Design and analysis of nested case–control studies for recurrent events subject to a terminal event

The process by which patients experience a series of recurrent events, such as hospitalizations, may be subject to death. In cohort studies, one strategy for analyzing such data is to fit a joint frailty model for the intensities of the recurrent event and death, which estimates covariate effects on the two event types while accounting for their dependence. When certain covariates are difficult to obtain, however, researchers may only have the resources to subsample patients on whom to collect complete data: one way is using the nested case–control (NCC) design, in which risk set sampling is performed based on a single outcome. We develop a general framework for the design of NCC studies in the presence of recurrent and terminal events and propose estimation and inference for a joint frailty model for recurrence and death using data arising from such studies. We propose a maximum weighted penalized likelihood approach using flexible spline models for the baseline intensity functions. Two standard error estimators are proposed: a sandwich estimator and a perturbation resampling procedure. We investigate operating characteristics of our estimators as well as design considerations via a simulation study and illustrate our methods using two studies: one on recurrent cardiac hospitalizations in patients with heart failure and the other on local recurrence and metastasis in patients with breast cancer.     

Regression analysis of discrete time survival data under heterogeneity

This paper concerns the regression analysis of discrete time survival data for heterogeneous populations by means of frailty models. We express the survival time for each individual as a sequence of binary variables that indicate if the individual survived at each time point. The main result is that the likelihood for these indicators can be factored into contributions that involve the conditional survival probabilities integrated over the frailty distribution of the risk set (population-averaged). We then model these population-averaged conditional probabilities as a function of covariates. The result justifies the practice of treating the failure indicators as independent Bernoulli trials and fitting binary regression models for the conditional failure probabilities at each time point. However, we must interpret the regression coefficients as population-averaged rather than subject-specific parameters. We apply the method to the Framingham Heart Study on risk factors for cardiovascular disease. © 1997 by John Wiley & Sons, Ltd.     

Modeling hazard functions in families

A genetic frailty model is presented for censored age of onset data in nuclear families where individuals carrying a genetic susceptibility gene have an increased risk of becoming affected. We use maximum likelihood via the EM algorithm to estimate the genetic relative risk and the allele frequency under a dominant susceptibility type and a proportional hazards model. When sampling is from a disease registry, likelihood corrections are necessary for reducing bias in the parameter estimates. In these biased samples, the full conditional likelihood is approximated by a likelihood conditional on the proband's age of onset. For unbiased samples, simulations show the distributions of the estimates are similar under both a semiparametric and the correctly specified parametric likelihoods. For biased samples, simulations under the approximate conditional likelihood show the median estimates of the allele frequency and genetic relative risk tend to under- and overestimate, respectively, the true values; however, the approximation is better for rarer allele frequencies (0.0033 vs. 0.01). In practice, large samples or more complex ascertainment corrections are recommended. Using the approximate conditional likelihood on familial breast cancer onset data collected as part of a case-control study at the Fred Hutchinson Cancer Research Center in Seattle, Washington, we estimate an allele frequency of 0.0009 (approximate 95% CI 0.0006–0.002) and a genetic relative risk of 104 (approximate 95% CI 55–181). Genet. Epidemiol. 15:147–171,1998. © 1998 Wiley-Liss, Inc.     

The prevalent cohort study and the acquired immunodeficiency syndrome

The acquired immunodeficiency syndrome (AIDS) is caused by a retrovirus, the human immunodeficiency virus (HIV). A rapid and convenient method to identify additional cofactors or risk modifiers and markers of disease progression is to study a cohort prevalent with HIV antibody. However, because the time of viral infection is usually unknown in the cohort, there are several potential sources of bias. Three sources of bias in a prevalent cohort study are identified assuming a proportional hazards model: onset confounding, differential length-biased sampling, and frailty selection. A number of problems in the interpretation of results on markers from a prevalent cohort also are considered. It is concluded that risk estimates derived from a prevalent cohort are not directly comparable to risk estimates derived from an incident cohort.     

Survival of hemodialysis patients: modeling differences in risk of dialysis centers.

OBJECTIVE: Dialysis is the most common renal replacement therapy for patients with end stage renal disease. This paper considers survival of dialysis patients, aiming to assess quality of renal replacement therapy at dialysis centers in Rio de Janeiro, Brazil, and to investigate differences in survival between health facilities. METHODS: A Cox proportional hazards model, allowing for time-varying covariates and prevalent data, was the basic method used to analyze the survival of 11,579 patients on hemodialysis in 67 health facilities in Rio de Janeiro State from January 1998 until August 2001, using data obtained from routine information systems. A frailty random effects model was applied to investigate differences in mortality between health centers not explained by measured characteristics. RESULTS: The individual variables associated with the outcome were age and underlying disease, with diabetes being the main isolated risk factor. Considering covariates of the health unit, two factors were associated with performance: bigger units had on average better survival times than smaller ones and units which offered cyclic peritoneal dialysis performed less well than those that did not. There were significant frailty effects among centers, with relative risks varying between 0.24 and 3.15, and an estimated variance of 0.43. CONCLUSIONS: Routine assessment based on health registries of the outcome of any high technology medical treatment is extremely important in maintaining quality of care and in estimating the impact of changes in therapies, units, and patient profiles. The frailty model allowed estimation of variation in risk between centers not attributable to any measured covariates. This can be used to guide more specific investigation and changes in health policies related to renal transplant therapies.     

Incorporation of nested frailties into semiparametric multi‐state models

Proportional hazards models are among the most popular regression models in survival analysis. Multi‐state models generalize them by jointly considering different types of events and their interrelations, whereas frailty models incorporate random effects to account for unobserved risk factors, possibly shared by clusters of subjects. The integration of multi‐state and frailty methodology is an interesting way to control for unobserved heterogeneity in the presence of complex event history structures and is particularly appealing for multicenter clinical trials. We propose the incorporation of correlated frailties in the transition‐specific hazard function, thanks to a nested hierarchy. We studied a semiparametric estimation approach based on maximum integrated partial likelihood. We show in a simulation study that the nested frailty multi‐state model improves the estimation of the effect of covariates, as well as the coverage probability of their confidence intervals. We present a case study concerning a prostate cancer multicenter clinical trial. The multi‐state nature of the model allows us to evidence the effect of treatment on death taking into account intermediate events. Copyright © 2015 JohnWiley & Sons, Ltd.     

Management of Frailty: A Systematic Review and Network Meta-analysis of Randomized Controlled Trials

ObjectiveTo analyze and determine the comparative effectiveness of interventions targeting frailty prevention or treatment on frailty as a primary outcome and quality of life, cognition, depression, and adverse events as secondary outcomes.DesignSystematic review and network meta-analysis (NMA).MethodsData sources—Relevant randomized controlled trials (RCTs) were identified by a systematic search of several electronic databases including MEDLINE, EMBASE, CINAHL, and AMED. Duplicate title and abstract and full-text screening, data extraction, and risk of bias assessment were performed. Data extraction—All RCTs examining frailty interventions aimed to decrease frailty were included. Comparators were standard care, placebo, or another intervention. Data synthesis—We performed both standard pairwise meta-analysis and Bayesian NMA. Dichotomous outcome data were pooled using the odds ratio effect size, whereas continuous outcome data were pooled using the standardized mean difference (SMD) effect size. Interventions were ranked using the surface under the cumulative ranking curve (SUCRA) for each outcome. The quality of evidence was evaluated using the GRADE approach.ResultsA total of 66 RCTs were included after screening of 7090 citations and 749 full-text articles. NMA of frailty outcome (including 21 RCTs, 5262 participants, and 8 interventions) suggested that the physical activity intervention, when compared to placebo and standard care, was associated with reductions in frailty (SMD –0.92, 95% confidence interval ?1.55, ?0.29). According to SUCRA, physical activity intervention and physical activity plus nutritional supplementation were probably the most effective intervention (100% and 71% likelihood, respectively) to reduce frailty. Physical activity was probably the most effective or the second most effective interventions for all included outcomes.Conclusion and implicationsPhysical activity is one of the most effective frailty interventions. The quality of evidence of the current review is low and very low. More robust RCTs are needed to increase the confidence of our NMA results and the quality of evidence.     

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).     

Fitting additive hazards models for case‐cohort studies: a multiple imputation approach

In this paper, we consider fitting semiparametric additive hazards models for case‐cohort studies using a multiple imputation approach. In a case‐cohort study, main exposure variables are measured only on some selected subjects, but other covariates are often available for the whole cohort. We consider this as a special case of a missing covariate by design. We propose to employ a popular incomplete data method, multiple imputation, for estimation of the regression parameters in additive hazards models. For imputation models, an imputation modeling procedure based on a rejection sampling is developed. A simple imputation modeling that can naturally be applied to a general missing‐at‐random situation is also considered and compared with the rejection sampling method via extensive simulation studies. In addition, a misspecification aspect in imputation modeling is investigated. The proposed procedures are illustrated using a cancer data example. Copyright © 2015 John Wiley & Sons, Ltd.     

Improving the efficiency of estimation in the additive hazards model for stratified case–cohort design with multiple diseases

The case–cohort study design has often been used in studies of a rare disease or for a common disease with some biospecimens needing to be preserved for future studies. A case–cohort study design consists of a random sample, called the subcohort, and all or a portion of the subjects with the disease of interest. One advantage of the case–cohort design is that the same subcohort can be used for studying multiple diseases. Stratified random sampling is often used for the subcohort. Additive hazards models are often preferred in studies where the risk difference, instead of relative risk, is of main interest. Existing methods do not use the available covariate information fully. We propose a more efficient estimator by making full use of available covariate information for the additive hazards model with data from a stratified case–cohort design with rare (the traditional situation) and non‐rare (the generalized situation) diseases. We propose an estimating equation approach with a new weight function. The proposed estimators are shown to be consistent and asymptotically normally distributed. Simulation studies show that the proposed method using all available information leads to efficiency gain and stratification of the subcohort improves efficiency when the strata are highly correlated with the covariates. Our proposed method is applied to data from the Atherosclerosis Risk in Communities study. Copyright © 2015 John Wiley & Sons, Ltd.     

BAYESIAN IMPUTATION OF PREDICTIVE VALUES WHEN COVARIATE INFORMATION IS AVAILABLE AND GOLD STANDARD DIAGNOSIS IS UNAVAILABLE

We suggest a conceptually simple Bayesian approach to inferences about the conditional probability of a specimen being infection-free given the outcome of a diagnostic test and covariate information. The approach assumes that the infection state of a specimen is not observable but uses the outcomes of a second test in conjuction with those of the first, that is, dual testing data. Dual testing procedures are often employed in clinical laboratories to assure that positive samples are not contaminated or to increase the likelihood of correct diagnoses. Using the CD4 count and a proxy for risk behaviour as covariates, we apply the method to obtain inferences about the conditional probability of an individual being HIV-1 infection-free given the individual's covariates and a negative outcome with the standard enzyme-linked immunoadsorbent assay/Western blotting test for HIV-1 detection. Inferences combine data from two studies where specimens were tested with the standard and with the more sensitive polymerase chain reaction test.     

Pseudo-likelihood methods for longitudinal binary data with non-ignorable missing responses and covariates

In this paper we consider longitudinal studies in which the outcome to be measured over time is binary, and the covariates of interest are categorical. In longitudinal studies it is common for the outcomes and any time-varying covariates to be missing due to missed study visits, resulting in non-monotone patterns of missingness. Moreover, the reasons for missed visits may be related to the specific values of the response and/or covariates that should have been obtained, i.e. missingness is non-ignorable. With non-monotone non-ignorable missing response and covariate data, a full likelihood approach is quite complicated, and maximum likelihood estimation can be computationally prohibitive when there are many occasions of follow-up. Furthermore, the full likelihood must be correctly specified to obtain consistent parameter estimates. We propose a pseudo-likelihood method for jointly estimating the covariate effects on the marginal probabilities of the outcomes and the parameters of the missing data mechanism. The pseudo-likelihood requires specification of the marginal distributions of the missingness indicator, outcome, and possibly missing covariates at each occasions, but avoids making assumptions about the joint distribution of the data at two or more occasions. Thus, the proposed method can be considered semi-parametric. The proposed method is an extension of the pseudo-likelihood approach in Troxel et al. to handle binary responses and possibly missing time-varying covariates. The method is illustrated using data from the Six Cities study, a longitudinal study of the health effects of air pollution.     

Analysis of childhood brain tumour data in New York City using frailty models.

This paper uses frailty models to analyse overall survival and progression-free survival times for children with a brain tumour. We are interested in how surgery resection affects survival times. We are also interested in how strong a child's progression-free survival correlates with his/her overall survival and if the association differs with age. Traditionally the frailty is modelled parametrically and a maximum likelihood approach is used to estimate the parameters of interest. However, the result is sensitive to the misspecification of the frailty distribution and the currently developed algorithms for the maximum likelihood approach do not allow the association parameter to depend on covariates. Xue formulates a Poisson regression model and applies an estimating equation approach to obtain a consistent estimate of the covariate effect on survival. This paper extends that approach to obtain consistent and efficient estimates of the association parameter as well as the covariate effect and to allow the association parameter to depend on the covariates. The approach does not require the specification of the frailty distribution. The performance of the method is evaluated through simulation studies. We apply this method to a childhood brain tumour data set in New York City.     

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster-specific random effect. Such models are often incorporated in more complicated joint models in survival analysis. If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time-dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows. We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time-dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real-world data analysis example.     

Evaluating bias from birth-cohort effects in the age-based cox proportional hazards model

The nested case-control design is frequently used to evaluate exposures and health outcomes within the confines of a cohort study. When incidence-density sampling is used to identify controls, the resulting data can be analyzed using conditional logistic regression (equivalent to stratified Cox proportional hazards regression). In these studies, exposure lagging is often used to account for disease latency. In light of recent criticism of incidence-density sampling, we used simulated occupational cohorts to evaluate age-based incidence-density sampling for lagged exposures in the presence of birth-cohort effects and associations among time-related variables. Effect estimates were unbiased when adjusted for birth cohort; however, unadjusted effect estimates were biased, particularly when age at hire and year of hire were correlated. When the analysis included an adjustment for birth cohort, the inclusion of lagged-out cases and controls (assigned a lagged exposure of zero) did not introduce bias.     

Frailty models for pneumonia to death with a left‐censored covariate

Frailty models are multiplicative hazard models for studying association between survival time and important clinical covariates. When some values of a clinical covariate are unobserved but known to be below a threshold called the limit of detection (LOD), naive approaches ignoring this problem, such as replacing the undetected value by the LOD or half of the LOD, often produce biased parameter estimate with larger mean squared error of the estimate. To address the LOD problem in a frailty model, we propose a flexible smooth nonparametric density estimator along with Simpson's numerical integration technique. This is an extension of an existing method in the likelihood framework for the estimation and inference of the model parameters. The proposed new method shows the estimators are asymptotically unbiased and gives smaller mean squared error of the estimates. Compared with the existing method, the proposed new method does not require distributional assumptions for the underlying covariates. Simulation studies were conducted to evaluate the performance of the new method in realistic scenarios. We illustrate the use of the proposed method with a data set from Genetic and Inflammatory Markers of Sepsis study in which interlekuin‐10 was subject to LOD. Copyright © 2015 John Wiley & Sons, Ltd.     

Assessing the goodness of fit of personal risk models

We describe a flexible family of tests for evaluating the goodness of fit (calibration) of a pre‐specified personal risk model to the outcomes observed in a longitudinal cohort. Such evaluation involves using the risk model to assign each subject an absolute risk of developing the outcome within a given time from cohort entry and comparing subjects’ assigned risks with their observed outcomes. This comparison involves several issues. For example, subjects followed only for part of the risk period have unknown outcomes. Moreover, existing tests do not reveal the reasons for poor model fit when it occurs, which can reflect misspecification of the model's hazards for the competing risks of outcome development and death. To address these issues, we extend the model‐specified hazards for outcome and death, and use score statistics to test the null hypothesis that the extensions are unnecessary. Simulated cohort data applied to risk models whose outcome and mortality hazards agreed and disagreed with those generating the data show that the tests are sensitive to poor model fit, provide insight into the reasons for poor fit, and accommodate a wide range of model misspecification. We illustrate the methods by examining the calibration of two breast cancer risk models as applied to a cohort of participants in the Breast Cancer Family Registry. The methods can be implemented using the Risk Model Assessment Program, an R package freely available at http://stanford.edu/~ggong/rmap/ . Copyright © 2014 John Wiley & Sons, Ltd.     

Multilevel model with random effects for clustered survival data with multiple failure outcomes

We present a multilevel frailty model for handling serial dependence and simultaneous heterogeneity in survival data with a multilevel structure attributed to clustering of subjects and the presence of multiple failure outcomes. One commonly observes such data, for example, in multi-institutional, randomized placebo-controlled trials in which patients suffer repeated episodes (eg, recurrent migraines) of the disease outcome being measured. The model extends the proportional hazards model by incorporating a random covariate and unobservable random institution effect to respectively account for treatment-by-institution interaction and institutional variation in the baseline risk. Moreover, a random effect term with correlation structure driven by a first-order autoregressive process is attached to the model to facilitate estimation of between patient heterogeneity and serial dependence. By means of the generalized linear mixed model methodology, the random effects distribution is assumed normal and the residual maximum likelihood and the maximum likelihood methods are extended for estimation of model parameters. Simulation studies are carried out to evaluate the performance of the residual maximum likelihood and the maximum likelihood estimators and to assess the impact of misspecifying random effects distribution on the proposed inference. We demonstrate the practical feasibility of the modeling methodology by analyzing real data from a double-blind randomized multi-institutional clinical trial, designed to examine the effect of rhDNase on the occurrence of respiratory exacerbations among patients with cystic fibrosis.     

One- and two-stage surgical revision of peri-prosthetic joint infection of the hip: a pooled individual participant data analysis of 44 cohort studies

One-stage and two-stage revision strategies are the two main options for treating established chronic peri-prosthetic joint infection (PJI) of the hip; however, there is uncertainty regarding which is the best treatment option. We aimed to compare the risk of re-infection between the two revision strategies using pooled individual participant data (IPD). Observational cohort studies with PJI of the hip treated exclusively by one- or two-stage revision and reporting re-infection outcomes were retrieved by searching MEDLINE, EMBASE, Web of Science, The Cochrane Library, and the WHO International Clinical Trials Registry Platform; as well as email contact with investigators. We analysed IPD of 1856 participants with PJI of the hip from 44 cohorts across four continents. The primary outcome was re-infection (recurrence of infection by the same organism(s) and/or re-infection with a new organism(s)). Hazard ratios (HRs) for re-infection were calculated using Cox proportional frailty hazards models. After a median follow-up of 3.7 years, 222 re-infections were recorded. Re-infection rates per 1000 person-years of follow-up were 16.8 (95% CI 13.6–20.7) and 32.3 (95% CI 27.3–38.3) for one-stage and two-stage strategies respectively. The age- and sex-adjusted HR of re-infection for two-stage revision was 1.70 (0.58–5.00) when compared with one-stage revision. The association remained consistently absent after further adjustment for potential confounders. The HRs did not vary importantly in clinically relevant subgroups. Analysis of pooled individual patient data suggest that a one-stage revision strategy may be as effective as a two-stage revision strategy in treating PJI of the hip.