二项分布比例的置信区间计算
之前一网友遇到类似问题，特查了相关文献，归结一下方法有二。
1.根据各方法的计算公式进行编辑公式，data 步就可以搞定。相关文献有：《A Comparison of Binomial Proportion Interval Estimation Methods 》、《Confidence Interval Calculation for Binomial Proportions》，这两篇文章都曾讨论比较了Wald、Wilson Score、Exact Clopper Pearson几种方法，另外《A SAS Program to Calculate and Plot Widths of (1- )100% Confidence Intervals for Binomial Proportions》一文将其中的Clopper Pearson法写成了宏。
2.PROC FREQ语句也提供了计算基于二项分布置信区间的方法。《Are you computing Confidence Interval for binomial proportion using PROC FREQ? Be Careful!!》和《Confidence Intervals for the Binomial Proportion with Zero Frequency》作了相关讨论，proc freq语句官方文档也作了相关说明（http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/viewer.htm#statug_freq_a0000000660.htm）
相关代码摘录如下。
/*********************************************************************** ** Program: Propci.sas ** Author: Keith Dunnigan ** ** Purpose: Calculate the confidence interval for one proportion. ** ** Description: Calculate Wald, Wilson Score, and exact Clopper-Pearson ** confidence intervals for a variety of n, p, and alpha ** combinations for either a one or two-sided interval. ** ***********************************************************************/ %macro runprog; data propci; set parms; if sided = 1 then do; zalpha=probit(1-(alpha)); end; else if sided = 2 then do; zalpha=probit(1-(alpha/2)); end; ** Wald; q = 1 - p; stder = sqrt(p*q/n); Wald_lcl = p - zalpha * stder; Wald_ucl = p + zalpha * stder; ** Wilson Score; part1 = p + ((zalpha**2)/(2*n)); part2 = sqrt( (p*q/n) + ((zalpha**2)/(4*n**2)) ); part3 = 1 + (zalpha**2)/n; Wilson_lcl = (part1 - (zalpha * part2))/ part3; Wilson_ucl = (part1 + (zalpha * part2))/ part3; ** Exact Clopper Pearson; x = round (n*p,0.1); * Calculate the lower limit.; v1 = 2*(n-x+1); v2 = 2*x; if sided = 1 then do; a = 1-(alpha); end; else if sided = 2 then do; a = 1-(alpha/2); end; coef = (n-x+1)/x; fscore = finv(a,v1,v2); exact_lcl = 1/(1+coef*fscore); * Calculate the upper limit.; v11 = 2*(x+1); v22 = 2*(n-x); fscore2 = finv(a,v11,v22); coef2 = (x+1)/(n-x); numer = coef2*fscore2; exact_ucl = numer/(1+numer); run; options nodate; title 'Confidence Intervals for a Single Proportion'; proc print data = propci split = '_' noobs; var n p alpha sided Wald_lcl Wald_ucl Wilson_lcl Wilson_ucl exact_lcl exact_ucl; label wald_lcl = 'LCL_(Wald)' wald_ucl = 'UCL_(Wald)' wilson_lcl = 'LCL_(Wilson_Score)' wilson_ucl = 'UCL_(Wilson_Score)' sided = 'Sides_on_CI' exact_lcl = 'LCL_(Exact)' exact_ucl = 'UCL_(Exact)'; run; %mend runprog; data parms; infile cards; input n p alpha sided; cards; 24 0.9 0.05 1 25 0.9 0.05 1 26 0.9 0.05 1 29 0.9 0.05 1 32 0.9 0.05 1 35 0.9 0.05 1 38 0.9 0.05 1 43 0.9 0.05 1 44 0.9 0.05 1 45 0.9 0.05 1 45 0.9 0.05 1 48 0.9 0.05 1 49 0.9 0.05 1 50 0.9 0.05 1 run; %runprog;
**Some programs for calculating the Exact Binomial P value and Confidence Bound;
**Using SAS Proc FREQ;
data temp1;
input batch bleeding $ count;
datalines;
1 have 2
1 no 3
2 have 3
2 no 7
3 have 4
3 no 11
run;
proc freq;
weight count;
tables bleeding / binomial (p=0.1) alpha=0.2 cl; **p=1 option indicates the standard rate to compare with,
here we assume the standard bleeding rate is 10%;
**Alpha=0.2 to obtain one side 90% lower limit;
exact binomial; *Obtain the exact p-value;
by batch;
run;
* This Program Calculates EXACT Confidence Bounds
* on proportion of "successes" in a Binomial Distribution
* See Sachs, Applied Statistics, p. 333
*
* Author: Mark DeHaan msd@inel.gov
* Date: 4/21/92
*
* Macro Variable Names & Definitions
* nsuccess= number of Successes found in sample
* sampsize= sample size
* twosided= two sided confidence bounds (0=no, 1=yes)
* conflev= confidence level for estimates (e.g. .95)
******************************************************;
%macro p_conf(nsuccess,sampsize,twosided,conflev);
Data null;
phat=&nsuccess/&sampsize;
if &twosided then do; *compute a two-sided conf. bound;
clevel=&conflev+((1-&conflev)/2);
put;put;
put "***Note: Two sided &conflev confidence bounds***";
end;
else do;
clevel=&conflev;
put;put;
put "***Note: One sided &conflev confidence bounds***";
end;
f_low=finv(clevel,2*(&sampsize-&nsuccess+1),2*&nsuccess);
f_up=finv(clevel,2*(&nsuccess+1),2*(&sampsize-&nsuccess));
low_phat=&nsuccess/(&nsuccess+(&sampsize-&nsuccess+1)*f_low);
up_phat=(&nsuccess+1)*f_up/(&sampsize-&nsuccess+(&nsuccess+1)*f_up);
put "Number of successes= &nsuccess Sample size= &sampsize";
put phat= low_phat= up_phat=;
put;put;
run;
%mend;
%p_conf(7,20,0,.975)
%p_conf(7,20,1,.95)
%p_conf(2,5,0,.90)
%p_conf(2,5,1,.80)
---------------------------------------------------------------
Date: Tue, 5 Jun 90 09:14:57 GMT
Reply-To: LESLIE DALY PhD <LDALY@IRLEARN>
Sender: "SAS(r) Discussion" <SAS-L@OHSTVMA>
From: LESLIE DALY PhD <LDALY@IRLEARN.BITNET>
Subject: POISSON (AND bINOMIAL) CONFIDENCE INTERVALS
To: Barry W Grau <U42054@UICVM.BITNET>
Stephen Hull asked for SAS code for a Poisson confidence interval. I
attach below two short MACROS to get confidence intervals for both the
binomial and Poisson cases. I have found these to be most useful. The
easiest way to use them is to copy this mail into a file and delete
all except the MACROS. %INCLUDE this file in the SAS program and
run the macro using %CIPOISS etc. The MACROS are commented in full.
%macro cibinom(CI,n,k,P,PL,PR);
*****************************************************************
* *
* MACRO FOR BINOMIAL CONFIDENCE INTERVALS *
* (Leslie Daly LDALY@IRLEARN.BITNET VERSION 1.1) *
* *
* This SAS Macro calculates the confidence interval for a *
* binomial distribution. The method is that given in the *
* Geigy Scientific Tables Vol 2 (Ed C Lentner) (Ciba-Geigy *
* Ltd, Basle, Switzerland) p221. It is one of the most *
* accurate approximations available in a closed form. *
* *
* This reference should be consulted for an explanation of *
* the code below. *
* *
* USAGE: %CIBINOM(CI,N,K,P,PL,PU); *
* *
* INPUT PARAMETERS: CI - Confidence level as a proportion *
* (ie for 95% confidence interval *
* CI should be 0.95) *
* N - Total number of trials *
* K - Number of successes *
* OUTPUT PARAMETERS; P - Observed prob. of success (K/N) *
* PL - Lower confidence limit *
* PU - Upper confidence limit *
* *
* *
*****************************************************************;
ALPHA = (1 - &CI)/2;
&P = &K/&N;
IF &K EQ 0 THEN DO;
&PL = 0;
&PR = 1 - 10**(LOG10(ALPHA)/&N);
END;
IF &K EQ &N THEN DO;
&PL = 10**(LOG10(ALPHA)/&N);
&PR = 1;
END;
IF &K GT 0 AND &K LT &N THEN DO;
calpha = probit(1 - ALPHA);
a = (calpha**4)/18 + (calpha**2)*(2*&N + 1)/6 + (&N + 1/3)**2 ;
AL = &K;
AR = &K + 1;
BL = (CALPHA**4)/18 + (CALPHA**2)*(4*(&N - AL) + 3)/6
+ 2*(AL*(3*&N + 1) - &N)/3 - 2/9;
CL = (CALPHA**4)/18 + (AL - 1/3)**2 - (CALPHA**2)*(2*AL - 1)/6;
&PL = BL/(2*A) - SQRT( (BL/(2*A))**2 - CL/A );
BR = (CALPHA**4)/18 + (CALPHA**2)*(4*(&N - AR) + 3)/6
+ 2*(AR*(3*&N + 1) - &N)/3 - 2/9;
CR = (CALPHA**4)/18 + (AR - 1/3)**2 - (CALPHA**2)*(2*AR - 1)/6;
&PR = BR/(2*A) + SQRT( (BR/(2*A))**2 - CR/A );
END;
%MEND CIBINOM;
%MACRO cipoiss(CI,K,KL,KR);
*****************************************************************
* *
* MACRO FOR POISSON CONFIDENCE INTERVALS *
* (Leslie Daly LDALY@IRLEARN.BITNET VERSION 1.1) *
* *
* This SAS Macro calculates the confidence interval for a *
* Poisson distribution. The confidence lilmits are exact *
* and are based on the relationship between the chisquare *
* distribution and the Poisson. *
* If CHI(df,P) represents the chisquuare value which on *
* df degrees of freedom cuts off an area P in the upper *
* tail of the distribution (ie CHI(1,0.05) = 3.84) then *
* for a count of x (observed) the 95% Confidence limits are: *
* Lower: 0.5*( CHI(2*X,0.975) *
* Upper: 0.5*( CHI(2*X +2,0.025) *
* with obvious adjustments for 99% limits etc. *
* *
* In SAS the chisquare value can be obtained using the *
* function GAMINV. In the notation above (watch the *
* probability levels) *
* *
* CHI(DF,P) = 2*GAMINV(1-P,0.5*DF) *
* *
* hence the code below. *
* *
* *
* USAGE: %CIPOISS(CI,K,KL,KR) *
* *
* *
* INPUT PARAMETERS: CI - Confidence level as a proportion *
* (ie for 95% confidence interval *
* CI should be 0.95) *
* K - Observed number of events *
* OUTPUT PARAMETERS; KL - Lower confidence limit *
* KU - Upper confidence limit *
* *
* *
*****************************************************************;
ALPHA = (1 - &CI)/2;
if &k ne 0 then do;
&KR = gaminv (1 - ALPHA,&K+1);
&KL = gaminv (ALPHA,&K);
END;
IF &K EQ 0 THEN DO;
&KR = -LOG(ALPHA);
&KL = 0;
END;
%MEND CIPOISS;