# Faq how RMSEC and RMSECV related to R2Y and Q2Y seen other software

### Issue:

How are RMSEC and RMSECV related to R2Y and Q2Y I see in other software?

### Possible Solutions:

In some software, the values "R2Y" and "Q2Y" are reported for regression models. The R2Y value is equivalent to the y-block cumulative variance captured (as reported in the 5th column of the variance captured table or the .detail.ssq field of a model).

${\displaystyle R2Y=1-{\frac {RMSEC^{2}}{{\frac {1}{m}}\sum _{i=1}^{n}\left(y_{i}-{\overline {y}}\right)^{2}}}}$

The "Q2Y" value is analogous to R2Y except it is based on the cross-validated results. It is related to the RMSECV values according to this equation :

${\displaystyle Q2Y=1-{\frac {RMSECV^{2}}{{\frac {1}{m}}\sum _{i=1}^{n}\left(y_{i}-{\overline {y}}\right)^{2}}}}$

where RMSECV is the root mean square error of cross-validation, m is the number of samples and yi is the actual (aka measured) y-value for sample #i. These relations are only true if the y-block is mean-centered before the model is built.

R2Y and Q2Y represent fractions of variance captured while the cumulative variance captured table and .detail.ssq field represent percentages. They are identical except for a factor of 100 difference between fraction and percentage.

Given a PLS model named "m" which used only mean centering or autoscaling on the univariate y-block, the following code calculates R2Y and Q2Y:

>> incl = m.detail.include{1,2};
>> y    = m.detail.data{2}.data(incl,:);
>> my   = length(incl);
>> R2Y = (1-(m.rmsec.^2)*my./sum(mncn(y).^2))
>> Q2Y = (1-(m.rmsecv.^2)*my./sum(mncn(y).^2))


The practical aspects of these statistics are:

R2Y and Q2Y generally increase towards 1 as a model's fit improves whereas RMSEC and RMSECV decrease to zero

RMSEC/CV are in units of the original y-block and can be interpreted as "error levels" (They are very similar to standard deviations) whereas R2Y and Q2Y are in fractional units

These values are available from:

- the Matlab command line

>> m.r2y
>> m.q2y


- an eigenvalue plot from the Analysis window

It is possible for Q2Y to exceed the 0 → 1 limit if the predicted y-values are particularly bad.)