Response multifactor

Experiments that will be used to estimate the behavior of a system should not be chosen in a whimsical or unplanned way, but rather, should be carefully designed with a view toward achieving a valid approximation to a region of the true response surface [Cochran and Cox (1950), Youden (1951), Wilson (1952), Mandel (1964), Fisher (1971)]. In the next several chapters, many of the important concepts of the design and analysis of experiments are introduced at an elementary level for the single-factor single-response case. In later chapters, these concepts will be generalized to multifactor, multiresponse systems. 

Figure 12.1 Two-factor, single-response system for discussion of multifactor experimentation.

One of the most useful models for approximating a region of a multifactor response surface is the full second-order polynomial model. For two factors, the model is of the form... 

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. 

Factorial designs are a popular class of experimental designs that are often used to investigate multifactor response surfaces. The word factorial does not have its usual mathematical meaning of an integer multiplied by all integers smaller than itself (e.g. 5 5x4/3/2 / 1) instead, it simply indicates that many... 

Methods for acute-to-chronic extrapolations have been developed and are available as computer programs such as the acute-to-chronic estimation (ACE Mayer et al. 1994 Ellersieck et al. 2003) software, which makes use of 3 methods — regression, multifactor probit analysis, and accelerated life testing — to consider the relationship between exposure concentration, degree of response, and time course of response (Mayer et al. 1994 Sun et al. 1995 Lee et al. 1995). All methods produce confidence intervals around the LC and/or EC percentage point estimate. 

Response Surfaces. 3. Basic Statistics. 4. One Experiment. 5. Two Experiments. 6. Hypothesis Testing. 7. The Variance-Covariance Matrix. 8. Three Experiments. 9. Analysis of Variance (ANOVA) for Linear Models. 10. A Ten-Experiment Example. 11. Approximating a Region of a Multifactor Response Surface. 12. Additional Multifactor Concepts and Experimental Designs. Append- ices Matrix Algebra. Critical Values of t. Critical Values of F, a = 0.05. Index. 

The expense of such multifactor experiments has led scientists to use process-based ecosystem models (see the discussion of terrestrial carbon models below) to predict the response of terrestrial ecosystems to future climates. When predicting the effects of CO2 alone, six global biogeochemical models showed a global terrestrial sink that began in the early part of the twentieth century and increased (with one exception) towards the year 2100 (Cramer et al., 2001). The maximum sink varied from 4 PgC yr to —10 PgC yr. Adding changes in climate (predicted by the Hadley Centre) to these models reduced the future sink (with one exception), and in one case reduced the sink to zero near the year 2100. 

Table 2.18 Schematic representation of factors and the response for a multiway (multifactor) analysis of variance at factor levels 1 -4.

By the method of mathematical multifactor planning of the experiment, equations were derived that describe the dependence of the mechanical properties of parent and weld metals on the chemical composition of the alloy. Table III shows the investigated factors, intervals, and levels of variation. Characteristics of mechanical properties and the index of alloy susceptibility to hot cracking of weld metal during welding served as functions of response. 

A syndrome called beriberi, characterized by edema and polyneuritis, has been known for a long time. Although beriberi is a multifactor deficiency, thiamine deficiency is mainly responsible for the disease. 

Injection molding processes using Stat-Ease Inc. response surface methods for process optimization are developed with design of experiments (DOE) and a multifactor linear constant (MLC) [10]. 

S. N. Deming and S. L. Morgan, in Experimental Design A Chemometric Approach, 2nd ed., Elsevier, Amsterdam, 1993, pp. 227-274. Approximating a Region of a Multifactor Response Surface. 

One of the following generie probit models (with or without some slight modifieation thereof) was used for fitting experimental response data via one-faetor MLE/probit analysis/multifactor probit analysis/etc. (see Table 9 2).36,38,48,55 binary or ordinal response (with a probit-link function)... 

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