What are Dynamic Predictions
In this post we will explain the concept of dynamic predictions and illustrate how these can be computed using the framework of joint models for longitudinal and survival data, and the R package JMbayes. The type of dynamic predictions we will discuss here are calculated in follow-up studies in which some sample units (e.g., patients) who are followed-up in time provide a set of longitudinal measurements. These longitudinal measurements are expected to be associated to events that the sample units may experience during follow-up (e.g., death, onset of disease, getting a child, dropout from the study, etc.). In this context, we would like to utilize the longitudinal information we have available up to particular time point t to predict the risk of an event after t. For example, for a particular patient we would like to use his available blood values up to year 5 to predict the chance that he will develop a disease before year 7 (i.e., within two years from his last available measurement). The dynamic nature of these predictions stems from the fact that each time we obtain a new a longitudinal measurement we can update the prediction we have previously calculated.
Joint models for longitudinal and survival data have been shown to be a valuable tool for obtaining such predictions. They allow to investigate which features of the longitudinal profiles are most predictive, while appropriately accounting for the complex correlations in the longitudinal measurements.
Fit a Joint Model
For this illustration we will be using the Primary Biliary Cirrhosis (PBC) data set collected by the Mayo Clinic from 1974 to 1984. For our analysis we will consider 312 patients who have been randomized to D-penicillamine and placebo. During follow-up several biomarkers associated with PBC have been collected for these patients. Here we focus on serum bilirubin levels, which is considered one of the most important ones associated with disease progression. In package JMbayes the PBC data are available in the data frames pbc2 and pbc2.id containing the longitudinal and survival information, respectively (i.e., the former is in the long format while the latter contains a single row per patient).
We start by fitting a joint model to the PBC data set. For the log-transformed serum bilirubin we use a linear mixed effects models with natural cubic splines in the fixed and random effects for time, and also correct in the fixed part for age and sex. For the time-to-death we use a Cox model with baseline covariates age, sex and their interaction and the underlying level of serum bilirubin as estimated from the mixed model. This joint model is fitted using the following piece of code:
Calculate Dynamic Predictions
The last available measurement of this patient was in year 8.83, and survfitJM() will by default produce estimates of event-free probabilities starting from this last time point to the end of the follow-up. The calculation of these probabilities is based on a Monte Carlo procedure, and in the output we obtain as estimates the mean and median over the Monte Carlo samples along with the 95% pointwise credible intervals. Hence, the probability that this patient will survive up to year 11.2 is 60%. A plot of these probabilities can be obtained using the plot() method for objects returned by survfitJM()