Discussion:
Confidence interval for difference between two non-independent proportions?
John Uebersax
2006-08-04 07:43:18 UTC
Is there a standard way to estimate the confidence interval
for the difference between two non-independent proportions?

For example, let:

N = no. of patients
p1 = proportion of asymptomatic (vs. symptomatic) patients
pre-treatment
p2 = proportion of asymptomatic (vs. symptomatic) patients
post-treatment

How would one get the 95% CI for (p2 - p1)?

The McNemar test can be used to test whether (p2 - p1) differs
significantly from 0, but it does not supply a means of getting
a confidence interval.

The only literature I've located is a note by CC Hsieh in
Stat Med (1985):

http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=pubmed&cmd=Retrieve&dopt=AbstractPlus&list_uids=3992071&query_hl=1&itool=pubmed_docsum

but so far I've been unable to see the article and don't know
what method it used.

--
John Uebersax PhD
m00es
2006-08-04 07:59:06 UTC
Post by John Uebersax
Is there a standard way to estimate the confidence interval
for the difference between two non-independent proportions?
N = no. of patients
p1 = proportion of asymptomatic (vs. symptomatic) patients
pre-treatment
p2 = proportion of asymptomatic (vs. symptomatic) patients
post-treatment
How would one get the 95% CI for (p2 - p1)?
See:

Tang, M.-L., Tang, N.-S., & Chan, I. S. F. (2005). Confidence interval
construction for proportion difference in small-sample paired studies.
Statistics in Medicine, 24(23), 3565-3579.

Best,

m00es
John Uebersax
2006-08-05 07:21:03 UTC
The OP asked about case 2 (with k = 2) and Tang, Tang, & Chan (2005)
applies. The website given above is for case 1.
A second website similarly seems to suggest that the formula for
*independent* proportions can be adapted to the problem of *dependent*
proportions in a 2x2 table.

See:

http://bmj.bmjjournals.com/collections/statsbk/6.html

I don't know if this method is making debatable assumptions, in what
way the method of Tang, Tang & Chan is better, or when the latter would
be considered necessary.

--
John Uebersax PhD
Eric Bohlman
2006-08-04 14:21:01 UTC
Post by John Uebersax
Is there a standard way to estimate the confidence interval
for the difference between two non-independent proportions?
N = no. of patients
p1 = proportion of asymptomatic (vs. symptomatic) patients
pre-treatment
p2 = proportion of asymptomatic (vs. symptomatic) patients
post-treatment
How would one get the 95% CI for (p2 - p1)?
See Agresti (1990), pp. 349-350
David Winsemius
2006-08-05 16:22:32 UTC
Post by John Uebersax
Is there a standard way to estimate the confidence interval
for the difference between two non-independent proportions?
N = no. of patients
p1 = proportion of asymptomatic (vs. symptomatic) patients
pre-treatment
p2 = proportion of asymptomatic (vs. symptomatic) patients
post-treatment
How would one get the 95% CI for (p2 - p1)?
The McNemar test can be used to test whether (p2 - p1) differs
significantly from 0, but it does not supply a means of getting
a confidence interval.
The only literature I've located is a note by CC Hsieh in
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=pubmed&cmd=Retrieve&do
pt=AbstractPlus&list_uids=3992071&query_hl=1&itool=pubmed_docsum
but so far I've been unable to see the article and don't know
what method it used.
"Exact confidence limits for the difference between two correlated proportions"
Chris J. Lloyd and Max V. Moldovan

"A New Confidence Interval for the Difference Between Two BinomialProportions
of Paired Data"
Xiao-Hua Zhou Gengsheng Qin
<http://www.bepress.com/cgi/viewcontent.cgi?article=1028&context=uwbiostat>

----citation to Agresti and Min----
Agresti, A. and Min, Y. (2005). Simple improved confidence intervals for comparing
matched proportions. Statistics in Medicine 24, 729-740.
(almost certainly available online to academicians)

--- a portion of a Google cached U.Fla web site, prob by Agresti----
R code for the adjustment of the Wald confidence interval for a difference
of proportions, with matched pairs. This is the interval called Wald+2 in
Agresti and Min, Statistics in Medicine, 2004, which adds 0.5 to each cell
before constructing the Wald CI. The CI is truncated when it overshoots the
boundary.

--------------------------------------------------------------------------
diffpropci <- function(b,c,n,conflev)
{
z <- qnorm(1-(1-conflev)/2)
diff <- (c-b)/(n+2)
sd <- sqrt((b+c+1)-(c-b)^2/(n+2))/(n+2)
ll <- diff - z*sd
ul <- diff + z*sd
if(ll < -1) ll = -1
if(ul > 1) ul = 1
c(ll, ul)
}
# Adjusted Wald interval for difference of proportions with matched pairs
# "conflev"=confidence coefficient, n=sample size, b,c = off-diag counts
------end extract from Agresti's site--------------------------------------
--
David Winsemius
John Uebersax
2006-08-08 06:09:44 UTC
Thanks David and the others who contributed to this thread!

--
John Uebersax PhD
John Uebersax
2006-08-08 06:55:04 UTC
Here's a chronological list of references I've compiled from the
various replies to my original message:

Tang ML, Tang NS, Chan ISF. (2005).
Confidence interval construction for proportion difference in
small-sample paired studies.
Statistics in Medicine, 24(23), 3565-3579.

Agresti, A. and Min, Y. (2005).
Simple improved confidence intervals for comparing matched proportions.

Statistics in Medicine 24, 729-740.

Berger, R.L. and Sidik, K. (2003)
Exact unconditional tests for a 2 × 2 matched-pairs design.
Statistical Methods in Medical Research 12, 91-108.

Newcombe, R.G. (1998)
Improved confidence intervals for the difference between binomial
proportions based on paired data.
Statistics in Medicine 17, 2635-2650.

Swinscow TDV (rev. by MJ Campbell).
Statistics at Square One, 9th Ed.
Ch. 6. Differences between percentages and paired
alternatives. London: Copyright BMJ Publishing Group, 1997.
http://bmj.bmjjournals.com/collections/statsbk/6.html

Suissa, S. and Shuster, J.J. (1991)
The 2 × 2 matched pairs trial: exact unconditional
design and analysis.
Biometrics 47, 361-372.

Agresti (1990), pp. 349-350.

Gardner MJ, Altman DG.
Confidence intervals rather than P values: estimation rather than
hypothesis testing.
Br Med J 1986;292:746-50.
Reprinted in Statistics with Confidence, 1st edition (eds. Gardner &