ci.impt {asbio} R Documentation

## 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

 y1 The number of successes associated with the first proportion. n1 The number of trials associated with the first proportion. y2 The number of successes associated with the second proportion. Not used if avail.known = TRUE. n2 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 alternative = "less" and alternative = "greater" provide lower and upper tailed CIs, respectively. 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.

### 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.

### Value

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

### Note

Method will perform poorly given unbalanced sample sizes.

Ken Aho

### References

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