Factor level

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... 

The response surfaces in Figure 14.2 are plotted for a limited range of factor levels (0 < A < 10, 0 < B < 10), but can be extended toward more positive or more negative values. This is an example of an unconstrained response surface. Most response surfaces of interest to analytical chemists, however, are naturally constrained by the nature of the factors or the response or are constrained by practical limits set by the analyst. The response surface in Figure 14.1, for example, has a natural constraint on its factor since the smallest possible concentration for the analyte is zero. Furthermore, an upper limit exists because it is usually undesirable to extrapolate a calibration curve beyond the highest concentration standard. 

The initial simplex is determined by choosing a starting point on the response surface and selecting step sizes for each factor. Ideally the step sizes for each factor should produce an approximately equal change in the response. For two factors a convenient set of factor levels is (a, b), a + s, h), and (a + 0.5sa, h + 0.87sb), where sa and sb are the step sizes for factors A and B. Optimization is achieved using the following set of rules ... 

Rule 4. Boundary conditions are a useful way to limit the range of possible factor levels. For example, it may be necessary to limit the concentration of a factor... 

Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

The terms Po, Pa, Pt, Pat, Paa, and Pt,t, are adjustable parameters whose values are determined by using linear regression to fit the data to the equation. Such equations are empirical models of the response surface because they have no basis in a theoretical understanding of the relationship between the response and its factors. An empirical model may provide an excellent description of the response surface over a wide range of factor levels. It is more common, however, to find that an empirical model only applies to the range of factor levels for which data have been collected. 

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 ... 

Example of Uncoded and Coded Factor Levels and Responses for a 2 Factorial Design... 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

To check the result we substitute the coded factor levels for the first run into the coded empirical model, giving... 

The experimental design for ruggedness testing is balanced in that each factor level is paired an equal number of times with the upper case and lower case levels... 

Condition in which coagulation factor level in a given patient is below the acceptable normal level. 

NOTE Code human factors design problems under HUMAN FACTORS (Level D)... 

What strategy should one follow In the classical experiment, one factor is varied at a time, usually over several levels, and a functional relationship between experimental response and factor level is established. The data analysis is carried out after the experiment(s). If several factors are at work, this approach is successful only if they are more or less independent, that is, do not strongly interact. The number of experiments can be sharply increased as in the brute-force approach, but this might be prohibitively expensive if a single production-scale experiment costs five- or six-digit dollar sums. Figure 3.4 explains the problem for the two-factor case. 

A problem with the simplex-guided experiment (right panel) is that it does not take advantage of the natural factor levels, e.g., molar ratios of 1 0.5, 1 1, 1 2, but would prescribe seemingly arbitrary factor combinations, even such ones that would chemically make no sense, but the optimum is rapidly approached. If the system can be modeled, simulation might help. The dashed lines indicate ridges on the complex response surface. The two figures are schematic. 

Form a complete factorial table consisting of all combinations of low (L) and high (H) values for each factor If there are n factors being considered, then the complete factorial table will consist of the 2 possible combinations This list of combinations of factor levels provides a guide for obtaining the information necessary to generate the experimental design ... 

O Intravenous factor replacement with recombinant or plasma-derived products to treat or prevent bleeding is the primary treatment hemophilia. Primary prophylaxis is defined as the regular administration of factor concentrates with the intention of preventing joint bleeds.4 The rationale for primary prophylaxis is that individuals with factor levels of greater than 0.02 unit/mL (2 IU/dL) rarely suffer from spontaneous bleeds and arthropathy. Therefore, to maintain a trough level above this might convert severe hemophilia to moderate disease, with the abolition of joint bleeds and the associated arthropathy.5... 

Soejima H, Ogawa H, Yasue H, et al. Angiotensin-converting enzyme inhibition reduces monocyte chemoattractant protein-1 and tissue factor levels in patients with myocardial infarction. J Am Coll Cardiol 1999 34(4) 983-988. 

G22. De Groote, M. A., Martin, M. A., Densen, P., Pfaller, M. A., and Wenzel, R. P., Plasma tumor necrosis factor levels in patients with presumed sepsis. JAMA 262,249-251 (1989). 

The macrophage-derived tumor necrosis factor level in plasma is elevated in thrombotic conditions associated with malignancy (93). Likewise, mast cell-derived platelet-activating factor levels in plasma are increased in thrombotic conditions accompanied by mast cell activation (93). 

Variance analysis should advantageously be carried out on the basis of balanced experiments where the number of observations per factor level is equal ( i = n2 =. .. = nm = n). 

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