Marginal regression approach for additive hazards models with clustered current status data

Current status data arise naturally from tumorigenicity experiments, epidemiology studies, biomedicine, econometrics and demographic and sociology studies. Moreover, clustered current status data may occur with animals from the same litter in tumorigenicity experiments or with subjects from the same family in epidemiology studies. Because the only information extracted from current status data is whether the survival times are before or after the monitoring or censoring times, the nonparametric maximum likelihood estimator of survival function converges at a rate of n ^1∕3 to a complicated limiting distribution. Hence, semiparametric regression models such as the additive hazards model have been extended for independent current status data to derive the test statistics, whose distributions converge at a rate of n ^1∕2 , for testing the regression parameters. However, a straightforward application of these statistical methods to clustered current status data is not appropriate because intracluster correlation needs to be taken into account. Therefore, this paper proposes two estimating functions for estimating the parameters in the additive hazards model for clustered current status data. The comparative results from simulation studies are presented, and the application of the proposed estimating functions to one real data set is illustrated. Copyright © 2013 John Wiley & Sons, Ltd.     

Practicable confidence intervals for current status data

Although confidence intervals (CIs) for binary isotonic regression and current status survival data have been well studied theoretically, their practical application has been limited, in part because of poor performance in small samples and in part because of computational difficulties. Ghosh, Banerjee, and Biswas (2008, Biometrics 64 , 1009‐1017) described three approaches to constructing CIs: (i) the Wald‐based method; (ii) the subsampling‐based method; and (iii) the likelihood‐ratio test (LRT)‐based method. In simulation studies, they found that the subsampling‐based method and LRT‐based method tend to have better coverage probabilities than a simple Wald‐based method that may perform poorly in realistic sample sizes. However, software implementing these approaches is currently unavailable. In this article, we show that by using transformations, simple Wald‐based CIs can be improved with small and moderate sample sizes to have competitive performance with LRT‐based method. Our simulations further show that a simple nonparametric bootstrap gives approximately correct CIs for the data generating mechanisms that we consider. We provide an R package that can be used to compute the Wald‐type and the bootstrap CIs and demonstrate its practical utility with two real data analyses. Copyright © 2012 John Wiley & Sons, Ltd.     

A nonparametric smoothing method for assessing GEE models with longitudinal binary data

Studies involving longitudinal binary responses are widely applied in the health and biomedical sciences research and frequently analyzed by generalized estimating equations (GEE) method. This article proposes an alternative goodness-of-fit test based on the nonparametric smoothing approach for assessing the adequacy of GEE fitted models, which can be regarded as an extension of the goodness-of-fit test of le Cessie and van Houwelingen (Biometrics 1991; 47:1267-1282). The expectation and approximate variance of the proposed test statistic are derived. The asymptotic distribution of the proposed test statistic in terms of a scaled chi-squared distribution and the power performance of the proposed test are discussed by simulation studies. The testing procedure is demonstrated by two real data. Copyright (c) 2008 John Wiley & Sons, Ltd.     

Statistical analysis of current status data with informative observation times

Current status data arise when each study subject is observed only once and the survival time of interest is known only to be either less or greater than the observation time. Such data often occur in, for example, cross-sectional studies, demographical investigations and tumorigenicity experiments and several semi-parametric and non-parametric methods for their analysis have been proposed. However, most of these methods deal only with the situation where observation time is independent of the underlying survival time completely or given covariates. This paper discusses regression analysis of current status data when the observation time may be related to the underlying survival time and inference procedures are presented for estimation of regression parameters under the additive hazards regression model. The procedures can be easily implemented and are applied to two motivating examples.     

Goodness‐of‐fit test for proportional subdistribution hazards model

This paper concerns using modified weighted Schoenfeld residuals to test the proportionality of subdistribution hazards for the Fine–Gray model, similar to the tests proposed by Grambsch and Therneau for independently censored data. We develop a score test for the time‐varying coefficients based on the modified Schoenfeld residuals derived assuming a certain form of non‐proportionality. The methods perform well in simulations and a real data analysis of breast cancer data, where the treatment effect exhibits non‐proportional hazards. Copyright © 2013 John Wiley & Sons, Ltd.     

Joint models for efficient estimation in proportional hazards regression models

In survival studies, information lost through censoring can be partially recaptured through repeated measures data which are predictive of survival. In addition, such data may be useful in removing bias in survival estimates, due to censoring which depends upon the repeated measures. Here we investigate joint models for survival T and repeated measurements Y, given a vector of covariates Z. Mixture models indexed as f (T/Z) f (Y/T,Z) are well suited for assessing covariate effects on survival time. Our objective is efficiency gains, using non-parametric models for Y in order to avoid introducing bias by misspecification of the distribution for Y. We model (T/Z) as a piecewise exponential distribution with proportional hazards covariate effect. The component (Y/T,Z) has a multinomial model. The joint likelihood for survival and longitudinal data is maximized, using the EM algorithm. The estimate of covariate effect is compared to the estimate based on the standard proportional hazards model and an alternative joint model based estimate. We demonstrate modest gains in efficiency when using the joint piecewise exponential joint model. In a simulation, the estimated efficiency gain over the standard proportional hazards model is 6.4 per cent. In clinical trial data, the estimated efficiency gain over the standard proportional hazards model is 10.2 per cent.     

Global goodness-of-fit tests in logistic regression with sparse data

The logistic regression model has become the standard analysing tool for binary responses in medical statistics. Methods for assessing goodness-of-fit, however, are less developed where this problem is especially pronounced in performing global goodness-of-fit tests with sparse data, that is, if the data contain only a small numbers of observations for each pattern of covariate values. In this situation it has been known for a long time that the standard goodness-of-fit tests (residual deviance and Pearson chi-square) behave unsatisfactorily if p-values are calculated from the chi(2)-distribution. As a remedy in this situation the Hosmer-Lemeshow test is frequently recommended; it relies on a new grouping of the observations to avoid sparseness, where this grouping depends on the estimated probabilities from the model. It has been shown, however, that the Hosmer-Lemeshow test also has some deficiencies, for example, it depends heavily on the calculating algorithm and thus different implementations might lead to different conclusions regarding the fit of the model. We present some alternative tests from the statistical literature which should also perform well with sparse data. Results from a simulation study are given which show that there exist some goodness-of-fit tests (for example, the Farrington test) that have good properties regarding size and power and that even outperform the Hosmer-Lemeshow test. We illustrate the various tests with an example from dermatology on occupational hand eczema in hairdressers.     

Testing goodness of fit of parametric models for censored data

We propose and study a goodness‐of‐fit test for left‐censored, right‐censored, and interval‐censored data assuming random censorship. Main motivation comes from dietary exposure assessment in chemical risk assessment, where the determination of an appropriate distribution for concentration data is of major importance. We base the new goodness‐of‐fit test procedure proposed in this paper on the order selection test. As part of the testing procedure, we extend the null model to a series of nested alternative models for censored data. Then, we use a modified AIC model selection to select the best model to describe the data. If a model with one or more extra parameters is selected, then we reject the null hypothesis. As an alternative to the use of the asymptotic null distribution of the test statistic, we define a bootstrap‐based procedure. We illustrate the applicability of the test procedure on data of cadmium concentrations and on data from the Signal Tandmobiel study and demonstrate its performance characteristics through simulation studies. Copyright © 2012 John Wiley & Sons, Ltd.     

Multinomial goodness-of-fit tests for logistic regression models.

We examine the properties of several tests for goodness-of-fit for multinomial logistic regression. One test is based on a strategy of sorting the observations according to the complement of the estimated probability for the reference outcome category and then grouping the subjects into g equal-sized groups. A g x c contingency table, where c is the number of values of the outcome variable, is constructed. The test statistic, denoted as Cg, is obtained by calculating the Pearson chi2 statistic where the estimated expected frequencies are the sum of the model-based estimated logistic probabilities. Simulations compare the properties of Cg with those of the ungrouped Pearson chi2 test (X2) and its normalized test (z). The null distribution of Cg is well approximated by the chi2 distribution with (g-2) x (c-1) degrees of freedom. The sampling distribution of X2 is compared with a chi2 distribution with n x (c-1) degrees of freedom but shows erratic behavior. With a few exceptions, the sampling distribution of z adheres reasonably well to the standard normal distribution. Power simulations show that Cg has low power for a sample of 100 observations, but satisfactory power for a sample of 400. The tests are illustrated using data from a study of cytological criteria for the diagnosis of breast tumors.     

Regression analysis of multivariate current status data with dependent censoring: application to ankylosing spondylitis data

Multivariate current‐status failure time data consist of several possibly related event times of interest, in which the status of each event is determined at a single examination time. If the examination time is intrinsically related to the event times, the examination is referred to as dependent censoring and needs to be taken into account. Such data often occur in clinical studies and animal carcinogenicity experiments. To accommodate for possible dependent censoring, this paper proposes a joint frailty model for event times and dependent censoring time. We develop a likelihood approach using Gaussian quadrature techniques for obtaining maximum likelihood estimates. We conduct extensive simulation studies for investigating finite‐sample properties of the proposed method. We illustrate the proposed method with an analysis of patients with ankylosing spondylitis, where the examination time may be dependent on the event times of interest. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian adaptive lasso for additive hazard regression with current status data

Variable selection is a crucial issue in model building and it has received considerable attention in the literature of survival analysis. However, available approaches in this direction have mainly focused on time-to-event data with right censoring. Moreover, a majority of existing variable selection procedures for survival models are developed in a frequentist framework. In this article, we consider additive hazards model in the presence of current status data. We propose a Bayesian adaptive least absolute shrinkage and selection operator procedure to conduct a simultaneous variable selection and parameter estimation. Efficient Markov chain Monte Carlo methods are developed to implement posterior sampling and inference. The empirical performance of the proposed method is demonstrated by simulation studies. An application to a study on the risk factors of heart failure disease for type 2 diabetes patients is presented.     

Quadratic inference functions in marginal models for longitudinal data

The quadratic inference function (QIF) is a new statistical methodology developed for the estimation and inference in longitudinal data analysis using marginal models. This method is an alternative to the popular generalized estimating equations approach, and it has several useful properties such as robustness, a goodness‐of‐fit test and model selection. This paper presents an introductory review of the QIF, with a strong emphasis on its applications. In particular, a recently developed SAS MACRO QIF is illustrated in this paper to obtain numerical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Two goodness-of-fit tests for logistic regression models with continuous covariates.

Assessing goodness-of-fit in logistic regression models can be problematic, in that commonly used deviance or Pearson chi-square statistics do not have approximate chi-square distributions, under the null hypothesis of no lack of fit, when continuous covariates are modelled. We present two easy to implement test statistics similar to the deviance and Pearson chi-square tests that are appropriate when continuous covariates are present. The methodology uses an approach similar to that incorporated by the Hosmer and Lemeshow goodness-of-fit test in that observations are classified into distinct groups according to fitted probabilities, allowing sufficient cell sizes for chi-square testing. The major difference is that the proposed tests perform this grouping within the cross-classification of all categorical covariates in the model and, in some situations, allow for a more powerful assessment of where model predicted and observed counts may differ. A variety of simulations are performed comparing the proposed tests to the Hosmer-Lemeshow test.     

Analysis of a nonsusceptible fraction with current status data

In studies involving subclinical events, times of events are often subject to interval censoring since their occurrence is only detected at inspection times. When individuals are event-free at an initial time and a single follow-up inspection is made, current status data are obtained. In many settings, however, the population comprised a susceptible and a nonsusceptible subpopulation, where only susceptible individuals will go on to experience the event. Then interest often lies primarily in identifying prognostic variables for susceptibility, and secondarily in the event time distribution among the susceptible individuals. We give a simple mixture model that facilitates estimation of the proportion of susceptible individuals, covariate effects on the odds of susceptibility, and the event time distribution under a current status observation scheme. Asymptotic relative efficiency of maximum likelihood estimators is considered based on the Fisher information for a variety of settings. EM algorithms are proposed for parametric, weakly parametric, and nonparametric estimation of the event time distribution. The methods are applied to motivating studies examining an immunological response to low molecular weight heparin in patients undergoing orthopedic surgery.     

A local likelihood proportional hazards model for interval censored data.

We discuss the use of local likelihood methods to fit proportional hazards regression models to right and interval censored data. The assumed model allows for an arbitrary, smoothed baseline hazard on which a vector of covariates operates in a proportional manner, and thus produces an interpretable baseline hazard function along with estimates of global covariate effects. For estimation, we extend the modified EM algorithm suggested by Betensky, Lindsey, Ryan and Wand. We illustrate the method with data on times to deterioration of breast cosmeses and HIV-1 infection rates among haemophiliacs.     

A Pearson-type goodness-of-fit test for stationary and time-continuous Markov regression models

Markov regression models describe the way in which a categorical response variable changes over time for subjects with different explanatory variables. Frequently it is difficult to measure the response variable on equally spaced discrete time intervals. Here we propose a Pearson-type goodness-of-fit test for stationary Markov regression models fitted to panel data. A parametric bootstrap algorithm is used to study the distribution of the test statistic. The proposed technique is applied to examine the fit of a Markov regression model used to identify markers for disease progression in psoriatic arthritis.     

Misspecification tests for binomial and beta-binomial models

The IOS test of Presnell and Boss (J. Am. Stat. Assoc. 2004; 99(465):216-227) is a general-purpose goodness-of-fit test based on the ratio of in-sample and out-of-sample likelihoods. For large samples, the IOS statistic can be approximated by a multiplicative contrast between two estimates of the information matrix, and in this way the IOS test is connected to White's (Econometrica 1982; 50:1-26) information matrix test, or IM test, which is based directly on the difference of two estimates of the information matrix. In this paper, we compare the performance of IOS to that of the IM test and of other goodness-of-fit tests for binomial and beta-binomial models, in both examples and simulations. Our findings suggest that IOS is strongly competitive, not only against the IM test but also against tests designed for specific binomial and beta-binomial models.     

Checking hazard regression models using pseudo-observations

Graphical methods for model diagnostics are an essential part of the model fitting procedure. However, in survival analysis, the plotting is always hampered by the presence of censoring. Although model specific solutions do exist and are commonly used, we present a more general approach that covers all the models using the same framework. The pseudo-observations enable us to calculate residuals for each individual at each time point regardless of censoring and provide methods for simultaneously checking all the assumptions of both the Cox and the additive model. We introduce methods for single as well as multiple covariate cases and complement them with corresponding goodness-of-fit tests. The methods are illustrated on simulated as well as real data examples.     

Regression analysis for current status data using the EM algorithm

We propose new expectation–maximization algorithms to analyze current status data under two popular semiparametric regression models: the proportional hazards (PH) model and the proportional odds (PO) model. Monotone splines are used to model the baseline cumulative hazard function in the PH model and the baseline odds function in the PO model. The proposed algorithms are derived by exploiting a data augmentation based on Poisson latent variables. Unlike previous regression work with current status data, our PH and PO model fitting methods are fast, flexible, easy to implement, and provide variance estimates in closed form. These techniques are evaluated using simulation and are illustrated using uterine fibroid data from a prospective cohort study on early pregnancy. Copyright © 2013 John Wiley & Sons, Ltd.     

Cox models with dynamic ridge penalties on time‐varying effects of the covariates

Analysis of long‐term follow‐up survival studies require more sophisticated approaches than the proportional hazards model. To account for the dynamic behaviour of fixed covariates, penalized Cox models can be employed in models with interactions of the covariates and known time functions. In this work, I discuss some of the suggested methods and emphasize on the use of a ridge penalty in survival models. I review different strategies for choosing an optimal penalty weight and argue for the use of the computationally efficient restricted maximum likelihood (REML)‐type method. A ridge penalty term can be subtracted from the likelihood when modelling time‐varying effects in order to control the behaviour of the time functions. I suggest using flexible time functions such as B‐splines and constrain the behaviour of these by adding proper penalties. I present the basic methods and illustrate different penalty weights in two different datasets. Copyright © 2013 John Wiley & Sons, Ltd.