O’Donnell et al. (2000) have studied the effect of the apolipoprotein E gene on the riskof recurrent lobar intracerebral hemorrhage in patients who have survived such a hemorrhage. Follow-up was obtained for 70 patients who had survived a lobar intracerebral hemorrhage and whose genotype was known. There are three common alleles for the apolipoprotein E gene:
ε2,ε3, and ε4. The genotype of all 70 patients was composed of these three alleles. Patients were classified as either being homozygous for ε3 (Group 1), or as having at least one of the other two alleles (Group 2).
Table 6.2 shows the follow-up for these patients. There were four recurrent hemorrhages among the 32 patients in Group 1 and 14 hemorrhages among the 38 patients in Group 2. Figure 6.4 shows the Kaplan–Meier disease free survival function for these two groups of patients. These curves were derived Table 6.2. Length of follow-up and fate for patients in the study by O’Donnell et al. (2000).
Patients are divided into two groups defined by their genotype for the apolipoprotein Egene.
Follow-up times marked with an asterisk indicate patients who had a recurrent lobar intracerebral hemorrhage at the end of follow-up. All other patients did not suffer this event during follow-up
Length of follow-up (months) Homozygousε3/ε3 (Group 1)
0.23∗ 1.051 1.511 3.055∗ 8.082 12.32∗ 14.69 16.72 18.46 18.66
19.55 19.75 24.77∗ 25.56 25.63 26.32 26.81 32.95 33.05 34.99
35.06 36.24 37.03 37.75 38.97 39.16 42.22 42.41 45.24 46.29
47.57 53.88
At least oneε2 orε4allele (Group 2)
1.38 1.413∗ 1.577 1.577∗ 3.318∗ 3.515∗ 3.548∗ 4.041 4.632 4.764∗
8.444 9.528∗ 10.61 10.68 11.86 13.27 13.60 15.57∗ 17.84 18.04
18.46 18.46 19.15∗ 20.11 20.27 20.47 24.87∗ 28.09∗ 30.52 33.61∗
37.52∗ 38.54 40.61 42.78 42.87∗ 43.27 44.65 46.88
208 6. Introduction to survival analysis
Probability of Hemorrhage-Free Survival
Months of Follow-up
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
Homozygous 3/ 3
At Least One
ε2 or ε4 Allele
Figure 6.4 Kaplan–Meier estimates of hemorrhage-free survival functions for patients who had previously survived a lobular intracerebral hemorrhage. Patients are subdi- vided according to their apolipoprotein Egenotype. Patients who were homo- zygous for theε3 allele of this gene had a much better prognosis than other patients (O’Donnell et al., 2000).
using equation (6.2). For example, suppose that we wish to calculate the disease free survival function at 15 months for the 32 patients in Group 1.
Three hemorrhages occurred in this group before 15 months at 0.23, 3.055 and 12.32 months. Therefore,
S[15]ˆ = p1p2p3 = 3 k=1
pk,
where pk is the probability of avoiding a hemorrhage on thekth day on which hemorrhages occurred. At 12.3 months there are 27 patients at risk;
two of the original 32 have already had hemorrhages and three have been censored. Hence,p3 =(27 –1)/27=0.9629. Similarly,p1 =(32−1)/32= 0.9688 and p2=(29−1)/29=0.9655. Therefore, ˆS[15]=0.9688× 0.9655×0.9629=0.9007.S[t] is constant and equals 0.9007 fromˆ t =12.32 until just before the next hemorrhage in Group 1, which occurs at time 24.77.
In Figure 6.4, the estimated disease free survival functions are constant over days when no hemorrhages are observed and drop abruptly on days when hemorrhages occur. If the time interval is short enough that there is rarely more than one death per interval, then the height of the drop at each death day indicates the size of the cohort remaining on that day. The accuracy of the survival curve gets less as we move towards the right, as it is based on fewer and fewer patients. Large drops in these curves are warnings
209 6.5. 95% Confidence intervals for survival functions
of decreasing accuracy of our survival estimates due to diminishing numbers of study subjects.
If there is no censoring and there areq death days before timetthen S(t)ˆ =
n1−d1
n1
n2−d2
n1−d1
. . . .
nq−dq
nq1−dq1
= nq −dq
n1 = m(t) n .
Hence the Kaplan–Meier survival curve reduces to the proportion of patients alive at timetif there is no censoring.
6.5. 95% Confidence Intervals for Survival Functions
The variance of ˆS(t) is estimated by Greenwood’s formula (Kalbfleisch and Prentice, 1980), which is
s2S(t)ˆ =S(t)ˆ 2
{k:tk<t}
dk
nk(nk−dk). (6.4)
A 95% confidence interval forS(t) could be estimated by ˆS(t)±1.96sS(tˆ ). However, this interval is unsatisfactory when ˆS(t) is near 0 or 1. This is because ˆS(t) has a skewed distribution near these extreme values. The true survival curve is never less than zero or greater than one, and we want our confidence intervals to never exceed these bounds. For this reason we calculate the statistic log[−log[ ˆS(t)]], which has variance
σˆ2(t)=
{k:tk<t}
dk
nk(nk−dk)
{k:tk<t}
log
(nk−dk) dk
2 (6.5)
(Kalbfleisch and Prentice, 1980). A 95% confidence interval for this statistic is
log[−log[ ˆS(t)]]±1.96 ˆσ(t). (6.6)
Exponentiating equation (6.6) twice gives a 95% confidence interval for S(t) ofˆ
S(t)ˆ exp(∓1.96 ˆσ(t)). (6.7)
Equation (6.7) provides reasonable confidence intervals for the entire range of values of ˆS(t). Figure 6.5 shows these confidence intervals plotted for the hemorrhage-free survival curve of patients with anε2 orε4 allele in
210 6. Introduction to survival analysis
Months of Follow-up
0 10 20 30 40 50
Probability of Hemorrhage-Free Survival
0.0 0.2 0.4 0.6 0.8 1.0 1
1 2 1
5 4
3
1
3 2 1
Figure 6.5 The thick black curve is the Kaplan–Meier survival function for patients with anε2 orε4 allele in the study by O’Donnell et al. (2000). The gray lines give 95% confidence intervals for this curve. The digits near the survival curve indi- cate numbers of patients who are lost to follow-up. Note that the confidence intervals widen with increasing time. This indicates the reduced precision of the estimated survival function due to the reduced numbers of patients with lengthy follow-up.
O’Donnell et al. (2000). If we return to the homozygous ε3 patients in Section 6.4 and lett=15 then
{k:tk<15}
dk
nk(nk−dk) = 1
32×31+ 1
29×28+ 1
27×26 =0.003 66 and
{k:tk<15}
log
(nk−dk) dk
=log
32−1 32
+log
29−1 29
+log
27−1 27
= −0.104 58.
Therefore, ˆσ2(15)=0.003 66/(−0.104 58)2 =0.335, and a 95% confi- dence interval for ˆS[15] is 0.9007exp(∓1.96×√0.335)=(0.722, 0.967). This in- terval remains constant from the previous Group 1 hemorrhage recurrence at time 12.32 until just before the next at time 24.77.