For clinical
trials with binary outcomes, the results can usually be presented as a 2x2
contingency table as below:
|
|
Responder
|
Non-responder
|
Total
|
|
Treatment
1
|
n11
|
n12
|
n1
|
|
Treatment
2
|
n21
|
n22
|
n2
|
We can
then calculate the proportion of responders for two treatment groups:
p1=n11/n1
p2=n21/n2
We have
two ways to compare two treatment groups:
- The difference between two proportions: p1-p2
- The ratio of two proportions: p1/p2
p1-p2 may
be called the absolute risk difference and p1/p2 is called relative risk (RR)
or risk ratio.
The
confidence interval can be constructed for the difference between two
proportions and for the relative risk.
For the
difference between two proportions, the
asymptotic confidence interval is ca1culated using the following formula:
(p1-p2)
+/- Z(alpha/2)*sqrt((p1 *(1-p1)/n1)+(p2*(1-p2)/n2))
Reference: Stokes, Davis, and Kock (2000) Categorical Data
Analysis using the SAS System, 2nd
edition
The
notations may be different in the reference book and in
SAS manual, but the results should be the same.
I had a posting a while ago about “Confidence Interval for Difference in Two Proportions” where I mentioned the corrections and the SAS codes.
For relative
risk, the asymptotic confidence interval is calculated using the following
formula:
Exp(log(RR) +/- Z(alpha/2) * sqrt((1-p1)/(n1*p1) +
(1-p2)/(n2*p2)))
Reference: Agresti A (2007) An Introduction to Categorical
Data Analysis, 2nd edition, JohnWiley & Sons, Inc.,
The
notations may be different in the reference book and in
SAS manual, but the results should be the same.
The
confidence interval for relative risk can be obtained from SAS Proc Freq and
can also be manually calculated using the formula above and the formula from
SAS manual.
Suppose we
have study results as below:
|
|
Success
|
Non-success
|
Total
|
|
Trt1
|
63
|
3
|
66
|
|
Trt2
|
56
|
13
|
69
|
data example;
length trt $8;
input trt $ success $ count;
datalines;
trt1
yes 63
trt1
no 3
trt2 yes 56
trt2 no 13
;
proc freq
data=example;
weight count;
tables trt*success/measures nopercent nocol;
title 'outputs from SAS Proc Freq';
run;
data agresti;
n11=63;
n21=56;
n1=66;
n2=69;
p1=n11/n1;
p2=n21/n2;
rr = p1/p2;
v = (1-p1)/(n1*p1) + (1-p2)/(n2*p2);
upper = exp(log(rr) - probit(0.025)*sqrt(v));
lower = exp(log(rr) + probit(0.025)*sqrt(v));
run;
proc print
data=agresti;
title "using the formula from Agresti's
book"
run;
data sasmanual;
n11=63;
n21=56;
n1=66;
n2=69;
p1=n11/n1;
p2=n21/n2;
rr = p1/p2;
v = (1-p1)/n11 + (1-p2)/n21;
upper = rr * exp(-probit(0.025)*sqrt(v));
lower = rr * exp(probit(0.025)*sqrt(v));
run;
proc print
data=sasmanual;
title "using the formula from SAS
manual";
run;
I recently read a paper by Fischer et al. The
confidence interval for relative risk was constructed using a method by
Koopman. In Koopman’s paper “Confidence
Intervals for the Ratio of Two Binomial Proportions”, a Chi-square method was
proposed and the method required using numerical procedure and the iterative computations.
There is no SAS program available for the calculation using Koopman's method.
There are other approaches proposed for computing confidence
intervals for the ratio of two proportions. However, the method
for calculating the asymptotic confidence interval adopted in SAS Proc Freq is
commonly used.
Further reading:
- Applied Statistics: Proportions, risk ratios and odds ratios
- An Introduction to Categorical Data Analysis by Agresti
- Confidence limits for the ratio of two rates based on likelihood scores: non-iterative method
- Discussions in Google forum about “Confidence interval for ratio of independent binomial proportions”
- Fagerland “Recommended confidence intervals for two independent binomial proportions”
- Dann and Koch “Review and evaluation of methods for computing confidence intervals for the ratio of two proportions and considerations for non-inferiority clinical trials”
- Test for comparing two proportions
