Multifactor

Can the relationship be approximated by an equation involving linear terms for the quantitative independent variables and two-factor interaction terms only or is a more complex model, involving quadratic and perhaps even multifactor interaction terms, necessary As indicated, a more sophisticated statistical model may be required to describe relationships adequately over a relatively large experimental range than over a limited range. A linear relationship may thus be appropriate over a narrow range, but not over a wide one. The more complex the assumed model, the more mns are usually required to estimate model terms. 

Calculation of dependence of o on the conducting filler concentration is a very complicated multifactor problem, as the result depends primarily on the shape of the filler particles and their distribution in a polymer matrix. According to the nature of distribution of the constituents, the composites can be divided into matrix, statistical and structurized systems [25], In matrix systems, one of the phases is continuous for any filler concentration. In statistical systems, constituents are spread at random and do not form regular structures. In structurized systems, constituents form chainlike, flat or three-dimensional structures. 

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. 

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. 

Asano, K., Clayton, J., Shalev, A., and Hinnebusch, A. G. (2000). A multifactor complex of eukaryotic initiation factors elPl, eIF2, eIF3, eIF5, and initiator tRNAMet is an important translation initiation intermediate in vivo. Genes Dev. 14, 2534—2546. 

Singh, C. R., He, H., Ii, M., Yamamoto, Y., and Asano, K. (2004). Efficient incorporation of eukaryotic initiation factor 1 into the multifactor complex is critical for formation of functional ribosomal preinitiation complexes in vivo.J. Biol. Chem. 279, 31910-31920. 

Valasek, L., Nielsen, K. H., and Hinnebusch, A. G. (2002). Direct eIF2—eIF3 contact in the multifactor complex is important for translation initiation in vivo. EMBO J. 21, 5886-5898. 

Daniel, C. and Wood, F., Fitting Equations to Data - Computer Analysis of Multifactor Data for Scientists and Engineers, 1st ed. (John Wiley Sons, 1971). 

Multidisciplinary and multifactor intervention programmes Systematic medication revision (lower dosages, withdrawals or change of drugs)... 

Finally hold in mind that falls are frequent in old age and can lead to severe consequences, including sufferings both for the individual and the next of kin. Also the cost for the society of falls and fractures are considerable. Falls increase with the use of fall-risk-increasing drugs and polypharmacy and can be prevented by a multifactor approach including reassessing the medications used by older people. 

Urinary incontinence can be caused and worsened by drug treatment for diverse chronic conditions and should be treated with a multifactor aim... 

U5. Utermann, G., Apolipoprotein polymorphism and multifactoral hyperlipidaemia. J. Inherited Metab. Dis. 11, Suppl. 1, 74-86 (1988). 

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. 

In previous chapters, many of the fundamental concepts of experimental design have been presented for single-factor systems. Several of these concepts are now expanded and new ones are introduced to begin the treatment of multifactor systems. 

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

In multifactor systems, it is possible that the effect of one factor will depend on the level of a second factor. For example, the slope of factor x might depend on the level of factor X2 in the following way ... 

The concept of interaction is fundamental to an understanding of multifactor systems. Much time can be lost and many serious mistakes can be made if interaction is not considered. 

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. 

In this chapter we discuss the multifactor concepts of confounding and randomization. The ideas underlying these concepts are then used to develop experimental designs for discrete or qualitative variables. 

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