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Ch6: The 2 k Factorial Design. Dr. Mohammed Alsayed. Introduction.
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Ch6: The 2k Factorial Design Dr. Mohammed Alsayed
Introduction • Ch5 presented general methods for the analysis of factorial designs. However, there are several special cases of the general factorial design that are important because they are widely used in the research work and also because they form the basis of other designs of considerable practical value. • The most important of these special cases is that of k factors, each at only of two levels. • These levels may be quantitative, such as temperature, pressure, or time; or they may be qualitative, such as two machines, two operators, the ‘’high’’ and ‘’low’’ levels of a factor, or perhaps the presence and absence of a factor. • A complete replicate of such a design requires 2 ˟ 2 ˟ …. ˟ 2 = 2k observations and is called a 2k factorial design.
Introduction • Throughout this chapter, we assume that (1) the factors are fixed, (2) the designs are completely randomized, and (3) the usual normality assumptions are satisfied. • The 2k design is practically useful in the early stages of experimental work when there are many factors to be investigated. • It is also widely used in factor screening experiments. • Because there are only two levels for each factor, we assume that the response is approximately linear over the range of the factor levels chosen.
The 2k Design • As an example, consider an investigation into the effect of the concentration of the reactant and the amount of the catalyst on the conversion (yield) in a chemical process. • Let reactant concentration be factor A, and let the two levels of interest be 15% and 25%. The catalyst is factor B, with the high level denoting the use of 2 pounds of the catalyst and the low level denoting the use of only 1 pound. The experiment is replicated three times.
The 2k Design • The data is summarized in the following table
The 2k Design • In 2k design, the low and high levels of A and B are denoted by ‘’-’’ and ‘’+’’ respectively.
The 2k Design • The effects of A, B, and AB can be calculated based on the average difference.
The 2k Design • So, for the example presented: • The effect of A (reactant concentration) is positive; this suggests that increasing A from the low level (15%) to the high level (25%) will increase the yield. • The effect of B is negative. • The interaction effect appears to be small relative to the main effects.
The 2k Design • In many experiments involving 2k designs, we will examine the magnitude and direction of the factor effects to determine which variables are likely to be important. • The analysis of variance can generally be used to confirm this interpretation.
The 2k Design • The analysis for the previous example is:
The 2k Design • ANOVA results are: • A: reactant concentration. • B: amount of catalyst.
The 2k Design • Consider again the sum of squares for A, B, and AB. Note that contrasts are used to estimate these SS’s: ContrastB ContrastAB
The 2k Design • This is referred as standard order, or (Yates order, for Dr. Frank Yates),. Using this order, we can see that the contrasts coefficients used in estimating the effects are
The 2k Design • To find the contrast for estimating any effect, simply multiply the signs in appropriate column of the table by the corresponding treatment combination and add. • For example, to estimate A, the contrast is: • –(1) + a – b + ab
The Regression Model • In a 2k factorial design, it is easy to express the results of experiment in terms of a regression model. • We could also use either an effects or means model, but the regression model approach is much more natural and intuitive. • For the chemical process experiment discussed earlier (reactant concentration and catalyst effects on yield), the regression model is: • Where x1 is a coded variable that represents the reactant concentration and x2 is a coded variable that represents the amount of catalyst and the β’s are regression coefficients.
The Regression Model • The relationship between the natural variable (reactant concentration and amount of catalyst), and the coded variables is: • Note that if you use the designed low or high values you will get -1 or +1.
The Regression Model • The fitted regression model is: • Where the intercept is the grand average of all 12 observations. • The regression coefficients β1 and β2 are one half the corresponding factor effect estimates. ᶺ ᶺ
Residuals and Model Adequacy • The regression model can be used to obtain the predicted or fitted value of y at the four points in the design. • The residuals are the differences between the observed and the fitted values of y. • For example, when the reactant concentration is at the low level (x1 = -1) and the catalyst is at the low level (x2 = -1), the predicted yield is: • There are three observations at this treatment combinations, and the residuals are:
Residuals and Model Adequacy • For the high level reactant concentration and low level of the catalyst, it is similar: • For the low level reactant concentration and high level of the catalyst: • For the high level of both:
Residuals and Model Adequacy • Figure 6-2 presents a normal probability plot of these residuals and a plot of residuals versus the predicted yield. These plots appear satisfactory, so we have no reason to suspect problems with the validity of our conclusions.
The Response Surface • The regression model is: • It can be used to generate response surface plots. • If it is desirable to construct these plots in terms of the natural factor levels, then we simply substitute the relationship between the natural and the coded variables:
The 23 Design • Suppose that three factors, A, B, and C, each at two levels, are of interest. • The design is called a 23 factorial design and the eight treatment combinations can now be displayed geometrically as a cube.
The 23 Design • There are seven degrees of freedom between the eight treatment combinations in the 23 design. • Three degree of freedom are associated with the main effects of A, B, and C. Four degrees of freedom are associated with interactions; one each with AB, AC, and BC and one with ABC.
The 23 Design • The A effect is just the average of the four runs where A is a the high level (ȳA+) minus the average of the four runs where A is ate the low level (ȳA-) • This equation can be rearranged as: • Similarly
The 23 Design • In the previous seven equations, the quantities between brackets are the contrasts in the treatment combinations. • Sums of squares for the effects are easily computed, because each effect has a corresponding single-degree-of freedom contrast. • In the 23 design with n replicates, the sum of squares for any effect is:
The 23 Design • Recall example 5.3, and suppose that only two levels of carbonation are used so that the experiment is a 23 factorial design with two replicates. • Example 5.3 was:
The 23 Design • The new experiment is:
The General 23 Design • A design of k factors each at two levels. • The statistical model for a 2k design include k main effects. • () two factor interactions. • three factor interactions. • And one k-factor interaction. • For a 2k design, the complete model would contain 2k – 1 effects.
The General 23 Design • The general approach to the statistical analysis of the 2k design is summarized in the table. • A computer software package is usually employed in the analysis process
The General 23 Design • It is obvious that you need a software package.
A single Replicate to the 23 Design • For even a moderate number of factors, the total number of treatment combinations in 2k factorial design is large. • For example, a 25 design has 32 treatment combinations, and a 26 design has 64 treatment combinations. • Because the resources are usually limited, number of replications may be restricted. • Frequently, available resources only allow a single replicate of the design to be run. Unless the experimenter wants to omit some of the original factors.
A single Replicate to the 23 Design • An obvious risk when conducting an experiment that has only one run at each test combination is that we may be fitting a model to noise. • If the response y is highly variable, misleading conclusions may result from the experiment.
