ci_prop_diff_mn_strata: Stratified Miettinen-Nurminen Confidence Interval for...

In cicalc: Calculate Confidence Intervals

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.

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:

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)

cicalc documentation built on Aug. 8, 2025, 7 p.m.