Assessing model fit in joint models of longitudinal and survival data with applications to cancer clinical trials

Abstract

Joint models for longitudinaland survival data now have along history of being used in clinical trials or other studies in which the goal is to assess a treatment effect while accounting for longitudinal assessments such as patient-reported outcomes or tumor response. Compared to using survival data alone,the joint modeling of survival and longitudinal data allows for estimation of direct and indirect treatment effects, thereby resulting in improved efficacy assessment. Although global fit indices such as AIC or BIC can be used to rank joint models, these measures do not provide separate assessments of each component of the joint model. In this paper, we develop a novel decomposition of AIC and BIC (i.e., AIC = AIC Long + AIC Surv|Long and BIC = BIC Long + BIC Surv|Long ) that allows us to assess the fit of each component of the joint model, and in particular to assess the fit of the longitudinal component of the model and the survival component separately. Based on this decomposition, we then propose ΔAIC Surv and ΔBIC Surv to determine the importance and contribution of the longitudinal data to the model fit of the survival data. Moreover, this decomposition, along with ΔAIC Surv and ΔBIC Surv , is also quite useful in comparing, for example, trajectory-based joint models and shared parameter joint models and deciding which type of model best fits the survival data. We examine a detailed case study in mesothelioma to apply our proposed methodology along with an extensive set of simulation studies.

JMFit: ASASMacro for Joint Models of Longitudinal and Survival Data

“…Following Zhang et al (2014), the joint likelihood given in (12) can be decomposed as $$L(bold\phi \mid \mathrm{bold-italicg},D_{\mathrm{obs}})=L_{\mathrm{Long}}({\phi }_1\mid \mathrm{bold-italicg},D_{\mathrm{obs}})L_{\mathrm{Surv}\mid \mathrm{Long}}({\phi }_2\mid \mathrm{bold-italicg},{\phi }_1,D_{\mathrm{obs}}),$$ where $L_{\mathrm{Long}}({\phi }_1\mid \mathrm{bold-italicg},D_{\mathrm{obs}})={\prod }_{i=1}^{n}L({\phi }_1\mid {\mathit{y}}_i,W_i)$ and $L_{\mathrm{Surv}\mid \mathrm{Long}}({\phi }_2\mid \mathrm{bold-italicg},{\phi }_1,D_{\mathrm{obs}})={\prod }_{i=1}^n\int L({\phi }_2\mid t_i,{\delta }_i,{\mathit{z}}_i,{\theta }_i,\mathrm{bold-italicg})f({\theta }_i\mid {\mathit{y}}_i,W_i,{\phi }_1)d{bold\theta }_i$. Using (13), the decomposition of the total Akaike Information Criterion (AIC) (Akaike 1973) developed in Zhang et al (2014) is given as $$\mathrm{AIC}={\mathrm{AIC}}_{\mathrm{Long}}+{\mathrm{AIC}}_{\mathrm{Surv}\mid \mathrm{Long}},$$ where $\mathrm{AIC}=-2\log L(\hat{\phi }\mid \mathrm{bold-italicg},D_{\mathrm{obs}})+2\dim \left(bold\phi \right)$, $$…”

“…Recently, Zhang et al (2014) derived a novel decomposition of the AIC and BIC criteria into additive components that will allow us to assess the goodness of fit for each component of the joint model. Such a decomposition leads to the development of ΔAIC and ΔBIC, which quantify the change of AIC and BIC in fitting the survival data with and without using the longitudinal data.…”

“…These newly developed model assessment criteria are not available in any of these packages or module mentioned before. We mention here that the methodology for ΔAIC and ΔBIC was fully developed in Zhang et al (2014), but our goal here is the novel implementation of this methodology into user-friendly software along with a class of joint models for jointly analyzing longitudinal and time-to-event data.…”

“…To fulfill this need, we develop a SAS macro, called JMFit. JMFit implements a variety of popular joint models and provides several model assessment measures including the decomposition of AIC and BIC as well as ΔAIC and ΔBIC recently developed in Zhang et al (2014). Examples with real and simulated data are provided to illustrate the use of JMFit.…”

%JM: A SAS Macro to Fit Jointly Generalized Mixed Models for Longitudinal Data and Time-to-Event Responses

“…More recently, Zhang, Chen, Ibrahim, Boye, Wang, and Shen (2014) have written several SAS macros based on the NLMIXED procedure that use the Riemann integral to compute the cumulative hazard function. This work is, though, limited to normally-distributed longitudinal responses that change over time following either a linear or a quadratic function, time-to-event responses with a piecewise constant baseline hazard, and the two most popular association structures, namely a trajectory current-value-dependent shared parameter and the use of the random effects coefficients as shared parameters.…”