# A short introduction to Aalen's additive regression model

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

*i*th individual is a member of the risk set at time

*t*, then the

*i*th row of

_{ }is the vector

_{ }where

_{ },

*j = 1,...,r*are, possibly time-varying, covariate values. If the

*i*th 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.

## References

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.