Blocking
Blocking is a technique used to mathematically remove the variation caused bysome identifiable change during the course of the experiment. For example, youmay need to use two different raw material batches to complete the experiment,or the experiment may take place over the course of several shifts or days. Foreach of these cases, the change may cause the response data to shift. Blockingremoves this shift and, in effect, “normalizes” the data.
The software provides various options for blocking, depending on how many runsyou choose to perform. The default of 1 block really means “no blocking.”
For example, in experiments with 16 runs, you may choose to carry out theexperiment in 2 or 4 blocks. Two blocks might be helpful if, for some reason,you must do half the runs on one day and the other half the next day. In thiscase, day to day variation may be removed from the analysis by blocking.
When you choose to block your design, one or more effects will no longer beestimable. You can look at the alias structure to see which effects have been“lost to blocks.” This is especially important when you have 4 or more blocks.In certain cases, a two-factor interaction may be lost and so then you will wantto make sure that the interaction is not one that you are interested in.
Another note about blocking - it is assumed that the block variable does notinteract with the factors. The effect must only be a linear shift, and not bedependent on the level of one or more of the factors under study.
Note
If you try to block on a factor, that factor will be aliased with theblock and you will not get any statistical details on the effect of that factor.Only block on things that you are NOT interested in studying.
Example: You are trying to determine the effects of factors in a coatingprocess such as speed, temperature, and pressure on your product’s tensile andelongation properties. Due to the number of runs involved, you will need to usetwo different batches of raw material. You expect that variations in the rawmaterial may have an effect on the response, but you are not interested instudying that effect at this time. Therefore, raw material is NOT a factor andyou should block on it instead. This will remove the effect of raw material ontensile and elongation from the ANOVA and allow you to better identify the otherfactor effects.
On the other hand, if you want to study the effect of raw material batchvariation, then it should be included as a factor and you should NOT set upblocks on this factor. Consider using a split-plot design instead of blocking inthese cases.
Center Points
A useful extension of two-level factorial and fractional factorial designsincorporates center points into the factorial structure. If you have at leastone numeric factor, you can choose to add center points to your design.
The software allows you to replicate the center point to random runs in thedesign to provide an estimate of pure error and test for curvature. Addingcenter points permits a statistical check for the goodness-of-fit of the planartwo-level factorial model. The average response value from the actual centerpoints is compared to the estimated value of the center point that comes fromaveraging all the factorial points. If there is curvature, the actual centerpoint value will be either higher or lower than predicted by the factorialdesign points. Curvature of the surface may indicate that the design is in theregion of an optimum.
Extra Details:
Two models are fit to the data: adjusted for curvature and unadjusted. Theadjusted model includes estimates a separate curvature term. The unadjustedattempts to fit the center points using an interaction model. The two methodsproduce different significance values and different summary statistics. Abenefit of this procedure is that the assumptions concerning normality andconstant variance can be checked by the adjusted model’s ANOVA even in thepresence of significant curvature. This allows problems to be identified duringthe data analysis that might otherwise be obscured by curvature inflating theresiduals.
If the curvature test is significant, we recommend the design be augmented to amore capable response surface design. Without the augment, the curvature willnot be fit by the model and will not be included for optimization.
Standardized and Normalized Factorial Effects
In general, an effect is the change in the response caused by changes in thefactors. A linear effect is the average value of a response at the high settingof a factor less the average value of a response at the low setting of thefactor. An interaction effect is an adjustment to the linear effect depending onthe setting of another factor.
Standardized and Normalized effects are seen on the half-normal plot and effectslist when analyzing factorial designs. The standardized effects are used fortwo-level factorial designs. The normal effects are used for multilevelcategoric designs.
Standardized effects are calculated by dividing the effect by the standard errorof estimating the associated coefficient and then multiplying this quotient bythe standard error of estimating the first linear coefficient in the model. Theeffects are standardized to the first alphabetical linear effect in the model.This stabilizes the effect estimates for non-orthogonal designs.
Normalized effects are calculated by subtracting the sum of squares for eachterm from the total sum of squares and dividing by the corrected total degreesof freedom less the number of degrees of freedom used to estimate the term’s sumof squares. This value is compared to a χ2 distribution to produce a provisionalp-value. The provisional p-value is converted to a standard normal score toproduce the normalized effect.
Factor and Block Generators
When you choose a factorial design that is either a fractional factorial or hasblocks, you can optionally (via checkbox on 1st page) select to view a page thatallows you to change the default generators. Normally, you shouldn’t changeanything on this screen, click Next to skip.
What is a generator? It is usually a high-order interaction with which a factoror block column is equated. For instance, the design for 4 factors in 8 runs hasa factor generator of D=ABC. This means that factor D is confounded, or aliasedwith, the ABC interaction. This is the basis for the remaining alias structurefor this design.
The default factor generators are those which correspond to minimum aberrationdesigns. This means that the least amount of aliasing will occur and it will bewith the highest order interactions. Generally, these designs are the same asthose found in standard textbooks. The exception will be some of the designsthat have blocks. The software uses blocking generators that provide the bestpossible alias structure and these designs may be different from thoserecommended in textbooks.
If you want to change the factor or block generators in order to match atextbook design, or a published case study, simply click on the box “Makegenerators editable” and then type in your generators. (The check box preventsinadvertent changes.) Check the alias structure on the next screen and decide ifyou want to continue with your new design.
Interpreting Models
Models are used to produce graphs, optimization and post analysis output. Thegraphs are easier to interpret than looking at the models directly. Thefollowing is a quick guide for how to interpret models.
Numeric Factors
A factorial model is composed of a list of coefficients multiplied by theirassociated factor levels.
The models include an intercept, main effects for the single factor effects, andinteractions where the effect of one factor depends on the settings of the otherfactors in the interaction. Curvature is a term that can be estimated if centerpoint runs are included.
The beta (β) coefficients in the above model are the slope indicating how muchchange is expected in the response (Y) when there is a one unit change in thefactor (A, B, C, …). When there are two or more factors in a term then it iseasiest to interpret the model by setting all but one to a fixed value. Multiplythe coefficient times the fixed factor settings to get a provisional coefficientfor the remaining, variable factor.
Categoric Factors
When a categoric factor has two levels, it interprets the same as a numericfactor. When a categoric factor has more than two levels, interpretation becomesa bit more difficult. Please see the Hints and FAQ topic on Interpretingthe Categoric Model.
Mixture Components
Mixture models are only readily interpretable when the mixture components all gofrom 0 to the total for the design. Most mixtures designs cover a moreconstrained space. Use the Model Graphs to better understand the models.
See the Scheffé Mix Model topic in Mixture Designs for moredetails.
