ci_prop_diff_mn: 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)

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

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:

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

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

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

Run this code

# 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)

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