Provides one and two-tailed confidence intervals for the true product of two proportions.
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)Returns a list of class = "ci". Printed results are the parameter estimate and confidence bounds.
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.
Logical. Are the proportions \(\pi_{2i}\) known? If avail.known = TRUE these proportions should specified in the pi.2 argument.
Proportions for \(\pi_{2i}\). Required if avail.known = TRUE.
Confidence level, i.e., 1 - \(\alpha\).
Logical. If true, estimate is multiplied by 100.
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 alternative = "less" and alternative = "greater" provide lower and upper tailed CIs, respectively.
Logical. If bonf = TRUE, and the number of requested intervals is greater than one, then Bonferroni-adjusted intervals are returned.
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.
Ken Aho
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 \(1 - \alpha\).
Aho, K., and Bowyer, T. 2015. Confidence intervals for a product of proportions: Implications for importance values. Ecosphere 6(11): 1-7.
ci.prat, ci.p