The correlated and shared gamma frailty model for bivariate current status data: An illustration for cross-sectional serological data

Retinal Optical Coherence Tomography Features Associated With Incident and Prevalent Parkinson Disease

BACKGROUND AND OBJECTIVES

Cadaveric studies have shown disease-related neurodegeneration and other morphological abnormalities in the retina of individuals with Parkinson disease (PD); however, it remains unclear whether this can be reliably detected with in vivo imaging. We investigated inner retinal anatomy, measured using optical coherence tomography (OCT), in prevalent PD and subsequently assessed the association of these markers with the development of PD using a prospective research cohort.

METHODS

This cross-sectional analysis used data from 2 studies. For the detection of retinal markers in prevalent PD, we used data from AlzEye, a retrospective cohort of 154,830 patients aged 40 years and older attending secondary care ophthalmic hospitals in London, United Kingdom, between 2008 and 2018. For the evaluation of retinal markers in incident PD, we used data from UK Biobank, a prospective population-based cohort where 67,311 volunteers aged 40-69 years were recruited between 2006 and 2010 and underwent retinal imaging. Macular retinal nerve fiber layer (mRNFL), ganglion cell-inner plexiform layer (GCIPL), and inner nuclear layer (INL) thicknesses were extracted from fovea-centered OCT. Linear mixed-effects models were fitted to examine the association between prevalent PD and retinal thicknesses. Hazard ratios for the association between time to PD diagnosis and retinal thicknesses were estimated using frailty models.

RESULTS

Within the AlzEye cohort, there were 700 individuals with prevalent PD and 105,770 controls (mean age 65.5 ± 13.5 years, 51.7% female). Individuals with prevalent PD had thinner GCIPL (-2.12 μm, 95% CI -3.17 to -1.07, p = 8.2 × 10-5) and INL (-0.99 μm, 95% CI -1.52 to -0.47, p = 2.1 × 10-4). The UK Biobank included 50,405 participants (mean age 56.1 ± 8.2 years, 54.7% female), of whom 53 developed PD at a mean of 2,653 ± 851 days. Thinner GCIPL (hazard ratio [HR] 0.62 per SD increase, 95% CI 0.46-0.84, p = 0.002) and thinner INL (HR 0.70, 95% CI 0.51-0.96, p = 0.026) were also associated with incident PD.

DISCUSSION

Individuals with PD have reduced thickness of the INL and GCIPL of the retina. Involvement of these layers several years before clinical presentation highlight a potential role for retinal imaging for at-risk stratification of PD.

A tutorial on frailty models

The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.

A general frailty model to accommodate individual heterogeneity in the acquisition of multiple infections: An application to bivariate current status data

The analysis of multivariate time-to-event (TTE) data can become complicated due to the presence of clustering, leading to dependence between multiple event times. For a long time, (conditional) frailty models and (marginal) copula models have been used to analyze clustered TTE data. In this article, we propose a general frailty model employing a copula function between the frailty terms to construct flexible (bivariate) frailty distributions with the application to current status data. The model has the advantage to impose a less restrictive correlation structure among latent frailty variables as compared to traditional frailty models. Specifically, our model uses a copula function to join the marginal distributions of the frailty vector. In this article, we considered different copula functions, and we relied on marginal gamma distributions due to their mathematical convenience. Based on a simulation study, our novel model outperformed the commonly used additive correlated gamma frailty model, especially in the case of a negative association between the frailties. At the end of the article, the new methodology is illustrated on real-life data applications entailing bivariate serological survey data.

A Gamma-frailty proportional hazards model for bivariate interval-censored data

Correlated survival data naturally arise from many clinical and epidemiological studies. For the analysis of such data, the Gamma-frailty proportional hazards (PH) model is a popular choice because the regression parameters have marginal interpretations and the statistical association between the failure times can be explicitly quantified via Kendall's tau. Despite their popularity, Gamma-frailty PH models for correlated interval-censored data have not received as much attention as analogous models for right-censored data. In this work, a Gamma-frailty PH model for bivariate interval-censored data is presented and an easy to implement expectation-maximization (EM) algorithm for model fitting is developed. The proposed model adopts a monotone spline representation for the purposes of approximating the unknown conditional cumulative baseline hazard functions, significantly reducing the number of unknown parameters while retaining modeling flexibility. The EM algorithm was derived from a data augmentation procedure involving latent Poisson random variables. Extensive numerical studies illustrate that the proposed method can provide reliable estimation and valid inference, and is moreover robust to the misspecification of the frailty distribution. To further illustrate its use, the proposed method is used to analyze data from an epidemiological study of sexually transmitted infections.

Correlated gamma frailty models for bivariate survival time data

Frailty models have been developed to quantify both heterogeneity as well as association in multivariate time-to-event data. In recent years, numerous shared and correlated frailty models have been proposed in the survival literature allowing for different association structures and frailty distributions. A bivariate correlated gamma frailty model with an additive decomposition of the frailty variables into a sum of independent gamma components was introduced before. Although this model has a very convenient closed-form representation for the bivariate survival function, the correlation among event- or subject-specific frailties is bounded above which becomes a severe limitation when the values of the two frailty variances differ substantially. In this article, we review existing correlated gamma frailty models and propose novel ones based on bivariate gamma frailty distributions. Such models are found to be useful for the analysis of bivariate survival time data regardless of the censoring type involved. The frailty methodology was applied to right-censored and left-truncated Danish twins mortality data and serological survey current status data on varicella zoster virus and parvovirus B19 infections in Belgium. From our analyses, it has been shown that fitting more flexible correlated gamma frailty models in terms of the imposed association and correlation structure outperforms existing frailty models including the one with an additive decomposition.

Measures for concordance and discordance with applications in disease control and prevention

Bivariate binary response data appear in many applications. Interest goes most often to a parameterization of the joint probabilities in terms of the marginal success probabilities in combination with a measure for association, most often being the odds ratio. Using, for example, the bivariate Dale model, these parameters can be modelled as function of covariates. But the odds ratio and other measures for association are not always measuring the (joint) characteristic of interest. Agreement, concordance, and synchrony are in general facets of the joint distribution distinct from association, and the odds ratio as in the bivariate Dale model can be replaced by such an alternative measure. Here, we focus on the so-called conditional synchrony measure. But, as indicated by several authors, such a switch of parameter might lead to a parameterization that does not always lead to a permissible joint bivariate distribution. In this contribution, we propose a new parameterization in which the marginal success probabilities are replaced by other conditional probabilities as well. The new parameters, one homogeneity parameter and two synchrony/discordance parameters, guarantee that the joint distribution is always permissible. Moreover, having a very natural interpretation, they are of interest on their own. The applicability and interpretation of the new parameterization is shown for three interesting settings: quantifying HIV serodiscordance among couples in Mozambique, concordance in the infection status of two related viruses, and the diagnostic performance of an index test in the field of major depression disorders.

Modelling time varying heterogeneity in recurrent infection processes: an application to serological data

Frailty models are often used in survival analysis to model multivariate time‐to‐event data. In infectious disease epidemiology, frailty models have been proposed to model heterogeneity in the acquisition of infection and to accommodate association in the occurrence of multiple types of infection. Although traditional frailty models rely on the assumption of lifelong immunity after recovery, refinements have been made to account for reinfections with the same pathogen. Recently, Abrams and Hens quantified the effect of misspecifying the underlying infection process on the basic and effective reproduction number in the context of bivariate current status data on parvovirus B19 and varicella zoster virus. Furthermore, Farrington, Unkel and their co‐workers introduced and applied time varying shared frailty models to paired bivariate serological data. In this paper, we consider an extension of the proposed frailty methodology by Abrams and Hens to account for age‐dependence in individual heterogeneity through the use of age‐dependent shared and correlated gamma frailty models. The methodology is illustrated by using two data applications.

Parametric overdispersed frailty models for current status data

Frailty models have a prominent place in survival analysis to model univariate and multivariate time-to-event data, often complicated by the presence of different types of censoring. In recent years, frailty modeling gained popularity in infectious disease epidemiology to quantify unobserved heterogeneity using Type I interval-censored serological data or current status data. In a multivariate setting, frailty models prove useful to assess the association between infection times related to multiple distinct infections acquired by the same individual. In addition to dependence among individual infection times, overdispersion can arise when the observed variability in the data exceeds the one implied by the model. In this article, we discuss parametric overdispersed frailty models for time-to-event data under Type I interval-censoring, building upon the work by Molenberghs et al. (2010) and Hens et al. (2009). The proposed methodology is illustrated using bivariate serological data on hepatitis A and B from Flanders, Belgium anno 1993-1994. Furthermore, the relationship between individual heterogeneity and overdispersion at a stratum-specific level is studied through simulations. Although it is important to account for overdispersion, one should be cautious when modeling both individual heterogeneity and overdispersion based on current status data as model selection is hampered by the loss of information due to censoring.

A Bivariate Mixture Model for Natural Antibody Levels to Human Papillomavirus Types 16 and 18: Baseline Estimates for Monitoring the Herd Effects of Immunization

Post-vaccine monitoring programs for human papillomavirus (HPV) have been introduced in many countries, but HPV serology is still an underutilized tool, partly owing to the weak antibody response to HPV infection. Changes in antibody levels among non-vaccinated individuals could be employed to monitor herd effects of immunization against HPV vaccine types 16 and 18, but inference requires an appropriate statistical model. The authors developed a four-component bivariate mixture model for jointly estimating vaccine-type seroprevalence from correlated antibody responses against HPV16 and -18 infections. This model takes account of the correlation between HPV16 and -18 antibody concentrations within subjects, caused e.g. by heterogeneity in exposure level and immune response. The model was fitted to HPV16 and -18 antibody concentrations as measured by a multiplex immunoassay in a large serological survey (3,875 females) carried out in the Netherlands in 2006/2007, before the introduction of mass immunization. Parameters were estimated by Bayesian analysis. We used the deviance information criterion for model selection; performance of the preferred model was assessed through simulation. Our analysis uncovered elevated antibody concentrations in doubly as compared to singly seropositive individuals, and a strong clustering of HPV16 and -18 seropositivity, particularly around the age of sexual debut. The bivariate model resulted in a more reliable classification of singly and doubly seropositive individuals than achieved by a combination of two univariate models, and suggested a higher pre-vaccine HPV16 seroprevalence than previously estimated. The bivariate mixture model provides valuable baseline estimates of vaccine-type seroprevalence and may prove useful in seroepidemiologic assessment of the herd effects of HPV vaccination.

A method for analyzing clustered interval-censored data based on Cox's model

Methods for analyzing interval-censored data are well established. Unfortunately, these methods are inappropriate for the studies with correlated data. In this paper, we focus on developing a method for analyzing clustered interval-censored data. Our method is based on Cox's proportional hazard model with piecewise-constant baseline hazard function. The correlation structure of the data can be modeled by using Clayton's copula or independence model with proper adjustment in the covariance estimation. We establish estimating equations for the regression parameters and baseline hazards (and a parameter in copula) simultaneously. Simulation results confirm that the point estimators follow a multivariate normal distribution, and our proposed variance estimations are reliable. In particular, we found that the approach with independence model worked well even when the true correlation model was derived from Clayton's copula. We applied our method to a family-based cohort study of pandemic H1N1 influenza in Taiwan during 2009-2010. Using the proposed method, we investigate the impact of vaccination and family contacts on the incidence of pH1N1 influenza.

Estimating the force of infection for HCV in injecting drug users using interval-censored data

Injecting drug users (IDUs) account for most new HCV infections. The objectives of this study were: to estimate the force of infection for hepatitis C virus in IDUs within the interval-censoring framework and to determine the impact of risk factors such as frequency of injection, drug injected, sharing of syringes and time of first injection on the time to HCV infection. We used data from the Amsterdam Cohort Study collected in The Netherlands and focused on those individuals who were HCV negative upon entry into the study. Based on the results, the force of infection was found to vary with time of first injection. The risk of infection was higher in the first 3 years of an IDU's career, implying estimates based on single cross-sectional studies could be biased. Frequency of injection and type of drug injected were found to be highly significant predictors, whereas sharing syringes was not.