Sum of squares due to purely experimental uncertainty

The matrix of purely experimental deviations, P, is obtained by subtracting J from Y. 

Zeros will appear in the P matrix for those experiments that were not replicated (J lt = ytl for these experiments). The sum of squares due to purely experimental uncertainty is easily calculated. 

The sum of squares due to purely experimental uncertainty has n—f degrees of freedom associated with it. 

The sum of squares due to lack of fit, SSlof, and the sum of squares due to purely experimental uncertainty, SSpe, add together to give the sum of squares of residuals, SSr. 

Summary of matrix operations used to calculate sums of squares. 

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If the model = 0 + r, does describe the true behavior of the system, we would expect replicate experiments to have a mean value of zero (y,- = 0) the sum of squares due to purely experimental uncertainty would be expected to be... 

If (and only if) replicate experiments have been carried out on a system, it is possible to partition the sum of squares of residuals, SS, into two components (see Figure 6.10) one component is the already familiar sum of squares due to purely experimental uncertainty, 55. the other component is associated with variation attributed to the lack of fit of the model to the data and is called the sum of squares due to lack of fit, SS. ... 

Assume the model = 0 + r, is used to describe the nine data points in Section 3.1. Calculate directly the sum of squares of residuals, the sum of squares due to purely experimental uncertainty, and the sum of squares due to lack of fit. How many degrees of freedom are associated with each sum of squares Do and SS add up to give SS l Calculate and What is the value of the Fisher F-ratio for lack of fit (Equation 6.27)7 Is the lack of fit significant at or above the 95% level of confidence ... 

In Section 6.4, it was shown for replicate experiments at one factor level that the sum of squares of residuals, SS can be partitioned into a sum of squares due to purely experimental uncertainty, SS, and a sum of squares due to lack of fit, SSi f. Each sum of squares divided by its associated degrees of freedom gives an estimated variance. Two of these variances, and were used to calculate a Fisher F-ratio from which the significance of the lack of fit could be estimated. 

Before discussing the sum of squares due to lack of fit and, later, the sum of squares due to purely experimental uncertainty, it is computationally useful to define a matrix of mean replicate responses, J, which is structured the same as the Y matrix, but contains mean values of response from replicates. For those experiments that were not replicated, the mean response is simply the single value of response. The J matrix is of the form... 

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