Abstract
We consider a parametric joint modelling of longitudinal measurements and survival times, motivated by a study conducted at the Heart Institute (Incor), São Paulo, Brazil, with the objective of evaluating the impact of B-type Natriuretic Peptide (BNP) collected at different instants on the survival of patients with Congestive Heart Failure (CHF). We employ a linear mixed model for the longitudinal response and a Birnbaum-Saunders model for the survival times, allowing the inclusion of subjects without longitudinal observations. We derive maximum likelihood estimators of the joint model parameters and conduct a simulation study to compare the true survival probabilities with dynamic predictions obtained from the fit of the proposed joint model and to evaluate the performance of the method for estimating the model parameters.The proposed joint model is applied to the cohort of 1609 patients with CHF, of which 1080 have no BNP measurements. The parameter estimates and their standard errors obtained via: i) the traditional approach, where only individuals with at least one measurement of the longitudinal response are included and ii) the proposed approach, which includes survival information from all individuals, are compared with those obtained via marginal (longitudinal and survival) models.
Highlights
In many studies, repeated measurements of one or more variables, time until the occurrence of one or more events and additional observations on explanatory variables are collected on a set of subjects in order to characterize their relationship
This is the case of a study conducted at the Heart Institute (Incor), São Paulo, Brazil, where data related to: i) longitudinal measurements of B-type Natriuretic Peptide (BNP) levels, ii) the time between admission to the study and the date of death or censoring, as well as iii) the values of basal covariates, were collected on patients with Congestive Heart Failure (CHF) to identify prognostic factors for the time to death
There are two scenarios in which it is more appropriate to perform a joint modelling: when interest is to analyze the behavior of the longitudinal response, considering a possible dependence of time to dropout, potentially informative, treated as the survival response (Hogan & Laird 1997a, Hogan & Laird 1997b, Diggle, Farewell & Henderson 2007) and when interest is to analyze the time-to-event considering the effect of the longitudinal response measurements
Summary
In many studies, repeated measurements of one or more variables (longitudinal responses), time until the occurrence of one or more events (survival responses) and additional observations on explanatory variables are collected on a set of subjects in order to characterize their relationship. Among many alternatives to describe survival times, Birnbaum-Saunders distributions seem appropriate in the context of CHF because in chronic cardiac diseases, a cumulative damage caused by several risk factors may lead to a degradation and to a consequent failure, an inherent feature of such models, as described in Galea, Leiva & Paula (2004), Leiva, Barros, Paula & Galea (2007), Barros, Paula & Leiva (2008), Balakrishnan, Leiva, Sanhueza & Vilca (2009) or Leiva, Athayde, Azevedo & Marchant (2011) With this in mind, we propose a joint modelling methodology that considers a linear mixed model to describe the longitudinal response and a Birnbaum-Saunders model to describe the survival response, allowing the inclusion of the survival data of subjects without longitudinal observations.
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