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ScoreYJall |
ScoreYJpn |
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ScoreYJmle
ScoreYJmle computes the likelihood ratio test for H_0=lambdaP=lambdaP0 and lambdaN=lambdaN0
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.
is the transformation parameter (scalar) for all the observations (positive and negative).
\lambda_P is the transformation parameter for positive observations.
\lambda_N is the transformation parameter for negative observations.
SSR is the residual sum of squares of the model which regresses z(λ) against X.
SSF is the residual sum of squares of the model which regresses z(\hat λ_{MLE}) against X where \lambda_{MLE} is the vector of length 2 which contains the MLE of \lambda_P and \lambda_N ScoreYJmle computes Num/Den where Num and Den are defined as follows: Num=(SSR-SSF)/2 and Den=SSF/(n-p-2), where p is the number of columns of matrix X (including intercept).
Example in which positive and negative observations require different lambda.outSC
=ScoreYJmle(y,
X,
Name, Value)
Examples
Example in which positive and negative obs require the same lambda.
Example in which positive and negative obs require the same lambda.
rng('default')
rng(100)
n=100;
yori=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(yori,[],la,'inverse',true);
% Start the analysis
X=ones(n,1);
[outSCmle]=ScoreYJmle(ytra,X,'intercept',false);
la=[-1 -0.5 0 0.5 1]';
Comb=[la outSCmle.Score(:,1)];
CombT=array2table(Comb,'VariableNames',{'la', 'FtestLR'});
disp(CombT)
la FtestLR
____ _______
-1 59.524
-0.5 21.104
0 3.6908
0.5 0.49713
1 13.49
Example in which positive and negative observations require different lambda.
Example in which positive and negative observations require different lambda.
rng(1000)
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);
la=[-1:0.25:1]';
[outSCmle]=ScoreYJmle(ytra,X,'intercept',false,'la',la);
Pval=fcdf(outSCmle.Score,2,n-2,'upper');
Comb=[la outSCmle.Score(:,1) Pval];
CombT=array2table(Comb,'VariableNames',{'la','FtestLR' 'Pvalues'});
disp(CombT)
disp('The test is significant for all values of lambda')
disp('This may indicate that the data need two separate lambdas for pos and neg observations')
la FtestLR Pvalues
_____ _______ __________
-1 350.59 2.1902e-45
-0.75 195.62 6.0722e-35
-0.5 106.44 2.711e-25
-0.25 54.822 1.0505e-16
0 25.305 1.3794e-09
0.25 9.5895 0.00015719
0.5 3.6966 0.028332
0.75 6.7847 0.0017393
1 21.293 2.0943e-08
The test is significant for all values of lambda
This may indicate that the data need two separate lambdas for pos and neg observations
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
