ScoreYJall
Computes all the 4 score tests for YJ transformation
Description
The transformations for negative and positive responses were determined by Yeo and Johnson (2000) by imposing the smoothness condition that the second derivative of zYJ(λ) with respect to y be smooth at y = 0. However some authors, for example Weisberg (2005), query the physical interpretability of this constraint which is often violated in data analysis. Accordingly, Atkinson et al (2019) and (2020) extend the Yeo-Johnson transformation to allow two values of the transformations parameter: λN for negative observations and λP for non-negative ones.
ScoreYJall computes: 1) the global t test associated with the constructed variable for λ=λP=λN.
2) the t test associated with the constructed variable computed assuming a different transformation for positive observations keeping the value of the transformation parameter for negative observations fixed. In short we call this test, "test for positive observations".
3) the t test associated with the constructed variable computed assuming a different transformation for negative observations keeping the value of the transformation parameter for positive observations fixed. In short we call this test, "test for negative observations".
4) the F test for the joint presence of the two constructed variables described in points 2) and 3).
Example in which positive and negative observations require the same lambda.outSC
=ScoreYJall(y,
X)
Example in which positive and negative observation require different lambda.outSC
=ScoreYJall(y,
X,
Name, Value)
Examples
Example in which positive and negative observations require the same lambda.
Example in which positive and negative observations require the same lambda.
rng('default')
rng(100)
n=100;
y=randn(n,1);
% Transform the value to find out if we can recover the true value of
% the transformation parameter.
la=0.5;
ytra=normYJ(y,[],la,'inverse',true);
% Start the analysis
X=ones(n,1);
[outSC]=ScoreYJall(ytra,X,'intercept',false);
la=[-1 -0.5 0 0.5 1]';
Sco=[la outSC.Score];
Scotable=array2table(Sco,'VariableNames',{'lambda','Tall','Tp','Tn','Ftest'});
disp(Scotable)
% Comment: if we consider the 5 most common values of lambda, the value
% of the score test when lambda=0.5 is the only one that is not
% significant. Both values of the score test for positive and negative
% observations and the Ftest confirm that this value of the
% transformation parameter is OK for both sides of the distribution.
lambda Tall Tp Tn Ftest
______ _______ _______ _______ ______
-1 21.177 32.887 14.266 575.2
-0.5 12.662 15.573 9.192 122.47
0 5.0377 5.3971 3.9745 14.416
0.5 -1.6988 -1.4896 -1.7737 1.5805
1 -8.4803 -7.4925 -9.3782 43.761
Example in which positive and negative observation require different lambda.
Example in which positive and negative observation require different lambda.
rng(2000)
n=100;
y=randn(n,1);
% Transform in a different way positive and negative values.
lapos=0;
ytrapos=normYJ(y(y>=0),[],lapos,'inverse',true);
laneg=1;
ytraneg=normYJ(y(y<0),[],laneg,'inverse',true);
ytra=[ytrapos; ytraneg];
% Start the analysis.
X=ones(n,1);
% also compute lik. ratio test based on MLE of laP and laN
scoremle=true;
[outSC]=ScoreYJall(ytra,X,'intercept',false,'scoremle',scoremle);
la=[-1 -0.5 0 0.5 1]';
Sco=[la outSC.Score];
Scotable=array2table(Sco,'VariableNames',{'lambda','Tall','Tp','Tn','FtestPN' 'FtestLR'});
disp(Scotable)
% Comment: if we consider the 5 most common values of lambda,
% the value of the score test when lambda=0.5 is the only one which is not
% significant. However, when lambda=0.5, the score test for negative
% observations is highly significant.
disp('Difference between the test for positive and the test for negative')
disp(abs(Scotable{4,3}-Scotable{4,4})),
% which is very large.
% This indicates that the two tails need a different value of the
% transformation parameter.
lambda Tall Tp Tn FtestPN FtestLR
______ _______ _______ _______ _______ _______
-1 36.467 55.13 25.519 1610.7 198.99
-0.5 20.184 22.481 16.391 250.17 61.484
0 7.8511 6.7795 8.197 33.341 12.708
0.5 -1.5618 -2.642 0.61876 14.23 3.0175
1 -11.647 -11.721 -9.2192 68.153 27.268
Difference between the test for positive and the test for negative
3.2608
Input Arguments
Output Arguments
References
Yeo, I.K. and Johnson, R. (2000), A new family of power transformations to improve normality or symmetry, "Biometrika", Vol. 87, pp. 954-959.
Atkinson, A.C. Riani, M., Corbellini A. (2019), The analysis of transformations for profit-and-loss data, Journal of the Royal Statistical Society, Series C, "Applied Statistics", https://doi.org/10.1111/rssc.12389
Atkinson, A.C. Riani, M. and Corbellini A. (2021), The Box–Cox Transformation: Review and Extensions, "Statistical Science", Vol. 36, pp. 239-255, https://doi.org/10.1214/20-STS778