Replication fractional

The degrees of freedom for lack of fit, f-p, must not be negative or the model cannot be fitted to the data (see Section 5.6 for example). However, it is possible to use all of the degrees of freedom from the factor combinations to estimate up to / parameters in a model. If p = /, there will be a perfect fit . 

The model usually fitted to data from a full 2 factorial design is 

Offset First-order effects Two-factor interactions Three-factor interaction Residual 

Statisticians seem to like the fact that p =/= 8 for this model and this design. The design is said to be saturated by the model (all of the available degrees of freedom are used up). 

If the two- and three-factor interactions are known or assumed to be negligible (in manufacturing or production, for example, where only a small portion of the larger response surface might be investigated), then the following model can be used  

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In this chapter we explore factorial-based experimental designs in more detail. We will show how these designs can be used in their full factorial form how factorial designs can be taken apart into blocks to minimize the effect of (or, if desired, to estimate the effect of) an additional factor and how only a portion of the full factorial design (a fractional replicate) can be used to screen many potentially useful factors in a very small number of experiments. Finally, we will illustrate the use of a Latin square design, a special type of fractionalized design. 

Fractional replication causes confusion among the factor effects. This confusion is called confounding or aliasing . To see this, compare the signs in the columns in the fractional factorial design ... 

As the number of factors increases, the economies of fractional replication become more evident. With six factors, a l/4 replicate of the 64 possible combinations is not unusual and a l/8 replicate of the 256 run eight-factor factorial is fairly common, Cochran and Cox [3] give a very extensive list of fractional factorial designs, including some in which factors are at three and four levels. 

Fractional replication a factorial experiment in which only a balanced fraction of the possible treatment combinations is run ... 

Factorial designs with more than two levels of the factors are quite common, and mixed factorial designs in which the several factors have different numbers of levels might fit certain experimental requirements. The fractional replication of designs of this type is somewhat hazardous, since balanced arrangements are hard to come by ... 

Fractional replicates of experimental designs in which all factors are at the same number of levels can be partially replicated in fractions whose denominators are multiples of the number of levels. These designs are the so-called Latin square designs. 

When some of the possible data points are omitted in factorial designs they are known as fractionally or partially replicated designs. The choice of points to be omitted is of considerable importance. There is no single fractional replicate which is best for any given complete factorial design. Usually the experimenter will have some idea as to the expected effects. When this sort of intuition is available, a particular design may be developed to fit a particular problem. 

Using the BH algorithm to estimate the effects, we note that column AB has the same signs as column C, column AC has those of column B, column BC those of column A and column ABC those of I. Hence the linear combination of observations in column A, 1a, can be used to estimate not only the main effect of A but also the BC interaction (1a = A + BC 1 a = A — BC if we select C = —AB as generator). Two (or more) effects that share this type of relationship are termed aliases". As a consequence, aliasing" is a direct result of fractional replication. In many practical situations, it will be possible to select a fraction of the experiments so that the main effect and the low-order interactions that are of interest become aliased (confounded) only with high-order interactions, which are probably negligible. 

Fractional replicate designs Categorical/qualitative and quantitative Screening of factors... 

Figure 3.34 The same 23 1 fractional replicate design (i+def) in the process factors d, e, and f set up at the three points of composition...

Wernimont reported an example of a nested design, which is an example of fractional replication. It was desired to evaluate an interlaboratory study of a method of acetyl determination. Results obtained by two analysts in each of eight laboratories were compared by having each analyst perform two tests on each of 3 days. The design was... 

As the number of levels of each variable is increased, the amount of information provided about the factors is increased but unfortunately, so also is the complexity of the design. If three levels of the six variables were employed (a 3 factorial) the result would be 729 combinations (determinations) containing information similar to the 2 design but of a more extensive nature. If a 4 design were desired, it would probably be wise to use fractional replication to reduce the labor and expense necessary to complete the work. Fractionally replicated designs have been discussed by Finney (1945), Fisher (1947), Kempthorne (1947, 1952), and Cochran and Cox (1950). 

Fractionally replicated designs find use in the examination of a relatively large number of factors where it is known (or assumed) that high-order interactions are nonexistent or negligible. Under these conditions it may be shown that an appropriate fraction of the total number of determinations indicated in the complete replicate may be used to yield information about the main treatment effects and about the two factor interactions. This procedure allows for a survey of the effects of all factors in a single experiment which usually involves less effort than any complete factorial design involving the same treatments. 

Kempthorne, O. 1947. A simple approach to confounding and fractional replication in factorial experiments. Biometrics 84 (Parts 3 and 4). 

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