# ScoreYJall

Computes all the 4 score tests for YJ transformation

## Syntax

• outSC=ScoreYJall(y,X)example
• outSC=ScoreYJall(y,X,Name,Value)example

## 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 oftern 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.

 outSC =ScoreYJall(y, X) Ex in which positive and negative observations require the same lambda.

 outSC =ScoreYJall(y, X, Name, Value) Ex in which positive and negative observation require different lambdas.

## Examples

expand all

### Ex 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 which 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



### Ex in which positive and negative observation require different lambdas.

rng(2000)
n=100;
y=randn(n,1);
% Tranform 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

### y — Response variable. Vector.

A vector with n elements that contains the response variable. It can be either a row or a column vector.

Data Types: single| double

### X — Predictor variables. Matrix.

Data matrix of explanatory variables (also called 'regressors') of dimension (n x p-1). Rows of X represent observations, and columns represent variables.

Missing values (NaN's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

Data Types: single| double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as  Name1,Value1,...,NameN,ValueN.

Example:  'intercept',false , 'la',[0 0.5] , 'scoremle',true , 'usefmin',true , 'nocheck',true 

### intercept —Indicator for constant term.true (default) | false.

Indicator for the constant term (intercept) in the fit, specified as the comma-separated pair consisting of 'Intercept' and either true to include or false to remove the constant term from the model.

Example:  'intercept',false 

Data Types: boolean

### la —transformation parameter.vector.

It specifies for which values of the transformation parameter it is necessary to compute the score test. Default value of lambda is la=[-1 -0.5 0 0.5 1]; that is the five most common values of lambda

Example:  'la',[0 0.5] 

Data Types: double

### scoremle —likelihood ratio test for the two different transformation parameters $\lambda_P$ and $\lambda_N$.boolean.

if scoremle is true it is possible to compute the likelihood ratio test. In this case the residual sum of squares of the null model bsaed on a single transformation parameter $\lambda$ is compared with the residual sum of squares of the model based on data transformed data using MLE of $\lambda_P$ and $\lambda_N$. If scoremle is true it is possible through following option usefmin, to control the parameters of the optmization routine.

Example:  'scoremle',true 

Data Types: logical

### usefmin —use solver to find MLE of lambda.boolean | struct.

if usefmin is true or usefmin is a struct it is possible to use MATLAB solvers fminsearch or fminunc to find the maximum likelihood estimates of $\lambda_P$ and $\lambda_N$. The default value of usefmin is false that is solver is not used and the likelihood is evaluated at the grid of points with steps 0.01.

If usefmin is a structure it may contain the following fields:

Value Description
MaxIter

Maximum number of iterations (default is 1000).

TolX

Termination tolerance for the parameters (default is 1e-7).

solver

name of the solver. Possible values are 'fminsearch' (default) and 'fminunc'. fminunc needs the optimization toolbox.

displayLevel

amount of information displayed by the algorithm. possible values are 'off' (displays no information, this is the default), 'final' (displays just the final output) and 'iter' (displays iterative output to the command window).

Example:  'usefmin',true 

Data Types: boolean or struct

### nocheck —Check input arguments.boolean.

If nocheck is equal to true no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=false.

Example:  'nocheck',true 

Data Types: boolean

## Output Arguments

### outSC — description Structure

Containing the following fields:

Value Description
Score

score tests. Matrix.

Matrix of size length(la)-by-5 which contains the value of the score test for each value of lambda specified in optional input parameter la. The first column refers to the global test, the second to the test for positive observations, the third refers to the test for negative observations and the fourth column refers to the F test for the joint presence of the two constructed variables.

If input option scoremle is true the fifth column will contain the exact likelihod ratio test based on the maximum likelihood estimates of the $\lambda_P$ and $\lambda_N$.

If la is not specified, the number of rows of outSc.Score is equal to 5 and will contain the values of the score tests for the 5 most common values of lambda.

laMLE

MLE of lambda. Vector.

Vector of dimension 2 which contains the value of maximum likelihood estimate of $\lambda_P$ and $\lambda_N$. This output is present only if input option scoremle is true.

## 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