Exercise 2 (the additive regression model of Aalen)

 

At the university hospital of the University of Massachusetts one has for a number of years studied the survival of patients admitted with acute myocardial infarction (AMI). One aim of the study has been to investigate whether the survival of AMI patients has improved over time. A number of covariates were measured at hospitalization. In addition to information on time of hospitalization (here given in five years periods), we will in this exercise restrict attention to the two demographic covariates age and sex and one covariate that is related to the seriousness of the AMI (amount of "heart enzyme"). Further we will in our analysis only consider patients with no previous AMI, and we will restrict our attention to the first 100 days after AMI (even though the data set contains follow-up information for 3 years). (Source: Hosmer & Lemeshow: Applied Survival Analysis: Regression Modeling of Time to Event Data, Wiley, 1998.)

You may read the AMI-data into R by the command:

ami=read.table("https://www.mn.uio.no/math/english/people/aca/borgan/iwsm2026/ami.txt",header=T)

(You may copy this command and the other commands below and paste them into the source editor of RStudio, possibly for further editing.)

The data are organized with one line for each of the 481 patients, and with the following variables in the seven columns:

Id: Patient number

days: Number of days from hospitalization to death or censoring

status: Indicator for death (1) or censoring (0)

per: Five year period (1 = 1975-79, 2 = 1980-84, 3 = 1985-89)

age: Age at hospitalization (in years)

sex: Sex (0 = male, 1 = female)

enzym: Amount of "heart enzyme" measured as "international units" divided by 100

prev: Information on earlier AMIs (0 = no earlier AMI, 1 = at least one earlier AMI).

 

We will study the effect of the four covariates per, age, sex, and enzym using the additive regression model. (In this exercise we will use the covariate prev only to select the patients with no previous AMI.) Our main interest lies in a simultaneous analysis of the effect of all four covariates. But we will start out by fitting additive regression models with one covariate at a time.

a) First we fit an additive model with sex as the only covariate and plot the estimated cumulative baseline hazard and the cumulative regression function for the covariate. This is achieved by the following commands. (The survival library has to be loaded.)

fit.sex=aareg(Surv(days,status)~factor(sex), data=ami, subset=(prev==0))

par(mfrow=c(1,2))

plot(fit.sex, maxtime=100)

Perform the commands and interpret the plot for the cumulative baseline hazard and the plot for the cumulative regression function for sex. In particular, what do the plots tell you about the variation over time of the baseline hazard and the regression function for sex?

b) Fit an additive regression model for the covariates per and interpret the plot for the cumulative baseline hazard and the plots for the cumulative regression functions for the periods 1980-84 and 1985-89.

c) We will then consider the numeric covariates age. In order to ease the interpretation of the cumulative baseline hazard, we will "center" age and consider the covariate agec, which is given as age minus 65 years. (65 years is close to the median age for patients with no previous AMI.) We may obtain "centered age" and fit the additive model with this covariate using the commands:

ami$agec=ami$age-65

fit.agec=aareg(Surv(days,status)~agec, data=ami, subset=(prev==0))

par(mfrow=c(1,2))

plot(fit.agec, maxtime=100)

Perform the commands and interpret the plot for the cumulative baseline hazard and the plot for the cumulative regression function for (centered) age.

d) The covariate enzym has a very skewed distribution, as can be seen by the histogram obtained by the command hist(ami$enzym,20). We will therefore define a log-transformed and "centered" covariate for enzym by the command

ami$enzymc=log2(ami$enzym/500)

(500 is somewhat below the median value for patients with no previous AMI.)

Fit an additive regression model with the "centered" covariate enzymc and interpret the plot for the cumulative baseline hazard and the plot for the cumulative regression function for (centered) enzym.

e) Fit an additive regression model with all four covariates and interpret the plots for the cumulative baseline hazard and the plots of the cumulative regression functions.

f) When analyzing censored survival data, Cox regression is commonly used. Discuss benefits and drawbacks of using the additive model (rather than the Cox model) for analyzing the AMI data.