Non-penalty shrinkage estimation of random effect models for longitudinal data with AR(1) errors

In this paper, we consider the non-penalty shrinkage estimation method of random effect models with autoregressive errors for longitudinal data when there are many covariates and some of them may not be active for the response variable. In observational studies, subjects are followed over equally or unequally spaced visits to determine the continuous response and whether the response is associated with the risk factors/covariates. Measurements from the same subject are usually more similar to each other and thus are correlated with each other but not with observations of other subjects. To analyse this data, we consider a linear model that contains both random effects across subjects and within-subject errors that follows autoregressive structure of order 1 (AR(1)). Considering the subject-specific random effect as a nuisance parameter, we use two competing models, one includes all the covariates and the other restricts the coefficients based on the auxiliary information. We consider the non-penalty shrinkage estimation strategy that shrinks the unrestricted estimator in the direction of the restricted estimator. We discuss the asymptotic properties of the shrinkage estimators using the notion of asymptotic biases and risks. A Monte Carlo simulation study is conducted to examine the relative performance of the shrinkage estimators with the unrestricted estimator when the shrinkage dimension exceeds two. We also numerically compare the performance of the shrinkage estimators to that of the LASSO estimator. A longitudinal CD4 cell count data set will be used to illustrate the usefulness of shrinkage and LASSO estimators.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

A Bayesian approach of joint models for clustered zero-inflated count data with skewness and measurement errors

Count data with excess zeros are widely encountered in the fields of biomedical, medical, public health and social survey, etc. Zero-inflated Poisson (ZIP) regression models with mixed effects are useful tools for analyzing such data, in which covariates are usually incorporated in the model to explain inter-subject variation and normal distribution is assumed for both random effects and random errors. However, in many practical applications, such assumptions may be violated as the data often exhibit skewness and some covariates may be measured with measurement errors. In this paper, we deal with these issues simultaneously by developing a Bayesian joint hierarchical modeling approach. Specifically, by treating intercepts and slopes in logistic and Poisson regression as random, a flexible two-level ZIP regression model is proposed, where a covariate process with measurement errors is established and a skew-t-distribution is considered for both random errors and random effects. Under the Bayesian framework, model selection is carried out using deviance information criterion (DIC) and a goodness-of-fit statistics is also developed for assessing the plausibility of the posited model. The main advantage of our method is that it allows for more robustness and correctness for investigating heterogeneity from different levels, while accommodating the skewness and measurement errors simultaneously. An application to Shanghai Youth Fitness Survey is used as an illustrate example. Through this real example, it is showed that our approach is of interest and usefulness for applications.     

A flexible approach for multivariate mixed-effects models with non-ignorable missing values

We propose a flexible model approach for the distribution of random effects when both response variables and covariates have non-ignorable missing values in a longitudinal study. A Bayesian approach is developed with a choice of nonparametric prior for the distribution of random effects. We apply the proposed method to a real data example from a national long-term survey by Statistics Canada. We also design simulation studies to further check the performance of the proposed approach. The result of simulation studies indicates that the proposed approach outperforms the conventional approach with normality assumption when the heterogeneity in random effects distribution is salient.     

The repair alert models with fixed and random effects

This article extends a random preventive maintenance scheme, called repair alert model, when there exist environmental variables that effect on system lifetimes. It can be used for implementing age-dependent maintenance policies on engineering devices. In other words, consider a device that works for a job and is subject to failure at a random time X, and the maintenance crew can avoid the failure by a possible replacement at some random time Z. The new model is flexible to including covariates with both fixed and random effects. The problem of estimating parameters is also investigated in details. Here, the observations are in the form of random signs censoring data (RSCD) with covariates. Therefore, this article generalizes derived statistical inferences on the basis of RSCD albeit without covariates in past literature. To do this, it is assumed that the system lifetime distribution belongs to the log-location-scale family of distributions. A real dataset is also analyzed on basis of the results obtained.     

Covariate Decomposition Methods for Longitudinal Missing‐at‐Random Data and Predictors Associated with Subject‐Specific Effects

Investigators often gather longitudinal data to assess changes in responses over time within subjects and to relate these changes to within‐subject changes in predictors. Missing data are common in such studies and predictors can be correlated with subject‐specific effects. Maximum likelihood methods for generalized linear mixed models provide consistent estimates when the data are ‘missing at random’ (MAR) but can produce inconsistent estimates in settings where the random effects are correlated with one of the predictors. On the other hand, conditional maximum likelihood methods (and closely related maximum likelihood methods that partition covariates into between‐ and within‐cluster components) provide consistent estimation when random effects are correlated with predictors but can produce inconsistent covariate effect estimates when data are MAR. Using theory, simulation studies, and fits to example data this paper shows that decomposition methods using complete covariate information produce consistent estimates. In some practical cases these methods, that ostensibly require complete covariate information, actually only involve the observed covariates. These results offer an easy‐to‐use approach to simultaneously protect against bias from both cluster‐level confounding and MAR missingness in assessments of change.     

Gaussian models for degradation processes-part I: Methods for the analysis of biomarker data

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.     

The use of auxiliary variables in capture-recapture modelling: An overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

Markov structure based logistic regression for repeated measures: An application to diabetes mellitus data

In longitudinal studies, as repeated observations are made on the same individual the response variables will usually be correlated. In analyzing such data, this dependence must be taken into account to avoid misleading inferences. The focus of this paper is to apply a logistic marginal model with Markovian dependence proposed by Azzalini [A. Azzalini, Logistic regression for autocorrelated data with application to repeated measures, Biometrika 81 (1994) 767–775] to the study of the influence of time-dependent covariates on the marginal distribution of the binary response in serially correlated binary data. We have shown how to construct the model so that the covariates relate only to the mean value of the process, independent of the association parameters. After formulating the proposed model for repeated measures data, the same approach is applied to missing data. An application is provided to the diabetes mellitus data of registered patients at the Bangladesh Institute of Research and Rehabilitation in Diabetes, Endocrine and Metabolic Disorders (BIRDEM) in 1984, using both time stationary and time varying covariates.     

The use of auxiliary variables in capture-recapture modelling: an overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

Semiparametric estimation of treatment effect with time-lagged response in the presence of informative censoring

In many randomized clinical trials, the primary response variable, for example, the survival time, is not observed directly after the patients enroll in the study but rather observed after some period of time (lag time). It is often the case that such a response variable is missing for some patients due to censoring that occurs when the study ends before the patient’s response is observed or when the patients drop out of the study. It is often assumed that censoring occurs at random which is referred to as noninformative censoring; however, in many cases such an assumption may not be reasonable. If the missing data are not analyzed properly, the estimator or test for the treatment effect may be biased. In this paper, we use semiparametric theory to derive a class of consistent and asymptotically normal estimators for the treatment effect parameter which are applicable when the response variable is right censored. The baseline auxiliary covariates and post-treatment auxiliary covariates, which may be time-dependent, are also considered in our semiparametric model. These auxiliary covariates are used to derive estimators that both account for informative censoring and are more efficient then the estimators which do not consider the auxiliary covariates.     

Analysis of two-phase sampling data with semiparametric additive hazards models

Under the case-cohort design introduced by Prentice (Biometrica 73:1–11, 1986), the covariate histories are ascertained only for the subjects who experience the event of interest (i.e., the cases) during the follow-up period and for a relatively small random sample from the original cohort (i.e., the subcohort). The case-cohort design has been widely used in clinical and epidemiological studies to assess the effects of covariates on failure times. Most statistical methods developed for the case-cohort design use the proportional hazards model, and few methods allow for time-varying regression coefficients. In addition, most methods disregard data from subjects outside of the subcohort, which can result in inefficient inference. Addressing these issues, this paper proposes an estimation procedure for the semiparametric additive hazards model with case-cohort/two-phase sampling data, allowing the covariates of interest to be missing for cases as well as for non-cases. A more flexible form of the additive model is considered that allows the effects of some covariates to be time varying while specifying the effects of others to be constant. An augmented inverse probability weighted estimation procedure is proposed. The proposed method allows utilizing the auxiliary information that correlates with the phase-two covariates to improve efficiency. The asymptotic properties of the proposed estimators are established. An extensive simulation study shows that the augmented inverse probability weighted estimation is more efficient than the widely adopted inverse probability weighted complete-case estimation method. The method is applied to analyze data from a preventive HIV vaccine efficacy trial.     

On proportional hazards assumption under the random effects models

The proportional hazards mixed-effects model (PHMM) was proposed to handle dependent survival data. Motivated by its application in genetic epidemiology, we study the interpretation of its parameter estimates under violations of the proportional hazards assumption. The estimated fixed effect turns out to be an averaged regression effect over time, while the estimated variance component could be unaffected, inflated or attenuated depending on whether the random effect is on the baseline hazard, and whether the non-proportional regression effect decreases or increases over time. Using the conditional distribution of the covariates we define the standardized covariate residuals, which can be used to check the proportional hazards assumption. The model checking technique is illustrated on a multi-center lung cancer trial.     

The analysis of incomplete data using stochastic covariates

In the development of many diseases there are often associated random variables which continuously reflect the progress of a subject towards the final expression of the disease (failure). At any given time these processes, which we call stochastic covariates, may provide information about the current hazard and the remaining time to failure. Likewise, in situations when the specific times of key prior events are not known, such as the time of onset of an occult tumour or the time of infection with HIV-1, it may be possible to identify a stochastic covariate which reveals, indirectly, when the event of interest occurred. The analysis of carcinogenicity trials which involve occult tumours is usually based on the time of death or sacrifice and an indicator of tumour presence for each animal in the experiment. However, the size of an occult tumour observed at the endpoint represents data concerning tumour development which may convey additional information concerning both the tumour incidence rate and the rate of death to which tumour-bearing animals are subject. We develop a stochastic model for tumour growth and suggest different ways in which the effect of this growth on the hazard of failure might be modelled. Using a combined model for tumour growth and additive competing risks of death, we show that if this tumour size information is used, assumptions concerning tumour lethality, the context of observation or multiple sacrifice times are no longer necessary in order to estimate the tumour incidence rate. Parametric estimation based on the method of maximum likelihood is outlined and is applied to simulated data from the combined model. The results of this limited study confirm that use of the stochastic covariate tumour size results in more precise estimation of the incidence rate for occult tumours.     

Hierarchical Generalized Linear Models and Frailty Models with Bayesian Nonparametric Mixing

This paper proposes Bayesian nonparametric mixing for some well-known and popular models. The distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term, a function of the observed covariates, and an additive or multiplicative random effects term. Typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family. This assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified. Another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution. As a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately. This paper therefore proposes a Bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates' effect on the distribution of the mixed effects.     

Evaluation of Bayesian multiple stage estimation under spatial CAR model variants

In this study, an evaluation of Bayesian hierarchical models is made based on simulation scenarios to compare single-stage and multi-stage Bayesian estimations. Simulated datasets of lung cancer disease counts for men aged 65 and older across 44 wards in the London Health Authority were analysed using a range of spatially structured random effect components. The goals of this study are to determine which of these single-stage models perform best given a certain simulating model, how estimation methods (single- vs. multi-stage) compare in yielding posterior estimates of fixed effects in the presence of spatially structured random effects, and finally which of two spatial prior models – the Leroux or ICAR model, perform best in a multi-stage context under different assumptions concerning spatial correlation. Among the fitted single-stage models without covariates, we found that when there is low amount of variability in the distribution of disease counts, the BYM model is relatively robust to misspecification in terms of DIC, while the Leroux model is the least robust to misspecification. When these models were fit to data generated from models with covariates, we found that when there was one set of covariates – either spatially correlated or non-spatially correlated, changing the values of the fixed coefficients affected the ability of either the Leroux or ICAR model to fit the data well in terms of DIC. When there were multiple sets of spatially correlated covariates in the simulating model, however, we could not distinguish the goodness of fit to the data between these single-stage models. We found that the multi-stage modelling process via the Leroux and ICAR models generally reduced the variance of the posterior estimated fixed effects for data generated from models with covariates and a UH term compared to analogous single-stage models. Finally, we found the multi-stage Leroux model compares favourably to the multi-stage ICAR model in terms of DIC. We conclude that the mutli-stage Leroux model should be seriously considered in applications of Bayesian disease mapping when an investigator desires to fit a model with both fixed effects and spatially structured random effects to Poisson count data.     

An Accelerated Model for Recurrence Data

Recurrence data usually come from sampling a system in different ages. It is common in areas of manufacturing, reliability, medicine, and risk analysis. In this article, an accelerated model for recurrence data is proposed. The model is based on the time between failures (TBFs) for an accelerated time regression method using the number of failures as a dummy covariate. Using this model, the repair (or cure) effect can also be studied. A graphical display of recurrence data created by plotting the log-TBFs versus the number of failures is proposed to detect any linear or nonlinear trend for log-TBFs. The model is then extended to incorporate covariates and/or time factors. Two data sets are used to demonstrate the usefulness of the proposed model.     

A consistent estimator for logistic mixed effect models

We propose a consistent and locally efficient method of estimating the model parameters of a logistic mixed effect model with random slopes. Our approach relaxes two typical assumptions: the random effects being normally distributed, and the covariates and random effects being independent of each other. Adhering to these assumptions is particularly difficult in health studies where, in many cases, we have limited resources to design experiments and gather data in long‐term studies, while new findings from other fields might emerge, suggesting the violation of such assumptions. So it is crucial to have an estimator that is robust to such violations; then we could make better use of current data harvested using various valuable resources. Our method generalizes the framework presented in Garcia & Ma (2016) which also deals with a logistic mixed effect model but only considers a random intercept. A simulation study reveals that our proposed estimator remains consistent even when the independence and normality assumptions are violated. This contrasts favourably with the traditional maximum likelihood estimator which is likely to be inconsistent when there is dependence between the covariates and random effects. Application of this work to a study of Huntington's disease reveals that disease diagnosis can be enhanced using assessments of cognitive performance. The Canadian Journal of Statistics 47: 140–156; 2019 © 2019 Statistical Society of Canada     

Nonparametric estimation for survival data with censoring indicators missing at random

In this paper, we consider the problem of hazard rate estimation in the presence of covariates, for survival data with censoring indicators missing at random. We propose in the context usually denoted by MAR (missing at random, in opposition to MCAR, missing completely at random, which requires an additional independence assumption), nonparametric adaptive strategies based on model selection methods for estimators admitting finite dimensional developments in functional orthonormal bases. Theoretical risk bounds are provided, they prove that the estimators behave well in term of mean square integrated error (MISE). Simulation experiments illustrate the statistical procedure.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.