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.