Additional Factorial Topics

Blocking

Blocking is a technique used to mathematically remove the variation caused by some identifiable change during the course of the experiment. For example, you may 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. For each of these cases, the change may cause the response data to shift. Blocking removes this shift and, in effect, “normalizes” the data.

The software provides various options for blocking, depending on how many runs you 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 the experiment 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 this case, 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 be estimable. 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 want to 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 not interact with the factors. The effect must only be a linear shift, and not be dependent on the level of one or more of the factors under study.

Example: You are trying to determine the effects of factors in a coating process such as speed, temperature, and pressure on your product’s tensile and elongation properties. Due to the number of runs involved, you will need to use two different batches of raw material. You expect that variations in the raw material may have an effect on the response, but you are not interested in studying that effect at this time. Therefore, raw material is NOT a factor and you should block on it instead. This will remove the effect of raw material on tensile and elongation from the ANOVA and allow you to better identify the other factor effects.

On the other hand, if you want to study the effect of raw material batch variation, then it should be included as a factor and you should NOT set up blocks on this factor. Consider using a split-plot design instead of blocking in these cases.

Center Points

A useful extension of two-level factorial and fractional factorial designs incorporates center points into the factorial structure. If you have at least one 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 the design to provide an estimate of pure error and test for curvature. Adding center points permits a statistical check for the goodness-of-fit of the planar two-level factorial model. The average response value from the actual center points is compared to the estimated value of the center point that comes from averaging all the factorial points. If there is curvature, the actual center point value will be either higher or lower than predicted by the factorial design points. Curvature of the surface may indicate that the design is in the region of an optimum.

Extra Details:

Two models are fit to the data: adjusted for curvature and unadjusted. The adjusted model includes estimates a separate curvature term. The unadjusted attempts to fit the center points using an interaction model. The two methods produce different significance values and different summary statistics. A benefit of this procedure is that the assumptions concerning normality and constant variance can be checked by the adjusted model’s ANOVA even in the presence of significant curvature. This allows problems to be identified during the data analysis that might otherwise be obscured by curvature inflating the residuals.

If the curvature test is significant, we recommend the design be augmented to a more capable response surface design. Without the augment, the curvature will not 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 the factors. A linear effect is the average value of a response at the high setting of a factor less the average value of a response at the low setting of the factor. An interaction effect is an adjustment to the linear effect depending on the setting of another factor.

Standardized and Normalized effects are seen on the half-normal plot and effects list when analyzing factorial designs. The standardized effects are used for two-level factorial designs. The normal effects are used for multilevel categoric designs.

Standardized effects are calculated by dividing the effect by the standard error of estimating the associated coefficient and then multiplying this quotient by the standard error of estimating the first linear coefficient in the model. The effects 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 each term from the total sum of squares and dividing by the corrected total degrees of freedom less the number of degrees of freedom used to estimate the term’s sum of squares. This value is compared to a χ2 distribution to produce a provisional p-value. The provisional p-value is converted to a standard normal score to produce the normalized effect.

Factor and Block Generators

When you choose a factorial design that is either a fractional factorial or has blocks, you can optionally (via checkbox on 1st page) select to view a page that allows you to change the default generators. Normally, you shouldn’t change anything on this screen, click Next to skip.

What is a generator? It is usually a high-order interaction with which a factor or block column is equated. For instance, the design for 4 factors in 8 runs has a factor generator of D=ABC. This means that factor D is confounded, or aliased with, the ABC interaction. This is the basis for the remaining alias structure for this design.

The default factor generators are those which correspond to minimum aberration designs. This means that the least amount of aliasing will occur and it will be with the highest order interactions. Generally, these designs are the same as those found in standard textbooks. The exception will be some of the designs that have blocks. The software uses blocking generators that provide the best possible alias structure and these designs may be different from those recommended in textbooks.

If you want to change the factor or block generators in order to match a textbook design, or a published case study, simply click on the box “Make generators editable” and then type in your generators. (The check box prevents inadvertent changes.) Check the alias structure on the next screen and decide if you want to continue with your new design.

Interpreting Models

Models are used to produce graphs, optimization and post analysis output. The graphs are easier to interpret than looking at the models directly. The following is a quick guide for how to interpret models.

Numeric Factors

A factorial model is composed of a list of coefficients multiplied by their associated factor levels.

The models include an intercept, main effects for the single factor effects, and interactions where the effect of one factor depends on the settings of the other factors in the interaction. Curvature is a term that can be estimated if center point runs are included.

The beta (β) coefficients in the above model are the slope indicating how much change is expected in the response (Y) when there is a one unit change in the factor (A, B, C, …). When there are two or more factors in a term then it is easiest to interpret the model by setting all but one to a fixed value. Multiply the coefficient times the fixed factor settings to get a provisional coefficient for the remaining, variable factor.

Categoric Factors

When a categoric factor has two levels, it interprets the same as a numeric factor. When a categoric factor has more than two levels, interpretation becomes a bit more difficult. Please see the Hints and FAQ topic on Interpreting the Categoric Model.

Mixture Components

Mixture models are only readily interpretable when the mixture components all go from 0 to the total for the design. Most mixtures designs cover a more constrained space. Use the Model Graphs to better understand the models.

See the Scheffé Mix Model topic in Mixture Designs for more details.