Variance-stabilising double arcsine transform of a proportion

A single arcsine blows up at zero and one; the double arcsine averages two neighbouring single arcsines so a study with no events still contributes, which is the case that matters in rare-event synthesis. The variance depends only on n, so the weights are fixed rather than estimated.

Usage

Mafrt(x, n)

Arguments

x

Event counts.

n

Sample sizes, elementwise against x.

Value

List with ft, var, se, k.

Details

Formula: FT = asin sqrt(x/(n+1)) + asin sqrt((x+1)/(n+1)), Var(FT) = 1/(n + 1/2).

References

Freeman, M. F. & Tukey, J. W. (1950). Ann Math Statist 21:607-611; Miller, J. J. (1978). Amer Statist 32:138.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Mafrt(V, V)
#> $ft
#> [1] 2.356194 2.526113 2.617994 2.677945 2.721058 2.753996 2.780226 2.801756
#> 
#> $var
#> [1] 0.6666667 0.4000000 0.2857143 0.2222222 0.1818182 0.1538462 0.1333333
#> [8] 0.1176471
#> 
#> $se
#> [1] 0.8164966 0.6324555 0.5345225 0.4714045 0.4264014 0.3922323 0.3651484
#> [8] 0.3429972
#> 
#> $k
#> [1] 8
#> 
#> $method
#> [1] "Freeman-Tukey double arcsine transform"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"