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In the discriminant analysis, the actual values of the five soil attributes (CEC, porosity, sand, TP, and Ca:Mg ratio) with high factor loadings retained in the four PCs (Table 5.3) were used. The discriminant function coefficients (Table 5.4) show that soil porosity followed by CEC and sand content are the best discriminators in the first function between group 1 (low SQ) and the combination of group 2 (medium SQ) and group 3 (high SQ), but Ca:Mg was least effective in discriminating these groups. The trend of the discriminant coefficients of these independent variables is similar to that of function 1 in function 2 (Table 5.4). This is because in function 2 soil porosity, CEC and sand are the variables with the largest standardized coefficients that discriminate best between the medium and high SQ category. This indicates that soil porosity and CEC in the PC1 factor, followed by the sand content in PC2, offers the greatest potential for monitoring changes in SQ variability with changes in land-use and soil management practices at catchment scale, as these are the most important for group separation in the discriminant function.

About 95% of the variance explained by the discriminant model is due to the first discriminant function, and the remaining 5% to the second function. This indicates that the variability between the low SQ group and the combination of the medium and high SQ groups is higher than that between the medium and high SQ groups. In addition, the relation of each group variable (dependent variables) with the independent variable as indicated by a discriminant function coefficient shows that soil porosity followed by sand content and CEC is the most influential in all the group variables (Table 5.4). But the size of prediction by the same independent variable is not the same in all the group variables. As a result, the R² of the independent variables in the low, medium and high SQ status as group variables is explained by 94, 88 and 94%, respectively. This percentage is analogous to the R² in the multiple regression analysis.

When we examine the relationship of the functions and the predictors, the coefficient of

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each independent variable defines the extent of the effect of that variable on the dependent variable and the sign of the coefficient the direction of the effect.

Table 5.4: Standardized and unstandardized coefficient functions of multiple discriminant analysis

Function ^a Constant Porosity CEC Sand TP Ca:Mg Model^b

1 -5.004 0.516 0.491 -0.435 0.341 0.086 (R² = 95%), P = 0.000 2 -5.622 0.991 -0.689 0.548 0.102 0.178 (R² = 5%), P = 0.008 Group Constant Porosity CEC Sand TP Ca:Mg Model Low SQ -32.843 1.029 0.289 0.561 0.204 0.112 (R² = 94%), P = 0.000 Medium SQ -50.101 1.457 0.465 0.476 0.389 0.167 (R² = 88%), P = 0.001 High SQ -53.973 1.503 1.503 0.352 0.524 0.219 (R² = 94%), P = 0.000

a Wilks’Lambda test of functions shows that the discriminant model was significant at probability P = 0.000 and 0.008, for function 1 and 2, respectively, indicating that these functions contributed more in the model.

b Coefficient of determination (R²) is optimal combination of the variables so that the functions provide the best overall discrimination between groups and prediction within groups.

Sand (%); total porosity (%); TP, total phosphorous (mg kg^-1 soil); Ca, exchangeable calcium (cmolc

kg^-1); Mg, magnesium (cmolc kg^-1); CEC, cation exchangeable capacity (cmolc kg^-1)

In addition to the discrimination function coefficients, visualization of the functions that discriminate the group variables by plotting the individual scores of each case is crucial (Figure 5.5). In this figure, the first discriminant function is shown to discriminate mainly between the group of low SQ and the combined groups (medium and high SQ categories) because low SQ falls to the left of the centre line (0), but the combined groups to the right of the centre line in function 1. In the vertical direction (function 2), some of the low SQ category points fall above the center line (0).

However, most medium SQ points are above the centre line of function 2. Most points of the high SQ category fall below the centre line (0) of function 2. The implication is that the second discrimination function discriminates between the medium and high SQ category (Figure 5.5).

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Figure 5.5: Discriminant functions separating the group variables as low, medium and high soil quality (SQ) category. Note: Group means are the centroids used as the cutting points for classifying cases to each group (SQ categories) Figure 5.5 and Table 5.5 show that for the original grouped cases, the discriminant analysis correctly classified 16, 15 and 16 of the 17 in each group as low, medium and high SQ categories with a 94.1, 88.2 and 94.1% correct classification rate, respectively.

In addition, in the cross-validated cases, 15 of the 17 cases in each group of the low and medium SQ category, the correct classification rate was 88.2%, which is similar in both groups. Of the 17 high SQ category group cases, 16 were correctly classified, i.e., a 94.1% correct classification rate in the cross-validated cases. Overall, about 92.1% of the original grouped cases and 90.2% of the cross-validated cases were correctly classified by the discriminant analysis method. This suggests that the overall prediction capability of the discriminant function analysis based on the independent variables can be accepted as more than 90% correct classification is adequate in discrimination of the SQ categories identified by the local farmers.

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Table 5.5: Classification of soil quality (SQ) categories (group variables) by discriminant analysis method

Case Actual group ^a

Discriminant classification of predicted group membership ^b

Low SQ Medium SQ High SQ Group classification rate (%)

Original group

Low SQ 16 1 0 94.1

Medium SQ 0 15 2 88.2

High SQ 0 1 16 94.1

Total 16 17 18 92.1^d

Cross-validated ^c

Low SQ 15 2 0 88.2

Medium SQ 0 15 2 88.2

High SQ 0 1 16 94.1

Total 15 18 18 90.2 ^e

a 17 weighted cases in each SQ category.

b Boldface figure in each group is number of cases correctly classified by the discriminant function analysis

c In cross-validation, each case is classified by the functions derived from all cases other than that case.

d Overall 92.1% of original grouped cases correctly classified.

e Overall 90.2% of cross-validated cases correctly classified.

5.3.6 Implication of evaluating farmer knowledge with scientific measurements