Factor effect estimates

P and P2 are one-half the corresponding factor effect estimates. The regression coefficient is one-half the effeet estimate because a regression coefficient measures the effect of a imit change in x on the mean of y, and the effect estimate is based on a two-unit change (from -1 to +1). 

The two-factor interaction effects and the dummy factor effects in FF and PB designs, respectively, are often considered negligible in robustness testing. Since the estimates for those effects are then caused by method variability and thus by experimental error, they can be used in the statistical analysis of the effects. Requirement is that enough two-factor interaction or dummy factor effects (>3) can be estimated to allow a proper error estimate (see Section VII.B.2.(b)). 

When a time effect or drift is present, the responses are corrected relative to the nominal result obtained before the design experiments. Otherwise possibly wrong decisions on the significance of the factor effects are drawn. These corrected responses, calculated with Equation (2), are then used to estimate the factor effects, which thus are estimated free from the drift effect. The correction of design results is also illustrated in Figure 5. 

Effects can be estimated from the measured or corrected design responses, depending on the absence or presence of drift in the considered response. When drift is absent, both effect estimates are similar, while in the presence of drift, they are different for the factors mostly affected by the drift. The effect of factor X, Ex, on a response Y is calculated with the following equation -i" i" " ... 

After estimation of the factor effects, they usually graphically and/or statistically are interpreted, to determine their significance. 

Another way to estimate (SE)e is using effects that are a priori considered negligible, such as two-factor interaction effects and dummy factor effects " in EE and PB designs, respectively (Equation (8)). Such effects are considered solely due to the experimental error of the method. ... 

Historically, factorial designs were introduced by Sir R. A. Fisher to counter the then prevalent idea that if one were to discover the effect of a factor, all other factors must be held constant and only the factor of interest could be varied. Fisher showed that all factors of interest could be varied simultaneously, and the individual factor effects and their interactions could be estimated by proper mathematical treatment. The Yates algorithm and its variations are often used to obtain these estimates, but the use of least squares fitting of linear models gives essentially identical results. 

Two-level factorial designs are useful for estimating first-order factor effects and interaction effects between factors, but they cannot be used to estimate additional second-order curvature effects such as those represented by the terms P, and P22 -in the model... 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

The design in Figure 13.4 is similar to the design in Figure 13.3, but the center point has been replicated a total of eight times, not four. This makes the design not only rotatable but also orthogonal in the coded factor space that is, the estimate of one factor effect (i.e pj, Pj, P , P22, or P j) is independent of the estimates of all other factor effects (see Section 12.10). 

Table 14.4 shows a typical regression analysis output for the 2 factorial design in Table 14.3. Most of the output is self-explanatory. For the moment, however, note the regression analysis estimates for the parameters of the model given by Equation 14.5 and compare them to the estimates obtained in Equations 14.8-14.15 above. The mean is the same in both cases, but the other non-zero parameters (the factor effects and interactions) in the regression analysis are just half the values of the classical factor effects and interaction effects How can the same data set provide two different sets of values for these effects ...

Finally, the results of the third pass are divided by the number of experiments going into the averages for that effect. The results are the numerical values of the estimated factor effects. 

In this comer design, if a mistake is made in the experiment at low levels of each factor, the estimation of all of the factor effects will be incorrect by the same amount. The fractional factorial design is generally considered to be superior to the comer design because of the averaging that takes place in calculating the factor effects in the fractional factorial design, an error in any one point will be distributed in smaller proportion over all factor effects. 

The parameter estimates a, Oj, and represent the noise factor effects. In this example, they represent how much influence the ambient humidity, source of raw material, and name of person running the process have on the viscosity. Ideally, for a rugged system, we would like Oi, a, and to be zero. 

In some applications, Latin square designs can be thought of as fractional three-level factorial designs that allow the estimation of one main factor effect while... 

For threshold effects, traditionally, a level of exposure below which it is believed that there are no adverse effects estimated, based on an approximation of the threshold termed the No-Observed-(Adverse)-Effect Level (NO(A)EL) and assessment factors this is addressed in detail in Chapter 5. This estimated level of exposure will in this book be termed tolerable exposure level. Examples, where this approach is used, include establishment of the Acceptable/Tolerable... 

The results show only a modest variation when the van der Waals radii are changed within reasonable bounds (Table 6.2). As the data were not refined with the aspherical atom formalism, the scale of the observed structure factors may be biased, an effect estimated on the basis of other studies (Stevens and Coppens 1975) to correspond to a maximal lowering of the scale by 2%. Values corrected for this effect are listed in the last two columns of Table 6.2. Since neutral TTF and TCNQ have, respectively, 72 and 52 valence electrons, the results imply a charge transfer close to 0.60 e. 

According to the vendor, the estimated price of remediation using a soil slurry-sequencing batch reactor system was 50 to 110/m of waste treated in 1995. Costs are usually 1.5 to 2 times less than excavation and inceration. The quantity of waste and initial contaminant concentration were cited as the most significant factors effecting price (D10036N, p. 15 D15328G, p. 7). 

When an unreplicated experiment is run, the error or residual sum of squares is composed of both experimental error and lack-of-fit of the model. Thus, formal statistical significance testing of the factor effects can lead to erroneous conclusions if there is lack-of-fit of the model. Therefore, it is recommended that the experiment be replicated so that an independent estimate of the experimental error can be calculated and both lack-of-fit and the statistical significance of the factor effects can be formally tested. 

Let us now consider three-factor interactions (e.g. ABC in Table 3.5) to give a general idea how these and higher-order interaction effects (four-, five-factor interaction effects, etc.) are derived. A three-factor interaction means that a two-factor interaction effect is different at the two levels of the third factor. Two estimates for the AB interaction are available from the experiments, one for each level of factor C. The AB interaction effect is estimated once with C at level (+) (represented by Eab,C(+)) nd once... 

As I have shown, the response given by the model equation (3.5) has an error term that includes the lack of fit of the model and dispersion due to the measurement (repeatability). For the three-factor example discussed above, there are four estimates of each effect, and in general the number of estimates are equal to half the number of runs. The variance of these estimated effects gives some indication of how well the model and the measurement bear up when experiments are actually done, if this value can be compared with an expected variance due to measurement alone. There are two ways to estimate measurement repeatability. First, if there are repeated measurements, then the standard deviation of these replicates (s) is an estimate of the repeatability. For N/2 estimates of the factor effect, the standard deviation of the effect is... 

Alternatively, the dummy effect can be taken as the repeatability of the factor effects. Recall that a dummy experiment is one in which the factor is chosen to have no effect on the result (sing the first or second verse of the national anthem as the -1 and +1 levels), and so whatever estimate is made must be due to random effects in an experiment that is free of bias. Each factor effect is the mean of N/2 estimates (here 4), and so a Student s t test can be performed of each estimated factor effect against a null hypothesis of the population mean = 0, with standard deviation the dummy effect. Therefore the t value of the ith effect is ... 

Factor Factor Range p -Value Effect Estimate (% Dissolution)... 

The Canadian provinces (with the exception of Quebec) have now pooled resources to create a common drug review (through the independent CEDAC operating under the not-for-profit CCOHTA). The review factors transparent (public) cost-effectiveness estimates, derived from systematic reviews, back to provincial cost-effectiveness... 

---