Type:	Package
Title:	Calculate Confidence Intervals
Version:	0.1.0
Description:	This calculates a variety of different CIs for proportions and difference of proportions that are commonly used in the pharmaceutical industry including Wald, Wilson, Clopper-Pearson, Agresti-Coull and Jeffreys for proprotions. And Miettinen-Nurminen (1985) <doi:10.1002/sim.4780040211>, Wald, Haldane, and Mee https://www.lexjansen.com/wuss/2016/127_Final_Paper_PDF.pdf for difference in proportions.
License:	Apache License (≥ 2)
URL:	https://gsk-biostatistics.github.io/cicalc/
Depends:	R (≥ 4.1.0)
Imports:	broom, cli, dplyr, forcats, glue, purrr, rlang, tidyr
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
Encoding:	UTF-8
RoxygenNote:	7.3.2
NeedsCompilation:	no
Packaged:	2025-07-17 09:28:42 UTC; christinafillmore
Author:	Christina Fillmore [aut, cre], GlaxoSmithKline Research & Development Limited [cph, fnd], Mike Sprys [aut], Dan Lythgoe [aut]
Maintainer:	Christina Fillmore <christina.e.fillmore@gsk.com>
Repository:	CRAN
Date/Publication:	2025-07-21 08:50:06 UTC

Agresti-Coull CI

Description

Calculates the Agresti-Coull interval (created by ⁠Alan Agresti⁠ and ⁠Brent Coull⁠) by (for 95% CI) adding two successes and two failures to the data and then using the Wald formula to construct a CI.

Usage

ci_prop_agresti_coull(x, conf.level = 0.95, data = NULL)

Arguments

Details

\left( \frac{\tilde{p} + z^2_{\alpha/2}/2}{n + z^2_{\alpha/2}} \pm z_{\alpha/2} \sqrt{\frac{\tilde{p}(1 - \tilde{p})}{n} + \frac{z^2_{\alpha/2}}{4n^2}} \right)

Value

An object containing the following components:

Clopper-Pearson CI

Description

Calculates the Clopper-Pearson interval by calling stats::binom.test(). Also referred to as the exact method.

Usage

ci_prop_clopper_pearson(x, conf.level = 0.95, data = NULL)

Arguments

Details

\left( \frac{k}{n} \pm z_{\alpha/2} \sqrt{\frac{\frac{k}{n}(1-\frac{k}{n})}{n} + \frac{z^2_{\alpha/2}}{4n^2}} \right) / \left( 1 + \frac{z^2_{\alpha/2}}{n} \right)

Value

An object containing the following components:

Haldane Confidence Interval for Difference in Proportions

Description

Haldane Confidence Interval for Difference in Proportions

Usage

ci_prop_diff_haldane(x, by, conf.level = 0.95, data = NULL)

Arguments

Details

The confidence interval is calculated by \theta^* \pm w where:

\theta^* = \frac{(\hat{p}_1 - \hat{p}_2) + z^2v(1-2\hat{\psi})}{1+z^2u}

where

w = \frac{z}{1+z^2u}\sqrt{u\{4\hat{\psi}(1-\hat{\psi})-(\hat{p}_1 - \hat{p}_2)^2\}+2v(1-2\hat{\psi})(\hat{p}_1-\hat{p}_2) +4z^2v^2(1-2\hat{\psi})^2 }

\hat{\psi} = \frac{\hat{p}_1 + \hat{p}_2}{2}

u = \frac{1/n_1 + 1/n_2}{4}

v = \frac{1/n_1 - 1/n_2}{4}

Value

An object containing the following components:

References

Constructing Confidence Intervals for the Differences of Binomial Proportions in SAS

Examples

responses <- expand(c(9, 3), c(10, 10))
arm <- rep(c("treat", "control"), times = c(10, 10))

# Calculate 95% confidence interval for difference in proportions
ci_prop_diff_haldane(x = responses, by = arm)

Jeffreys-Perks Confidence Interval for Difference in Proportions

Description

Jeffreys-Perks Confidence Interval for Difference in Proportions

Usage

ci_prop_diff_jp(x, by, conf.level = 0.95, data = NULL)

Arguments

Details

The confidence interval is calculated by \theta^* \pm w where:

\theta^* = \frac{(\hat{p}_1 - \hat{p}_2) + z^2v(1-2\hat{\psi})}{1+z^2u}

where

w = \frac{z}{1+z^2u}\sqrt{u\{4\hat{\psi}(1-\hat{\psi})-(\hat{p}_1 - \hat{p}_2)^2\}+2v(1-2\hat{\psi})(\hat{p}_1-\hat{p}_2) +4z^2v^2(1-2\hat{\psi})^2 }

\hat{\psi} = \frac{1}{2}\left(\frac{x_1 + 1/2}{n_1+1}+\frac{x_2 + 1/2}{n_2+1}\right)

u = \frac{1/n_1 + 1/n_2}{4}

v = \frac{1/n_1 - 1/n_2}{4}

Value

An object containing the following components:

References

Constructing Confidence Intervals for the Differences of Binomial Proportions in SAS

Examples

responses <- expand(c(9, 3), c(10, 10))
arm <- rep(c("treat", "control"), times = c(10, 10))

# Calculate 95% confidence interval for difference in proportions
ci_prop_diff_jp(x = responses, by = arm)

Mee Confidence Interval for Difference in Proportions

Description

Mee Confidence Interval for Difference in Proportions

Usage

ci_prop_diff_mee(x, by, conf.level = 0.95, delta = NULL, data = NULL)

Arguments

Details

The confidence interval is calculated by \theta^* \pm w where:

\theta^* = \frac{(\hat{p}_1 - \hat{p}_2) + z^2v(1-2\hat{\psi})}{1+z^2u}

where

w = \frac{z}{1+z^2u}\sqrt{u\{4\hat{\psi}(1-\hat{\psi})-(\hat{p}_1 - \hat{p}_2)^2\}+2v(1-2\hat{\psi})(\hat{p}_1-\hat{p}_2) +4z^2v^2(1-2\hat{\psi})^2 }

\hat{\psi} = \frac{1}{2}\left(\frac{x_1 + 1/2}{n_1+1}+\frac{x_2 + 1/2}{n_2+1}\right)

u = \frac{1/n_1 + 1/n_2}{4}

v = \frac{1/n_1 - 1/n_2}{4}

Value

An object containing the following components:

References

Constructing Confidence Intervals for the Differences of Binomial Proportions in SAS

Examples

responses <- expand(c(9, 3), c(10, 10))
arm <- rep(c("treat", "control"), times = c(10, 10))

# Calculate 95% confidence interval for difference in proportions
ci_prop_diff_mee(x = responses, by = arm)

Miettinen-Nurminen Confidence Interval for Difference in Proportions

Description

Calculates the Miettinen-Nurminen (MN) confidence interval for the difference between two proportions. This method can be more accurate than traditional methods, especially with small sample sizes or proportions close to 0 or 1.

Usage

ci_prop_diff_mn(x, by, conf.level = 0.95, delta = NULL, data = NULL)

Arguments

Details

The function implements the Miettinen-Nurminen method to compute confidence intervals for the difference between two proportions. This approach:

• Calculates the Miettinen-Nurminen score test statistic for different possible values of the proportion difference (delta)

• Identifies the delta values where the test statistic equals the critical value corresponding to the desired confidence level

• Returns these boundary values as the confidence interval limits

The method uses a score test with a small-sample correction factor, making it more accurate than normal approximation methods, especially for small samples or extreme proportions. The equation for the test statistics is as follows:

H_0: \hat{d}-\delta <= 0 \qquad \text{vs.} \qquad H_1: \hat{d}-\delta > 0

T_\delta = \frac{\hat{p_x} - \hat{p_y} - \delta}{\sigma_{mn}(\delta)}

where \hat{p_*} = s_*/n_* represent the observed number of successes divided by the number of participant in that group. The \sigma_{mn}(\delta) is a function of the delta values and is create with the following equation" \tilde{p_*} represent the MLE of the proportions.

\sigma_{mn}(\delta) = \sqrt{\left[\frac{\tilde{p_y}(1-\tilde{p_y})}{n_x}+\frac{\tilde{p_x}(1-\tilde{p_x})}{n_y} \right]\left(\frac{N}{N-1}\right)}

\tilde{p_x} = 2p\cdot{cos(a)} - \frac{L_2}{3L_3} and \tilde{p_y} = \tilde{p_x} + \delta where:

• p = \pm \sqrt{\frac{L_2^2}{(3L_3)^2} - \frac{L_1}{3L_3}}

• a = 1/3[\pi + cos^{-1}(q/p^3)]

• q = \frac{L_2^3}{(3L_3)^3} - \frac{L_1L_2}{6L_3^2} + \frac{L_0}{2L_3}

• L_3 = n_x + n_y

• L_2 = (n_x + 2 n_y)\delta - N - (s_x + s_y)

• L_1 = (n_y\delta - L_3 - 2s_y)\delta + s_x + s_y

• L_0 = s_y\delta(1-\delta)

For more information about these equations see Miettinen (1985)

Value

An object containing the following components:

If delta is not provided statistic and p.value will be NULL

References

Miettinen, O. S., & Nurminen, M. (1985). Comparative analysis of two rates. Statistics in Medicine, 4(2), 213-226.

Examples

# Generate binary samples
responses <- expand(c(9, 3), c(10, 10))
arm <- rep(c("treat", "control"), times = c(10, 10))

# Calculate 95% confidence interval for difference in proportions
ci_prop_diff_mn(x = responses, by = arm)

# Calculate 99% confidence interval
ci_prop_diff_mn(x = responses, by = arm, conf.level = 0.99)

# Calculate the p-value under the null hypothesis delta = -0.1
ci_prop_diff_mn(x = responses, by = arm, delta = -0.1)

# Calculate from a data.frame
data <- data.frame(responses, arm)
ci_prop_diff_mn(x = responses, by = arm, data = data)

Stratified Miettinen-Nurminen Confidence Interval for Difference in Proportions

Description

Calculates Stratified Miettinen-Nurminen (MN) confidence intervals and corresponding point estimates for the difference between two proportions

Usage

ci_prop_diff_mn_strata(
  x,
  by,
  strata,
  method = c("score", "summary score"),
  conf.level = 0.95,
  delta = NULL,
  data = NULL
)

Arguments

Details

The function implements the stratified Miettinen-Nurminen method to compute confidence intervals for the difference between two proportions across multiple strata.

H_0: \hat{d}-\delta <= 0 \qquad \text{vs.} \qquad H_1: \hat{d}-\delta > 0

The "score" method is a weighted MN score first described in the original 1985 paper. The formula is:

• Calculates weights for each stratum as w_i = \frac{n_{xi} \cdot n_{yi}}{n_{xi} + n_{yi}}

• Computes the overall weighted difference \hat{d} = \frac{\sum w_i \hat{p}_{xi}}{\sum w_i} - \frac{\sum w_i \hat{p}_{yi}}{\sum w_i}

• Uses the stratified test statistic:

  Z_{\delta} = \frac{\hat{d} - \delta} {\sqrt{\sum_{i=1}^k \left(\frac{w_i}{\sum w_i}\right)^2 \cdot \hat{\sigma}_{mn}^2({d})}}

• Finds the range of all values of \delta for which the stratified test statistic (Z_\delta) falls in the acceptance region \{ Z_\delta < z_{\alpha/2}\}

The \hat{\sigma}_{mn}^2(\hat{d}) is the Miettinen-Nurminen variance estimate. See the details of ci_prop_diff_mn() for how \hat{\sigma}_{mn}^2(\delta) is calculated.

The "summary score" method follows the meta-analyses proposed in Agresti 2013 and is consistent with the "Summary Score Confidence Limits" method used in SAS. The formula is:

• The point estimate of the stratified risk difference is a weighted average of the midpoints of the within-stratum MN confidence intervals:

  \hat{d}_{\text{S}} = \sum_i \hat{d}_i w_i

• Define s_i as the width of the CI for the ith stratum divided by 2 \times z_{\alpha/2} and then stratum weights are given by

  w_i = \left( \frac{1}{s_i^2} \right) \bigg/ \sum_i \left( \frac{1}{s_i^2} \right)

• The variance of \hat{d}_{\text{S}} is computed as

  \widehat{\text{Var}}(\hat{d}_{\text{S}}) = \frac{1}{\sum_i \left( \frac{1}{s_i^2} \right) }

• Confidence limits for the stratified risk difference estimate are

  \hat{d}_{\text{S}} \pm \left( z_{\alpha /2} \times \widehat{\text{Var}}(\hat{d}_{\text{S}}) \right)

Value

An object containing the following components:

If delta is not provided statistic and p.value will be NULL

References

Miettinen, O. S., & Nurminen, M. (1985). Comparative analysis of two rates. Statistics in Medicine, 4(2), 213-226.

Common Risk Difference :: Base SAS(R) 9.4 Procedures Guide: Statistical Procedures, Third Edition

Agresti, A. (2013). Categorical Data Analysis. 3rd Edition. John Wiley & Sons, Hoboken, NJ

Examples

# Generate binary samples with strata
responses <- expand(c(9, 3, 7, 2), c(10, 10, 10, 10))
arm <- rep(c("treat", "control"), 20)
strata <- rep(c("stratum1", "stratum2"), times = c(20, 20))

# Calculate stratified confidence interval for difference in proportions
ci_prop_diff_mn_strata(x = responses, by = arm, strata = strata)

# Using the summary score method
ci_prop_diff_mn_strata(x = responses, by = arm, strata = strata,
                      method = "summary score")

# Calculate 99% confidence interval
ci_prop_diff_mn_strata(x = responses, by = arm, strata = strata,
                      conf.level = 0.99)

# Calculate p-value under null hypothesis delta = 0.2
ci_prop_diff_mn_strata(x = responses, by = arm, strata = strata,
                      delta = 0.2)

Wald Confidence Interval for Difference in Proportions

Description

Calculates the Wald interval by following the usual textbook definition for a difference in proportions confidence interval using the normal approximation.

Usage

ci_prop_diff_wald(x, by, conf.level = 0.95, correct = FALSE, data = NULL)

Arguments

Details

(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1}+\frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}

Value

An object containing the following components:

Examples

responses <- expand(c(9, 3), c(10, 10))
arm <- rep(c("treat", "control"), times = c(10, 10))

# Calculate 95% confidence interval for difference in proportions
ci_prop_diff_wald(x = responses, by = arm)

Jeffreys CI

Description

Calculates the Jeffreys interval, an equal-tailed interval based on the non-informative Jeffreys prior for a binomial proportion.

Usage

ci_prop_jeffreys(x, conf.level = 0.95, data = NULL)

Arguments

Details

\left( \text{Beta}\left(\frac{k}{2} + \frac{1}{2}, \frac{n - k}{2} + \frac{1}{2}\right)_\alpha, \text{Beta}\left(\frac{k}{2} + \frac{1}{2}, \frac{n - k}{2} + \frac{1}{2}\right)_{1-\alpha} \right)

Value

An object containing the following components:

Wald CI

Description

Calculates the Wald interval by following the usual textbook definition for a single proportion confidence interval using the normal approximation.

Usage

ci_prop_wald(x, conf.level = 0.95, correct = FALSE, data = NULL)

Arguments

Details

\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}

Value

An object containing the following components:

Examples

# example code
x <- c(
TRUE, TRUE, TRUE, TRUE, TRUE,
FALSE, FALSE, FALSE, FALSE, FALSE
)

ci_prop_wald(x, conf.level = 0.9)

Wilson CI

Description

Calculates the Wilson interval by calling stats::prop.test(). Also referred to as Wilson score interval.

Usage

ci_prop_wilson(x, conf.level = 0.95, correct = FALSE, data = NULL)

Arguments

Details

\frac{\hat{p} + \frac{z^2_{\alpha/2}}{2n} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n} + \frac{z^2_{\alpha/2}}{4n^2}}}{1 + \frac{z^2_{\alpha/2}}{n}}

Value

An object containing the following components:

Stratified Wilson CI

Description

Calculates the stratified Wilson confidence interval for unequal proportions as described in Xin YA, Su XG. Stratified Wilson and Newcombe confidence intervals for multiple binomial proportions. Statistics in Biopharmaceutical Research. 2010;2(3).

Usage

ci_prop_wilson_strata(
  x,
  strata,
  weights = NULL,
  conf.level = 0.95,
  max.iterations = 10L,
  correct = FALSE,
  data = NULL
)

Arguments

Details

\frac{\hat{p}_j + \frac{z^2_{\alpha/2}}{2n_j} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_j(1 - \hat{p}_j)}{n_j} + \frac{z^2_{\alpha/2}}{4n_j^2}}}{1 + \frac{z^2_{\alpha/2}}{n_j}}

Value

An object containing the following components:

Examples

# Stratified Wilson confidence interval with unequal probabilities

set.seed(1)
rsp <- sample(c(TRUE, FALSE), 100, TRUE)
strata_data <- data.frame(
  x = sample(c(TRUE, FALSE), 100, TRUE),
  "f1" = sample(c("a", "b"), 100, TRUE),
  "f2" = sample(c("x", "y", "z"), 100, TRUE),
  stringsAsFactors = TRUE
)
strata <- interaction(strata_data)
n_strata <- ncol(table(rsp, strata)) # Number of strata

ci_prop_wilson_strata(
  x = rsp, strata = strata,
  conf.level = 0.90
)

# Not automatic setting of weights
ci_prop_wilson_strata(
  x = rsp, strata = strata,
  weights = rep(1 / n_strata, n_strata),
  conf.level = 0.90
)

Function to combine strata via interaction if strata is passed as a vector

Description

Function to combine strata via interaction if strata is passed as a vector

Usage

combine_strata(x, strata)

Expand Count Data into Binary Vectors

Description

Converts count data (number of successes and total sample size) into a binary vector of TRUE/FALSE values. This is useful for converting summary statistics back into raw data format for analysis functions that require individual-level data.

Usage

expand(x, n)

Arguments

Details

For each pair of values in x and n, the function creates a vector with x TRUE values followed by n-x FALSE values. If multiple pairs are provided, the resulting vectors are concatenated in order.

Value

A logical vector where TRUE represents a success and FALSE represents a failure. The length of the vector equals the sum of all sample sizes.

Examples

# Convert 4 successes out of 13 participants to binary data
expand(4, 13)

# Convert multiple groups of data
# Group 1: 9 successes out of 10
# Group 2: 3 successes out of 10
expand(c(9, 3), c(10, 10))

To get the n's and response totals with out without strata

Description

To get the n's and response totals with out without strata

Usage

get_counts(x, by, strata = 1)