Table 2 Some equations and parameters used for the model validation

Internal validation

Friedman Lack-Of-Fit (LOF)

\({\text{LOF}} = \frac{{{\text{SEE}}}}{{\left( {1 - \frac{c + d \times p}{M}} \right)^{2} }}\)

\({\text{SEE}} = \sqrt {\frac{{\left( {Y_{\exp } - Y_{{{\text{pred}}}} } \right)^{2} }}{N - P - 1}}\)

4

Allows for the best fitness score to be obtained

–

Correlation coefficient (R²)

\(R^{2} = 1 - \left[ {\frac{{\sum \left( {Y_{\exp } - Y_{{{\text{pred}}}} } \right)^{2} }}{{\sum \left( {Y_{\exp } - \overline{Y}_{{{\text{training}}}} } \right)^{2} }}} \right]\)

5

Measures the degree of fitness of the regression equation

 ≥ 0.6

Adjusted R²

\(R_{{{\text{adj}}}}^{2} = \frac{{R^{2} - p\left( {n - 1} \right)}}{n - p + 1}\)

6

Ensures model’s stability and reliability

 ≥ 0.5

Cross-validation regression coefficient (Q²cv)

\(Q_{{{\text{cv}}}}^{2} = 1 - \left[ {\frac{{\sum \left( {Y_{{{\text{pred}}}} - Y_{\exp } } \right)^{2} }}{{\sum (Y_{\exp } - \overline{Y}_{{{\text{training}}}} )^{2} }}} \right]\)

7

Indicates a high internal predictive power

 ≥ 0.5

The coefficient of determination (\(cR_{{\text{p}}}^{2}\)) of Y-Randomization

\(cR_{{\text{p}}}^{2} = R X [R^{2} - \left( {R_{r} } \right)^{2} ]^{2}\)

8

This is for a confirmation that the QSAR model built is strong and not created by chance

\(cR_{{\text{p}}}^{2}\) > 0.50

External validation

Predicted R² (R² test)

\(R_{{{\text{test}}}}^{2} = 1 - \frac{{\sum \left( {Y{\text{pred}}_{{{\text{test}}}} - Y\exp_{{{\text{test}}}} } \right)^{2} }}{{\sum \left( {Y{\text{pred}}_{{{\text{test}}}} - \overline{Y}_{{{\text{training}}}} } \right)^{2} }}\)

9

Measures the ability of the model to predict activity values of external set of compounds

 ≥ 0.6

Golbraikh and Tropsha acceptable model criteria

\(\left| {r_{o}^{2} - r_{o}^{^{\prime}2} } \right|\)

\(\left| {r^{2} - \frac{{r_{o}^{{{^{\prime}}2}} }}{{r^{2} }}} \right|\)

kʹ (threshold value)

–

Assess the robustness and stability of the model

 < 0.3

 < 0.1

0.85 ≤ k′ ≤ 1.15