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
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:
C
opyright © 2012 SciRes. OJS

E. DEMIDENKO 306
 
21
=1 ,zz

 
12
=2.zz


2 22
= ,z z




We solve this system using Newton’s algorithm by
updating the z-values as follows:
11 1
=,zz

where

12
1
zz
2
12
2
=,
z
zz







12
2
zz
1
12
2
=,
z
zz







21
=1,zz

 
=zz starting from the standard values, 11/2

 and
21/2   where

=,zz

denotes the density of the stand-
ard normal variable. Our practice showed that a only
three-four iterations are required to guarantee the con-
vergence up to . After 1 are 2 are determined,
the CI for OR is computed as
8z10 z

1  

12
ˆ
zz

%100
,ee
ˆ

 
z
z
.
In Figure 1, we show the 95% lower (1) and upper
limit (2) z-values as a function of the standard error of
the log OR estimate, .

The dashed horizontal line
corresponds to the standard procedure of CI computation
(11/2 =zz


=).zz

and 21/2
  The shortest width CI
uses smaller z-values. The percent OR width reduction is
computed as st
ˆˆ
1.96 1.96
st =We e
optst ,WWW100 where

 

 
ˆˆ
12
=zz e
is the relative width of the 95%
standard CI and opt We

 
 is the relative
width of the optimal (shortest) CI. As one can see from
the right plot, if the ML estimate has small variance the
difference is not substantial. However, when

in-
creases the optimal CI may be up to 25% narrower.
3. Why Shortest Confidence Interval?
When constructing a confidence interval, besides cover-
age probability which concerns the probability of cover-
ing the true parameter value (in our case OR), one has to
take into account the probability of covering the “wrong”
parameter value. In a way, this consideration is similar to
computation of the type II error of a statistical test. We
assert that the OR CI developed in this article has a
smaller probability of covering of the wrong parameter
value in the area of interest then the standard CI yet
having the same coverage probability of the true OR.
Since the distribution of the log OR is normal the
probability of the coverage the wrong value wrong
(shortly, wrong coverage) for any is computed
as
OR
21
<zz Figure 1. The shortest CI for OR with the confidence level
95% as a function of σ. For large variation in the MLE the
% width reduction c a n be substantial, up to 25%.

E. DEMIDENKO 307


wrong
wrong true
wrong true
PrOR< OR< OR
ln OROR
=
ln(OROR )
LU










true
1
2
OR= OR
.
z
z







For the standard CI we have

1
=2 


1 and z

1
=12, 

 z z
=0.25
2 and for optimal CI 1 and 2 are
computed via iterations as a solution to an optimization
problem.
z
The result of comparison of wrong coverage proba-
bilities for standard and optimized 95% CI is depicted in
Figure 2. Two scenarios are used: one with

and another with =0.4;

in both cases true
When the wrong OR approaches 1.2 the wrong coverage
is
OR= 1.2.
1= 0.95.

 When wrong increases the wrong
coverage monotonically vanishes. On the entire range of
OR values the coverage of the wrong OR is smaller for
the shortest width CI—the shortest CI is preferable.
OR
100
4. Simulations
In this section, we describe a statistical simulation study
to confirm that CI for OR in logistic regression
developed in the previous section has a shorter width on
average in finite sample (n = 100 compared with the
traditional Wald CI. We simulated 5000 normally distri-
buted samples

2
~0,
1,,
x
xx


i
y
with . The
22
=2
x


binary has the probability 22
1
x
x
ii
e
 

OR= e
e
 where

(the intercept = −2). For each sample, the Wald
and the shortest width CIs were computed; coverage
Figure 2. The probability of false coverage for the tradi-
tional Wald CI and the CI with shorest width. The opti-
mized CI has a smaller coverage of the false OR value over
the entire range of OR > 1.2.
probability was computed as the proportion of simulated
samples for which CI covers the true OR; the CI width is
computed as the median of 5000 widths (we prefer median
over mean to reduce the unwanted effect of outliers in
case of false convergence, especially in the case of large
OR values). The results of our simulations are depicted
in Figure 3. The Shortest Width CI has the width con-
Figure 3. The coverage probability and the width of two CIs
for OR in logistic regression from a simulation study (the
number of experiments = 5000; the nominal coverage pro-
bability = 95%). Both methods have coverage probability
close to the nominal level. However, the “shortest width” CI
has the width shorter than the traditional one on average
(the width is computed as the median to avoid possible out-
liers). This difference increases with the value of the true
R. O

E. DEMIDENKO 308
Table 1. Odds ratios and their confidence intervals for child birth/parity as a preventive factor against bladder cancer in
women computed via the traditional way and the shortest-width CI in eight studies.
Study OR σ Lower CI
standard
Upper CI
standard
Lower CI
shortest
Upper CI
shortest
% width
reduction
Cantor 1992 0.67 0.201 0.45 0.99 0.43 0.96 1.9
LaVecchia 1993 1.08 0.315 0.60 2.06 0.51 1.87 6.8
Cantwell 2006 0.70 0.221 0.45 1.07 0.43 1.04 1.6
McGrath 2006 0.78 0.188 0.54 1.13 0.52 1.10 1.7
Prizment 2007 0.66 0.240 0.41 1.05 0.38 1.01 1.6
Davis-Dao 2009 0.66 0.160 0.48 0.90 0.47 0.88 2.4
Huang 2009 0.43 0.386 0.20 0.91 0.16 0.83 5.6
Dietrich current 0.71 0.293 0.40 1.26 0.36 1.18 4.7
delta2 = (d1 + d2*dnorm(z1))/den sistently smaller that the regular CI although for this par-
ticular simulation set up the gain is not very substantial. if(abs(delta1) + abs(delta2) < eps)
break
5. Example z1 = z1 + delta1
z2 = z2 – delta2
We illustrate the computation of the shortest width CI for
OR using a recently published article on the meta-ana-
lysis of preventive and risk factors for bladder cancer in
women [5]. Table 1 presents the results of eight case-
control studies where the bladder cancer occurrence was
correlated with woman’s parity. In most studies, it was
found that child birth is a statistically significant pre-
ventive factor against bladder cancer. Traditional and
shortest width CIs for OR are presented. The percent
width reduction is in the range from 1.6 to 6.8. Note that
the shortest width CI tends to reduce the upper limit.
}
return(c(z1,z2))
}
7. Acknowledgements
This work was supported by a grant from NIH/NCI R01
CA130880.
REFERENCES
[1] A. Agresti, “Categorical Data Analysis,” 3d Edition, Wiley,
New York, 2002.
6. The R Function
[2] B. Rosner, “Fundamentals of Biostatistics,” 7th Edition,
Pacific Grove, Duxbury, 2010.
The following function implements the Newton’s itera-
tions described in the previous section. For example z1z2
(sigma = 0.201) returns values for and as
−2.199928 1.797928.
[3] R. C. Dahiya and I. Guttman, “Shortest Confidence and
Prediction Intervals for the Log-Normal,” Canadian Jour-
nal of Statistics, Vol. 10, No. 4, 1982, pp. 277-291.
doi:10.2307/3556194
1
z2 z
z1z2 = function(sigma,alpha = 0.05,
eps = 0.000001,maxit = 100) [4] P. D. Wilson and P. Langenberg, “Usual and Shortest Con-
fidence Intervals on Odds Ratios from Logistic Regres-
sion,” The American Statistician, Vol. 53, No. 4, 1999, pp.
332-335.
{
z1 = qnorm(alpha/2)
z2 = –z1
for(it in 1:maxit) [5] K. Dietrich, E. Demidenko, A. Schned, M. S. Zens, J.
Heaney and M. R. Karagas, “Parity, Early Menopause and
the Incidence of Bladder Cancer in Women: A Case—
Control Study and Meta-Analysis,” European Journal of
Cancer, Vol. 47, No. 4, 2011, pp. 592-599.
doi:10.1016/j.ejca.2010.10.007
{
den = dnorm(z1) + dnorm(z2)
d1 = pnorm(z2) – pnorm(z1) – 1 + alpha
d2 = z1 + z2 + 2*sigma
delta1 = (d1 – d2*dnorm(z2))/den