Stratified Miettinen-Nurminen Confidence Interval for Difference in Proportions

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.

Value

A list 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

Details

The function implements the stratified Miettinen-Nurminen method to compute confidence intervals for the difference between two proportions across multiple strata. $$H_0: \hat{d}-\delta <= 0 \qquad \text{vs.} \qquad H_1: \hat{d}-\delta > 0$$

For the "score" method, the approach:

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.

For the "summary score" method, the function:

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

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

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)
#> 
#> ── Stratified Score Miettinen-Nurminen Confidence Interval ─────────────────────
#> • 12/20 - 9/20
#> • Weights: stratum1 = 5, stratum2 = 5
#> • Estimate: 0.15
#> • 95% Confidence Interval:
#>   (-0.1606, 0.4338)

# Using the summary score method
ci_prop_diff_mn_strata(x = responses, by = arm, strata = strata,
                      method = "summary score")
#> 
#> ── Stratified Summary Score Miettinen-Nurminen Confidence Interval ─────────────
#> • 12/20 - 9/20
#> • Weights: stratum1 = 0.511, stratum2 = 0.489
#> • Estimate: -0.126
#> • 95% Confidence Interval:
#>   (-0.4113, 0.1586)

# Calculate 99% confidence interval
ci_prop_diff_mn_strata(x = responses, by = arm, strata = strata,
                      conf.level = 0.99)
#> 
#> ── Stratified Score Miettinen-Nurminen Confidence Interval ─────────────────────
#> • 12/20 - 9/20
#> • Weights: stratum1 = 5, stratum2 = 5
#> • Estimate: 0.15
#> • 99% Confidence Interval:
#>   (-0.2509, 0.5072)

# Calculate p-value under null hypothesis delta = 0.2
ci_prop_diff_mn_strata(x = responses, by = arm, strata = strata,
                      delta = 0.2)
#> 
#> ── Stratified Score Miettinen-Nurminen Confidence Interval ─────────────────────
#> • 12/20 - 9/20
#> • Weights: stratum1 = 5, stratum2 = 5
#> • Estimate: 0.15
#> • 95% Confidence Interval:
#>   (-0.1606, 0.4338)
#> 
#> ── Delta 
#> • At 0.2 the statistic is -0.319 and the p-value is 0.375