Source degree of freedom (df) Groups I - 1 Error N - I Total N- 1 Source Sum of Squares (SS) Groups SS(R) = [mathematical expression not reproducible] Error SS(E) = [mathematical expression not reproducible] Total SS(Total) = [mathematical expression not reproducible] Source Mean Squre (MS) Groups MS(R) = SS(R)/(I - 1) Error MS(K) = SS(E)/(N - I) Total In the ANOVA table above, SS(R) is called the regression sum of squares, SS(E) is called the

error sum of squares, SS(Total) is called the total sum of squares, MS(R) is called the mean regression sum of squares, and SS(E)is called the mean

error sum of squares.

(4) We compare the original quadratic PLS with the proposed robust PLS using the explained variance, as well as the predictive mean squared error and the predicted residual

error sum of squares (PRESS).

One is for the group sum of squares in the numerator and one is for the

error sum of squares in the denominator.

The

error sum of squares [SS.sub.e], defining the distribution of Y estimates around the regression function, i.e.

The optimum numbers of terms in the PCA models were indicated by the lowest number of factors that gave the minimum value of the prediction residual

error sum of squares (PRESS) in cross validation in order to avoid over fitting in the models.

Total Sum of Squares (TSS) =

Error Sum of Squares (ESS) + Regression Sum of Squares (RSS) (15)

Coefficient of determinations ([r.sup.2]), mean square error (MSE), root mean square error (RMSE),

error sum of squares (SSE) and prediction sum of squares (PRESS) were calculated.

Error sum of squares [SS.sub.error] = SST - [R.sub.1] - [R.sub.2].

We tested the internal validity of the models using the coefficient of determination ([R.sup.2]), mean square error (MSE),

error sum of squares (SSE) and predicted residual

error sum of squares (PRESS) as described in Ghoreishi et al.

The matching data is found by minimizing the

error sum of squares. The unknown data is obtained by the least squares method simply.

The prediction

error sum of squares PRESS is 8.35, and consequently [R.sub.2] for prediction in our experiment amounts to 0.89.

To identify an optimal grouping of participants in the clustering hierarchy, the agglomeration schedule was examined to find a late stage in the hierarchy, with a relatively small number of participant clusters, where in which the

error sum of squares coefficients increased dramatically at subsequent stages in the hierarchy, after relatively small increases at previous stages (Berven & Hubert, 1977).