[R] proportions confidence intervals

[Spencer Graves]

      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
>
>