Applied Survival Analysis: Regression Modeling of
Time-to-Event Data, Second Edition
by David W. Hosmer, Stanley Lemeshow and Susanne May
Copyright 2008 John Wiley & Sons, Inc.
APPENDIX 1
The Delta Method
Deriving an expression for an estimator of the variance of the estimator is one
problem faced by statisticians when developing an estimator of a parameter. Both
estimators are needed for confidence interval estimation and/or hypothesis testing.
Statisticians use a procedure commonly called the delta method to obtain an
estimator of the variance when the estimator is not a simple sum of observations.
The basic idea is to use a method from calculus called a Taylor series expansion to
derive a linear function that approximates the more complicated function. We
refer the reader to any introductory calculus text for a discussion of the Taylor
series expansion.
To apply the delta method, the function must be one that can be approximated
by a Taylor series and, in general, this means that it is a "smooth" function, with
no "corners." Consider such a function of a random variable X denoted as f(X).
To apply the delta method, we use the first two terms of a Taylor series expansion
about the mean of the variable to approximate the value of the function as
f{X)sf{n) + {X-n)f'{n),
(A.l)
where
is the derivative of the function with respect to X evaluated at the mean of X. It
follows from (A. 1 ) that the variance of the function is approximately
Var[/(X)] = V a r ( X - ^ x [ / ' ( / / ) ] 2
^2x[/'(/l)]\
where a2 is the variance of X. The delta method estimator of the variance of the
function is obtained when we use the estimators of /I and a2 in (A.2) as follows
355
356
APPENDIX 1
var[/(*)] = a 2 x [ / ' ( ) ] \
(A.3)
As an example, consider the function ln(X). The expansion from (A.l) is
ln(X) = ln(/i) + ( X - / i ) - .
(A.4)
The delta method estimator of the variance from (A.3) is
Var[ln(X)]s(7 2 4r,
where a2 and fi denote estimators of cr and p..
As a second example, we provide the details for the development of the delta
method estimator of the variance of the log of the Kaplan-Meier estimator of the
survival function shown in (2.3) and that of the Kaplan-Meier estimator itself in
(2.5). The estimator as shown in (2.1 ) is
5(0 = 1 1 ^ .
and its log is
ln[s(r)]=Iln
j
n, - a,
= Sln().
where p = (, d )fn . The first key assumption in the development of the variance estimator is that the observations of survival among the n subjects at risk are
independent Bernoulli trials with constant probability, p,. Under this assumption,
the estimator of the constant probability is
p,
with variance estima-
tor p. ( 1 - p, )/ni. The Taylor series expansion for the log function in (A.4) yields
ln(p,)=ln(/j,) + ( - p , ) ,
Pi
and from (A.3) the delta method variance estimator is
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THE DELTA METHOD
Var[ln(/?,)J = i
-
d,
n,(n,-d,)
'
The second key assumption is that observations in different risk sets are independent. Thus the delta method estimator of the variance of the log of the KaplanMeier estimator, as shown in (2.3) is
vr{ln[5(i)]}svr[ln()]
d,
-1/,s, " , ( / -
i)
The estimator of the variance of the Kaplan-Meier estimator comes from a
second application of the delta method. In this application the function is
/ ( X ) = exp(X),
e.g., S(f) = expjlnl S ( r ) l | . It follows from (A.l) that the series expansion is
exp(x)sexp(/i) + (X- i u)exp( i u)
and, from (A.2) the approximate variance is
V ^ [ e x p ( X ) ] = <r[exp()]\
(A.5)
Application of the approximation in (A.5) yields the Greenwood estimator in (2.5),
namely
vt[(.)].[S(,)JZ;nd,i_j
The confidence interval estimator for the Kaplan-Meier estimator discussed
in Chapter 2 is based on the log-log survival function, that is, lnj-ln 5(i) \>. The
variance estimator of this function requires a second application of the expansion
of the log function. In this case, X = ln[ S(t)\ and application of the approxima-
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APPENDIX 1
tion to the variance of the log of a random variable yields the estimator shown in
(2.6),
1 L
/JJ
[ln(s(t))] /(.-4)
The results presented in this appendix provide a brief introduction to the use
of the delta method within the specific context of deriving an estimator of the variance of the Kaplan-Meier estimator, or functions of it. The technique is quite
general and has been used in a variety of settings |see Agresti (1990) for applications in general categorical data models].
