# NAG Library Routine Document

## 1Purpose

g07aaf computes a confidence interval for the argument $p$ (the probability of a success) of a binomial distribution.

## 2Specification

Fortran Interface
 Subroutine g07aaf ( n, k, pl, pu,
 Integer, Intent (In) :: n, k Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: clevel Real (Kind=nag_wp), Intent (Out) :: pl, pu
#include nagmk26.h
 void g07aaf_ (const Integer *n, const Integer *k, const double *clevel, double *pl, double *pu, Integer *ifail)

## 3Description

Given the number of trials, $n$, and the number of successes, $k$, this routine computes a $100\left(1-\alpha \right)%$ confidence interval for $p$, the probability argument of a binomial distribution with probability function,
 $fx= n x px1-pn-x, x=0,1,…,n,$
where $\alpha$ is in the interval $\left(0,1\right)$.
Let the confidence interval be denoted by [${p}_{l},{p}_{u}$].
The point estimate for $p$ is $\stackrel{^}{p}=k/n$.
The lower and upper confidence limits ${p}_{l}$ and ${p}_{u}$ are estimated by the solutions to the equations;
 $∑x=kn n x plx 1-pl n-x =α/2 ,$
 $∑x= 0k n x pux 1-pu n-x =α /2 .$
Three different methods are used depending on the number of trials, $n$, and the number of successes, $k$.
1. If $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)<{10}^{6}$.
The relationship between the beta and binomial distributions (see page 38 of Hastings and Peacock (1975)) is used to derive the equivalent equations,
 $pl = βk,n-k+1,α/2, pu = βk+1,n-k,1-α/2,$
where ${\beta }_{a,b,\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the beta distribution with arguments $a$ and $b$. These beta deviates are computed using g01fef.
2. If $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)\ge {10}^{6}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)\le 1000$.
The binomial variate with arguments $n$ and $p$ is approximated by a Poisson variate with mean $np$, see page 38 of Hastings and Peacock (1975).
The relationship between the Poisson and ${\chi }^{2}$-distributions (see page 112 of Hastings and Peacock (1975)) is used to derive the following equations;
 $pl = 12n χ2k,α/22, pu = 12n χ2k+2,1-α/22,$
where ${\chi }_{\delta ,\nu }^{2}$ is the deviate associated with the lower tail probability, $\delta$, of the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom.
In turn the relationship between the ${\chi }^{2}$-distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)) yields the following equivalent equations;
 $pl = 12n γk,2;α/2, pu = 12n γk+1,2;1-α/2,$
where ${\gamma }_{\alpha ,\beta \text{;}\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the gamma distribution with shape argument $\alpha$ and scale argument $\beta$. These deviates are computed using g01fff.
3. If $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)>{10}^{6}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)>1000$.
The binomial variate with arguments $n$ and $p$ is approximated by a Normal variate with mean $np$ and variance $np\left(1-p\right)$, see page 38 of Hastings and Peacock (1975).
The approximate lower and upper confidence limits ${p}_{l}$ and ${p}_{u}$ are the solutions to the equations;
 $k-npl npl1-pl = z1-α/2, k-npu npu1-pu = zα/2,$
where ${z}_{\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the standard Normal distribution. These equations are solved using a quadratic equation solver (c02ajf).

## 4References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of trials.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of successes.
Constraint: $0\le {\mathbf{k}}\le {\mathbf{n}}$.
3:     $\mathbf{clevel}$ – Real (Kind=nag_wp)Input
On entry: the confidence level, $\left(1-\alpha \right)$, for two-sided interval estimate. For example ${\mathbf{clevel}}=0.95$ will give a $95%$ confidence interval.
Constraint: $0.0<{\mathbf{clevel}}<1.0$.
4:     $\mathbf{pl}$ – Real (Kind=nag_wp)Output
On exit: the lower limit, ${p}_{l}$, of the confidence interval.
5:     $\mathbf{pu}$ – Real (Kind=nag_wp)Output
On exit: the upper limit, ${p}_{u}$, of the confidence interval.
6:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{k}}<0$, or ${\mathbf{n}}<{\mathbf{k}}$, or ${\mathbf{clevel}}\le 0.0$, or ${\mathbf{clevel}}\ge 1.0$.
${\mathbf{ifail}}=2$
When using the relationship with the gamma distribution to calculate one of the confidence limits, the series to calculate the gamma probabilities has failed to converge. Both pl and pu are set to zero. This is a very unlikely error exit and if it occurs please contact NAG.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For most cases using the beta deviates the results should have a relative accuracy of $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\text{0.5E−12},50.0×\epsilon \right)$ where $\epsilon$ is the machine precision (see x02ajf). Thus on machines with sufficiently high precision the results should be accurate to $12$ significant figures. Some accuracy may be lost when $\alpha /2$ or $1-\alpha /2$ is very close to $0.0$, which will occur if clevel is very close to $1.0$. This should not affect the usual confidence levels used.
The approximations used when $n$ is large are accurate to at least $3$ significant digits but usually to more.

## 8Parallelism and Performance

g07aaf is not threaded in any implementation.

None.

## 10Example

The following example program reads in the number of deaths recorded among male recipients of war pensions in a six year period following an initial questionnaire in 1956. We consider two classes, non-smokers and those who reported that they smoked pipes only. The total number of males in each class is also read in. The data is taken from page 216 of Snedecor and Cochran (1967). An estimate of the probability of a death in the six year period in each class is computed together with 95% confidence intervals for these estimates.

### 10.1Program Text

Program Text (g07aafe.f90)

### 10.2Program Data

Program Data (g07aafe.d)

### 10.3Program Results

Program Results (g07aafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017