Multifactor experiments

Two major areas are likely to be the focus of expert systems in the scientific software area assisting users without extensive statistical training in starting to use statistics, and helping design multifactor experiments. 

Optimality of using the factor space for an adequate multifactor experiment means an increase in experiment efficiency proportional to the increase in the number of its factors. 

Sample size and treatment choice are key design questions for general multifactor experiments. Authors have proposed the use of standard factorial experiments in completely randomized designs, block designs, or Latin squares (see, for example, Chapter 6 and Churchill, 2003). However, the unusual distribution of gene expression data makes one question the relevance of standard orthogonal factorial experiments in this context. 

Chipman, H. A. and Hamada, M. S. (1996). Comment on Follow-up designs to resolve confounding in multifactor experiments. Technometrics, 38, 317-320. 

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. 

DOE is an acronym for design of experiments, also called experimental design or multifactor experiments. These experimental approaches provide rich veins of data that can be mined for information that cannot be found any other way. 

The linear models use at data processing of the multifactor experiments... 

At the planning of appraisal of the sensor s error experiments, the test officer usually knows the list of the influencing factors and their boundary conditions. From the theory of experiment point of view, the optimal test setup would be the multifactor experiment at which both all influencing factors and measuring gas concentrations would be varied within all ranges with following appraisal of the dispersion of the output signal. However, it is impossible to implement the optimal test setup in practice. 

The first restriction results in the consecutive plan of experiments, and the second one in resolution of the multifactor experiment on some sequence of single-factor and/or two-factor experiments. 

In the present example, we have examined only one factor the component concentration. In the case of studying several factors, randomization is similarly necessary. In multifactor experiments, the experimental design must be run in a randomized order, as we will see. 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

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. 

Table 2.102 offers a researcher several very useful two-level fractional factor designs with effects that can be estimated (under assumptions that three-factor and multifactor interactions are negligible). The design of experiments matrix consist of trials that are given for each FRFE but in a completely random sequence. 

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. 

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. 

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]. 

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