Package 'cicalc'

Title: Calculate Confidence Intervals
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 proportions. 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.
Authors: Christina Fillmore [aut, cre] (ORCID: <https://orcid.org/0000-0003-0595-2302>), GlaxoSmithKline Research & Development Limited [cph, fnd], Mike Sprys [aut], Dan Lythgoe [aut] (ORCID: <https://orcid.org/0000-0002-4682-772X>)
Maintainer: Christina Fillmore <[email protected]>
License: Apache License (>= 2)
Version: 0.2.1
Built: 2026-05-08 15:21:29 UTC
Source: https://github.com/gsk-biostatistics/cicalc

Help Index

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

(p~+zα/22/2n+zα/22±zα/2p~(1−p~)n+zα/224n2)\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:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Type of method used

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

(1+N−n+1nF[α2;2n,2(N−n+1)])−1,(1+N−n(n+1)F[1−α2;2(n+1),2(N−n)])−1\left(1 + \frac{N-n+1}{nF[\frac{\alpha}{2};2n,2(N-n+1)]} \right)^{-1}, \left( 1 + \frac{N-n}{(n+1)F[1-\frac{\alpha}{2};2(n+1),2(N-n)]} \right)^{-1}

Value

An object containing the following components:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Type of method used

Anderson-Hauck Confidence Interval for Difference in Proportions

Description

Anderson-Hauck Confidence Interval for Difference in Proportions

Usage

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

Arguments

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

The confidence interval is given by:

(p^1−p^2)±[12min⁡(n1,n2)+zp^1(1−p^1)n1−1+p^2(1−p^2)n2−1](\hat{p}_1 - \hat{p}_2) \pm \left[ \frac{1}{2 \min(n_1, n_2)} + z \sqrt{ \frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1 - 1} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2 - 1} } \right]

.

Value

An object containing the following components:

n

The number of responses for each group
N

The total number in each group
estimate

The point estimate of the difference in proportions
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Anderson-Hauck Confidence Interval

References

Hauck WW, Anderson S. (1986) A comparison of large-sample confidence interval methods for the difference of two binomial probabilities The American Statistician 40(4). p.318-322. 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_ha(x = responses, by = arm)

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

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

θ∗=(p^1−p^2)+z2v(1−2ψ^)1+z2u\theta^* = \frac{(\hat{p}_1 - \hat{p}_2) + z^2v(1-2\hat{\psi})}{1+z^2u}

where

w=z1+z2uu{4ψ^(1−ψ^)−(p^1−p^2)2}+2v(1−2ψ^)(p^1−p^2)+4z2v2(1−2ψ^)2w = \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 }

ψ^=p^1+p^22\hat{\psi} = \frac{\hat{p}_1 + \hat{p}_2}{2}

u=1/n1+1/n24u = \frac{1/n_1 + 1/n_2}{4}

v=1/n1−1/n24v = \frac{1/n_1 - 1/n_2}{4}

Value

An object containing the following components:

n

The number of responses for each group
N

The total number in each group
estimate

The point estimate of the difference in proportions (theta*)
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Haldane Confidence Interval

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

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

θ∗=(p^1−p^2)+z2v(1−2ψ^)1+z2u\theta^* = \frac{(\hat{p}_1 - \hat{p}_2) + z^2v(1-2\hat{\psi})}{1+z^2u}

where

w=z1+z2uu{4ψ^(1−ψ^)−(p^1−p^2)2}+2v(1−2ψ^)(p^1−p^2)+4z2v2(1−2ψ^)2w = \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 }

ψ^=12(x1+1/2n1+1+x2+1/2n2+1)\hat{\psi} = \frac{1}{2}\left(\frac{x_1 + 1/2}{n_1+1}+\frac{x_2 + 1/2}{n_2+1}\right)

u=1/n1+1/n24u = \frac{1/n_1 + 1/n_2}{4}

v=1/n1−1/n24v = \frac{1/n_1 - 1/n_2}{4}

Value

An object containing the following components:

n

The number of responses for each group
N

The total number in each group
estimate

The point estimate of the difference in proportions (theta*)
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Jeffreys-Perks Confidence Interval

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
delta

(numeric)
Optionally a single number or a vector of numbers between -1 and 1 (not inclusive) to set the difference between two groups under the null hypothesis. If provided, the function returns the test statistic and p-value under the delta hypothesis.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

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

θ∗=(p^1−p^2)+z2v(1−2ψ^)1+z2u\theta^* = \frac{(\hat{p}_1 - \hat{p}_2) + z^2v(1-2\hat{\psi})}{1+z^2u}

where

w=z1+z2uu{4ψ^(1−ψ^)−(p^1−p^2)2}+2v(1−2ψ^)(p^1−p^2)+4z2v2(1−2ψ^)2w = \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 }

ψ^=12(x1+1/2n1+1+x2+1/2n2+1)\hat{\psi} = \frac{1}{2}\left(\frac{x_1 + 1/2}{n_1+1}+\frac{x_2 + 1/2}{n_2+1}\right)

u=1/n1+1/n24u = \frac{1/n_1 + 1/n_2}{4}

v=1/n1−1/n24v = \frac{1/n_1 - 1/n_2}{4}

Value

An object containing the following components:

n

The number of responses for each group
N

The total number in each group
estimate

The point estimate of the difference in proportions (p1-p2)
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Mee Confidence Interval

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)

Mantel-Haenszel Common Risk Difference Confidence Interval

Description

Calculates the confidence interval for the Mantel-Haenszel estimate of the common risk difference across multiple 2x2 tables (strata), using the Sato or Independent Binomial variance estimator.

Usage

ci_prop_diff_mh_strata(
  x,
  by,
  strata,
  conf.level = 0.95,
  sato_var = TRUE,
  data = NULL
)

Arguments

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
strata

(numeric)
A vector specifying the stratum for each observation. It needs to be the length of x or a multiple of x if multiple levels of strata are present. Can also be a column name (or vector of column names NOT quoted) if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
sato_var

(logical)
Use Sato variance estimate
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

The Mantel-Haenszel common risk difference is computed as:

δ^MH=∑kwkδ^k∑kwk\hat{\delta}_{MH} = \frac{\sum_{k} w_k \hat{\delta}_k }{\sum_{k} w_k}

where wk=nxknykNkw_k = \frac{n_{xk} n_{yk}}{N_k}, δ^k=sxk/nxk−yyk/nyk\hat{\delta}_k = s_{xk}/n_{xk} - y_{yk}/n_{yk}, Nk=nxk+nykN_k = n_{xk} + n_{yk}, sxks_{xk} and syks_{yk} are the number of events in each group, and nxkn_{xk}, and nykn_{yk} are the group sizes in stratum kk.

The Sato variance is:

σ^2(δ^MH)=d^MH∑kPk+∑kQk(∑kwk)2\hat{\sigma}^2(\hat{\delta}_{MH}) = \frac{\hat{d}_{MH} \sum_{k}{P_k} + \sum_k Q_k}{\left( \sum_k w_k \right)^2}

where Pk=nxk2syk−nyk2sxk+nxknyk(nyk−nxk)/2Nk2P_k = \frac{n_{xk}^2 s_{yk} - n_{yk}^2 s_{xk} + n_{xk} n_{yk} (n_{yk} - n_{xk})/2}{N_k^2} and Qk=sxk(nyk−syk)+syk(nxk−sxk)2NkQ_k = \frac{s_{xk}(n_{yk} - s_{yk}) + s_{yk}(n_{xk} - s_{xk})}{2 N_k}.

The Cochran Independent Binomial variance is:

σ^2(δ^C)=∑kwk2[p^1k(1−p^1k)n1k+p^2k(1−p^2k)n2k]\hat{\sigma}^2(\hat{\delta}_{C}) = \sum_{k} w_k^2 \left[ \frac{\hat{p}_{1k}(1 - \hat{p}_{1k})}{n_{1k}} + \frac{\hat{p}_{2k}(1 - \hat{p}_{2k})}{n_{2k}} \right]

where p^1k=sxknxk\hat{p}_{1k} = \frac{s_{xk}}{n_{xk}} and p^2k=syknyk\hat{p}_{2k} = \frac{s_{yk}}{n_{yk}}.

The confidence interval is then δ^MH±z1−α/2σ^2(d^MH)\hat{\delta}_{MH} \pm z_{1-\alpha/2} \sqrt{\hat{\sigma}^2(\hat{d}_{MH})}.

Value

An object containing the following components:

estimate

The Mantel-Haenszel estimated common risk difference
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
variance

Variance estimate
statistic

Z-Statistic under the null hypothesis, assuming a common risk difference of 0
p.value

p-value under the null hypothesis, assuming a common risk difference of 0
method

Description of the method used ("Mantel-Haenszel Confidence Interval, Sato Variance") or ("Mantel-Haenszel Confidence Interval, Independent Binomial")

References

Agresti, A. (2013). Categorical Data Analysis. 3rd Edition. John Wiley & Sons, Hoboken, NJ p. 231 Cochran, W.G. (1954). The Combination of estimates from different experiments. Biometrics, 10(1), p.101-129

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 common risk difference
ci_prop_diff_mh_strata(x = responses, by = arm, strata = strata)
# Calculate risk difference with independent binomial variance
ci_prop_diff_mh_strata(x = responses, by = arm, strata = strata, sato_var = FALSE)

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
delta

(numeric)
Optionally a single number or a vector of numbers between -1 and 1 (not inclusive) to set the difference between two groups under the null hypothesis. If provided, the function returns the test statistic and p-value under the delta hypothesis.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

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:

H0:d^−δ<=0vs.H1:d^−δ>0H_0: \hat{d}-\delta <= 0 \qquad \text{vs.} \qquad H_1: \hat{d}-\delta > 0

Tδ=px^−py^−δσmn(δ)T_\delta = \frac{\hat{p_x} - \hat{p_y} - \delta}{\sigma_{mn}(\delta)}

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

σmn(δ)=[py~(1−py~)nx+px~(1−px~)ny](NN−1)\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)}

px~=2p⋅cos(a)−L23L3\tilde{p_x} = 2p\cdot{cos(a)} - \frac{L_2}{3L_3} and py~=px~+δ\tilde{p_y} = \tilde{p_x} + \delta where:

  • p=±L22(3L3)2−L13L3p = \pm \sqrt{\frac{L_2^2}{(3L_3)^2} - \frac{L_1}{3L_3}}

  • a=1/3[π+cos−1(q/p3)]a = 1/3[\pi + cos^{-1}(q/p^3)]

  • q=L23(3L3)3−L1L26L32+L02L3q = \frac{L_2^3}{(3L_3)^3} - \frac{L_1L_2}{6L_3^2} + \frac{L_0}{2L_3}

  • L3=nx+nyL_3 = n_x + n_y

  • L2=(nx+2ny)δ−N−(sx+sy)L_2 = (n_x + 2 n_y)\delta - N - (s_x + s_y)

  • L1=(nyδ−L3−2sy)δ+sx+syL_1 = (n_y\delta - L_3 - 2s_y)\delta + s_x + s_y

  • L0=syδ(1−δ)L_0 = s_y\delta(1-\delta)

For more information about these equations see Miettinen (1985)

Value

An object containing the following components:

estimate

The point estimate of the difference in proportions (p_x - p_y)
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
delta

delta value(s) used
statistic

Z-Statistic under the null hypothesis based on the given 'delta'
p.value

p-value under the null hypothesis based on the given 'delta'
method

Description of the method used ("Miettinen-Nurminen Confidence Interval")

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
strata

(numeric)
A vector specifying the stratum for each observation. It needs to be the length of x or a multiple of x if multiple levels of strata are present. Can also be a column name (or vector of column names NOT quoted) if a data frame provided in the data argument.
method

(string)
Specifying how the CIs should be calculated. It must equal either 'score' or 'summary score'. See details for more information about the implementation differences.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
delta

(numeric)
Optionally a single number or a vector of numbers between -1 and 1 (not inclusive) to set the difference between two groups under the null hypothesis. If provided, the function returns the test statistic and p-value under the delta hypothesis.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

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

H0:d^−δ<=0vs.H1:d^−δ>0H_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 wi=nxi⋅nyinxi+nyiw_i = \frac{n_{xi} \cdot n_{yi}}{n_{xi} + n_{yi}}

  • Computes the overall weighted difference d^=∑wip^xi∑wi−∑wip^yi∑wi\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:

    Zδ=d^−δ∑i=1k(wi∑wi)2⋅σ^mn2(d)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 δ\delta for which the stratified test statistic (ZδZ_\delta) falls in the acceptance region {Zδ<zα/2}\{ Z_\delta < z_{\alpha/2}\}

The σ^mn2(d^)\hat{\sigma}_{mn}^2(\hat{d}) is the Miettinen-Nurminen variance estimate. See the details of ci_prop_diff_mn() for how σ^mn2(δ)\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:

    d^S=∑id^iwi\hat{d}_{\text{S}} = \sum_i \hat{d}_i w_i

  • Define sis_i as the width of the CI for the iith stratum divided by 2×zα/22 \times z_{\alpha/2} and then stratum weights are given by

    wi=(1si2)/∑i(1si2)w_i = \left( \frac{1}{s_i^2} \right) \bigg/ \sum_i \left( \frac{1}{s_i^2} \right)

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

    Var^(d^S)=1∑i(1si2)\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

    d^S±(zα/2×Var^(d^S))\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:

estimate

The point estimate of the difference in proportions (p_x - p_y)
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
delta

delta value(s) used
statistic

Z-Statistic under the null hypothesis based on the given 'delta'
p.value

p-value under the null hypothesis based on the given 'delta'
method

Description of the method used ("Stratified {method} Miettinen-Nurminen Confidence Interval")

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)

Newcombe Confidence Interval for Difference in Proportions

Description

Newcombe Confidence Interval for Difference in Proportions

Usage

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

Arguments

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
correct

(logical)
apply continuity correction.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

The Wilson (Score) confidence limits without continuity correction for each individual binomial proportion, pi=xi/nip_i = x_i / n_i, for i=1,2i = 1, 2, are given by:

(2nip^i+z2)±z4nip^i(1−p^i)+z22(ni+z2)\frac{ (2 n_i \hat{p}_i + z^2) \pm z \sqrt{ 4 n_i \hat{p}_i (1 - \hat{p}_i) + z^2 } }{ 2 (n_i + z^2) }

Denote the lower and upper Wilson (Score) confidence limits for pip_i as LiL_i and UiU_i, respectively.

Then, the Newcombe (Score) confidence limits for the difference in proportions (p1−p2p_1 - p_2) are given by:

Lower limit: (p^1−p^2)−(p^1−L1)2+(U2−p^2)2\text{Lower limit: } (\hat{p}_1 - \hat{p}_2) - \sqrt{ (\hat{p}_1 - L_1)^2 + (U_2 - \hat{p}_2)^2 }

Upper limit: (p^1−p^2)+(U1−p^1)2+(p^2−L2)2\text{Upper limit: } (\hat{p}_1 - \hat{p}_2) + \sqrt{ (U_1 - \hat{p}_1)^2 + (\hat{p}_2 - L_2)^2 }

The confidence intervals with continuity correction for each individual binomial proportion are obtained using the Wilson (Score) confidence limits with continuity correction.

For each binomial proportion pi=xi/nip_i = x_i / n_i, where i=1,2i = 1, 2, the confidence intervals are given by:

2nip^i+z22(ni+z2)  ±  z2(ni+z2)z2−2ni+4p^i[ni(1−p^i)+1]\frac{ 2 n_i \hat{p}_i + z^2 }{ 2 (n_i + z^2) } \; \pm \; \frac{ z }{ 2 (n_i + z^2) } \sqrt{ z^2 - \frac{2}{n_i} + 4 \hat{p}_i \left[ n_i (1 - \hat{p}_i) + 1 \right] }

Value

An object containing the following components:

n

The number of responses for each group
N

The total number in each group
estimate

The point estimate of the difference in proportions
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Anderson-Hauck Confidence Interval

References

Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions: Comparison of eleven methods. Statistics in Medicine, 17(8), 873–890. 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_nc(x = responses, by = arm)

Stratified Newcombe Common Risk Difference Confidence Interval

Description

Calculates the stratified Newcombe confidence interval for unequal proportions as described in Yan X, Su XG. Stratified Wilson and Newcombe confidence intervals or multiple binomial proportions. Weights are estimated using CMH or Wilson methods.

Usage

ci_prop_diff_nc_strata(
  x,
  by,
  strata,
  conf.level = 0.95,
  correct = FALSE,
  weights_method = c("wilson", "cmh"),
  data = NULL
)

Arguments

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
strata

(numeric)
A vector specifying the stratum for each observation. It needs to be the length of x or a multiple of x if multiple levels of strata are present. Can also be a column name (or vector of column names NOT quoted) if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
correct

(scalar logical)
include the continuity correction. For further information, see for example [ci_prop_diff_nc())].

[ci_prop_diff_nc())]: R:ci_prop_diff_nc())
weights_method

(character)
Can be either "wilson" or "cmh" and directs the way weights are estimated.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

L=d^MH−zα/2∑hwh2L2h(1−L2h)+∑hwh2U1h(1−U1h)L = \hat{d}_{\rm MH} - z_{\alpha/2} \sqrt{ \sum_h w_h^2 L_{2h} (1 - L_{2h}) + \sum_h w_h^2 U_{1h} (1 - U_{1h}) }

U=d^MH+zα/2∑hwh2L2h(1−L2h)+∑hwh2U1h(1−U1h)U = \hat{d}_{\rm MH} + z_{\alpha/2} \sqrt{ \sum_h w_h^2 L_{2h} (1 - L_{2h}) + \sum_h w_h^2 U_{1h} (1 - U_{1h}) }

Where:

  • d^MH\hat{d}_{\rm MH}: Mantel-Haenszel common risk difference

  • zα/2z_{\alpha/2}: standard normal critical value

  • whw_h: stratum weights

  • L2hL_{2h}, U1hU_{1h}: Wilson-type CI limits for stratum h

Value

An object containing the following components:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
weights

Weights of each strata calculated as per the specified "weights_method" argument.
method

Type of method used

Examples

set.seed(1)
rsp <- sample(c(TRUE, FALSE), 100, TRUE)
grp <- sample(c("Placebo", "Treatment"), 100, TRUE)
strata_data <- data.frame(
  "f1" = sample(c("a", "b"), 100, TRUE),
  "f2" = sample(c("x", "y", "z"), 100, TRUE),
  stringsAsFactors = TRUE
)
strata <- interaction(strata_data)

ci_prop_diff_nc_strata(
  x = rsp, by = grp, strata = strata, weights_method = "cmh",
  conf.level = 0.95
)

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
correct

(logical)
apply continuity correction.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

(p^1−p^2)±zα/2p^1(1−p^1)n1+p^2(1−p^2)n2(\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:

n

Number of responses in each by group
N

Total number in each by group
estimate

The point estimate of the difference in proportions (p_1 - p_2)
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Type of method used

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

(Beta(k2+12,n−k2+12)α,Beta(k2+12,n−k2+12)1−α)\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:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Type of method used

Mid-P CI

Description

Calculates the exact mid-p CI for binomial proportions by inverting two one-sided binomial tests that include the mid-p tail. This is calculated by finding the PLP_L and PUP_U that satisfies the following equations:

∑x=n1+1n(nx)PLx(1−PL)n−x+12(nn1)PLn1(1−PL)n−n1=α/2\sum _{x=n_1+1}^{n} \binom {n}{x} P_{L}^{x}(1-P_{L})^{n-x} + \frac{1}{2} \binom{n}{n_1} P_{L}^{n_1}(1-P_{L})^{n-n_1} = \alpha /2

∑x=0n1−1(nx)PUx(1−PU)n−x+12(nn1)PUn1(1−PU)n−n1=α/2\sum _{x=0}^{n_1-1} \binom {n}{x} P_{U}^{x}(1-P_{U})^{n-x} + \frac{1}{2} \binom{n}{n_1} P_{U}^{n_1}(1-P_{U})^{n-n_1} = \alpha /2

Usage

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

Arguments

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Value

An object containing the following components:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Type of method used

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
correct

(logical)
apply continuity correction.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

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

Value

An object containing the following components:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Type of method used

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
correct

(logical)
apply continuity correction.
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

p^+zα/222n±zα/2p^(1−p^)n+zα/224n21+zα/22n\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:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
method

Type of method used

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

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
strata

(numeric)
A vector specifying the stratum for each observation. It needs to be the length of x or a multiple of x if multiple levels of strata are present. Can also be a column name (or vector of column names NOT quoted) if a data frame provided in the data argument.
weights

(numeric)
weights for each level of the strata. If NULL, they are estimated using the iterative algorithm that minimizes the weighted squared length of the confidence interval.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
max.iterations

(positive integer)
maximum number of iterations for the iterative procedure used to find estimates of optimal weights.
correct

(scalar logical)
include the continuity correction. For further information, see for example stats::prop.test().
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

p^j+zα/222nj±zα/2p^j(1−p^j)nj+zα/224nj21+zα/22nj\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:

n

Number of responses
N

Total number
estimate

The point estimate of the proportion
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
weights

Weights of each strata, will be the same as the input unless unspecified, then it will be the dynamically calculated weights.
method

Type of method used

Examples

# Stratified Wilson confidence interval with unequal probabilities

set.seed(1)
rsp <- sample(c(TRUE, FALSE), 100, TRUE)
strata_data <- data.frame(
  "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
)

Mantel-Haenszel Stratified Relative Risk Confidence Interval

Description

Calculates the confidence interval for the Mantel-Haenszel estimate of the common relative risk across multiple 2x2 tables (strata)

Usage

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

Arguments

x

(binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by

(string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
strata

(numeric)
A vector specifying the stratum for each observation. It needs to be the length of x or a multiple of x if multiple levels of strata are present. Can also be a column name (or vector of column names NOT quoted) if a data frame provided in the data argument.
conf.level

(⁠scalar numeric⁠)
a scalar in (0,1) indicating the confidence level. Default is 0.95
data

(data.frame)
Optional data frame containing the variables specified in x and by.

Details

The Mantel-Haenszel relative risk difference is computed as:

RRMH=∑ksxk nyk/Nk∑ksyk nxk/NkRR_{MH} = \frac{\sum_{k} s_{xk}~n_{yk}/N_k}{\sum_{k} s_{yk}~n_{xk}/N_k}

The variance is:

σ^2=Var^(log(RRMH))=∑k(nxk nyk (sxk+syk)−sxk syk Nk)/Nk2(∑ksxk nyk/Nk)(∑ksyk nxk/Nk)\hat{\sigma}^2 = \hat{Var}(log(RR_{MH})) = \frac{\sum_{k}(n_{xk}~n_{yk}~(s_{xk}+s_{yk}) - s_{xk}~s_{yk}~N_k)/N_k^2} {(\sum_{k}s_{xk}~n_{yk}/N_k)(\sum_{k}s_{yk}~n_{xk}/N_k)}

The confidence interval is then (RRMH×exp(−z1−α/2σ^2,RRMH×exp(z1−α/2σ^2)\left(RR_{MH}\times exp(-z_{1-\alpha/2} \sqrt{\hat{\sigma}^2}, RR_{MH}\times exp(z_{1-\alpha/2} \sqrt{\hat{\sigma}^2}\right).

Value

An object containing the following components:

estimate

The Mantel-Haenszel estimated common risk difference
conf.low

Lower bound of the confidence interval
conf.high

Upper bound of the confidence interval
conf.level

The confidence level used
variance

Mantel-Haenszel variance estimate Var(log(RRMH))Var(log(RR_MH))
method

Description of the method used ("Mantel-Haenszel Common Relative Risk Confidence Interval")

References

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 common risk difference
ci_rel_risk_cmh_strata(x = responses, by = arm, strata = 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

x

Integer (or vector of integers) representing the number of successes.
n

Integer (or vector of integers) representing the total number of participants.

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))