Pure error sum of squares

An F-test for lack of fit is based on the ratio of the lack of fit sum to the pure error sum of squares divided by their corresponding degrees of freedom ... 

If there are n replications at q different settings of the independent variables, then the pure-error sum of squares is said to possess (n — 1) degrees of freedom (1 degree of freedom being used to estimate y) while the lack-of-fit sum of squares is said to possess N — p — q(n — 1) degrees of freedom, i.e., the difference between the degrees of freedom of the residual sum of squares and the pure-error sum of squares. 

This is, then, the regression sum of squares due to the first-order terms of Eq. (69). Then, we calculate the regression sum of squares using the complete second-order model of Eq. (69). The difference between these two sums of squares is the extra regression sum of squares due to the second-order terms. The residual sum of squares is calculated as before using the second-order model of Eq. (69) the lack-of-fit and pure-error sums of squares are thus the same as in Table IV. The ratio contained in Eq. (68) still tests the adequacy of Eq. (69). Since the ratio of lack-of-fit to pure-error mean squares in Table VII is smaller than the F statistic, there is no evidence of lack of fit hence, the residual mean square can be considered to be an estimate of the experimental error variance. The ratio... 

The suitability of the regression model should be proven by a special statistical lack-of-fit-test, which is based on an analysis of variance (ANOVA). Here the residual sum of squares of regression is separated into two components the sum of squares from lack-of-fit (LOF) and the pure error sum of squares (PE, pure errors)... 

The residual sum of squares (SSR) contains contributions of a pure error PE due to pure experimental errors, and a lack-of-fit LF due to the inadequacy of the model. The pure error sum of squares PE can be obtained by, for example nc replicated experiments at a number of, at least one, experimental settings. The relationships that hold are given in eq 52 ... 

Here yt are the averaged values of the data for replicates. Equation (7-180) is valid if there are n replicate experiments and the pure error sum of squares (PESS) is known. Without replicates,... 

Since certain experiments have been replicated (in this case all of the experiments, but the treatment is the same if only some of the experiments are repeated) the residual sum of squares may be divided further, into two parts pure error, and lack-of-fit. The pure error sum of squares is given by ... 

The pure error sum of squares SSg/yi is therefore the sum of squares of the differences between the response for each replication and their mean value ... 

The residuals - yi) that are combined to give SS-pe have ni-l degrees of freedom at each level i. Taking the sum over all levels we obtain the number of degrees of freedom of the pure error sum of squares ... 

Since no model can reproduce the pure error sum of squares, the maximum explainable variation is the total sum of squares minus SSpe- In our case, SSt - SSpe = 8930.00 - 45.00 = 8885.00, which corresponds to 8885.00/8930.00 = 99.50% of the total sum of squares. This percentage is close to 100%, because the pure error contribution is relatively small, but it is with this new value that we should compare the variation explained by the regression, 77.79%. The model inadequacy appears clearly in the two first graphs of Fig. 5.8. Once again the residuals are distributed in a curved pattern. 

The lack-of fit test can be apvphed only if the design contains replicate experiments which permit estimation of the so-called pure error, i.e. an error term that is free from modelling errors. Assuming that the replicate experiments are included in regression, the calculations are carried out according to the following equations. First calculate the pure error sum of squares SS ... 

Various sums of square are used to test models. Figure 2.6.3.1-1 shows the partitioning of the total sum of square into its components. The model adequacy can be tested when the lack of fit sum of squares and the pure error sum of squares are available. The latter can be calculated when replicated experiments have been performed. An estimate of the pure error variance is obtained from... 

When replicates are not available and the pure error sum of squares is not known, a different F-test can be applied. It is based upon the regression sum of squares and the residual sum of squares ... 

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