Normal probability plot of residuals

To verify the adequacy of the developed models of solvent-resistance in THF, chloroform, and MEK, normal probability plots were evaluated. Typical normal probability plots of residuals should be close to a straight line as shown in Fig. 19.7 because the underlying error distribution is expected to be normal.47 This means that the normality assumption is valid for the proposed model. Residuals that were intensified in the middle of straight line indicated that data were normally distributed. Also, there were no outliers in the model as indicated by absence of significant deviations from the straight line. A combination of the normal distribution of the model residuals (Fig. 19.7) and the very high values of adjusted R2 demonstrated a good quality of the model. 

Prediction of the log reduction of an inoculated organism as a function of acid concentration, time, and temperature can also be done by a mathematical model developed for this purpose, using the second-order polynomial equation to fit the data. The following tests justified the reliability of the model the analysis of variance for the response variable indicated that the model was significant (P < 0.05 and R2 = 0.9493) and had no significant lack of fit (P > 0.05). Assumptions underlying the ANOVA test were also investigated and it was demonstrated that with the normal probability plot of residuals, plot of residuals versus estimated values for the responses, and plot of residuals versus random order of runs, that the residuals satisfied the assumptions of normality, independence, and randomness (Jimenez et al., 2005). 

Fig. 12.4 Normal probability plot of residuals in tbe enamine study.

Figure 4. Normal probability plot of the studentized residuals from the plasma etching experiment.

To check the assumptions of the model, Bartlett s or Levene s tests can be used to assess the assumption of equality of variance, and the normal probability plot of the residuals (etj = Xij - Xj) to assess the assumption of normality. If either equality or normality are inappropriate, we can transform the data, or we can use the nonparametric Kruskal-Wallis test to compare the k groups. In any case, the ANOVA procedure is insensitive to moderate departures from the assumptions (Massart et al. 1990). 

A normal probability plot of the residuals is shown in Fig. 12.4. There are evidently some abnormally high errors associated with the replicated center point experiments. Maybe they were run on a different occasion. The remaining residuals... 

Fig. 9 shows the fitted and CV-predicted production values and the corresponding residual normal probability plots of models 1-3. By cross-validation, the model 2, i.e. y = bg +b X +bj2 i 2 the best one. Finally, Fig. 10 shows the contour plot of the best model, model 2. 

Fig. 9. Cross-validation of models 1-3. Left panel Production vs. the number of experiment black circles data blue triangles fitted values red pluses cross-validated leave-one-out prediction. Right panel Normal probability plots of the cross-validated leave-one-out residuals.

Tests for Normal Distribution The most common method to test normality is to plot a normal probability plot of the residuals. The points should lie along a straight line. Examples of good and bad normal probability plots are shown in Table 3.2. Alternatively, more advanced methods that consider the correlatiOTi properties of normally distributed errors can be used. 

The normal probability plot of the residuals is shown in Fig. 3.3. There seem to be some mild deviations from normality in the central region. Overall, given the small sample, there is not much that can be concluded with this particular sample. 

Fig. 3.4 (Jop) Normal probability plots of the residuals and bottom) residuals as a function of temperature for (left) linearised and right) nonlinear models...

Fig. 3.9 Normal probability plot of the residuals for the quadratic case...

It can be noted that adding a quadratic term has improved the size of the parameter estimate confidence intervals, as well as increasing R. Furthermore, the normal probability plot of the results seems to suggest that there could be some problems due to the clustering of values. The residuals as a function of temperature plot does not show any real issues. There does seem to be a small increase in variability of the values as the temperature increases. There are no discernible parameters that would improve the fit. 

Plot the residuals as a function of weight percent and as a function of the freezing point. Include a normal probability plot of the residuals. Are there any issues with the model assumptions ... 

Fig. 4.6 (Top) Normal probability plot of the residuals and (bottom) time series plot of the residuals with the different replicates clearly shown...

Fig. 5.21 (Left) Residual analysis for the final temperature model autocorrelation plot of the residuals and (right) normal probability plot of the residuals...

The normal probability plot of the residuals and the autocorrelation of the residuals are shown in Fig. 5.21. Both results show that the residuals are normally distributed and white. In the normal probability plot, the tails deviate a bit from what would be desirable, but given that this is real data, such behaviour is inevitable. 

Fig. 7.1 Linear regression example MATLAB plots of the (top, left) normal probability plot of the residuals, (top, centre) residuals as a function of y, (top, right) residuals as a function of the first regressor, Xj, (bottom, left) residuals as a function of x, (bottom, centre) residuals as a function of y, and (bottom, right) a time series plot of the residuals...

The variance ratio (F value) is not readily calculated because replicated data are not available to allow the residual error term to be evaluated. However, it is usual practice to use the interaction data in such instances if the normal probability plot has shown them to be on the linear portion of the graph. By grouping the interaction terms from Table 7 as an estimate of the residual error,... 

Small departures of normality do not significantly influence the use of the calibration model in residue analysis. However, major departures of normality are mostly related to analytical or instrumental problems. The use of an inappropriate calibration model can give rise to nonnormality of the residuals. In this case also, one or more of the other four basic assumptions have been violated. Normality can be evaluated by means of several statistical tests (i.e., Kolgomorov-Smirnov, Shapiro-Wilk W) or by constructing normal probability plots [8]. 

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