Determination Coefficient

Determination coefficient (R2) usually regarded as the key factor that reflects the quality of regression model describing relationships between dependent and independent variables in a model. R2 is calculated by the formula:

R2 shows what fraction of variance of the observed variable can be explained by the created model, that is, the value of determination coefficient determines the fraction (in percents) of changes caused by factor characteristics in the general variability of the output characteristic (property).

The R2 value must lie within the range from zero to one: 0 ≤ R2 ≤ 1. A model is considered to be more reliable if the value of the coefficient of determination is around 1. If R2=1, the empiric points (xi; yi) lie strictly in the regression line, and there is a linear functional dependency between the Y and X variables. If R2=0, the variance of the dependent variables is caused only by the factors not accounted for in the model.

The uncentered determination coefficient () is used to compare the quality of models with a constant and without a constant. The uncentered determination coefficient is calculated differently for the models with a constant and without a constant:

The constant value is estimated	The constant is not used	The constant value is defined manually

Where:

yi. Values of the observed variable.
. Average value of the experimental (observed) data.
. Model values based on estimated parameters.
Cfix. Fixed constant value.

Thus, the value always lies within the range from 0 to 1.

However, it is incorrect to use determination coefficient to compare quality of models that include different number of regressors, because R2 grows as the number of regression factors increases. Adding a new characteristic to a model cannot decrease the value of R2, because each next variable can only add to rather than reduce the information explaining the behavior of the output variable. To eliminate growth effect, determination coefficient is adjusted by the number of factors. This new coefficient is named adjusted determination coefficient.