Table 6 Validation parameters for each model using multilinear regression (MLR)
S/NO | Validation parameters | Formula | Threshold | Model |
|---|---|---|---|---|
Internal validation | ||||
1 | Friedman lack of fit (LOF) | \( \frac{\mathrm{SEE}}{{\left(1-\frac{w+q\times j}{N}\right)}^2} \) | Significantly low | 0.1802 |
2 | R-squared | \( 1-\left[\frac{\sum {\left({Y}_{\mathrm{obs}\kern0.5em -{Y}_{\mathrm{pred}}}\right)}^2}{\sum {\left({Y}_{\mathrm{obs}\kern0.5em -{\overline{Y}}_{\mathrm{training}}}\right)}^2}\right] \) | R2 > 0.6 | 0.7759 |
3 | Adjusted R-squared | \( \frac{R^2-P\ \left(N-1\right)}{N-p+1} \) | \( {R}_{\mathrm{adj}}^2>0.6 \) | 07381 |
4 | Cross-validated R-squared (\( {Q}_{cv}^2\Big) \) | \( 1-\left[\frac{\sum {\left({Y}_{\mathrm{pred}\kern0.5em -{Y}_{\mathrm{obs}}}\right)}^2}{\sum {\left({Y}_{\mathrm{obs}\kern0.5em -{\overline{Y}}_{\mathrm{training}}}\right)}^2}\right] \) | Q2 > 0.6 | 0.6954 |
5 | Significant regression | Yes | ||
6 | Significance-of-regression F value | 13.42 | ||
7 | Critical SOR F value (95%) | \( \frac{\sum {\left({Y}_{\mathrm{pred}\kern0.5em -{Y}_{\mathrm{obs}}}\right)}^2}{p}/\frac{\sum {\left({Y}_{\mathrm{pred}\kern0.5em -{Y}_{\mathrm{obs}}}\right)}^2}{N-p-1} \) | F(test) > 2.09 | 2.7294 |
8 | Replicate points | 0 | ||
9 | Computed observed error | 0 | ||
10 | Min expt. error for non-significant LOF (95%) | 0.4120 | ||
Model randomization | ||||
11 | Average of the correlation coefficient for randomized data (\( {\overline{\boldsymbol{R}}}_{\boldsymbol{r}} \)) | \( \overline{R}<0.5 \) | 0.3642 | |
12 | Average of determination coefficient for randomized data (\( {\overline{\boldsymbol{R}}}_{\boldsymbol{r}}^{\mathbf{2}}\Big) \) | \( {\overline{R}}_r^2<0.5 \) | 0.1823 | |
13 | Average of leave one out cross-validated determination coefficient for randomized data ( \( {\overline{\boldsymbol{Q}}}_{\boldsymbol{r}}^{\mathbf{2}} \) ) | \( {\overline{Q}}_r^2<0.5 \) | − 0.3915 | |
14 | Coefficient for Y-randomization (c\( {R}_p^2\Big) \) | \( {R}^2\times \left(1-\sqrt{\left|{R}^2-{\overline{R}}_{\mathrm{r}}^2\right|}\ \right) \) | c\( {R}_p^2>0.6 \) | 0.9229 |
External validation | ||||
15 | \( /{\boldsymbol{r}}_{\mathbf{0}}^{\mathbf{2}}-{{\boldsymbol{r}}^{\prime}}_{\mathbf{0}}^{\mathbf{2}}/ \) | < 0.3 | 0.1591 | |
16 | \( \frac{{\boldsymbol{r}}^{\mathbf{2}}-{\boldsymbol{r}}_{\mathbf{0}}^{\mathbf{2}}}{{\boldsymbol{r}}^{\mathbf{2}}} \) | < 0.1 | 0.0023 | |
17 | \( \frac{{\boldsymbol{r}}^{\mathbf{2}}-{{\boldsymbol{r}}^{\prime}}_{\mathbf{0}}^{\mathbf{2}}}{{\boldsymbol{r}}^{\mathbf{2}}} \) | < 0.1 | 0.0136 | |
18 | \( {\boldsymbol{R}}_{\mathbf{test}}^{\mathbf{2}} \) | \( {R}_{test}^2=1-\frac{\sum {\left(Y{\mathrm{pred}}_{\mathrm{test}}-{Y}_{{\mathrm{obs}}_{\mathrm{test}}}\right)}^2}{\sum {\left(Y{\mathrm{pred}}_{\mathrm{test}}-{\overline{Y}}_{\mathrm{training}}\ \right)}^2} \) | >0.6 | 0.6550 |