g07aac (ci_binomial) : NAG Library CL Interface, Mark 27

3 Description

Given the number of trials,

n

, and the number of successes,

k

, this function computes a

100 (1 - α) %

confidence interval for

p

, the probability parameter of a binomial distribution with probability function,

f (x) = (\begin{matrix} n \\ x \end{matrix}) p^{x} {(1 - p)}^{n - x}, x = 0, 1, \dots, n,

where

α

is in the interval

(0, 1)

Let the confidence interval be denoted by [

p_{l}, p_{u}

The point estimate for

p

\hat{p} = k / n

The lower and upper confidence limits

p_{l}

and

p_{u}

are estimated by the solutions to the equations;

\sum_{x = k}^{n} (\begin{matrix} n \\ x \end{matrix}) p_{l}^{x} {(1 - p_{l})}^{n - x} = α / 2,

\sum_{x = 0}^{k} (\begin{matrix} n \\ x \end{matrix}) p_{u}^{x} {(1 - p_{u})}^{n - x} = α / 2 .

Three different methods are used depending on the number of trials,

n

, and the number of successes,

k

1.If $\max (k, n - k) < 10^{6}$ .
The relationship between the beta and binomial distributions (see page 38 of Hastings and Peacock (1975)) is used to derive the equivalent equations,

$\begin{array}{lcl} p_{l} & = & β_{k, n - k + 1, α / 2}, \\ p_{u} & = & β_{k + 1, n - k, 1 - α / 2}, \end{array}$

where $β_{a, b, δ}$ is the deviate associated with the lower tail probability, $δ$ , of the beta distribution with parameters $a$ and $b$ . These beta deviates are computed using g01fec.
2.If $\max (k, n - k) \geq 10^{6}$ and $\min (k, n - k) \leq 1000$ .
The binomial variate with parameters $n$ and $p$ is approximated by a Poisson variate with mean $n p$ , see page 38 of Hastings and Peacock (1975).

The relationship between the Poisson and $χ^{2}$ -distributions (see page 112 of Hastings and Peacock (1975)) is used to derive the following equations;

$\begin{array}{lcl} p_{l} & = & \frac{1}{2 n} χ_{2 k, α / 2}^{2}, \\ p_{u} & = & \frac{1}{2 n} χ_{2 k + 2, 1 - α / 2}^{2}, \end{array}$

where $χ_{δ, ν}^{2}$ is the deviate associated with the lower tail probability, $δ$ , of the $χ^{2}$ -distribution with $ν$ degrees of freedom.

In turn the relationship between the $χ^{2}$ -distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)) yields the following equivalent equations;

$\begin{array}{lcl} p_{l} & = & \frac{1}{2 n} γ_{k, 2; α / 2}, \\ p_{u} & = & \frac{1}{2 n} γ_{k + 1, 2; 1 - α / 2}, \end{array}$

where $γ_{α, β; δ}$ is the deviate associated with the lower tail probability, $δ$ , of the gamma distribution with shape parameter $α$ and scale parameter $β$ . These deviates are computed using g01ffc.
3.If $\max (k, n - k) > 10^{6}$ and $\min (k, n - k) > 1000$ .
The binomial variate with parameters $n$ and $p$ is approximated by a Normal variate with mean $n p$ and variance $n p (1 - p)$ , 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;

$\begin{array}{lcl} \frac{k - n p_{l}}{\sqrt{n p_{l} (1 - p_{l})}} & = & z_{1 - α / 2}, \\ \frac{k - n p_{u}}{\sqrt{n p_{u} (1 - p_{u})}} & = & z_{α / 2}, \end{array}$

where $z_{δ}$ is the deviate associated with the lower tail probability, $δ$ , of the standard Normal distribution. These equations are solved using a quadratic equation solver .

5 Arguments

n

– Integer Input

On entry:

n

, the number of trials.

Constraint:

n \geq 1

k

– Integer Input

On entry:

k

, the number of successes.

Constraint:

0 \leq k \leq n

clevel

– double Input

On entry: the confidence level,

(1 - α)

, for two-sided interval estimate. For example

clevel = 0.95

will give a

95 %

confidence interval.

Constraint:

0.0 < clevel < 1.0

pl

– double * Output

On exit: the lower limit,

p_{l}

, of the confidence interval.

pu

– double * Output

On exit: the upper limit,

p_{u}

, of the confidence interval.

fail

– NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument

⟨ value ⟩

had an illegal value.

NE_CONVERGENCE

When using the relationship with the gamma distribution the series to calculate the gamma probabilities has failed to converge. Both pl and pu are set to zero. This is an unlikely error exit.

NE_INT

On entry,

k = ⟨ value ⟩

.
Constraint:

k \geq 0

On entry,

n = ⟨ value ⟩

.
Constraint:

n \geq 1

NE_INT_2

On entry,

n = ⟨ value ⟩

and

k = ⟨ value ⟩

.
Constraint:

n \geq k

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

NE_REAL

On entry,

clevel = ⟨ value ⟩

.
Constraint:

0.0 < clevel < 1.0

7 Accuracy

For most cases using the beta deviates the results should have a relative accuracy of

\max (0.5e−12, 50.0 \times ε)

where

ε

is the machine precision (see X02AJC). Thus on machines with sufficiently high precision the results should be accurate to

12

significant figures. Some accuracy may be lost when

α / 2

1 - α / 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.

10 Example

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.

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g07aac (ci_binomial)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interfaceg07aac (ci_​binomial)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g07aac (ci_binomial)