Statistical Methods in Medical Research 4ed

574 Survival analysis
otherwise the life-table calculations represent quantities with no simple interpretation.
In Table 17.2 the calculations could have been continued beyond 10 years.
Suppose, however, that d10 and w10 had both been zero, as they would have been
if no patients had been observed for more than 10 years. Then n10 would have
been zero, no values of q10 and p10 could have been calculated and, in general, no
value of l11 would have been available unless l10 were zero (as it would be if any
one of p0, p1, ..., p9 were zero), in which case l11 would also be zero. This point
can be put more obviously by saying that no survival information is available for
periods of follow-up longer than the maximum observed in the study. This
means that the expectation of life (which implies an indefinitely long followup)
cannot be calculated from follow-up studies unless the period of follow-up,
at least for some patients, is sufficiently long to cover virtually the complete span
of survival. For this reason the life-table survival rate (column (8) of Table 17.2)
is a more generally useful measure of survival. Note that the value of x for which
lx ˆ 50% is the median survival time; for a symmetric distribution this would be
equal to the expectation of life.
For further discussion of life-table methods in follow-up studies, see Berkson
and Gage (1950), Merrell and Shulman (1955), Cutler and Ederer (1958) and
Newell et al. (1961).
17.4 Samplingerrors in the life-table
Each of the values of px in a life-table calculation is subject to sampling variation.
Were it not for the withdrawals the variation could be regarded as
binomial, with a sample size nx. The effect of withdrawals is approximately the
same as that of reducing the sample size to n 0 x . The variance of lx is given
approximately by the following formula due to Greenwood (1926), which can
be obtained by taking logarithms in (17.6) and using an extension of (5.20).
var…lx† ˆl 2 Px 1
di
x n
iˆ0
0 i…n0 i
In Table 17.2, for instance, where l4 ˆ 35 0%,
var…l4† ˆ…35 0† 2
di† : …17:7†
90
…374†…284† ‡
76
…284†…208† ‡
51
…208†…157† ‡
25
…151†…126†
ˆ 6 14
p
so that SE…l4† ˆ 6 14 ˆ 2 48, and approximate 95% confidence limits for l4 are
35 0 …1 96†…2 48† ˆ30 1 and 39 9: