Solutions to the practice problems. - UCLA Biostatistics

(e) The printout below shows the log-rank test comparing these two groups. Explain what hypotheses are
being tested and give your real-world conclusions.
Solution: The log rank test tests whether the hazards rates for the two groups are the same over a given
interval (generally up to the time of the last observed event which for us occurred at 28.65 years. The
hypotheses are therefore:
H 0 : h 0 (t) = h 1 (t) for all t ≤ 28.65–the two groups have the same hazard function (or equivalently survival
function, distribution of survival times, etc.) over the period covered by our study.
H A : For at least some t, h 0 (t) ≠ h 1 (t)–the two hazard functions, survival curves, etc. are not identical.
From the printout we see that the p-value for the test is .0189 which is less than our usual significance level
of α = .05 so we reject the null hypothesis and conclude that there is a difference in the hazard functions,
at least over some part of the 0-30 year time interval. Given what we have seen in the K-M plot of the two
survival curves this isn’t too surprising–the curves look very different. The reason our results have been so
close is that our sample is very small.
Log-rank test for equality of survivor functions
| Events Events
group | observed expected
------+-------------------------
0 | 7 10.47
1 | 7 3.53
------+-------------------------
Total | 14 14.00
chi2(1) = 5.51
Pr>chi2 = 0.0189
(f) Suppose none of the observations had been censored. What would your estimates of the 3-year survival
probabilities have been and why
Solution: If there is no censoring then the Kaplan-Meier estimate of the survival curve is simply the
proportion of people who have survived up to that time. For 3-year survival if all our observations were
uncensored we’d have ˆp 0 = 7/10 = .7 surviving in group 0 and ˆp 1 = 5/19 = .5 surviving in group 1.
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