Association models for periodontal disease progression: A comparison of methods for clustered binary data

We investigate population-averaged (PA) and cluster-specific (CS) associations for clustered binary logistic regression in the context of a longitudinal clinical trial that investigated the association between tooth-specific visual elastase kit results and periodontal disease progression within 26 weeks of follow-up. We address estimation of population-averaged logistic regression models with generalized estimating equations (GEE), and conditional likelihood (CL) and mixed effects (ME) estimation of CS logistic regression models. Of particular interest is the impact of clusters that do not provide information for conditional likelihood methods (non-informative clusters) on inferences based upon the various methodologies. The empirical and analytical results indicate that CL methods yield smaller test statistics than ME methods when non-informative clusters exist, and that CL estimates are less efficient than ME estimates under certain conditions. Moreover, previously reported relationships between population-averaged and cluster-specific parameters appear to hold for the corresponding estimates in the presence of these clusters.     

POPULATION-AVERAGED AND CLUSTER-SPECIFIC MODELS FOR CLUSTERED ORDINAL RESPONSE DATA

We compare population-averaged and cluster-specific models for clustered ordinal data. We consider generalized estimating equations and constrained equations maximum likelihood estimation of population-averaged cumulative logit regression models, and mixed effects estimation of cluster-specific cumulative logit regression models. A previously reported relationship between population-averaged and cluster-specific parameters for the binary logistic link appears to hold for analogous parameters under the cumulative logit link. We address these issues in the context of data from two cross-over clinical trials.     

Repeated events survival models: the conditional frailty model

Repeated events processes are ubiquitous across a great range of important health, medical, and public policy applications, but models for these processes have serious limitations. Alternative estimators often produce different inferences concerning treatment effects due to bias and inefficiency. We recommend a robust strategy for the estimation of effects in medical treatments, social conditions, individual behaviours, and public policy programs in repeated events survival models under three common conditions: heterogeneity across individuals, dependence across the number of events, and both heterogeneity and event dependence. We compare several models for analysing recurrent event data that exhibit both heterogeneity and event dependence. The conditional frailty model best accounts for the various conditions of heterogeneity and event dependence by using a frailty term, stratification, and gap time formulation of the risk set. We examine the performance of recurrent event models that are commonly used in applied work using Monte Carlo simulations, and apply the findings to data on chronic granulomatous disease and cystic fibrosis.     

Bayesian transformation cure frailty models with multivariate failure time data

We propose a class of transformation cure frailty models to accommodate a survival fraction in multivariate failure time data. Established through a general power transformation, this family of cure frailty models includes the proportional hazards and the proportional odds modeling structures as two special cases. Within the Bayesian paradigm, we obtain the joint posterior distribution and the corresponding full conditional distributions of the model parameters for the implementation of Gibbs sampling. Model selection is based on the conditional predictive ordinate statistic and deviance information criterion. As an illustration, we apply the proposed method to a real data set from dentistry.     

Effects and non-effects of paired identical observations in comparing proportions with binary matched-pairs data

Binary matched-pairs data occur commonly in longitudinal studies, such as in cross-over experiments. Many analyses for comparing the matched probabilities of a particular outcome do not utilize pairs having the same outcome for each observation. An example is McNemar's test. Some methodologists find this to be counterintuitive. We review this issue in the context of subject-specific and population-averaged models for binary data, with various link functions. For standard models and inferential methods, pairs with identical outcomes may affect the estimated size of the effect and its standard error, but they have negligible, if any, effect on significance. We also discuss extension of this result to matched sets.     

Maximum likelihood regression methods for paired binary data

We discuss maximum likelihood methods for analysing binary responses measured at two times, such as in a cross-over design. We construct a 2 x 2 table for each individual with cell probabilities corresponding to the cross-classification of the responses at the two times; the underlying likelihood for each individual is multinomial with four cells. The three dimensional parameter space of the multinomial distribution is completely specified by the two marginal probabilities of success of the 2 x 2 table and an association parameter between the binary responses at the two times. We examine a logistic model for the marginal probabilities of the 2 x 2 table for individual i; the association parameters we consider are either the correlation coefficient, the odds ratio or the relative risk. Simulations show that the parameter estimates for the logistic regression model for the marginal probabilities are not very sensitive to the parameters used to describe the association between the binary responses at the two times. Thus, we suggest choosing the measure of association for ease of interpretation.     

A comparison of frailty models for multivariate survival data

This paper reviews some of the main approaches to the analysis of multivariate censored survival data. Such data typically have correlated failure times. The correlation can be a consequence of the observational design, for example with clustered sampling and matching, or it can be a focus of interest as in genetic studies, longitudinal studies of recurrent events and other studies involving multiple measurements. We assume that the correlation between the failure or survival times can be accounted for by fixed or random frailty effects. We then compare the performance of conditional and mixture likelihood approaches to estimating models with these frailty effects in censored bivariate survival data. We find that the mixture methods are surprisingly robust to misspecification of the frailty distribution. The paper also contains an illustrative example on the times to onset of chest pain brought on by three endurance exercise tests during a drug treatment trial of heart patients.     

Coping with time and space in modelling malaria incidence: a comparison of survival and count regression models

To study the effect of a mega hydropower dam in southwest Ethiopia on malaria incidence, we have set up a longitudinal study. To gain insight in temporal and spatial aspects, that is, in time (period = year–season combination) and location (village), we need models that account for these effects. The frailty model with periodwise constant baseline hazard (a constant value for each period) and a frailty term that models the clustering in villages provides an appropriate tool for the analysis of such incidence data. Count data can be obtained by aggregating for each period events at the village level. The mixed Poisson regression model can be used to model the count data. We show the similarities between the two models. The risk factor in both models is the distance to the dam, and we study the effect of the risk factor on malaria incidence. In the frailty model, each subject has its own risk factor, whereas in the Poisson regression model, we also need to average the risk factors of all subjects contributing to a particular count. The power loss caused by using village averaged distance instead of individual distance is studied and quantified. The loss in the malaria data example is rather small. In such a setting, it might be advantageous to use less labor‐intensive sampling schemes than the weekly individual follow‐up scheme used in this study; the proposed alternative sampling schemes might also avoid community fatigue, a typical problem in such research projects. Copyright © 2013 John Wiley & Sons, Ltd.     

Statistical analysis of failure times in total joint replacement

Time to revision is an important criterion describing the quality of implants in total joint surgery. Estimates of failure probabilities are required to inform a patient about the risk of suffering a reoperation. Also, regression models are used for comparing different prosthesis designs. Typically, patients dying before a revision are considered as censored for time to prosthesis failure. We argue that this technique is inadequate for estimation of failure probabilities and insufficient for comparison of different designs. We propose a new approach based on a competing risk model to account for concurrent mortality. We describe differences in the estimation of failure probabilities and in the interpretation of regression models for implant failure. Additionally, we introduce a random effects term in the regression model to account for potential dependencies in the failure times of bilaterally treated patients. The new approach is illustrated with fictitious data and data from an observational study conducted at a specialized hospital in Switzerland.     

Assessing heterogeneity and correlation of paired failure times with the bivariate frailty model.

We consider bivariate survival times for heterogeneous populations, where heterogeneity induces deviations in an individual's risk of an event as well as associations between survival times. The heterogeneity is characterized by a bivariate frailty model. We measure the heterogeneity effects through deviations associated with hazard functions and an association function defined through the conditional hazard functions: the cross-ratio function proposed by Oakes. We show how the deviation and association measures are determined by the frailty distribution. A Gibbs sampling method is developed for Bayesian inferences on regression coefficients, frailty parameters and the heterogeneity measures. The method is applied to a mental health care data set.     

Methodology for analyzing censored correlated data: application of marginal and frailty approaches in human genetics. The European Community Alport Syndrome Concerted Action Group (ECASCA)

BACKGROUND: Statistical analysis for correlated censored data allows to study censored events in clustered structure designs. Considering a possible correlation among failure times of the same group, standard methodology is no longer applicable. We investigated proposed models in this context to study familial data about a genetic disease, Alport syndrome. Alport syndrome is a severe hereditary disease due to abnormal collagenous chains. Renal failure is the main symptom of the disease. It progresses toward end-stage renal failure (IRT) according to a high time variability. As shown by genetic studies, mutations of COL4A5 gene are involved in the X-linked Alport Syndrome. Due to the large range of the mutation types, the aim of this study was to search for a possible genetic origin of the heterogeneity of the disease severity. METHODS: Marginal survival models and mixed effects survival models (so-called frailty models) were proposed to take into account the possible non independence of the observations. In this study, time until end-stage renal failure is a rightly censored end point. Possible intra-familial correlations due to shared environmental and/or genetic factors could induce dependence among familial failure times. In this paper, we fit marginal and frailty proportional hazards models to evaluate the effect of mutation type on the risk of IRT and an interfamilial heterogeneity of failure times. RESULTS: In this study, the use of these models allows to show the presence of an interfamilial heterogeneity of the failure times to IRT. Moreover, the results suggest that some mutation types are linked to a higher risk of fast evolution to IRT, which explains partially the interfamilial heterogeneity of the failure times. CONCLUSIONS: This paper shows the interest of marginal and frailty models to evaluate the heterogeneity of censored responses and to study relationships between a censored criterion and covariables. This study puts forward the importance of characterizing the mutation at a molecular level to understand the relationship between genotype and phenotype.     

Strategies for imputing missing covariates in accelerated failure time models

Missing covariates often occur in biomedical studies with survival outcomes. Multiple imputation via chained equations (MICE) is a semi‐parametric and flexible approach that imputes multivariate data by a series of conditional models, one for each incomplete variable. When applying MICE, practitioners tend to specify the conditional models in a simple fashion largely dictated by the software, which could lead to suboptimal results. Practical guidelines for specifying appropriate conditional models in MICE are lacking. Motivated by a study of time to hip fractures in the Women's Health Initiative Observational Study using accelerated failure time models, we propose and experiment with some rationales leading to appropriate MICE specifications. This strategy starts with specifying a joint model for the variables involved. We first derive the conditional distribution of each variable under the joint model, then approximate these conditional distributions to the extent which can be characterized by commonly used regression models. We propose to fit separate models to impute incomplete variables by the failure status, which is key to generating appropriate MICE specifications for survival outcomes. The proposed strategy can be conveniently implemented with all available imputation software that uses fully conditional specifications. Our simulation results show that some commonly used simple MICE specifications can produce suboptimal results, while those based on the proposed strategy appear to perform well and be robust toward model misspecifications. Hence, we warn against a mechanical use of MICE and suggest careful modeling of the conditional distributions of variables to ensure proper performance.     

Parameter estimation in Cox models with missing failure indicators and the OPPERA study

In a prospective cohort study, examining all participants for incidence of the condition of interest may be prohibitively expensive. For example, the “gold standard” for diagnosing temporomandibular disorder (TMD) is a physical examination by a trained clinician. In large studies, examining all participants in this manner is infeasible. Instead, it is common to use questionnaires to screen for incidence of TMD and perform the “gold standard” examination only on participants who screen positively. Unfortunately, some participants may leave the study before receiving the “gold standard” examination. Within the framework of survival analysis, this results in missing failure indicators. Motivated by the Orofacial Pain: Prospective Evaluation and Risk Assessment (OPPERA) study, a large cohort study of TMD, we propose a method for parameter estimation in survival models with missing failure indicators. We estimate the probability of being an incident case for those lacking a “gold standard” examination using logistic regression. These estimated probabilities are used to generate multiple imputations of case status for each missing examination that are combined with observed data in appropriate regression models. The variance introduced by the procedure is estimated using multiple imputation. The method can be used to estimate both regression coefficients in Cox proportional hazard models as well as incidence rates using Poisson regression. We simulate data with missing failure indicators and show that our method performs as well as or better than competing methods. Finally, we apply the proposed method to data from the OPPERA study. Copyright © 2015 John Wiley & Sons, Ltd.     

Obtaining marginal estimates from conditional categorical repeated measurements models with missing data

The most commonly used models for categorical repeated measurement data are log-linear models. Not only are they easy to fit with standard software but they include such useful models as Markov chains and graphical models. However, these are conditional models and one often also requires the marginal probabilities of responses, for example, at each time point in a longitudinal study. Here a simple method of matrix manipulation is used to derive the maximum likelihood estimates of the marginal probabilities from any such conditional categorical repeated measures model. The technique is applied to the classical Muscatine data set, taking into account the dependence of missingness on previous observed values, as well as serial dependence and a random effect.     

MULTIVARIABLE PROGNOSTIC MODELS: ISSUES IN DEVELOPING MODELS,EVALUATING ASSUMPTIONS AND ADEQUACY,AND MEASURING AND REDUCING ERRORS

Multivariable regression models are powerful tools that are used frequently in studies of clinical outcomes. These models can use a mixture of categorical and continuous variables and can handle partially observed (censored) responses. However, uncritical application of modelling techniques can result in models that poorly fit the dataset at hand, or, even more likely, inaccurately predict outcomes on new subjects. One must know how to measure qualities of a model's fit in order to avoid poorly fitted or overfitted models. Measurement of predictive accuracy can be difficult for survival time data in the presence of censoring. We discuss an easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities. Both types of predictive accuracy should be unbiasedly validated using bootstrapping or cross-validation, before using predictions in a new data series. We discuss some of the hazards of poorly fitted and overfitted regression models and present one modelling strategy that avoids many of the problems discussed. The methods described are applicable to all regression models, but are particularly needed for binary, ordinal, and time-to-event outcomes. Methods are illustrated with a survival analysis in prostate cancer using Cox regression.     

Bootstrap analysis of multivariate failure time data

Multivariate failure time data often arise in research. Cox proportional hazards modelling is a widely used method of analysing failure time data for independent observations. However, when failure times are correlated the Cox proportional hazards model does not yield valid estimates of standard errors or significance tests. Many methods for the analysis of multivariate failure time data have been proposed. These methods commonly test hypotheses about the regression parameters, a practice which averages the treatment effect across time. The purpose of this paper is to examine the bootstrap method for obtaining standard errors in the multivariate failure time case, particularly when the focus is the survival probability or the treatment effect at a single time point such as in a surgical trial. Our motivating example comes from the Asymptomatic Carotid and Atherosclerosis Study (ACAS) in which the outcome of stroke or perioperative complications could be observed for either or both carotid arteries within each patient. Extensive simulation studies were conducted to examine the bootstrap procedure for analysing correlated failure time data under a variety of conditions including a range of treatment effects, cluster sizes, intercluster correlation values and for both proportional and non-proportional data. We found that the bootstrap method was able to estimate the standard error adequately for survival probabilities at a specific time and the standard error for the survival difference and the relative risk at a specific time. We illustrated the bootstrap method for calculating the standard error for the survival probability and statistical testing at a specific time value by analysing the two arteries per patient from the ACAS study.     

Estimating and testing for center effects in competing risks

The problems of fitting Gaussian frailties proportional hazards models for the subdistribution of a competing risk and of testing for center effects are considered. In the analysis of competing risks data, Fine and Gray proposed a proportional hazards model for the subdistribution to directly assess the effects of covariates on the marginal failure probabilities of a given failure cause. Katsahianbiet al. extended their model to clustered time to event data, by including random center effects or frailties in the subdistribution hazard. We first introduce an alternate estimation procedure to the one proposed by Katsahian et al. This alternate estimation method is based on the penalized partial likelihood approach often used in fitting Gaussian frailty proportional hazards models in the standard survival analysis context, and has the advantage of using standard survival analysis software. Second, four hypothesis tests for the presence of center effects are given and compared via Monte-Carlo simulations. Statistical and numerical considerations lead us to formulate pragmatic guidelines as to which of the four tests is preferable. We also illustrate the proposed methodology with registry data from bone marrow transplantation for acute myeloid leukemia (AML).     

THE ROLE OF FRAILTY MODELS AND ACCELERATED FAILURE TIME MODELS IN DESCRIBING HETEROGENEITY DUE TO OMITTED COVARIATES

In survival analysis, deviations from proportional hazards may sometimes be explained by unaccounted random heterogeneity, or frailty. This paper recalls the literature on omitted covariates in survival analysis and shows in a case study how unstably frailty models might behave when asked to account for unobserved heterogeneity in standard survival analysis with no replications per heterogeneity unit. Accelerated failure time modelling seems to avoid these difficulties and also to yield easily interpretable results. We propose that it would be advantageous to upgrade the accelerated failure time approach alongside the hazard modelling approach to survival analysis. © 1997 by John Wiley & Sons, Ltd.     

Proportional hazards models with frailties and random effects

We discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival. One of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure, their so-called frailty, additional to any effects we wish to quantify via regression. Although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model, we argue that the concept is of limited practical value. Individual random effects (frailties), whenever detected, can be made to disappear by elementary model transformation. In consequence, unless we are to take some model form as unassailable, beyond challenge and carved in stone, and if we are to understand the term 'frailty' as referring to individual random effects, then frailty models have no value. Random effects models on the other hand, in which groups of individuals share some common effect, can be used to advantage. Even in this case however, if we are prepared to sacrifice some efficiency, we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model. Stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models, a view that gives further insight. The added structure of a random effects model, viewed as a stratified proportional hazards model with some added distributional constraints, will, for group sizes of five or more, provide no more than modest efficiency gains, even when the additional assumptions are exactly true. On the other hand, for moderate to large numbers of very small groups, of sizes two or three, the study of twins being a well known example, the efficiency gains of the random effects model can be far from negligible. For such applications, the case for using random effects models rather than the stratified model is strong. This is especially so in view of the good robustness properties of random effects models. Nonetheless, the simpler analysis, based upon the stratified model, remains valid, albeit making a less efficient use of resources.     

A maximum likelihood method for secondary analysis of nested case‐control data

Many epidemiological studies use a nested case‐control (NCC) design to reduce cost while maintaining study power. Because NCC sampling is conditional on the primary outcome, routine application of logistic regression to analyze a secondary outcome will generally be biased. Recently, many studies have proposed several methods to obtain unbiased estimates of risk for a secondary outcome from NCC data. Two common features of all current methods requires that the times of onset of the secondary outcome are known for cohort members not selected into the NCC study and the hazards of the two outcomes are conditionally independent given the available covariates. This last assumption will not be plausible when the individual frailty of study subjects is not captured by the measured covariates. We provide a maximum‐likelihood method that explicitly models the individual frailties and also avoids the need to have access to the full cohort data. We derive the likelihood contribution by respecting the original sampling procedure with respect to the primary outcome. We use proportional hazard models for the individual hazards, and Clayton's copula is used to model additional dependence between primary and secondary outcomes beyond that explained by the measured risk factors. We show that the proposed method is more efficient than weighted likelihood and is unbiased in the presence of shared frailty for the primary and secondary outcome. We illustrate the method with an application to a study of risk factors for diabetes in a Swedish cohort. Copyright © 2014 John Wiley & Sons, Ltd.