The additive hazards model (Aalen's additive model): how to stratify?

[Martina Andersson]

Dear Statalist members,

I am trying to conduct a stratified additive hazards model (Aalen’s additive model).
I am using Stata 12.1.

The stata syntax for the Aalen’s additive hazards model is stlh. I have pooled a number of cohorts and would like to calculate the coefficients for risk factors for the outcome and let the intercept vary by sex and cohort.

However, nowhere in previous literature can I find information on how to stratify this model, ie, running the model with a function corresponding to the ”strata” command in the stcox.

I would be very thankful for any type of input on this.

Best,
Martina

[Steve Samuels]

Martina,

You are asked in the FAQ to state where you got a contributed program. stlh, by David Hosmer and Patrick Royston, can be located by findit and installed by clicking on the "st0024" link. Be sure to read, if you haven't already done so, the accompanying Stata Journal article that can be downloaded from http://ageconsearch.umn.edu/bitstrea...art_st0024.pdf.

The model which would be most analogous to a stratified Cox models is one in which there is a non-parametric baseline hazard function in each cohort-gender category and a linear model with constant coefficients for the risk factors.

In a Cox model with stratification, there is a different baseline hazard function in each stratum. Call it \(h_i(t)\). With covariates, the model for the log hazard in stratum i can be written:

\[
\text{log}(h_i(t, x)) = \text{log}(h_i(t)) + \beta_1 x_1 + \beta_2 x_2 + \dots +\beta_p x_p
\]

For comparison, the general form of Aalen's linear hazard model is:

$$
h(t, x) = \beta_0(t) + \beta_1(t)x_1 + \beta_2(t)x_2 + \dots + \beta_p(t)x_p
$$

Note that every coefficient is a function of t. This makes it possible to easily study possible non-proportionality of hazard ratios.

Define \(Z_i(t)\) to be a 0-1 indicator variable for stratum i. Then the model with a separate baseline hazard in each stratum can be written:

$$
h(t, x) = \beta_0(t) + \sum_{i=2}^I \alpha_i(t) Z_i + \beta_1(t)x_1 + \beta_2(t)x_2 + \dots + \beta_p(t)x_p
$$

The baseline hazard for stratum \(i\) is

\[
\begin{align}
h_i(t) & = & \beta_0(t) , i= 1 \\
& = & \beta_0(t) + \alpha_i(t), i>1
\end{align}
\]

Here is suggested Stata code to fit this model; add the options you require. I use the old "xi" prefix because stph was written before Stata introduced factor variables.

egen stratum = group(sex cohort)
xi: stlh i.stratum x1 x2 x3

or

xi: stlh i.cohort*i.sex x1 x2 x3

stlh was written 12 years ago and has some limitations; for instance you cannot constrain some coefficients to be constant; nor can you model data in which hazard ratios are proportional over part of a time range and non-proportional elsewhere. To fit such models you need the timereg package in R (Martinussen and Scheike, 2002, 2006; Scheike, 2014).

References:

Hosmer, David, and Patrick Royston. 2002. Using Aalen’s linear hazards model to investigate time-varying effects in the proportional hazards regression model. The Stata Journal 2, 331-350

Martinussen, T., & Scheike, T. H. (2002). A flexible additive multiplicative hazard model. Biometrika, 89(2), 283-298.

Martinussen T, Scheike T. (2006) Dynamic regression models for survival data. New York: Springer.

T Scheike (2014) http://cran.r-project.org/web/packag...eg/timereg.pdf