Proportional Rates/Means Models for Recurrent Events
Let
be the number of events experienced by a subject over the time interval
. Let
be the increment of the counting process
over
. The rate function is given by
where
is an unknown continuous function. If the
are time independent, the rate model is reduced to the mean model
The partial likelihood for
independent triplets
, of counting, at-risk, and covariate processes is the same as that of the multiplicative hazards model. However, a robust sandwich estimate is used for the covariance matrix of the parameter estimator instead of the model-based estimate.
Let
be the
th event time of the
th subject. Let
be the censoring time of the
th subject. The at-risk indicator and the failure indicator are, respectively,
Denote
Let
be the maximum likelihood estimate of
, and let
be the observed information matrix. The robust sandwich covariance matrix estimate is given by
where
For a given realization of the covariates
, the Nelson estimator is used to predict the mean function
with standard error estimate given by
where
Since the cumulative mean function is always nonnegative, the log transform is used to compute confidence intervals. The
% pointwise confidence limits for
are
where
is the upper
percentage point of the standard normal distribution.