Influence diagnostics for random effect survival models: Application to a recurrent infection study for kidney patients on portable dialysis

In modeling multivariate failure time data, a class of survival model with random effects is applicable. It incorporates the random effect terms in the linear predictor and includes various random effect survival models as special cases, such as the random effect model assuming Cox's proportional hazards, with Weibull baseline hazards and with power family of transformation in the relative risk function. Residual maximum likelihood (REML) estimation of parameters is achieved by adopting the generalised linear mixed models (GLMM) approach. Accordingly, influence diagnostics are developed as sensitivity measures for the REML estimation of model parameters. A data set of recurrent infections of kidney patients on portable dialysis illustrates the usefulness of the influence diagnostics. A simulation study is carried out to examine the performance of the proposed influence diagnostics.     

Multilevel cumulative logistic regression model with random effects: Application to British social attitudes panel survey data

A multilevel model for ordinal data in generalized linear mixed models (GLMM) framework is developed to account for the inherent dependencies among observations within clusters. Motivated by a data set from the British Social Attitudes Panel Survey (BSAPS), the random district effects and respondent effects are incorporated into the linear predictor to accommodate the nested clusterings. The fixed (random) effects are estimated (predicted) by maximizing the penalized quasi likelihood (PQL) function, whereas the variance component parameters are obtained via the restricted maximum likelihood (REML) estimation method. The model is employed to analyze the BSAPS data. Simulation studies are conducted to assess the performance of estimators.     

Random effects in promotion time cure rate models

In this paper, a survival model with long-term survivors and random effects, based on the promotion time cure rate model formulation for models with a surviving fraction is investigated. We present Bayesian and classical estimation approaches. The Bayesian approach is implemented using a Markov chain Monte Carlo (MCMC) based on the Metropolis-Hastings algorithms. For the second one, we use restricted maximum likelihood (REML) estimators. A simulation study is performed to evaluate the accuracy of the applied techniques for the estimates and their standard deviations. An example on an oropharynx cancer study is used to illustrate the model and the estimation approaches considered in the study.     

Stacked Laplace-EM algorithm for duration models with time-varying and random effects

An extension of the Cox proportional hazards model for clustered survival data is proposed. This allows both general random effects (frailties) and time-varying regression coefficients, the latter being smooth functions of time. The model is fitted using a mixed-model representation of penalized spline smoothing which offers a unified framework for estimation of the baseline hazard, the smooth effects and the random effects. The estimator is computed using a stacked laplace-EM (SLaEM) algorithm. More specifically, the smoothing parameters are integrated out in the log likelihood via a Laplace approximation. The approximation itself involves an integrated log-likelihood over the random cluster effects, for which the EM algorithm is used. A marginal Akaike information criterion is developed for selection among possible candidate models. The time-varying and mixed effects model is applied to unemployment data taken from the German Socio-Economic Panel. The duration of unemployment is modeled in a flexible way including smooth covariate effects and individual random effects.     

Semiparametric estimation of a nested random effects model for the analysis of multi-level clustered failure time data

Multi-level clustered failure time data arise when the clustering of data occurs at more than one level. It is of interest to estimate the relative risks of covariates and clustering effect of failure times at each level. We consider a nested random effect proportional hazards model, where a subcluster-specific frailty operates multiplicatively on the conditional hazard model, and its distribution function depends on a cluster-specific random frailty. Under this model, we propose a Monte-Carlo EM-based semiparametric estimation procedure to estimate regression coefficients, nonparametric baseline cumulative hazard and the association parameters. In addition, we derive a covariance matrix of the parameter estimates. We illustrate the proposed method using clustered survival data collected from a vitamin A supplementation trial in Nepal, where it is of scientific interest to assess the clustering effect of mortality within households and within villages. We use simulations to study the performance of the proposed estimation procedure.     

Robust diagnostics for the heteroscedastic regression model

The assumption of equal variance in the normal regression model is not always appropriate. Cook and Weisberg (1983) provide a score test to detect heteroscedasticity, while Patterson and Thompson (1971) propose the residual maximum likelihood (REML) estimation to estimate variance components in the context of an unbalanced incomplete-block design. REML is often preferred to the maximum likelihood estimation as a method of estimating covariance parameters in a linear model. However, outliers may have some effect on the estimate of the variance function. This paper incorporates the maximum trimming likelihood estimation ( [Hadi and Luce?o, 1997] and [Vandev and Neykov, 1998]) in REML to obtain a robust estimation of modelling variance heterogeneity. Both the forward search algorithm of Atkinson (1994) and the fast algorithm of Neykov et al. (2007) are employed to find the resulting estimator. Simulation and real data examples are used to illustrate the performance of the proposed approach.     

Semiparametric analysis of clustered interval-censored survival data with a cure fraction

A generalization of the semiparametric Cox’s proportional hazards model by means of a random effect or frailty approach to accommodate clustered survival data with a cure fraction is considered. The frailty serves as a quantification of the health condition of the subjects under study and may depend on some observed covariates like age. One single individual-specific frailty that acts on the hazard function is adopted to determine the cure status of an individual and the heterogeneity on the time to event if the individual is not cured. Under this formulation, an individual who has a high propensity to be cured would tend to have a longer time to event if he is not cured. Within a cluster, both the cure statuses and the times to event of the individuals would be correlated. In contrast to some models proposed in the literature, the model accommodates the correlations among the observations in a more natural way. A multiple imputation estimation method is proposed for both right-censored and interval-censored data. Simulation studies show that the performance of the proposed estimation method is highly satisfactory. The proposed model and method are applied to the National Aeronautics and Space Administration’s hypobaric decompression sickness data to investigate the factors associated with the occurrence and the time to onset of grade IV venous gas emboli under hypobaric environments.     

Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal

Stochastic leaky integrate-and-fire (LIF) neuronal models are common theoretical tools for studying properties of real neuronal systems. Experimental data of frequently sampled membrane potential measurements between spikes show that the assumption of constant parameter values is not realistic and that some (random) fluctuations are occurring. In this letter, we extend the stochastic LIF model, allowing a noise source determining slow fluctuations in the signal. This is achieved by adding a random variable to one of the parameters characterizing the neuronal input, considering each interspike interval (ISI) as an independent experimental unit with a different realization of this random variable. In this way, the variation of the neuronal input is split into fast (within-interval) and slow (between-intervals) components. A parameter estimation method is proposed, allowing the parameters to be estimated simultaneously over the entire data set. This increases the statistical power, and the average estimate over all ISIs will be improved in the sense of decreased variance of the estimator compared to previous approaches, where the estimation has been conducted on each individual ISI. The results obtained on real data show good agreement with classical regression methods.     

Hierarchical-likelihood approach for nonlinear mixed-effects models

The restricted maximum likelihood (REML) procedure is useful for inferences about variance components in linear mixed models (LMMs). However, its extension to nonlinear mixed models (NLMMs) is often hampered by analytically intractable integrals. For NLMMs various estimation methods have been suggested, but they have suffered from unsatisfactory biases. In this paper we propose a statistically and computationally efficient REML procedure, based upon hierarchical likelihood. Numerical studies show that the proposed method reduces the biases in the existing methods that we studied. We also study how the current REML procedure for LMMs can be modified to compute the proposed estimators.     

Mixture cure models for multivariate survival data

Mixture cure models (MCMs) have been widely used to analyze survival data with a cure fraction. The MCMs postulate that a fraction of the patients are cured from the disease and that the failure time for the uncured patients follows a proper survival distribution, referred to as latency distribution. The MCMs have been extended to bivariate survival data by modeling the marginal distributions. In this paper, the marginal MCM is extended to multivariate survival data. The new model is applicable to the survival data with varied cluster size and interval censoring. The proposed model allows covariates to be incorporated into both the cure fraction and the latency distribution for the uncured patients. The primary interest is to estimate the marginal parameters in the mean structure, where the correlation structure is treated as nuisance parameters. The marginal parameters are estimated consistently by treating the observations within the cluster as independent. The variances of the parameters are estimated by the one-step jackknife method. The proposed method does not depend on the specification of correlation structure. Simulation studies show that the new method works well when the marginal model is correct. The performance of the MCM is also examined when the clustered survival times share common random effect. The MCM is applied to the data from a smoking cessation study.     

Likelihood approach for count data in longitudinal experiments

In many cancer studies and clinical research, repeated observations of response variables are taken over time on each individual in one or more treatment groups. In such cases the repeated observations of each vector response are likely to be correlated and the autocorrelation structure for the repeated data plays a significant role in the estimation of regression parameters. A random intercept model for count data is developed using exact maximum-likelihood estimation via numerical integration. A simulation study is performed to compare the proposed methodology with the traditional generalized linear mixed model (GLMM) approach and with the GLMM when penalized quasi-likelihood method is used to perform maximum-likelihood estimation. The methodology is illustrated by analyzing data sets containing longitudinal measures of number of tumors in an experiment of carcinogenesis to study the influence of lipids in the development of cancer.     

Mixed model-based inference in geoadditive hazard regression for interval-censored survival times

Mixed model-based estimation of additive or geoadditive regression models has become popular throughout recent years. It provides a unified and modular framework that facilitates joint estimation of nonparametric covariate effects and the corresponding smoothing parameters. Therefore, extensions of mixed model-based inference to a Cox-type regression model for the hazard rate are considered, allowing for a combination of general censoring schemes for the survival times and a flexible, geoadditive predictor. In particular, the proposed methodology allows for arbitrary combinations of right, left, and interval censoring as well as left truncation. The geoadditive predictor comprises time-varying effects, nonparametric effects of continuous covariates, spatial effects, and potentially a number of extensions such as cluster-specific frailties or interaction surfaces. In addition, all covariates are allowed to be piecewise constant time-varying. Nonlinear and time-varying effects as well as the baseline hazard rate are modeled by penalized splines. Spatial effects can be included based on either Markov random fields or stationary Gaussian random fields. Estimation is based on a reparametrization of the model as a variance component mixed model. The variance parameters, corresponding to inverse smoothing parameters, can then be determined using an approximate marginal likelihood approach. An analysis on childhood mortality in Nigeria serves as an application, where the interval censoring framework additionally allows to deal with the problem of heaped survival times. The effect of ignoring the impact of interval-censored observations is investigated in a simulation study.     

Likelihood approach for count data in longitudinal experiments

In many cancer studies and clinical research, repeated observations of response variables are taken over time on each individual in one or more treatment groups. In such cases the repeated observations of each vector response are likely to be correlated and the autocorrelation structure for the repeated data plays a significant role in the estimation of regression parameters. A random intercept model for count data is developed using exact maximum-likelihood estimation via numerical integration. A simulation study is performed to compare the proposed methodology with the traditional generalized linear mixed model (GLMM) approach and with the GLMM when penalized quasi-likelihood method is used to perform maximum-likelihood estimation. The methodology is illustrated by analyzing data sets containing longitudinal measures of number of tumors in an experiment of carcinogenesis to study the influence of lipids in the development of cancer.     

Estimation in the probit normal model for binary outcomes using the SAEM algorithm

Generalized linear mixed models (GLMM) form a very general class of random effects models for discrete and continuous responses in the exponential family. They are useful in a variety of applications. The traditional likelihood approach for GLMM usually involves high dimensional integrations which are computationally intensive. In this work, we investigate the case of binary outcomes analyzed under a two stage probit normal model with random effects. First, it is shown how ML estimates of the fixed effects and variance components can be computed using a stochastic approximation of the EM algorithm (SAEM). The SAEM algorithm can be applied directly, or in conjunction with a parameter expansion version of EM to speed up the convergence. A procedure is also proposed to obtain REML estimates of variance components and REML-based estimates of fixed effects. Finally an application to a real data set involving a clinical trial is presented, in which these techniques are compared to other procedures (penalized quasi-likelihood, maximum likelihood, Bayesian inference) already available in classical softwares (SAS Glimmix, SAS Nlmixed, WinBUGS), as well as to a Monte Carlo EM (MCEM) algorithm.     

SAS macros for estimation of the cumulative incidence functions based on a Cox regression model for competing risks survival data

When considering competing risks survival data, the cause specific hazard functions are often modelled by the proportional hazards Cox regression model. First, we present how to estimate the parameters in this model when some of the covariates are allowed to have exactly the same effect on several causes of failure. In many cases, the focus is not on the parameter estimates, but rather on the probability of observing a failure from a specific cause for individuals with specified covariate values. These probabilities, the cumulative incidences, are not simple functions of the parameters and they are, so far, not provided by the standard statistical software packages. We present two SAS macros: a SAS macro named CumInc for estimation of the cumulative incidences and a SAS macro named CumIncV for estimation of the cumulative incidences and the variances of the estimated cumulative incidences. The use of the macros is demonstrated through an example.     

A random walk approach for quantifying uncertainty in group sequential survival trials

The development of group sequential methods has produced multiple criteria that are used to guide the decision of whether a clinical trial should be stopped early given the data observed at the time of an interim analysis. However, the potential for time-varying treatment effects should be considered when monitoring survival endpoints. In order to quantify uncertainty in future treatment effects it is necessary to consider future alternatives which might reasonably be observed conditional upon data collected up to the time of an interim analysis. A method of imputation of future alternatives using a random walk approach that incorporates a Bayesian conditional hazards model and splits the prior distribution for model parameters across regions of sampled and unsampled support is proposed. By providing this flexibility, noninformative priors can be used over regions of sampled data while providing structure to model parameters over future time intervals. The result is that inference over areas of sampled support remains consistent with commonly used frequentist statistics while a rich class of predictive distributions of treatment effect over the maximal duration of a trial are generated to assess potential treatment effects which may be plausibly observed if the trial were to continue. Selected operating characteristics of the proposed method are investigated via simulation and the approach is applied to survival data stemming from trial 002 of the Community Programs for Clinical Research on AIDS (CPCRA) study.     

Analysis of zero-inflated clustered count data: A marginalized model approach

Min and Agresti (2005) proposed random effect hurdle models for zero-inflated clustered count data with two-part random effects for a binary component and a truncated count component. In this paper, we propose new marginalized models for zero-inflated clustered count data using random effects. The marginalized models are similar to Dobbie and Welsh’s (2001) model in which generalized estimating equations were exploited to find estimates. However, our proposed models are based on a likelihood-based approach. A Quasi-Newton algorithm is developed for estimation. We use these methods to carefully analyze two real datasets.     

Regression models for grouped survival data: Estimation and sensitivity analysis

In this study, regression models are evaluated for grouped survival data when the effect of censoring time is considered in the model and the regression structure is modeled through four link functions. The methodology for grouped survival data is based on life tables, and the times are grouped in k intervals so that ties are eliminated. Thus, the data modeling is performed by considering the discrete models of lifetime regression. The model parameters are estimated by using the maximum likelihood and jackknife methods. To detect influential observations in the proposed models, diagnostic measures based on case deletion, which are denominated global influence, and influence measures based on small perturbations in the data or in the model, referred to as local influence, are used. In addition to those measures, the local influence and the total influential estimate are also employed. Various simulation studies are performed and compared to the performance of the four link functions of the regression models for grouped survival data for different parameter settings, sample sizes and numbers of intervals. Finally, a data set is analyzed by using the proposed regression models.     

Parametric and semiparametric methods for mapping quantitative trait loci

Most statistical methods for mapping quantitative trait loci (QTLs) have been extensively developed for normally distributed and completely observed phenotypes. A survival model such as Cox’s proportional hazards model, or an accelerated failure time model, is the natural choice for regressing a phenotype onto a maker-type when the primary phenotype belongs to failure time. This study proposes parametric and semiparametric methods based on accelerated failure time models for interval mapping. In the parametric model, the EM algorithm (expectation maximization algorithm) is adopted to estimate regression parameters and search the entire chromosome for QTL. In the semiparametric model, the error distribution is left unspecified, and a rank-based inference is developed for the effect and location of QTL. Simulated data are used to compare the mapping performance of the semiparametric methodology with that of the parametric method, which separately selects the correct and incorrect error distribution. Analytical results reveal that the parametric estimators may be more efficient in determining the effect and location of QTL, but have obvious bias when selecting the incorrect error distribution. In contrast, the semiparametric inference is robust to the error distribution.     

Comparison of different estimation procedures for proportional hazards model with random effects

Proportional hazards models with multivariate random effects (frailties) acting multiplicatively on the baseline hazard are a topic of intensive research. Several estimation procedures have been proposed to deal with this type of models. Four procedures used to fit these models are compared in two real-life datasets and in a simulation study. The performance of the four methods is investigated in terms of the bias of point estimates, their empirical variability and the bias of the estimation of the variability.