| A {ANOPA} | R Documentation |
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 (Laurencelle and Cousineau 2023). It was originally proposed by (Zubin 1935) and formalized by (Anscombe 1948).
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
Anscombe FJ (1948).
“The transformatin of poisson, binormial and negative-binomial data.”
Biometrika, 35, 246–254.
doi:10.1093/biomet/35.3-4.246.
Laurencelle L, Cousineau D (2023).
“Analysis of proportions using arcsine transform with any experimental design.”
Frontiers in Psychology, 13, 1045436.
doi:10.3389/fpsyg.2022.1045436.
Zubin J (1935).
“Note on a transformation function for proportions and percentages.”
Journal of Applied Psychology, 19, 213–220.
doi:10.1037/h0057566.
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) )