Experimental variance-reproducibility

Figure 5.119 shows that a number of standards lie very close to each other. This implies that the model will have a difficult time distinguishing between these samples. However, keep in mind that this plot shows only two of the six dimensions used in the model. The scores plot in Figure 5.120 shows the location of the samples in three dimensions (representing 99.98% of the spectral variance). The threcHiimensional view has been rotated to look down on the lines formed by varying temperature. Each cluster of points (noted by the number on the graph) contains all spectra collected on one standard. This view of the scores reproduces the experimental design (i.e., the standards are in the same position relative to each other in the scores plot as in the concentration plot, see Figure 5-42). This gives confidence that the measurements and the model accurately reflect the variation in the concentrations. Niuner-ous other scores plots can be examined for this rank six model, but they are not shown here.

An increase in the number of replicated trials causes a decrease in reproducibility variance or experimental error as well as in the associated variances of regression coefficients. Design points-trials can be replicated in all points of the experiment or in some of them. An upgrade of the design of experiment may be realized by a shift from fractional to full factorial experiment, a switch to bigger replica (from 1/6 to 1 /2 replica), a switch to second-order design (when the optimum region is dose by), etc. 

Based on the outcomes in the experimental center, these reproducibility variances have been determined Sp =4.466 with this degree of freedom f=n0-l=5. Regression coefficients have these values ... 

The basic levels and variation intervals are given in Table 2.170. It is known from previous design points that the optimum is within the studied factor space. Orthogonal design has therefore been done to obtain the regression model, Table 2.171. Reproducibility variance of the experiment is determined from four additional design points in the experimental center (y01=61.8 y02=59.3 y03=58.7 and y04=69.0). 

Six design points in the experimental center had to be done due to Table 2.164. For the sake of economy only one design point in the experimental center was set in this example, since the reproducibility variance was obtained in the basic experiment. By processing all 15 design points, the following regression coefficients for second-order model were obtained ... 

CCRD for k=3 has been analyzed in Example 2.44. The design included twenty trials, six of which in null point (six replications under identical conditions in the experimental center), eight FUFE and six trials in starlikC points (Table 2.139). The reproducibility error or reproducibility variance was determined from replications in the design center. 

N gives the total number of experiments in the plan. When we use a complete second order plan, it is not necessary to have parallel trials to calculate the reproducibility variance, because it is estimated through the experiments carried out at the centre of the experimental plan. The model adequacy also has to be examined with the next procedure ... 

Table 5.39 also contains the indications and calculations required to verify the zero hypothesis. This hypothesis considers the equality of the variance containing the effect of the factor on the process response (Sj) with the variance that shows the experimental reproducibility (s ). 

When the investigated process shows a small residual variance we can consider that the variance results from the action of small random factors. At the same time, this small variance is a good indication of an excellent reproducibility of the experimental measurements. Conversely, a great residual variance can show that the measurements are characterized by poor reproducibility. However, this situation can also result from one or more unexpected or unconsidered factors this situation can be encountered when the interactions between the factors (parameters) have been neglected. In these cases, the variance of the interactions represents an important part of the overall residual variance. 

In [132] the second moment of the variance coefficient M = (a/c) was calculated for a single phase (air) turbulent flow in a pipe with a Tee mixer with the CFD code (Phoenix, version 1.6) for two distances x/D = 3 and 5 from the addition point and different values of momentum length/pipe diameter ud/ yD The K-e model utilized contained two additional laws of conservation for the mean kinetic energy K and the dissipation s. The parameter range extended over 0.026 model used reproduces well the existing experimental data and only a few adjustments are necessary to the already existing process relationship, whose constant is ca. 70% lower than the newly acquired one. 

The experimental results are also listed in table 9.14. The data were analysed according to the two models, equations 9.9 and 9.10. Since some measurements of solubility were in duplicate, we can estimate the reproducibility of the experimental technique. It is possible to estimate the model with the data for the duplicated points 1-6, and then validate it with the test points 7-12. Instead we estimate it by multilinear regression over all the data and then test by analysis of variance. 

At variance with the endo/exo selectivity, the solvent effects of Table 4 do not explain the observed increase of the B-regioadducts due to the increased solvent polarity. We show in Table 5 that the inclusion of electron correlation in the evaluation of the solvent effect succeeds in reproducing the experimental trend of reaction rates, of endo/exo selectivities and of regioselectivities as a function of the increasing dielectric constant of the solvent. 

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