Select the method of your choice.

- Adjusted Blaker exact confidence interval and test methods
- Clopper-Pearson exact confidence interval method
- Test that corresponds to the Clopper-Pearson exact confidence interval
- Wilson-score confidence interval method
- Score test
- Agresti-Coull confidence interval and test methods
- Confidence interval for Wald normal approximation (Web app)

The adjusted Blaker exact method produces two-sided confidence intervals for the
proportion of events and produces p-values for the alternative hypothesis of
*p* ≠ *p*_{0}. Blaker^{1}^{2} provides an
exact, two-sided confidence interval by inverting the p-value function of an exact
test. The Clopper-Pearson intervals are wider and always contain the Blaker
confidence intervals. Intervals from the Blaker exact method are nested. This
property means that confidence intervals with higher confidence levels contain
confidence intervals with lower confidence levels. For example, an exact, two-sided
Blaker 95% confidence interval contains the corresponding 90% confidence interval.

Blaker's original exact method has 2 limitations. One limitation is that the numerical algorithm to calculate the confidence intervals is slow, particularly when the sample size is large. Another limitation is that for some data, the original Blaker exact method produces an interval that covers a hypothesized proportion when the p-value is less than the significance level that corresponds to the confidence level. The limitation also arises when the confidence interval does not contain a hypothesized proportion when the p-value is greater than the significance level that corresponds to the confidence level.

To overcome these limitations, the analysis in Minitab Statistical Software produces
the confidence interval and p-value using the algorithm from Klaschka and
Reiczigel.^{3} The name of this method is the adjusted Blaker exact method.
This numerical algorithm is faster to calculate and produces confidence intervals
and tests that agree in general. The adjusted Blaker confidence intervals are also
exact and nested.

For an alternative hypothesis with less than or with greater than, the analysis uses the Clopper-Pearson exact method.

The interval (*P*_{L}, *P*_{U}) is a two-sided, 100(1 – *α*)%
confidence interval of *p*. When the sample has no events, then the lower limit is 0.
When the sample has events only, then the upper limit is 1.

Term | Description |
---|---|

v_{1} | 2x |

v_{2} | 2(n – x + 1) |

x | number of events |

n | number of trials |

F | lower α/2 point of F distribution with v_{1} and v_{2} degrees of freedom |

Term | Description |
---|---|

v_{1} | 2(x + 1) |

v_{2} | 2(n – x) |

x | number of events |

n | number of trials |

F | upper α/2 point of F distribution with v_{1} and v_{2} degrees of freedom |

The sample (*X*) comes from binomial distribution with parameters *n* and *p*.
The p-values depend on the alternative hypothesis.

*H*:_{a}*p*≠*p*_{0}- p-value =
*H*:_{a}*p*>*p*_{0}- p-value = P{
*X*≥*x*|*p*=*p*_{o}} *H*:_{a}*p*<*p*_{0}- p-value = P{
*X*≤*x*|*p*=*p*_{o}}

Term | Description |
---|---|

p_{0}
| hypothesized proportion |

n | number of trials |

p | probability of an event |

x | number of events |

Wilson^{4} inverts the score test to obtain
confidence intervals that Minitab Statistical Software names Wilson-score confidence
intervals. Wilson-score intervals have two forms, one without a continuity
correction and one with a continuity correction. The coverage of the intervals
without the correction is sometimes below the nominal confidence level. The actual
confidence level of the intervals with the correction is at least the nominal
confidence level. For both methods when the sample has no events, then the lower
limit is 0. When the sample has events only, then the upper limit is 1.

The two-sided, 100(1 – *α*)% confidence interval has the following formula:

The one-sided, 100(1 – *α*)% lower limit has the following formula:

The one-sided, 100(1 – *α*)% upper limit has the following formula:

The lower bound of the two-sided 100(1 – *α*)% interval has the following
formula:

The upper bound of the two-sided, 100(1 – *α*)% interval has the following
formula:

The one-sided, 100(1 – *α*)% lower limit has the following formula:

The one-sided, 100(1 – *α*)% upper limit has the following formula:

Term | Description |
---|---|

observed probability, = x / n | |

x | number of events |

n | number of trials |

z_{γ}
| the upper percentile point of the standard normal distribution at γ |

α | 1 – confidence level/100 |

The test that corresponds to the Wilson-score confidence interval and the normal approximation method (Web app) is the well-known score test. The score test statistic has the following equation:

The p-value for the test depends on the alternative hypothesis.

*H*:_{a}*p*≠*p*_{0}- p-value =
*H*:_{a}*p*>*p*_{0}- p-value =
*H*:_{a}*p*<*p*_{0}- p-value =

The test statistic and p-value for the procedure with a continuity correction depend on the alternative hypothesis.

*H*:_{a}*p*≠*p*_{0}*H*:_{a}*p*>*p*_{0}*H*:_{a}*p*<*p*_{0}

Term | Description |
---|---|

observed probability, x/n | |

x | number of events |

n | number of trials |

p_{0} | hypothesized proportion |

cumulative distribution function of the standard normal distribution at
y |

Agresti and Coull^{5} provide an
adjustment to the Wald method for confidence intervals that improves the coverage
properties. For a two-sided, 95% confidence interval, the adjustment approximately
adds 2 events and 2 non-events, then calculates the confidence intervals from the
formulas for the Wald confidence interval formulas. When the sample has no events,
then the lower limit is 0. When the sample has events only, then the upper limit is
1.

The two-sided, 100(1 – *α*)% interval has the following formula:

where

and

The one-sided, 100(1 – *α*)% lower limit has the following formula:

The one-sided, 100(1 – *α*)% upper limit has the following formula:

For the one-sided limits, use in the definition of and :

The analysis calculates the p-value for the test by inverting the confidence interval procedure.

Term | Description |
---|---|

x
| number of events |

n
| number of trials |

z_{γ}
| the upper percentile point of the standard normal distribution at
γ |

α
| 1 – confidence level/100 |

Term | Description |
---|---|

observed probability, = x / n | |

x | observed number of events in n trials |

n | number of trials |

z_{α}_{/2} | inverse cumulative probability of the standard normal distribution at 1–α/2 |

α | 1 – confidence level/100 |