  Predicting patient survival probabilities based on observed covariates is an
important assessment in clinical practice. These patient-specific covariates
are often measured over multiple follow-up appointments. It is then of interest
to predict survival based on the history of these longitudinal measurements,
and to update predictions as more observations become available. The standard
approaches to these so-called `dynamic prediction' assessments are joint models
and landmark analysis. Joint models involve high-dimensional parametrisations,
and their computational complexity often prohibits including multiple
longitudinal covariates. Landmark analysis is simpler, but discards a
proportion of the available data at each `landmark time'. In this work we
propose a `retarded kernel' approach to dynamic prediction that sits somewhere
in between the two standard methods in terms of complexity. By conditioning
hazard rates directly on the covariate measurements over the observation time
frame, we define a model that takes into account the full history of covariate
measurements but is more practical and parsimonious than joint modelling.
Time-dependent association kernels describe the impact of covariate changes at
earlier times on the patient's hazard rate at later times. Under the
constraints that our model (i) reduces to the standard Cox model for
time-independent covariates, and (ii) contains the instantaneous Cox model as a
special case, we derive two natural kernel parameterisations. Upon application
to three clinical data sets, we find that the predictive accuracy of the
retarded kernel approach is comparable to that of the two existing standard
methods.
