Significance testing normal probability plots

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]. 

Significant effects, i.e. effects that are significantly larger than could be due to experimental variability, can be identified by means of both graphical and statistical methods. The graphical method that is used most often is the normal probability plot explained in the preceding section (Fig. 6.9). The statistical tests are often based on a /-test, where the test statistic can be written as... 

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). 

The significant variables will be identified as those for which the corresponding coefficient in the model is significantly larger than the experimental error. The significance can be assessed by statistical tests, such as r-tests, F-tests on ANOVA tables, or from cumulative normal probability plots. 

The optimal model obtained by SROV, using the quadratic model with transformed variables, includes 12 explanatory variables (13 parameters), where all the confidence intervals are significantly different from zero. The variance is the smallest of all the models tested and is the closest to one. The condition number is very small in comparison to the other models, thus this solution can be considered highly accurate. The linearity of the normal probability plot for this case indicates normal distribution of the residuals. 

Finally, some brief comments regarding the different methods are needed. Firstly, it can be noted that the normal probability plot method can be used irrespective of whether replicates were performed. In this particular case, the method underestimates the number of significant parameters in the model. On the other hand, the F-test method produces a larger model, and it would seem a more accurate model, but it does require that replicates be available. 

---