Coded level

The linear regression calculations for a 2 factorial design are straightforward and can be done without the aid of a sophisticated statistical software package. To simplify the computations, factor levels are coded as +1 for the high level, and -1 for the low level. The relationship between a factor s coded level, Xf, and its actual value, Xf, is given as... 

The factor A has coded levels of +1 and -1 with an average factor level of 100, and d equal to 5. What are the actual factor levels ... 

A 2 factorial design with two factors requires four runs, or sets of experimental conditions, for which the uncoded levels, coded levels, and responses are shown in Table 14.4. The terms Po> Po> Pfc> and Pafc in equation 14.4 account for, respectively, the mean effect (which is the average response), first-order effects due to factors A and B, and the interaction between the two factors. Estimates for these parameters are given by the following equations... 

A 2 factorial design was used to determine the equation for the response surface in problem lb. The uncoded levels, coded levels, and the responses are shown in the following table. 

If Xj is temperature and two experiments are carried out, one at a coded level of -1 and one at a coded level of -t-1, and we get a classical factor effect of -i-3.6% yield, it tells us that working at the -i-l coded temperature gives more yield than working at the -1 coded temperature. But this classical factor effect by itself doesn t tell us very much about how sensitive the reaction is to temperature because 5x, isn t included in the factor effect. 

Thus, in modem research using interval and ratio scales the 5x usually shouldn t be ignored. Let s add 8x, to the calculation to obtain b, as would be done with regression analysis. Because Xj went from a coded level of -1 to a coded level of -t-1, 5x, = 2. Thus, b (the factor effect in the coded factor space) = 8y,/8xJ = -i-3.6% per 2 coded units = -i-1.8% per coded unit. The fact that 8x is equal to 2 with this system of coding is why regression analysis of coded data gives results that are smaller by Vi from the results obtained from the classical approach ... 

Coding is sometimes used as a way of sweeping these considerations under the rug, of ignoring this effect of units. It is often assumed that the researcher will use an experimentally relevant domain or meaningful coded levels of the factors so that the bare 8y will show the effect of the factor at the extremes of this domain. If this is understood explicitly, then no harm comes of it. But if the experimenter is unaware of these influences of units and coding, misinterpretation of the results is easily possible. 

FIGURE 2.8 Coded levels for the experiments required by a two-level full factorial design with three factors. 

Calculations for factorial designs are often greatly simplified if coding levels is employed (see Section 8.5). For the present example, setting c and dr = [Pg.192]

Coded Levels and Corresponding Actual Values Employed... 

A central composite design for three factors was used to generate 20 combinations. The effects of independent variables—acid/glycerol molar ratio (R), temperature (T), and enzyme concentration (E)—on the response (i.e., the monolaurin molar fraction at 4 h) were investigated. The upper and lower limits of each variable were chosen based on published data and preliminary studies (12,13). Actual independent variables or factors and their corresponding coded levels are presented in Table 1. 

Central Composite Design of Factors in Coded Levels with Nisin and Lactic Acid Concentration as Response... 

The restricted CP algorithm can be summarized as follows for two-level designs with coded levels +1 and —1 (Li and Nachtsheim, 2000, Appendix B). 

The coefficient b0 is very different. In die current calculation it represents die predicted response in the centre of the design, where the coded levels of the diree factors are (0, 0, 0). In the calculation in Section 2.2.3 it represents die predicted response at 0 pH units, 0 °C and 0 mM, conditions that cannot be reached experimentally. Note also that this approximates to die mean of the entire dataset (21.518) and is close to the average over the six replicates in die central point (17.275). For a perfect fit, with no error, it wdl equal the mean of the entire dataset, as it will for designs centred on the point (0, 0, 0) in which the number of experiments equals the number of parameters in the model such as a factorial designs discussed in Section 2.6. 

The following is a table of responses of eight experiments at coded levels of three... 

Peterson, S., Sigman-Grant, M., and Achterberg, C., Development of a food code level sorting procedure using the 1989-91 CSFII database, Earn. Econ. Nutr. Rev., 10, 36, 1997. 

If the factors are quantitative, they may take any number of levels. Only two-level designs are described here. Qualitative levels are set arbitrarily at the coded levels. If, for example, the screening method was one of the factors tested, wet screening could be set at 1 and dry screening at -bl (or vice versa). 

The variable X3 takes 3 levels, so it was chosen to correspond to the stirring rate. The normal rate for the dissolution testing of the formulation was 75 rpm, and the extremes to be tested were 75 25 rpm. The coded levels 0.866 correspond therefore to 50 and 100 rpm. (It would also have been possible to set the levels 1 to 50 and 100 rpm, in which case the levels coded 0.866 would have corresponded to 53 and 97 rpm.) If each experiment were carried out twice, these experiments could be done in two dissolution runs. We assume here that the run to run reproducibility is sufficiently good. There remain 7 experiments at the level zero of stirring speed (75 rpm). These experiments also were carried out in duplicate. This allowed the whole experimental design to be carried out using only 4 dissolution runs as shown in table 5.23. 

The objective was to reduce the turbidity as far as possible, and to obtain a solution with a cloud point less than 70°C, at a level of invert sucrose that is as high as possible (in spite of its deleterious effect on the cloud point). Figure 6.4 shows slices taken in the propylene glycol, sucrose plane (Xj, X3) at different levels of polysorbate X, (coded levels x, = - /4, 0, Vi, 1). Examination of the response surfaces indicates an optimum compromise formulation at 60 mL sucrose medium, 4.3% polysorbate 80 and 23% polyethylene glycol. 

Since the variation in the level of drug substance is so small in relation to that of the other variable constituents, it may be treated as an independent variable Z5 (see chapter 9, section IV) and set at coded levels -1 to -fl, corresponding to 1% and 2%. Two possible models are proposed, a first-order model ... 

This is the value given earlier for the standard deviation of an optimum matrix, with each factor at two levels. To estimate the standard error of each coefficient estimate f>, we therefore need an estimate of the experimental standard deviation. Note that this equation applies to coded levels set at 1. For levels set... 

The results of the design are given in Table 3 below (Xi and X2 refer to aeration and agitation in coded levels, respectively). 

Based on the mathematical model, the response surfaces can be explored graphically. An example plot of the response rate in dependence on PPD concentration and pH is shown in Figure 4.15a. The curved dependences in the direction of both factors lead to a maximum rate at coded levels of PPD of about 0.4 and of pH at 0.2. This relates to decoded levels of 16.6 mM PPD and a pH value of 5.95. Maxima are best found from the contour plots as represented in Figure 4.15b. 

Coding of Factor Levels To convert factor levels between original and coded levels, the corresponding formulas will be given here. The coded level in the interval -1 to +1, x, is calculated from the original factor level, a , by... 

Generate the initial simplex according to the coded levels of the factors as given in Table 4.15. 

The initial simplex is chosen according to the scheme in Table 4.15 in coded levels. The responses for the initial simplex are as follows ... 

After contraction, the simplex is reflected to point 6 and again contracted to vertex 7 (cf. Figure 4.19). If the calculation is continued, the simplex moves in the direction of the optimum as given in the figure. After 20 iterations, the following optimum is found in coded levels ... 

Retrmrk In Table 32, for the factor HLB, the coded level ctHresponding to the mean real value S.OS should in fact be 0.1080. 

Real value (%1 Coded level Real value (%) Coded level Particle size (nm) SD (ntn) Polydiqrersity... 

Accidents that are not anticipated to occur during the lifetime of the facility. Natural phenomena of this probability class include Uniform Building code-level earthquake, 100-y flood, maximum wind gust, etc. Frequency between one in 100 y and once in 10,000 operating years ... 

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