Lack of fit mean square

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. 

If the nonlinear estimation procedure is carefully applied, a minimum in the sums-of-squares surface can usually be achieved. However, because of the fitting flexibility generally obtainable with these nonlinear models, it is seldom advantageous to fit a large number of models to a set of data and to try to eliminate inadequate models on the basis of lack of fit (see Section IV). For example, thirty models were fitted to the alcohol dehydration data just discussed (K2). As is evident from the residual mean squares of Table II, approximately two-thirds of the models exhibit an acceptable fit of the data... 

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... 

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... 

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... 

Matrix of mean replicate responses and sum of squares due to lack of fit. Calculate the J matrix for Problem 9.6. Calculate the corresponding L matrix and... 

Spreadsheet 8.2. Calculations for test of linear range of data shown in spreadsheet 8.1. Range tested, 3.1-50.0 nM N=24 data points number of concentrations, k, = 6. ME = measurement error, LOF = lack of fit, SS = sum of squares, MS = mean square. 

Insignificant, there is no reason to doubt lack of fit of the obtained regression model, and mean sums of squares of experimental error and lack of fit can be used for variance estimate a2. 

Estimate ofthe experimental error Replication of the center point experiment gives an independent estimate, s of the experimental error variance, o, which can be used to asses the significance of the model. It can be used to evaluate the lack of fit by comparison with the residual mean square, as weU as to assess the significance of the individual terms in the model. (2) Check of curvature If a linear or a second-order interaction model is adequate, the constant term ho will correspond to the expected response, y(0), at the center point. If the difference y(0) - should be significantly greater than the standard deviation of the experimental error as determined by the r-statistic... 

Sometimes it is seen that the residual mean square / (N - p) is compared to an estimate of the pure error from the replicated center point experiments. This is not quite correct, but will reveal a highly significant lack-of-fit. 

Within each of the assays A, B, C, and D, least squares linear regression of observed mass will be regressed on expected mass. The linear regression statistics of intercept, slope, correlation coefficient (r), coefficient of determination (r ), sum of squares error, and root mean square error will be reported. Lack-of-fit analysis will be performed and reported. For each assay, scatter plots of the data and the least squares regression line will be presented. 

Sum of squares Degrees of freedom Mean square Regression Residual Lack of fit, Pure error. 

The null hypothesis that both models perform equally well was tested by comparing the mean square lack-of-fit values (Whitmore, 1991) ... 

Here TV is the number of measurements, P is the number of parameters of the model, and Cj meas and cFcalc are measured and calculated solute concentrations for the /th observation, respectively. The presence of the number of parameters in the denominator makes the mean square lack-of-fit to be an unbiased estimator of the model s standard error (Whitmore, 1991). To test the null hypothesis, one has to compare the / -ratio of the mean of lack-of fit squares F=st ade2 st,fade2 to the critical value of the Fisher s statistic Fn pADE n pfade> where PADE = 2, and PFADE = 3. The null hypothesis can be rejected if F > Fn pADE> n-pfade- Data in Table 2-3 show that the F ratio exceeds the critical value taken at the 0.05 significance level, so that the FADE performs better. 

The ratio = MS QpIMSp p = 1.69 is not significantly high, showing that there is no significant lack-of-fit, and the mean squares for the lack-of-fit and the pure error are comparable. Thus, the residual mean square MS psiD can be used as our estimate for the experimental variance. Taking its square root, the experimental standard deviation is estimated as 0.69, with 10 degrees of freedom. 

In conclusion, despite the indication of the test point 7, going from a quadratic to a reduced cubic model does not improve the model. There is a substantial and statistically significant lack of fit of the model to the data. The probability that the lack of fit is due to random error is less than 0.1 %. Values of the F ratio are therefore calculated using the pure error mean square. 

Dividing each of these sums of squares by their respective numbers of degrees of freedom we obtain three mean squares, whose values we can compare to evaluate the model s possible lack of fit. 

With the partitioning of the residual sum of squares into contributions from lack of fit and pure error, the ANOVA table gains two new lines and becomes the complete version (Table 5.8). The pure error mean square. 

Since there is no lack of fit, both MSiof and MS estimate a. We can take advantage of this fact to obtain a variance estimate with a larger number of degrees of freedom, summing SSiof and SS and dividing the total by (viof + Vpe). With this operation, we again calculate the residual mean square, which now becomes a legitimate estimate of the pure error. 

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