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Miettinen-Nurminen Confidence Interval for Difference in Proportions

ci_prop_diff_mn.Rd

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.

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 ("Miettinen-Nurminen Confidence Interval")

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

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)

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)
#> 
#> ── Miettinen-Nurminen Confidence Interval ──────────────────────────────────────
#> • 9/10 - 3/10
#> • Estimate: 0.6
#> • 95% Confidence Interval:
#>   (0.17, 0.8406)

# Calculate 99% confidence interval
ci_prop_diff_mn(x = responses, by = arm, conf.level = 0.99)
#> 
#> ── Miettinen-Nurminen Confidence Interval ──────────────────────────────────────
#> • 9/10 - 3/10
#> • Estimate: 0.6
#> • 99% Confidence Interval:
#>   (0.0218, 0.8792)

# Calculate the p-value under the null hypothesis delta = -0.1
ci_prop_diff_mn(x = responses, by = arm, delta = -0.1)
#> 
#> ── Miettinen-Nurminen Confidence Interval ──────────────────────────────────────
#> • 9/10 - 3/10
#> • Estimate: 0.6
#> • 95% Confidence Interval:
#>   (0.17, 0.8406)
#> 
#> ── Delta 
#> • At -0.1 the statistic is 3.115 and the p-value is 9e-04

# Calculate from a data.frame
data <- data.frame(responses, arm)
ci_prop_diff_mn(x = responses, by = arm, data = data)
#> 
#> ── Miettinen-Nurminen Confidence Interval ──────────────────────────────────────
#> • 9/10 - 3/10
#> • Estimate: 0.6
#> • 95% Confidence Interval:
#>   (0.17, 0.8406)