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In the period 1962-77 a total of \(205\) patients with malignant melanoma (cancer of the skin) were operated at Odense University hospital in Denmark. A number of covariates were recorded at operation, and the patients were followed up until death or censoring at the end of the study at December 31, 1977. We will study death from malignant melanoma considering death from other causes as censorings. (Source: Andersen, Borgan, Gill & Keiding, Springer, 1993.)
Straight slope => constant hazard independent of gender
Make Nelson-Aalen plots for patients with ulceration present and absent and interpret the plots. (Ulceration is “present” if the surface of the tumor viewed in a microscope show signs of ulcers and “absent” otherwise.)
In this exercise, we will use the Kaplan-Meier estimator and the log-rank test to study survival for the melanoma patients.
We will consider Kaplan-Meier estimates for the mortality from malignant melanoma treating death from other causes as censoring.
We may compute and plot the Kaplan-Meier estimate of the survival distribution for male patients by the commands (you need to load the survival-library)
fit.m=survfit(Surv(lifetime,status==1)~1,data=melanoma, subset=(sex==2), conf.type="plain")plot(fit.m, mark.time=FALSE, xlab="Years after operation", main ="Males")abline(h =0.75, col="red")abline(h =0.5, col="blue")
To obtain a summary of the results, you may give the command:
Perform these commands and interpret the Kaplan-Meier plot. Determine the lower quartile of the survival distribution for males with 95% confidence limits using the output from the summary-command. (Note that the lower quartile corresponds to 75% survival probability.)
Look at the first time the Survival probability falls below \(75\%\)
We may obtain the quartiles of the survival distribution for males by the command
quantile(fit.m)
$quantile
25 50 75
3.364384 NA NA
$lower
25 50 75
2.172603 6.542466 NA
$upper
25 50 75
5.761644 NA NA
Perform this command and compare with the result you obtained in a).
Make a Kaplan-Meier plot for females, and determine the lower quartile for females with 95% confidence limits (if possible). Compare with the results for males.
fit.f=survfit(Surv(lifetime,status==1)~1,data=melanoma, subset=(sex==1), conf.type="plain")plot(fit.f, mark.time=FALSE, xlab="Years after operation", main ="Females")
quantile(fit.f)
$quantile
25 50 75
8.334247 NA NA
$lower
25 50 75
5.29589 NA NA
$upper
25 50 75
NA NA NA
Use the log-rank test to test the null hypothesis that males and females have the same mortality from malignant melanoma:
Call:
survdiff(formula = Surv(lifetime, status == 1) ~ sex, data = melanoma)
N Observed Expected (O-E)^2/E (O-E)^2/V
sex=1 126 28 37.1 2.25 6.47
sex=2 79 29 19.9 4.21 6.47
Chisq= 6.5 on 1 degrees of freedom, p= 0.01
What may you conclude from the test?
Make Kaplan-Meier plots for patients with ulceration present and absent and interpret the results. Is it possible to estimate the lower quartile for both ulceration groups? Estimate the lower quartile with confidence limits if possible. Is there a significant difference in cancer mortality for patients with ulceration present and absent?
plot(surv.ulcer, mark.time=FALSE, lty =1:2, xlab="Years after operation", main ="Ulceration")legend("bottomleft", c("Present","Absent"), lty=1:2)
quantile(surv.ulcer)
$quantile
25 50 75
strata(ulcer)=ulcer=1 2.690411 8.334247 NA
strata(ulcer)=ulcer=2 NA NA NA
$lower
25 50 75
strata(ulcer)=ulcer=1 2.060274 4.728767 NA
strata(ulcer)=ulcer=2 7.621918 NA NA
$upper
25 50 75
strata(ulcer)=ulcer=1 4.126027 NA NA
strata(ulcer)=ulcer=2 NA NA NA
# NO STRATA IN THE FORMULA NEEDED!survdiff(formula =Surv(lifetime, status ==1) ~ ulcer, data = melanoma)
Call:
survdiff(formula = Surv(lifetime, status == 1) ~ ulcer, data = melanoma)
N Observed Expected (O-E)^2/E (O-E)^2/V
ulcer=1 90 41 21.2 18.5 29.6
ulcer=2 115 16 35.8 10.9 29.6
Chisq= 29.6 on 1 degrees of freedom, p= 5e-08
Make Kaplan-Meier plots for the three thickness groups 0-1 mm, 2-4 mm, 5+ mm and interpret the plots. Estimate the lower quartile with confidence limits if possible. Is there a significant difference in cancer mortality between the three thickness groups?
