Pure error

A number of replications under at least one set of operating conditions must be carried out to test the model adequacy (or lack of fit of the model). An estimate of the pure error variance is then calculated from ... 

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 ... 

Here, y is the average of all of the replicated data points. If the residual sum of squares is the amount of variation in the data as seen by the model, and the pure-error of squares is the true measure of error in the data, then the inability of the model to fit the data is given by the difference between these two quantities. That is, the lack-of-fit sum of squares is given by... 

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. 

The sums of squares of the individual items discussed above divided by its degrees of freedom are termed mean squares. Regardless of the validity of the model, a pure-error mean square is a measure of the experimental error variance. A test of whether a model is grossly adequate, then, can be made by acertaining the ratio of the lack-of-fit mean square to the pure-error mean square if this ratio is very large, it suggests that the model inadequately fits the data. Since an F statistic is defined as the ratio of sum of squares of independent normal deviates, the test of inadequacy can frequently be stated... 

In some cases when estimates of the pure-error mean square are unavailable owing to lack of replicated data, more approximate methods of testing lack of fit may be used. Here, quadratic terms would be added to the models of Eqs. (32) and (33), the complete model would be fitted to the data, and a residual mean square calculated. Assuming this quadratic model will adequately fit the data (lack of fit unimportant), this quadratic residual mean square may be used in Eq. (68) in place of the pure-error mean square. The lack-of-fit mean square in this equation would be the difference between the linear residual mean square [i.e., using Eqs. (32) and (33)] and the quadratic residual mean square. A model should be rejected only if the ratio is very much greater than the F statistic, however, since these two mean squares are no longer independent. 

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... 

This model has previously been shown (Hll, K12) to have a residual mean square comparing favorably with that expected from pure error, as discussed in Section IV. It is to be noted that we have been led logically from one model to another within the small class of models for which n — 3 by the above analysis. For these data, adsorbed methane is not required however, for data with higher methane concentrations, the adsorbed-methane term may be needed. 

The amount data corresponding to the response values in 1 above were transformed by the same general family of power transformations until linearity was obtained. The F-test statistic that relates lack of fit and pure error was used as the criterion for linearity. 

F =MSLF/MSPE, based on the ratio mean square for lack of fit (MSLF) over the mean square for pure error (MSPE) ( 31 ). F follows the F distribujfion with (r-2) and (N-r) degrees of freedom. A value of F regression equation. Since the data were manipulated by transforming the amount values jfo obtain linearity, i. e., to achieve the smallest lack of fit F statistic, the significance level of this test is not reliable. 

Formal tests are also available. The ANOVA lack-of-fit test ° capitalizes on the decomposition of the residual sum of squares (RSS) into the sum of squares due to pure error SSs and the sum of squares due to lack of fit SSiof. Replicate measurements at the design points must be available to calculate the statistic. First, the means of the replicates (4=1,. .., m = number of different design points) at all design points are calculated. Next, the squared deviations of all replicates U — j number of replicates] from their respective mean... 

Most textbooks refer to as the variance due to pure error , or the pure error variance . In this textbook, is called the variance due to purely experimental uncertainty , or the purely experimental uncertainty variance . What assumptions might underlie each of these systems of naming See Problem 6.14. [See, also, pages 123-127 in Mandel (1964).]... 

The book has been written around a framework of linear models and matrix least squares. Because we authors are so often involved in the measurement aspects of investigations, we have a special fondness for the estimation of purely experimental uncertainty. The text reflects this prejudice. We also prefer the term purely experimental uncertainty rather than the traditional pure error , for reasons we as analytical chemists believe should be obvious. 

The response is not only affected by the tolerances of x, but also the measurement error (or pure error) in the response disturbs the response, of which the following assumptions are made ... 

The procedure of calculating the experimental error variance or the so-called pure error consists of ... 

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 idea of the lack-of-fit test is to compare the pure error of the regression line with the error due to the use of an inappropriate regression model. The MSLOf, which is a measure for the spread of the mean response per concentration from the regression line, is divided by the MSpE, which is a measure for the spread of the instrument response due to experimental variation. The obtained F-value (F = MS[ oii/MS i) is compared with the F-distribution with k 2 and n k d.f. 

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