# diffprop: Difference Between Two Proportions Distribution In tolerance: Statistical Tolerance Intervals and Regions

## Description

Density (mass), distribution function, quantile function, and random generation for the difference between two proportions. This is determined by taking the difference between two independent beta distributions.

## Usage

 ```1 2 3 4 5 6 7``` ```ddiffprop(x, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5, log = FALSE, ...) pdiffprop(q, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5, lower.tail = TRUE, log.p = FALSE, ...) qdiffprop(p, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5, lower.tail = TRUE, log.p = FALSE, ...) rdiffprop(n, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5) ```

## Arguments

 `x, q` Vector of quantiles. `p` Vector of probabilities. `n` The number of observations. If `length>1`, then the length is taken to be the number required. `k1, k2` The number of successes drawn from groups 1 and 2, respectively. `n1, n2` The sample sizes for groups 1 and 2, respectively. `a1, a2` The shift parameters for the beta distributions. For the fiducial approach, we know that the lower and upper limits are set at `a1 = a2 = 0` and `a1 = a2 = 1`, respectively, for the true `p1` and `p2`. While computations can be performed on real values outside the unit interval, a `warning` message will be returned if such values are specified. For practical purposes, the default value of 0.5 should be used for each parameter. `log, log.p` Logical vectors. If `TRUE`, then the probabilities are given as `log(p)`. `lower.tail` Logical vector. If `TRUE`, then probabilities are P[X≤ x], else P[X>x]. `...` Additional arguments passed to the Appell `F1` function.

## Details

The difference between two proportions distribution has a fairly complicated functional form. Please see the article by Chen and Luo (2011), who corrected a typo in the article by Nadarajah and Kotz (2007), for the functional form of this distribution.

## Value

`ddiffprop` gives the density (mass), `pdiffprop` gives the distribution function, `qdiffprop` gives the quantile function, and `rdiffprop` generates random deviates.

## References

Chen, Y. and Luo, S. (2011), A Few Remarks on 'Statistical Distribution of the Difference of Two Proportions', Statistics in Medicine, 30, 1913–1915.

Nadarajah, S. and Kotz, S. (2007), Statistical Distribution of the Difference of Two Proportions, Statistics in Medicine, 26, 3518–3523.

`runif` and `.Random.seed` about random number generation.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```## Randomly generated data from the difference between ## two proportions distribution. set.seed(100) x <- rdiffprop(n = 100, k1 = 2, k2 = 10, n1 = 17, n2 = 13) hist(x, main = "Randomly Generated Data", prob = TRUE) x.1 <- sort(x) y <- ddiffprop(x = x.1, k1 = 2, k2 = 10, n1 = 17, n2 = 13) lines(x.1, y, col = 2, lwd = 2) plot(x.1, pdiffprop(q = x.1, k1 = 2, k2 = 10, n1 = 17, n2 = 13), type = "l", xlab = "x", ylab = "Cumulative Probabilities") qdiffprop(p = 0.20, k1 = 2, k2 = 10, n1 = 17, n2 = 13, lower.tail = FALSE) qdiffprop(p = 0.80, k1 = 2, k2 = 10, n1 = 17, n2 = 13) ```