Get a proportion back out of the double-arcsine scale

The double arcsine stabilises the variance of a proportion, which is what makes it poolable, but the pooled value is then on a scale nobody can read. Back-transforming is not a matter of inverting the formula twice: the transform depends on the sample size, and Miller's inverse substitutes the harmonic mean of the study sizes for it. Using the arithmetic mean instead is a known way to get a proportion outside [0, 1].

Usage

Mafrti(ft, n_harmonic)

Arguments

ft

Value(s) on the double-arcsine scale.

n_harmonic

Harmonic mean of the study sample sizes, positive.

Value

List with p, n_harmonic, clamped.

Details

Formula: \(p = 0.5 (1 - sgn(cos t) sqrt(1 - (sin t + (sin t - 1/sin t)/n)^2))\) for t the double-arcsine value and n the harmonic mean sample size – Miller (1978).

References

Miller, J. J. (1978). The American Statistician 32(4):138. doi:10.1080/00031305.1978.10479283 .

Examples

Mafrti(ft = c(1, 2, 3, 4, 5, 6, 7, 8), n_harmonic = 5L)
#> $p
#> [1] 0.18224107 0.74545803 0.50000000 0.88255767 0.33239310 0.03760570 0.06245596
#> [8] 0.58605238
#> 
#> $n_harmonic
#> [1] 5
#> 
#> $clamped
#> [1] 0
#> 
#> $method
#> [1] "Freeman-Tukey double arcsine back-transformation"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"