Sum of squares corrected for the mean

This matrix may be used to calculate another useful sum of squares, the sum of squares corrected for the mean, SS, sometimes called the sum of squares about the mean or the corrected sum of squares. 

The sum of squares corrected for the mean has n - degrees of freedom associated with it. 

Figure 9.4 emphasizes the relationship among three sums of squares in the ANOVA tree - the sum of squares due to the factors as they appear in the model, SSf (sometimes called the sum of squares due to regression, SS ) the sum of squares of residuals, SS, and the sum of squares corrected for the mean, (or the total sum of squares, SSj, if there is no Pq term in the model). 

It is important to realize that an / or r value (instead of an or value) might give a false sense of how well the factors explain the data. For example, the R value of 0.956 arises because the factors explain 91.4% of the sum of squares corrected for the mean. An R value of 0.60 indicates that only 36% of 55 has been explained by the factors. Although most regression analysis programs will supply both R (or r) and R (or r ) values, researchers seem to prefer to report the coefficients of correlation R and r) simply because they are numerically larger and make the fit of the model look better. 

Use the C matrix of Equation 9.5 to calculate the sum of squares corrected for the mean, SScorr> for the nine responses in Section 3.1 (see Equation 9.6). How many degrees of freedom are associated with this sum of squares ... 

Although the partitioning of the total sum of squares into a sum of squares due to the mean and a sum of squares corrected for the mean may be carried out for any data set, it is meaningful only for the treatment of models containing a / 0 term. In effect, the /30 term provides the degree of freedom necessary for offsetting the responses so the mean of the corrected responses can be equal to zero. 

It is important to realize that R might give a false sense of how well the factors explain the data. For example, the R value of 0.956 arises because the factors explain 91.4% of the sum of squares corrected for the mean. An R value of 0.60 indicates that only 36% of SScori has been explained by the factors. 

The total variance, expressed as the sum of squares of deviations from the grand mean, is partitioned into the variances within the different groups and between the groups. This means that the sum of squares corrected for the mean, is obtained from... 

For the total sum of squares, Eq. (2.45) is again valid, that is, the total sum of squares corrected for the mean is obtained as sum... 

The coefficient of determination describes the fraction of the sum of squares due to the factors in relation to the sum of squares corrected for the mean. The square root of the coefficient of determination reveals the multiple correlation coefficient ... 

Table 6.4 provides the results for the ANOVA. The sum of squares corrected for the mean is explained by 95.41% (coefficient of determination Eq. (6.17)) due to the factors, here the wavelength. The F test for goodness-of-fit is also given in Table 6.4. Based on a significance level of a = 0.05, the goodness-of-fit test is significant, since the p level is smaller than 0.05. This means that the parameters of the linear calibration model are significantly different from zero.

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