Residuals statistical test

The first criterion compares the hazardous waste residues to waste residues that would be found if the BIF were not burning hazardous waste at all. A statistical test describes methods that should be used when comparing the waste-derived residues with these baseline levels to determine... 

Over time, statisticians have devised many tests for the distributions of data, including one that relies on visual inspection of a particular type of graph. Of course, this is no more than the direct visual inspection of the data or of the calibration residuals themselves. However, a statistical test is also available, this is the x2 test for distributions, which we have previously described. This test could be applied to the question, but shares many of the disadvantages of the F-test and other tests. The main difficulty is the practical one this test is very insensitive and therefore requires a large number of samples and a large departure from linearity in order for this test to be able to detect it. Also, like the F-test it is not specific for nonlinearity, false positive indication can also be triggered by other types of defects in the data. 

Examination to ascertain if the residuals are represented by a normal distribution (so that statistical tests can be applied). 

Figure 9.5 emphasizes the relationships among three other sums of squares in the ANOVA tree - the sum of squares due to lack of fit, SS f , the sum of squares due to purely experimental uncertainty, SS and the sum of squares of residuals, 55,. Two of the resulting variances, and were used in Section 6.5 where a statistical test was developed for estimating the significance of the lack of fit of a model to a set of data. The null hypothesis... 

Residuals should not be serially correlated, as identified by visual inspection or by an appropriate statistical test (for example a Run s test). 

Rpl =0.179, Rp2 = 0.202, Ral =0.207, and Ra2 = 0.249. Statistical tests show that model a2 can be rejected in favor of aj. The choice between the parallel models pj and p2 is more difficult but examination of model p2 shows that this model cannot be fully hydrogen bonded, and hence p2 is rejected in favor of pj, which also gives better x-ray agreement. Thus models pj and aj were taken as the most likely parallel and antiparallel models for further refinement. At this point the unobserved data were included, calculating weighted R and R" where w = l for observed and w=l/2 for unobserved reflections. F(hkl) for an unobserved reflection was set at two thirds an assigned threshold and was included only if the calculated structure amplitude exceeded the threshold. The final residuals for the two models were Rpi = 0.233, R = 0.299, and Rj = 0.215, R j = 0.270. Application of the Hamilton statistical test (13) to these data indicates that the a model can be rejected at the 99.5% level. 

The following protocol was proposed and consisted of 4 measuring days. Each day, four (or six at day 1) standards and four samples are analyzed. The calibration curves are constructed by least squares regression analysis and statistically tested for nonlinearity by means of an F-test on the residuals. The amount of cortisol in the serum samples is obtained by linear interpolation on the daily calibration curve. Preliminary experiments were also set up to determine the influence of the use of peak height or peak area ratios. For the cortisol measurement, some separation takes place between syn and anti isomers, therefore the use of peak heights is less favorable. 

The linearity of (a part of) the range should be evaluated to check the appropriateness of the straight-line model. This can be achieved by a graphical evaluation of the residual plots or by using statistical tests. It is strongly recommended to use the residual plots in addition to the statistical tests. Mostly, the lack-of-fit test and Mandel s fitting test are used to evaluate the linearity of the regression line [8, 10]. The ISO 8466 describes in detail the statistical evaluation of the linear calibration function [11]. 

Heteroskedasticity can be evaluated by means of several statistical tests (modified Levene, White, or Breusch-Pagan test) or by looking at the residual plots [20-22],... 

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]. 

We now assume that the least-squares refinement has converged satisfactorily, that any necessary rejection of discordant data has taken place before the final cycles were carried out, and that statistical tests on the weighted residuals have given reassuring results. It is now appropriate to estimate the uncertainties in the determined values of the adjustable parameters a,.f... 

Obtain the reduced chi-square value xl given by Eq. (31b) and carry out the appropriate statistical tests for goodness of fit, including inspection of the weighted residuals for systematic trends (see Goodness of Fit). 

In Equations 4 and 5, r is the multiple correlation coefficient, r2 is the percent correlation, SE is the standard error of the equation (i.e the error in the calculated error squares removed by regression to the mean sum of squares of the error residuals not removed by regression. The F-values were routinely used in statistical tests to determine the goodness of fit of the above and following equations. The numbers in parentheses beneath the fit parameters in each equation denote the standard error in the respective pa-... 

F-ralio test. Among the most well-known statistical tests, this is defined as the ratio between the model sum of squares MSS and the residual sum of squares RSS ... 

After outliers have been purged from the data and a model has been evaluated visually and/or by, e.g. residual plots, the model fit should also be tested by appropriate statistical methods [2, 6, 9, 10, 14], The fit of unweighted regression models (homoscedastic data) can be tested by the ANOVA lack-of-fit test [6, 9]. A detailed discussion of alternative statistical tests for both unweighted and weighted calibration models can be found in Ref. [16]. The widespread practice to evaluate a calibration model via its coefficients of correlation or determination is not acceptable from a statistical point of view [9]. 

After the PLS model is developed with the in-control (calibration) data set, the statistics for the residuals are computed for setting the null hypothesis. A test sample block of size ut x p is taken from the process measurements. The residual statistics for the test sample are then generated by using the PLS model developed. The statistical test compares the residual statistics of the test sample with the statistics of the calibration for detecting any significant departures. 

Figure 8.4. Plots of residuals statistics for test data, (a) Residuals means, (b) residuals variances, (c) residuals variance ratios to detection limits. Variable numbers are based on Table 8.1.

---