Transformations: Transformation functions

In ANOPA: Analyses of Proportions using Anscombe Transform

Transformation functions

Description

The transformation functions A() performs the Anscombe transformation on a pair {number of success; number of trials} = {s; n} (where the symbol ";" is to be read "over". The function varA() returns the theoretical variance from the pair {s; n}. Both functions are central to the ANOPA \insertCitelc23ANOPA. It was originally proposed by \insertCitez35ANOPA and formalized by \insertCitea48ANOPA.

Usage

A(s, n)

varA(s, n)

Atrans(v)

SE.Atrans(v)

var.Atrans(v)

CI.Atrans(v, gamma)

prop(v)

CI.prop(v, gamma)

Arguments

s	a number of success;
n	a number of trials.
v	a vector of 0s and 1s.
gamma	a confidence level, default to .95 when omitted.

Details

The functions A() and varA() take as input two integers, s the number of success and n the number of observations. The functions Atrans(), SE.Atrans(), var.Atrans(), CI.Atrans(), prop() and CI.prop() take as input a single vector v of 0s and 1s from which the number of success and the number of observations are derived.

Value

A() returns a score between 0 and 1.57 where a s of zero results in A(0,n) tending to zero when the number of trials is large, and where the maximum occurs when s equals n and are both very large, so that for example A(1000,1000) = 1.55. The midpoint is always 0.786 irrespective of the number of trials A(0.5 * n, n) = 0.786. The function varA() returns the theoretical variance of an Anscombe transformed score. It is exact as n gets large, and overestimate variance when n is small. Therefore, a test based on this transform is either exact or conservative.

References

\insertAllCited

Examples

# The transformations from number of 1s and total number of observations:
A(5, 10)
 
varA(5, 10)
 
# Same with a vector of observations:
Atrans( c(1,1,1,1,1,0,0,0,0,0) )
 
var.Atrans( c(1,1,1,1,1,0,0,0,0,0) )
 

ANOPA documentation built on Aug. 19, 2025, 1:11 a.m.