denote
the hazard rate of individual i, n the number of individuals and
r the number of covariates in the analysis. Consider the column vector
of
hazard rates
,i
= 1,...,n.
The additive regression is an alternative (or supplement) to the Cox model. It results in plots that are informative regarding the effect of covariates on survival.
Assume that one observes the possibly censored life times of a number of individuals. Let
denote
the hazard rate of individual i, n the number of individuals and
r the number of covariates in the analysis. Consider the column vector
of
hazard rates
,i
= 1,...,n.
The additive model is given by
where
the n
(r+1) matrix
is
constructed as follows: If the ith individual is a member of the risk
set at time t, then the ith row of
is
the vector
where
,j = 1,...,r are, possibly time-varying, covariate values. If the
ith individual is not in the risk set at time t, then the
corresponding row of
contains only zeros.
The vector
,
contains the regression information: The first element is a baseline function;
while the remaining elements, called regression functions, measure the influence of the respective covariates. These
functions are allowed to vary freely over time.
It is easiest to estimate the cumulative regression functions defined by
.
Let
be the column vector with elements
,
j = 0,...,r. This is estimated by an approach which is similar to that
for ordinary linear models, resulting in the following estimator:
Here
are the ordered event times, while
is a column vector consisting of zeros except for a one in the place corresponding to the subject who experiences an event at time
.
The estimator is only defined over the time interval where
has full rank. The matrix
is a generalized inverse of
and
will normally be defined by the ordinary least squares inverse:
The components of
are
to be plotted against time and give information about effects of covariates.
Notice that the regression functions are the derivatives of the cumulative
functions, and so it is the slopes of the plots that are informative.
For further information on the methodology, see the references below:
Aalen, O.O. (1989). A linear regression model for the analysis of life times. Statistics in Medicine 8 907-925.
Aalen, O.O. (1993). Further results on the non - parametric linear regression model in survival analysis. Statistics in Medicine 12 1569-1588.