Uniform precision

Rotatability implies that the value of the variance function dj in a point Xj is only dependent on its distance r to the center point. As dj depends on the whole design used to fit the model, it will also depend on the number of experiments at the center. 

It is possible to select the number of center point experiments so that dj is fairly constant in the whole experimental domain. This will give almost the same precision of the predicted response, for all possible settings of the experimental variables in the explored domain. Such a design is called a uniform precision design.[4] It is evident that this is a desirable property if the model is to be used for simulations. 

The number of center points that should be included to obtain a uniform precision design is given in Table 12.3. 

One important conclusion from this is, that it is always useful to include at least one experiment at the center point when a factorial or fractional factorial design is run. This will not cause any difficulties if tbe design is evaluated by hand-calculation A least squares fit of the linear coefficients, hj, and the interaction coefficients, bjj, are detennined as usual from the factorial points. The intercept, b, is computed as the average of all experiments. The standard error of the estimated coefficient will, however, be slightly different. Let N be the total number of experiments, and let Np be the number of factorial points. If is the experimental enor variance, the standard error of the intercept b will be aNN and for the linear coefficients and interaction coefficients the standard enor will be aNNp. 

---

High-temperature materials. Uniform, precise coatings for applications like microminiature circuits. 

A Theoretical Model Chemistry /18,24/ has an underlying approximation scheme which is of potentially uniform precision for all the phenomena it wants to describe, so that the viability of the model can be tested by an appeal to the experimental results. The criteria that a theoretical model should satisfy are the following ... 

Uniform precision-. It is possible to establish experimental designs for determining second order quadratic response surface models, so that the prediction variance is fairly constant in the explored domain, i.e. d-, < constant. This criterion is linked to rotatability. 

The experimental domain is given in Table 12.4. A uniform precision central composite rotatable design (5 center point experiments) was used. The design and the yields of enamine obtained after 15 minutes are given in Table 12.5. 

The equiradial designs are rotatable. It is possible to adjust the number of center point experiments to achieve uniform precision as well as near-orthogonal properties. Uniform precision for the pentagon and the hexagon is obtained with three center point experiments. Orthogonal properties are obtained with five center point experiments for the pentagon, and with six center point experiments for the hexagon. 

It is clear that if we want uniform precision, that is a variance function that remains constant within the domain, we first of all require a rotatable design. Thus a must be set to V p as described above, so that d varies only with the distance from the centre. This variation in precision with distance may be modulated, and "evened out" by adjusting A/p, the number of experiments at the centre. We illustrate this with the 2 factor composite design. 

A completely constant variance of prediction is not possible. Mathematically it can be shown that the closest to uniform precision can be obtained for = 5. However, if we compare the isovariance curves (figure 5.9) for Aq = 5 with those for = 3, we see that the difference may not be great enough to justify carrying out the extra experiments if these are very costly. (Note that the variance function may be transformed to a relative standard deviation by taking the square root.) On the other hand a single experiment at the centre appears to be insufficient. 

In table 5.16, we give theoretical values of giving uniform precision along with what we consider to be the minimum number of experiments necessary to give an adequate quality in prediction. 

Figure 5.9 Uniform precision effect of the number of centre points on the variance function for a 2 factor composite design, (a-b) = 1 (c) = 3 (d) = 5.

The property of quasi-orthogonality, by putting The property of uniform precision, by putting... 

Table 19 Rules for the Construction of the Composite Experimental Designs Having the Property of Isovariance by Rotation and the Propeny of Orthogonality or the Prt rty of Uniform Precision...

A greater number of points is necessary at (he center to respect the property of near-orthogonality than to respect the propeny of uniform precision. It is easy to understand why. since we have the choice, we prefer the property of uniform precision, it must be noted however that this propeny corresponds better to our objectives to know the value of the response studied at any point of the experimental domain. It is. however, preferable to know this estimated response with the same precision in all the experimental domain of interest. 

Even if the property of uniform precision is respected, the number of points at the center U relatively great. 

The Doehlert designs have none of the conventional properties of the response surface experimental designs (isovariance by rotation, cmhogonality, uniform precision, etc.). It can indeed be shown by calculation that ... 

The design includes several statistical properties such as orthogonality that makes the calculation of [X] [X] term simple and rotability that insures the uniform precision of the predicted value. 

Faster inactivation of enzymes fouling (proteins) and corrosion (mainly at low frequencies) electrical conductivity temperature dependent (temperature runaway possible) material with non-conductive parts (particles/(fat)globules) food with high consistency may not be heated uniformly precise temperature/mass flow control required ... 

The equiradial designs describe a circular experimental domain with the points distributed along the circle periphery and at least one point in the center of the circle. Equiradial designs are rotatable and, depending on the number of replicates of the center point, uniform precision and nearorthogonality can be achieved. 

The main advantages of two-shot molding are high geometric design freedom and suitability for high-volume production with uniform precision and low costs per unit. A combination of electric circuitry and mechanicai functionality is the typical requirements list for a two-shot MID such as a snap hook that enables the electrical part to connect to other components. 

---