In this case, the Z-statistic based on the expected proportion (the null hypothesis H₀: proportion is equal to the expected proportion) is:

      If the Full Wald Output box is checked, then the missing parts of the Wald output, i.e. one- and two-tailed probabilities based on the sample standard error and the confidence limits under H₀ are also displayed. The user should take care with the interpretation of this extended output.

Wilson (score): The confidence limits without continuity correction are:

      Earlier versions of UNISTAT report these confidence limits for the Asymptotic without Continuity Correction case.

Wilson with Continuity Correction:

      Earlier versions of UNISTAT report these confidence limits for the Asymptotic with Continuity Correction case.

Agresti-Coull: This is similar to Wald interval but  (i.e. half of the square of normal critical value) is added to numbers of successes and failures:

      where:

      If the Full Wald Output box is checked, then the following Z-statistic and its one- and two-tailed probabilities are also displayed:

Agresti-Coull (+2): This similar to the Agresti-Coull interval except that 2 (a crude approximation to ) is added to the numbers of successes and failures:

      If the Full Wald Output box is checked, then the following Z-statistic and its one- and two-tailed probabilities are also displayed:

Jeffreys: The confidence limits are defined as the following critical values from the inverse beta distribution:

Clopper-Pearson (exact): The exact one- and two-tailed binomial probabilities are reported. The exact confidence interval is calculated as:

Example 1

Table I on p. 861 from Newcombe, R. G. (1998) where examples with five confidence intervals supported by UNISTAT are given. The following group sizes are given for the second column of the results table.

Select Statistics 1 → Nonparametric Tests (1-2 Samples) → Binomial Proportion and select the data option 3 Cell Frequencies are Given. Enter the above group sizes and check only the Binomial Test output option to obtain the following results:

Binomial Proportion

Data option: Test Statistics are Given

Binomial Test

Example 2

Example 4.7 on p. 119 from Armitage, P. & G. Berry (1994). Patients’ preference for two analgesic drugs, X and Y is recorded. The null hypothesis “the ratio of preferences is not different from 50%” is tested.

Select Statistics 1 → Nonparametric Tests (1-2 Samples) → Binomial Proportion and select the data option 3 Cell Frequencies are Given. Enter values in the above table and check only the Binomial Test output option to obtain the following results:

Binomial Proportion

Data option: Test Statistics are Given

Binomial Test

This result is significant at the 1% level. Hence reject the null hypothesis “the patients have no significant preference for a particular analgesic drug”.

6.4.3.3. Noninferiority Test

The null hypothesis tested is that “the expected proportion is worse than the expected proportion by a given margin δ”. The alternative hypothesis is “the observed proportion is not inferior.”

The noninferiority test similar to Binomial Test with the exception that the expected proportion is reduced by the noninferiority margin δ. Also, the Z‑statistic is based on the observed proportion (unlike the Binomial Test where it is based on the expected proportion H₀). The confidence limits are reported at 1 ‑ 2α level, rather than the usual 1 ‑ α.

Wald: By default, the Z-statistic and the confidence interval are both based on the sample standard error:

      where:

      is the observed proportion and:

      and:

      If the Full Wald Output box is checked, then on the next line, the Z-statistic and confidence interval based on the noninferiority limit are also reported:

      where:

Wald with Continuity

Clopper-Pearson (exact): The exact one- and two-tailed binomial probabilities and the exact confidence interval are reported.