[R] proportions confidence intervals
Peter Flom
flom at ndri.org
Mon Jul 12 21:17:06 CEST 2004
There's also an article by Agresti and Coull
author = {Alan Agresti and B. A. Coull},
title = {Approximate is better than "exact" for interval estimation of
binomial proportions},
journal = {American Statistician},
year = {1998},
volume = {52},
number = {},
pages = {119-126},
which may be of interest.
Peter
Peter L. Flom, PhD
Assistant Director, Statistics and Data Analysis Core
Center for Drug Use and HIV Research
National Development and Research Institutes
71 W. 23rd St
www.peterflom.com
New York, NY 10010
(212) 845-4485 (voice)
(917) 438-0894 (fax)
>>> Spencer Graves <spencer.graves at pdf.com> 7/12/2004 3:07:00 PM >>>
According to Brown, Cai and DasGupta (cited below), the "exact"
confidence intervals are hyperconservative, as they are designed to
produce actual coverage probabilities at least the nominal. Thus for a
95% confidence interval, the actual coverage could be 98% or more,
depending on the true but unknown proportion; please check their
papers
for exact numbers. They report that the Wilson procedure performs
reasonably well, as does the asymptotic logit procedure. I can't say
without checking, but I would naively expect that confint.glm would
likely also be among the leaders.
By the way, confint.glm is independent of the parameterization,
assuming 2*log(likelihood ratio) is approximately chi-square. It is
therefore subject to intrinsic nonlinearity but is at least free of
parameter effects (see, e.g., Bates and Watts (1988) Nonlinear
Regression Analysis and Its Applications (Wiley)). To check this,
consider the following:
fit10c <- glm(y~1, family=binomial(link=cloglog), data=DF10,
weights=size)
fit100c <- glm(y~1, family=binomial(link=cloglog), data=DF100,
weights=size)
(CI10c <- confint(fit10c))
(CI100c <- confint(fit100c))
> 1-exp(-exp(CI10c))
2.5 % 97.5 %
0.005989334 0.371562793
> 1-exp(-exp(CI100c))
2.5 % 97.5 %
0.05140762 0.16875918
These are precisely the number reported below with the default
binomial link = logit.
hope this helps. spencer graves
Chuck Cleland wrote:
> Darren also might consider binconf() in library(Hmisc).
>
> > library(Hmisc)
>
> > binconf(1, 10, method="all")
> PointEst Lower Upper
> Exact 0.1 0.002528579 0.4450161
> Wilson 0.1 0.005129329 0.4041500
> Asymptotic 0.1 -0.085938510 0.2859385
>
> > binconf(10, 100, method="all")
> PointEst Lower Upper
> Exact 0.1 0.04900469 0.1762226
> Wilson 0.1 0.05522914 0.1743657
> Asymptotic 0.1 0.04120108 0.1587989
>
> Spencer Graves wrote:
>
>> Please see:
>> Brown, Cai and DasGupta (2001) Statistical Science, 16:
101-133
>> and (2002) Annals of Statistics, 30: 160-2001
>> They show that the actual coverage probability of the standard
>> approximate confidence intervals for a binomial proportion are quite
>> poor, while the standard asymptotic theory applied to logits
produces
>> rather better answers.
>> I would expect "confint.glm" in library(MASS) to give decent
>> results, possibly the best available without a very careful study of
>> this particular question. Consider the following:
>> library(MASS)# needed for confint.glm
>> library(boot)# needed for inv.logit
>> DF10 <- data.frame(y=.1, size=10)
>> DF100 <- data.frame(y=.1, size=100)
>> fit10 <- glm(y~1, family=binomial, data=DF10, weights=size)
>> fit100 <- glm(y~1, family=binomial, data=DF100, weights=size)
>> inv.logit(coef(fit10))
>>
>> (CI10 <- confint(fit10))
>> (CI100 <- confint(fit100))
>>
>> inv.logit(CI10)
>> inv.logit(CI100)
>>
>> In R 1.9.1, Windows 2000, I got the following:
>>
>>> inv.logit(coef(fit10))
>>
>>
>> (Intercept)
>> 0.1
>>
>>>
>>> (CI10 <- confint(fit10))
>>
>>
>> Waiting for profiling to be done...
>> 2.5 % 97.5 %
>> -5.1122123 -0.5258854
>>
>>> (CI100 <- confint(fit100))
>>
>>
>> Waiting for profiling to be done...
>> 2.5 % 97.5 %
>> -2.915193 -1.594401
>>
>>>
>>> inv.logit(CI10)
>>
>>
>> 2.5 % 97.5 %
>> 0.005986688 0.371477058
>>
>>> inv.logit(CI100)
>>
>>
>> 2.5 % 97.5 %
>> 0.0514076 0.1687655
>>
>>>
>>> (naiveCI10 <- .1+c(-2, 2)*sqrt(.1*.9/10))
>>
>>
>> [1] -0.08973666 0.28973666
>>
>>> (naiveCI100 <- .1+c(-2, 2)*sqrt(.1*.9/100))
>>
>>
>> [1] 0.04 0.16
>
>
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