CI on the Difference between Two Proportions

The Confidence Interval on the Difference between Two Proportions uses X and Y, the number of successes found in the two samples, and n_x and n_y, the size of the samples, to estimate the population parameters. This implementation uses the Agresti-Coull Interval method, which is appropriate for any sample size. mctad.confidenceIntervalOnTheDifferenceBetweenTwoProportions() returns an Array of lower, upper values for the (1 - α/2)interval or a Number for the lower or upper (1 - α) boundary.

Assumptions

X and Y are the number of successes found in the two samples, n_x and n_y are the size of the samples, and 0.0 < α < 1.0. By default, the confidence interval is two-tailed; this may be changed by specifying type as either 'l' for lower, or 'u' for upper. Any other value for type generates a two-tailed confidence interval.

Use

mctad.confidenceIntervalOnTheDifferenceBetweenTwoProportions(X, Y, n_x, n_y, α, 'u')

Inline Comments

mctad.confidenceIntervalOnTheDifferenceBetweenTwoProportions = function (X, Y, n_x, n_y, α, type) { if (typeof X !== 'number' || typeof Y !== 'number' || !mctad.isInteger(n_x) || !mctad.isInteger(n_y) || α <= 0.0 || α >= 1.0) { return undefined; }

Apply the Agresti-Coull Interval transformation.

var n_tilde_x = n_x + 2, n_tilde_y = n_y + 2, p_tilde_x = (X + 1) / n_tilde_x, p_tilde_y = (Y + 1) / n_tilde_y, p_tilde_x_minus_p_tilde_y = p_tilde_x - p_tilde_y, σ_p_tilde_x_p_tilde_y = Math.sqrt(p_tilde_x * (1.0 - p_tilde_x) / n_tilde_x + p_tilde_y * (1.0 - p_tilde_y) / n_tilde_y);

Return the upper confidence bound of a one-tailed confidence interval.

if (typeof type !== 'undefined' && type.toLowerCase() === 'u') { return p_tilde_x_minus_p_tilde_y + mctad.z(1 - α) * σ_p_tilde_x_p_tilde_y; } else {

Return the lower confidence bound of a one-tailed confidence interval.

if (typeof type !== 'undefined' && type.toLowerCase() === 'l') { return p_tilde_x_minus_p_tilde_y - mctad.z(1 - α) * σ_p_tilde_x_p_tilde_y; } else {

Return both bounds of a two-tailed confidence interval.

return [ p_tilde_x_minus_p_tilde_y - mctad.z(1 - α / 2) * σ_p_tilde_x_p_tilde_y, p_tilde_x_minus_p_tilde_y + mctad.z(1 - α / 2) * σ_p_tilde_x_p_tilde_y ]; } } };