Open Journal of Statistics, 2012, 2, 305-308

http://dx.doi.org/10.4236/ojs.2012.23037 Published Online July 2012 (http://www.SciRP.org/journal/ojs)

The Shortest Width Confidence Interval for Odds Ratio

in Logistic Regression

Eugene Demidenko

Section of Biostatistics and Epidemiology, Geisel School of Medicine at Dartmouth, Hanover, USA

Email: eugened@dartmouth.edu

Received May 16, 2012; revised June 18, 2012; accepted July 2, 2012

ABSTRACT

The shortest width confidence interval (CI) for odds ratio (OR) in logistic regression is developed based on a theorem

proved by Dahiya and Guttman (1982). When the variance of the logistic regression coefficient estimate is small, the

shortest width CI is close to the regular Wald CI obtained by exponentiating the CI for the regression coefficient esti-

mate. However, when the variance increases, the optimal CI may be up to 25% narrower. It is demonstrated that the

shortest width CI is favorable because it has a smaller probability of covering the wrong OR value compared with the

standard CI. The closed-form iterations based on the Newton’s algorithm are provided, and the R function is supplied.

A simulation study confirms the superior properties of the new CI for OR in small sample. Our method is illustrated

with eight studies on parity as a preventive factor against bladder cancer in women.

Keywords: Bladder Cancer; Coverage Probability; Logistic Regression; Newton’s Algorithm

1/2 1/2

ˆˆ

,

zz

1. Introduction

Odds ratio, as the exponentiated logistic regression co-

efficient, is a popular measure of association in medicine,

epidemiology and biostatistics. Routinely, the confidence

interval (CI) for odds ratio (OR) in logistic regression is

computed by exponentiating the CI for the beta-co-

efficient (log OR, hereafter denoted as

), [1,2]. While

it is true that if a CI for

has coverage probability

1

the exponentiated CI for OR has the same

coverage probability, such CI does not have the shortest

width and therefore can be improved. The goal of this

note is to demonstrate how to compute the shortest CI for

OR using a theorem proved in [3]. Previously, [4] sug-

gested to find the shortest confidence interval for OR

using the same approach but their procedure of minimi-

zation of the interval’s width was just an approximate

solution. In this paper, we find the exact minimum via

Newton’s iterations.

2. The Method

Let the coefficient of logistic regression

be estimated

by maximum likelihood (ML) so that

2

ˆ,

OR =

in

large sample. We want to construct the shortest CI for

e

based on ˆ

assuming that its variance 2

is known. In practice, this variance is not known but

usually the sample size is large enough, so that one can

assume that 2

is fixed. Routinely, one first constructs

the CI for

100 1

%

as

and then exponentiates it to ob-

100 1%

CI for OR as tain the

1/2 1/2

ˆˆ

,,

zz

ee

1/2

z

where is the

12

th

quantile of the standard normal cdf,

1

=12,z

=0.05

1/2

where is the cdf of the stand-

ard normal distribution. For example, if

we

have 1/2

This CI will be refered to as the

(traditional) Wald CI with symmetric z-values.

=1.96.z

1,

The idea of the shortest CI is to chose asymmetric

z-values such that the coverage probability is the same,

but the length of the CI is minimum. Thus we

seek CI for OR in the form

12

ˆˆ

,

zz

ee

12

<zz

(1)

where are such that

21

=1zz .

21

=zz

(2)

. Clearly, the standard CI has the form (1) with

Since the width of interval (1) is OR we

21

,

zz

ee

arrive at the following optimization problem:

21

min zz

ee

(3)

under restriction (2). As was shown by Dahiya and Guttman

(1982), this optimization problem reduces to the solution

of the following system of equations for z1 and z2:

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