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Applied Survival Analysis: Regression Modeling of Time to Event Data, 2nd Edition by Stanley Lemeshow, Susanne May, David W. Hosmer

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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 ƒ (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

(A.1)app1

where

app2

is the derivative of the function with respect to X evaluated at the mean of X. It follows from (A.l) that the variance of the function is approximately

(A.2)

where σ2 is the variance of X. The delta ...

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