsatisfactory power for departures from that model where the difference between T and P via the T:T and P:P groups is not similar to that via the T:P and P:T groups, as might occur when T has a systemic effect in a bilateral trial. For this situation, more robust test statistics have identification, but there is recognition that the parallel groups design with only the T:T and P:P groups may be more useful than the bilateral design with four sequence groups. [ABSTRACT FROM AUTHOR]

This article investigates the problem of model diagnostics for systems described by nonlinear ordinary differential equations (ODEs). I propose modeling lack of fit as a time-varying correction to the right-hand side of a proposed differential equation. This correction can be described as being a set of additive forcing functions, estimated from data. Representing lack of fit in this manner allows us to graphically investigate model inadequacies and to suggest model improvements. I derive lack-of-fit tests based on estimated forcing functions. Model building in partially observed systems of ODEs is particularly difficult and I consider the problem of identification of forcing functions in these systems. The methods are illustrated with examples from computational neuroscience. [ABSTRACT FROM AUTHOR]

The Cox hazards model ( Cox, 1972 , Journal of the Royal Statistical Society, Series B 34, 187–220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model ( Aalen, 1980 , in Lecture Notes in Statistics-2, 1–25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon–Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model ( Martinussen and Scheike, 2002 , Biometrika 89, 283–298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set. [ABSTRACT FROM AUTHOR]

Nonparametric smoothing methods are used to model longitudinal data, but the challenge remains to incorporate correlation into nonparametric estimation procedures. In this article, we propose an efficient estimation procedure for varying-coefficient models for longitudinal data. The proposed procedure can easily take into account correlation within subjects and deal directly with both continuous and discrete response longitudinal data under the framework of generalized linear models. The proposed approach yields a more efficient estimator than the generalized estimation equation approach when the working correlation is misspecified. For varying-coefficient models, it is often of interest to test whether coefficient functions are time varying or time invariant. We propose a unified and efficient nonparametric hypothesis testing procedure, and further demonstrate that the resulting test statistics have an asymptotic chi-squared distribution. In addition, the goodness-of-fit test is applied to test whether the model assumption is satisfied. The corresponding test is also useful for choosing basis functions and the number of knots for regression spline models in conjunction with the model selection criterion. We evaluate the finite sample performance of the proposed procedures with Monte Carlo simulation studies. The proposed methodology is illustrated by the analysis of an acquired immune deficiency syndrome (AIDS) data set. [ABSTRACT FROM AUTHOR]

This article presents a Bayesian analysis of mark-recapture-recovery data on Soay sheep. A reversible jump Markov chain Monte Carlo technique is used to determine age classes of common survival, and to model the survival probabilities in those classes using logistic regression. This involves environmental and individual covariates, as well as random effects. Auxiliary variables are used to impute missing covariates measured on individual sheep. The Bayesian approach suggests different models from those previously obtained using classical statistical methods. Following model averaging, features that were not previously detected, and which are of ecological importance, are identified. [ABSTRACT FROM AUTHOR]

Conventional measures of model fit for indexed data (e.g., time series or spatial data) summarize errors iny, for instance by integrating (or summing) the squared difference between predicted and measured values over a range ofx. We propose an approach which recognizes that errors can occur in thex-direction as well. Instead of just measuring the difference between the predictions and observations at each site (or time), we first“deform” the predictions, stretching or compressing along thex-direction or directions, so as to improve the agreement between the observations and the deformed predictions. Error is then summarized by (a) the amount of deformation inx, and (b) the remaining difference inybetween the data and the deformed predictions (i.e., the residual error inyafter the deformation). A parameter,λ, controls the tradeoff between (a) and (b), so that asno deformation is allowed, whereas forthe deformation minimizes the errors iny. In some applications, the deformation itself is of interest because it characterizes the (temporal or spatial) structure of the errors. The optimal deformation can be computed by solving a system of nonlinear partial differential equations, or, for a unidimensional index, by using a dynamic programming algorithm. We illustrate the procedure with examples from nonlinear time series and fluid dynamics. [ABSTRACT FROM AUTHOR]

Cox's regression model is the standard regression tool for survival analysis in most applications. Often, however, the model only provides a rough summary of the effect of some covariates. Therefore, if the aim is to give a detailed description of covariate effects and to consequently calculate predicted probabilities, more flexible models are needed. In another article, Scheike and Zhang (2002, Scandinavian Journal of Statistics 29,75-88), we suggested a flexible extension of Cox's regression model, which aimed at extending the Co model only for those covariates where additional flexibility are needed. One important advantage of the suggested approach is that even though covariates are allowed a nonparametric effect, the hassle and dificulty of finding smoothing parameters are not needed. We show how the extended model also leads to simple formulae for predicted probabilities and their standard errors, for example, in the competing risk framework. [ABSTRACT FROM AUTHOR]

Summary. This is in response to Garrett and Zeger (2000, Biometrics 56, 1055–1067) who, within the Bayesian framework, developed mainly graphical methods for latent class model diagnosis. Possible problems with this approach, and with its application to both generated and empirical data, are pointed out. The impact of the proposed tools cannot be understood by their reader, as no comparisons are made to results obtainable using established methods for latent class model diagnosis; this applies especially to overall goodness-of-fit tests, for which alternatives (bootstrap, Rudas-Clogg-Lindsay index of fit) are mentioned. Further, in one case of generated data, the methods proposed by Garrett and Zeger seem to give problematic results as to identifiability; in the case of the empirical data on major depression, they lead to accepting a suboptimal three-class model. In the latter case, one can be rather sure that an identifiable, well-fitting latent class model could have been identified—if Garrett and Zeger had also considered restricted latent class models. [ABSTRACT FROM AUTHOR]