A Joint Model for Survival and Longitudinal Data Using Shape Invariant Models
Open AccessIt is common to collect information on both survival outcomes and longitudinal measurements in medical studies. The traditional approach treated longitudinal measures as time dependent covariates in survival models. There are some drawbacks of this approach. In practice, the longitudinal measurements are not always available at the same time when the event of interest was ascertained. One can use missing data imputing methods, such as Last Observation Carried Forward (LOCF), to overcome the issue. However, LOCF may introduce bias to the analysis. In addition, using the observed longitudinal data is subject to measurement error which can cause the estimated risk parameters in the survival model to be biased toward the null. In this dissertation, we develop a two-stage approach to model the two disease processes together using a shape invariant model to represent the longitudinal measures. In the first stage, we model the longitudinal data. The population are assumed to have a common trajectory and individual trajectories are obtained by shifting and scaling the common curve. We use splines to describe the population common trajectory to allow flexibility and apply three spline methods: truncated polynomial spline, B-spline and natural cubic spline. Individual variations are described by two subject-specific terms: ‘intercept’ and ‘speed’ which have straightforward interpretations. The term ‘intercept’ represents a vertical shift up or down in the y axis and ‘speed’ represents the shrink or stretch according to time on the x axis. Covariates of interest can be included in the two subject-specific terms in order to estimate their effects on the longitudinal outcome. The shape invariant model can be fitted using nonlinear mixed effects model approaches. We treat the coefficients of the spline basis functions as fixed effects and the two subject-specific terms as random effects. In the second stage, we model the survival data using either the Cox proportional hazards model or the accelerated failure model which include the estimated subject specific terms obtained from the first stage as time-independent covariates. The association between the longitudinal and the survival data is characterized by the coefficients of the two subject-specific terms. This two-stage approach can be easily implemented with existing statistical software for nonlinear mixed effects models and survival models. Comparing with other joint model approaches using tradition linear mixed effects model, the shape invariant model is more flexible when fitting curves with complex shapes. Also, by summarizing the individual curves into two subject-specific terms which are easily interpretable, this approach promotes the communication with non-statistical audiences.Extensive simulation studies are conducted to explore the performance of the model under various scenarios. The parameters can be estimated accurately as long as the sample size is not too small and the measurement error is not too large. The estimates of the subject-specific speed are more sensitive to large measurement errors and small sample sizes than the estimates of the subject-specific intercept. The three spline methods are shown to have similar performance. We apply this method to explore the longitudinal trajectory of Hemoglobin A1c (HbA1c) post diabetes and its association with nephropathy using the Diabetes Prevention Program Outcomes Study (DPPOS) data. In the longitudinal submodel, the fitted population mean curve of HbA1c fluctuates in the first 5 years and becomes strictly increasing afterwards. The progress is accelerated by larger values of baseline body mass index (BMI) while taking lifestyle intervention, metformin and other glucose lower medications decelerates the progression. In the survival submodel, we did not find statistically significant association between the two subject-specific terms and development of nephropathy. To compare, we applied Cox models using the observed HbA1c as a time-dependent variable which produced very similar results with slightly lower concordance statistics.
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