Pure error 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. 

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 standard deviation is the square root of the pure error mean square. Taking its antilogarithm we obtain 1.081, indicating an overall error of about 8%. 

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. 

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. 

Surely you noticed that the response for one of the three replicates of run no. 6 is much lower than the other two. If this value is removed from the data as a suspected outlier, the pure error mean square drops from 17.4 to only 3.6. The downside of this decision is that in comparison with the smaller error value even the special cubic model shows lack of fit. This suggests that a more extensive model is needed to adequately represent these data. 

Dividing the sum of squares by the number of degrees of freedom for the respective case yields the, values for the mean squares, MS eg, MSloF, MSpg, MS,ot,corr- The pure error mean square, MSpg, is an estimate of that is valid both when there is a lack of fit in the model and when there isn t. 

Obtained by dividing the regression mean square (3, 5 degrees of freedom). square by the pure error mean ... 

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. 

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... 

The NRL tight-binding method has been used to address the adsorption of 02 on Pt(l 1 1) [99]. The Pt-Pt interactions were taken from a large data base of TB parameter for the elements which are posted on the world wide web [100]. These parameters were obtained from a fit to DFT bulk calculations. Still, it has been demonstrated that the pure Pt surface is also well-described by this parametrization [42], For the Pt-O and the 0-0 TB parameters a new fit had to be performed. They were adjusted in order to reproduce the GGA-DFT results of the 02/Pt(l 1 1) potential energy surface [91, 92], The root mean square error of the fit is below 0.1 eV [41] which is in the range of the error of the GGA-DFT calculations. The spin state of the oxygen molecule was not explicitly considered in the... 

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. 

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

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 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. 

SSpe, sum of squares pure error = (y,y — which reflects the vari ability of the replicate values about the mean of those values, yj. 

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. 

Since there is practically no lack of fit for the quadratic model, we can take the square root of the residual mean square as an estimate of pure error ... 

This method was in fact carried out around two decades ago [30, 31]. However, it was applied only in the fermentation of pure microbial cultures. In a recent report by Acros-Hernandez and coworkers [32], infrared spectroscopy was applied to quantify the PHA produced in microbial mixed cultures. Around 122 spectra from a wide range of production systems were collected and used for calibrating the partial least squares (PLS) model, which relates the spectra with the PHA content (0.03-0.58 w/w) and 3-hydroxyvalerate monomer (0-63 mol%). The calibration models were evaluated by the correlation between the predicted and measured PHA content (R ), root mean square error of calibration, root mean square error of cross validation and root mean square error of prediction (RMSEP). The results revealed that the robust PLS model, when coupled with the Fourier-Transform infrared spectrum, was found to be applicable to predict the PHA content in microbial mixed cultures, with a low RMSEP of 0.023 w/w. This is considered to be a reliable method and robust enough for use in the PHA biosynthesis process using mixed microbial cultures, which is far more complex. 

In Figure 4 the root mean square error (RMS) and the root mean square error of cross validation (RMSECV) of different data processing methods and parameters are shown. As expected, the RMSECV is larger than the RMS for each method. The larger errors of the IHM are due to the non-perfect description of the pure spectra. Interestingly, CPR shows for a set of ranks (number of components used for description of the spectra) and power coefficients the lowest errors. In this example of the mixture of water and oil, this is attributed to the fact, that CPR not only considers the correlation, but also the variance with a power coefficient. 

The mean square for pure error is calculated from the center points as follows ... 

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