Factor-effect plots

Figure 3. Single Factor Effect Plots and Signal to Noise Ratios. A. 0 is the effect of passes on vesicle to ribosome ratio. A is the signal to noise ratios for the data points. B. e is the effect of pressure on vesicle to ribosome ratio. A is the signal to noise ratios for the data points. C. b is the effect of passes on yield. A is the signal to noise ratios for the data points. D. o is the effect of pressure on yield. A is the signal to noise ratios for the data points.

Factor effects plots were generated from the oxygen permeability results as well as the logs of the standard deviations and signal/noise (S/N) ratios. These plots are not included here but can be found in the complete report on this work [29]. 

The factor-effect plots for the Taguchi L9 fractional factorial design of experiments are given in Figure 8.13 depicting the individual and interaction factor effects on the measured value of G02-1 thin fihn resistivity. 

Figure 8.13 Factor-effect plots for resistivity of thin GO Him samples.

One of the most effective ways to think about optimization is to visualize how a system s response changes when we increase or decrease the levels of one or more of its factors. A plot of the system s response as a function of the factor levels is called a response surface. The simplest response surface is for a system with only one factor. In this case the response surface is a straight or curved line in two dimensions. A calibration curve, such as that shown in Figure 14.1, is an example of a one-factor response surface in which the response (absorbance) is plotted on the y-axis versus the factor level (concentration of analyte) on the x-axis. Response surfaces can also be expressed mathematically. The response surface in Figure 14.1, for example, is... 

Experiments were conducted with air through micro-channel A = 319 (friction factor. The relative surface roughness was low k /H = 0.001) and Kn < 0.001, thus the experiments were effectively isolated from the influence of surface roughness and rarefaction. The local friction factor is plotted versus Ma in Fig. 2.25 for air. The experimental A increases about 8% above the theoretical A as Ma increases to 0.35. 

Figure S.3S. Effectiveness factor e plotted as a function of the Thiele diffusion modulus Og. The effective factor is well approximated by 3/Og for Og > 10.

FIGURE 15 An example of a main effect plot.The average responses at low level and high level of the factors are plotted. 

FIGURE 15 Example of a main effect plot (for the critical resolution) clearly showing the extent of the effects relative to each other. Factor temperature appears to be the most important factor on the resolution. Reprinted with permission from reference 18. 

In references 82-86, the results were treated statistically. Main effects and standard errors were calculated. In references 83, 85, and 86 also a graphical interpretation by means of bar plots was performed. Both positive and negative effects were seen on these plots, but all effects between levels [—1,0] are negative, while all those between [0,4-1] are positive. Possibly, the length of the bars represents the absolute value of the factor effects, and all effects for the interval [—1,0] seem to be given a negative sign, while all those for [0,4-1] a positive. However, the above are assumptions since no details were provided. In references 83 and 86, critical effects are drawn on the bar plots. 

In Figure 7.4 the effectiveness factor is plotted against the Thiele modulus for spherical catalyst particles. For low values of 0, Ef is almost equal to unity, with reactant transfer within the catalyst particles having little effect on the apparent reaction rate. On the other hand, Ef decreases in inverse proportion to 0 for higher values of 0, with reactant diffusion rates limiting the apparent reaction rate. Thus, decreases with increasing reaction rates and the radius of catalyst spheres, and with decreasing effective diffusion coefficients of reactants within the catalyst spheres. 

The main effects plot represents the response (vertical axis) versus the levels of the factor under consideration (horizontal axis, see Figure 2.2). Each point represents the mean response obtained at each level. For our example (see Figure 2.2), the mean responses (yield) for factor A at the low and high levels are 82.5% and 86.75%, respectively. That is to say, the increase in yield (effect of A, pH) is 4.25% when we change the pH from 3 to 5. 

Since there is a non-negligible interaction between temperature (B) and time (C), the effects of these factors must be considered jointly. An interaction effect plot is a simple way to carry out this analysis. In Figure 2.3, the vertical axis represents, again, the response whereas the horizontal axis shows the levels of one of the factors involved in the interaction. For each of these levels we represent the mean response obtained at each combination of levels of the other factor. As presented in Figure 2.3, when time (C) is set at its low level, a change from the low to the high level of temperature (B) decreases the response (yield)... 

Fig. 12 a Sergeants and soldiers effect plots of anisotropy factor g versus the mole fraction of chiral 19 for helical supramolecular assembly of chiral 19 and achiral 21 in M-butanol at 10-5 (O) and KT4 M ( ). (Reproduced with permission from [71]. Copyright 2001 Springer.) b Majority rule effect plots of g versus the enantiomeric excess of 18a and 18b for helical supramolecular assembly of chiral 18a and 18b in n-octane at 20 ( ) and 50 °C (O). (Reproduced with permission from [72]. Copyright 2005 American Chemical Society)... 

Figure 3. Fialf-normal probability plot of the factor effects from the 2s factorial experiment on hue.

Figure 6.11 holds for a slab. Similar figures can be obtained for other catalyst geometries. This is illustrated in Figure 6.13 where the effectiveness factor is plotted versus 8 for zeroth-order kinetics in an infinite slab, infinite cylinder and a sphere. Figure 6.13 has been constructed on the basis of the formulae given in Table 6.6. Hence, the discussion that follows is not restricted to a slab, but holds for any arbitrary catalyst geometry. 

From now on the two-parameter model is used because it is almost as accurate as the three-parameter model and it gives a better insight. For example, the curves which were drawn by Weisz and Hicks [2] for different values of a and s (Figure 6.4) reduce to one. This is illustrated in Figure 7.1 where the effectiveness factor is plotted versus An0 for several values of C, and for a first-order reaction occurring in a slab. Notice that all the curves in Figure 7.1 coincide in the low ij region, since ij is plotted versus An0. The formulae used for Ana now follow. 

This is illustrated in Figure 7.4 where the effectiveness factor is plotted versus the low ij Aris number An0 for a bimolecular reaction with (1,1) kinetics, and for several values of/ . P lies between 0 and 1, calculations were made with a numerical method. Again all curves coincide in the low tj region, because rj is plotted versus An0. For p = 0, the excess of component B is very large and the reaction becomes first order in component A. For p = 1, A and B match stoichiometrically and the reaction becomes pseudosecond order in component A (and B for that matter). Hence the rj-An0 graphs for simple first- and second-order reactions are the boundaries when varying p. 

Main Effects Plot is used to plot data when there are multiple faetors. The points in the plot are the means of the response variable at the various levels of eaeh factor, with a reference line drawn at the grand mean of the response data. 

Fluorescence-based measurements are already very sensitive and widely used in bio-medical analysis. However, the metallic nanostructures provide further improvement on the sensitivity and limit of detections through the enhancement of the local field. Therefore, a large number of researchers are dedicated to developing substrates for SEFS [46-52]. The effect of the geometrical parameter of the nanostructure on the efficiency of the SEFS is well illustrated in Fig. 9. In this case, the SEFS enhancement factor (SEFS enhancement factor) is plotted against the periodicity of the arrays of nanoholes in gold films. The experiments were realized by spin-coating the arrays of nanoholes with a polystyrene film doped with the oxazine 720 [48]. 

Fig. 6.12. Effect plots of five factors on a combined response formed by the resolution and the analysis time (adapted from [46]).

Because of the lack of randomization in time, the variation in the data observed from time point to time point within a reaction (whole plot) is likely to be less than that observed from reaction to reaction. In effect, two process variance terms have been introduced, one reflecting the within-reaction variation, and a second reflecting the variation from reaction to reaction. This has consequences for how to determine which factor is having effects beyond the inherent process variation. Fundamentally, any factor effect measures need to be compared against the correct comprehensive variance estimate. 

The analysis utilized the approach of normal probability plots, as no reliable historical estimate of variation was available. For each primary response the observed factor effects were plotted on normal probability paper in rank order (Fig. 2). Here, factor effects are estimated as the difference of the average response at the high level (-I-) from the average response of the low level (-). On these plots, values that deviate markedly from the general trend line indicate significant effects. The largest factor effects (in absolute value) are labeled. 

To better rmderstand the magnitude and shape of the factor effects it is helpful to examine effects plots. These plots depict a response (y-axis) for factor(s) of interest (x-axis) averaged over the remaining factors of the design. Figure 3 provides key effect plots for yield and the formation of D and E at the 12-hour hold point. 

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