Factorial design

Factorial designs are used in experiments where the effects and interactions of different factors on the experimental outcome are to be studied simultaneously (Bolton, 1997b). This is undertaken by describing the effect o f changing factor levels or factor combinations on the response variable. Particularly useful for this purpose are fraetionated faetorial designs, which require only a fraetion of all possible combinations o f a full factorial design. Once the potential interaetion has been detected with a fractionated design, a more accurate analysis can be performed for the particular factor combinations to ensure and characterize these interactions.

A factor is an assigned variable, which is chosen depending on the experimental objectives. A factor can be quantitative or qualitative, sueh as deseribing starch concentration, film thickness or film forming temperature, as in this work. The factor levels are the values assigned to the faetor, which in the abovementioned example would correspond to the different coneentrations, film thicknesses or temperatures. The experiments in this work were 3-faetorial and 2-faetorial with 5 factor levels each tested as a composite design, whieh will be explained later.

The effeet of a factor is the change in response resulting from varying the levels of the faetor while the main effeet is the effect o f a factor averaged over all levels of the other factors (Bolton, 1997b).

The eombination o f all factor levels of all faetors makes up the number of runs in one experimental series.

Interaction in statistical terms is known as laek of additivity o f factor effects. It can be defined as the differenee between the two effeets o f one faetor at the two different levels of the other faetor (Edwards, 1979). This means, more specifieally, that the effect measured at one factor level is different from the effect measured at the second level. The interactions possible are graphieally depicted in Figures 1.10.1 to 1.10.3.

Chapter 1 F ig u r e s 1.1 0 .1 - 1 .1 0 .3 : G r a p h s o f tw o -fa c t o r in te r a c tio n s o f fa c to r s A a n d B . T h e 2 le v e ls o f A a r e d e p ic te d a s • a n d ♦ w h ile th o s e o f B a r e d e n o te d B i an d B%. (m o d ifie d a fte r E d w a r d s , 1 9 7 9 ) F ig u r e 1 .1 0 .1 : D is o r d in a l in te r a c tio n . F ig u r e 1 .1 0 .2 : N o in te r a c tio n . F ig u r e 1 .1 0 .3 : O r d in a l in te r a c tio n .

Here, a line represents the effect at each factor level. If the lines are parallel, as in Figure 1.10.2, no interaction exists or, in other words, both effects are the same and the system is therefore ‘additive’. Lack of parallelism, on the other hand, suggests interaction, which is the case in Figures 1.10.1 and 1.10.3. In Figure 1.10.1 the two lines intersect owing to a reversal in the rank order of the effects at the two factor levels. This type of interaction is referred to as ‘disordinal’. By contrast, in Figure 1.10.3 the two lines do not cross indicating that the rank order of the effects is equal for both levels, despite the fact that the difference between the two effects is not the same for the two factor levels. This interaction is called ‘ordinal’

Chapter 1

The advantages of factorial designs are manifold: first, they provide maximum efficiency in estimating main effects in the absence of interaction. Second, if interactions exist, they are the necessary tool for revealing and identifying these interactions. Third, the conclusions drawn from factorial-design experiments are applicable to a range of conditions, since the factor effects are measured over varying levels o f other factors. Fourth, maximum use is made of the data by calculating all main effects and interactions from the entire data set, hence preserving its statistical power. Finally, factorial designs are orthogonal in that all estimated effects and interactions are independent of effects of other factors.

Lastly, factorial designs can be composed of several or only one factor, where the latter fits a one-way ANOVA design.

Composite design

Composite designs are effective designs to estimate second- or third-order terms. In addition to allowing an estimate of curvature, which is the deviation o f the response from the predicted value, composite designs deliver orthogonal estimates of the polynomial coefficients, and allow for the possibility o f proceeding with the experiment in a stepwise fashion rather than performing the entire experiment at once (Bolton, 1997b). Composite designs are very efficient in providing much information on experiment variable effects and overall experimental error in a minimum number o f required runs.

A so-called Central Composite Design (Box et a l, 1978a) contains an imbedded factorial or fractional factorial design with a centre point or experiment around which a group of further factorial points or experiments are arranged in order to allow estimation of the curvatures. If the distance between the centre and a factorial point is ±1 unit for each factor, the distance from the centre to a factorial point is ±a with |a| >

1. The precise value of a depends on which properties are desired for the design as well as on the number of factors to be studied.

Chapter 1

F ig u re 1.11: R e p r e se n ta tio n o f a c e n tr a l c o m p o site d esign fo r 2 fa cto rs.

mirii maXi

min m ax

F ig u re 1.12: R ep resen ta tio n o f a c e n tr a l c o m p o site d esig n fo r 3 fa cto rs. O u te r m in im u m ex trem e (m in ), in n er m in im u m ex tr e m e (m iuj), o u te r m axim u m e x tr e m e (m a x ) an d in n er m axim u m ex tr e m e ( m a x j , ce n tr e ex p e r im e n t (c). (a fte r B ox et ai, 19 7 8 b )

A central composite design always contains twice as many factorial points as there are factors in the design. These represent extreme values for each factor in the design, and are named ‘outer minimum extreme’ (min), ‘inner minimum extreme’ (mini), ‘outer maximum extreme’ (max) and ‘inner maximum extreme’ (max;).

A central composite design of this type, which is more accurately referred to as ‘circumscribed central composite design’, has circular, spherical, or hyperspherical symmetry and requires 5 levels for each factor.

Chapter 1