Flexible parametric joint modelling of longitudinal and survival data

Stat Med. 2012 Dec 30;31(30):4456-71. doi: 10.1002/sim.5644. Epub 2012 Oct 4.


The joint modelling of longitudinal and survival data is a highly active area of biostatistical research. The submodel for the longitudinal biomarker usually takes the form of a linear mixed effects model. We describe a flexible parametric approach for the survival submodel that models the log baseline cumulative hazard using restricted cubic splines. This approach overcomes limitations of standard parametric choices for the survival submodel, which can lack the flexibility to effectively capture the shape of the underlying hazard function. Numerical integration techniques, such as Gauss-Hermite quadrature, are usually required to evaluate both the cumulative hazard and the overall joint likelihood; however, by using a flexible parametric model, the cumulative hazard has an analytically tractable form, providing considerable computational benefits. We conduct an extensive simulation study to assess the proposed model, comparing it with a B-spline formulation, illustrating insensitivity of parameter estimates to the baseline cumulative hazard function specification. Furthermore, we compare non-adaptive and fully adaptive quadrature, showing the superiority of adaptive quadrature in evaluating the joint likelihood. We also describe a useful technique to simulate survival times from complex baseline hazard functions and illustrate the methods using an example data set investigating the association between longitudinal prothrombin index and survival of patients with liver cirrhosis, showing greater flexibility and improved stability with fewer parameters under the proposed model compared with the B-spline approach. We provide user-friendly Stata software.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Anti-Inflammatory Agents / therapeutic use
  • Biomarkers
  • Biostatistics / methods
  • Computer Simulation
  • Humans
  • Kaplan-Meier Estimate
  • Likelihood Functions
  • Linear Models
  • Liver Cirrhosis / drug therapy*
  • Longitudinal Studies / statistics & numerical data*
  • Prednisone / therapeutic use*
  • Prognosis
  • Proportional Hazards Models*
  • Randomized Controlled Trials as Topic
  • Time Factors


  • Anti-Inflammatory Agents
  • Biomarkers
  • Prednisone