ci.impt: Confidence interval for the product of two proportions

Description

Provides one and two-tailed confidence intervals for the true product of two proportions.

Usage

ci.impt(y1, n1, y2 = NULL, n2 = NULL, avail.known = FALSE, pi.2 = NULL, 
conf = .95, x100 = TRUE, alternative = "two.sided", bonf = TRUE, wald = FALSE)

Arguments

The number of successes associated with the first proportion.

The number of trials associated with the first proportion.

The number of successes associated with the second proportion. Not used if avail.known = TRUE.

The number of trials associated with the first proportion. Not used if avail.known = TRUE.

avail.known

Logical. Are the proportions $\pi_{2i}$ known? If avail.known = TRUE these proportions should specified in the pi.2 argument.

pi.2

Proportions for $\pi_{2i}$. Required if avail.known = TRUE.

conf

Confidence level, i.e., 1 - $\alpha$.

x100

Logical. If true, estimate is multiplied by 100.

alternative

One of "two.sided", "less", "greater". Allows lower, upper, and two-tailed confidence intervals. If alternative = "two.sided" (the default), then a conventional two-sided confidence interval is given. The specifications

bonf

Logical. If bonf = TRUE, and the number of requested intervals is greater than one, then Bonferroni-adjusted intervals are returned.

wald

Logical. If avail.known = TRUE one can apply one of two standard error estimators. The default is a delta-derived estimator. If wald = TRUE is specified a modified Wald standard error estimator is used.

Value

Returns a list of class = "ci". Printed results are the paramter estimate and confidence bounds.

Details

Let $Y_1$ and $Y_2$ be multinomial random variables with parameters $n_1$, $\pi_{1i}$ and $n_2$, $\pi_{2i}$, respectively; where $i = 1,2,\dots, r$. Under delta derivation, the log of the products of $\pi_{1i}$ and $\pi_{2i}$ (or the log of a product of $\pi_{1i}$ and $\pi_{2i}$ and a constant) is asymptotically normal with mean $log(\pi_{1i} \times \pi_{2i})$ and variance $(1 - \pi_{1i})/\pi_{1i}n_1 + (1 - \pi_{2i})/ \pi_{2i}n_2$. Thus, an asymptotic $(1 - \alpha)100$ percent confidence interval for $\pi_{1i} \times \pi_{2i}$ is given by: $$\hat{\pi}_{1i} \times \hat{\pi}_{2i} \times exp(\pm z_{1-(\alpha/2)}\hat{\sigma}_i)$$ where: $\hat{\sigma}^2_i = \frac{(1 - \hat{\pi}_{1i})}{\hat{\pi}_{1i}n_1} + \frac{(1 - \hat{\pi}_{2i})}{\hat{\pi}_{2i}n_2}$ and $z_{1-(\alpha/2)}$ is the standard normal inverse CDF at probability $1 - \alpha$.

Examples

Run this code

ci.impt(30,40, 25,40)