The models for Plackett-Burman and factorial designs are:
The models for mixture designs are:
In terms of results, the ANOVA table provides information about how well the model fits the responses. The ANOVA table also provides information about the participation or effect of the different design variables and their possible interaction, as well as the significance of these effects. The analysis sequence is then to first look at the model p-value and R². A p-value below 5% indicates a good model and a R² close to 1 indicates a good correlation between the predicted response value and the actual response value. Consideration must then be given to the value of the individual effects or model terms and their sign. Consideration should also be given to the size of the corresponding p-values. Each effect with a p-value < 5% is considered significant; if the p-value is < 1% it is highly significant. A p-value between 5 and 10% indicates a possibly significant effect.A p-value > 10%, indicates that an effect is not considered as significant. A p-value for an interaction term effect between 5 to 10% is considered to be significant. The summary table provides data for the effects with the values and associated p-values for all response variables.
|Sum of Square (SS)||Degree of Freedom (DF)||Mean Square||F-ratio||p-value|
|B||1.250 e+03||1||1.250 e+03||416.667||0.0000|
In this example the model is valid (p-value=0.0001) and all effects are significant (p-values < 0.05). The most significant effect is B as it has the smallest p-value.
The error sum of square and degree of freedom can be calculated either on the design samples or on the replicated center samples if the design is saturated.
Note: A saturated design is a design that has the number of experimental design samples equal to the number of model terms (including B0 if necessary). This type of design uses all the degree of freedom to calculate the model terms.
Other checks that can be applied after analyzing the ANOVA table include the detection of curvature effects. These can be found by plotting the main effects plot. If a nonlinear trend is detected when checking the position of the center sample, one may consider a possible curvature effect and include the square term of the effect in the model.
Main effect plot with curvature
When a variable is category, it is necessary to check which effects are significant and also if they are significantly different.
The multiple comparison test provides this type of information. It is based on a comparison of the averages of the response variable at the different levels. If the difference between two averages is greater than the critical limit the two levels are significantly different. If not they have a similar effect. If no level has an effect all levels will have a statistically similar effect and the averages for the response variables at the different levels will be non-significantly different.
In The Unscrambler®, there are three specific outputs for the multiple comparison test:
The models for CC and BB designs are:
The models for mixture designs are:
In addition one can also study the response surface. Maximum, minimum or saddle points can be detected on the response surface by varying the conditions to plot the response surface. More information on how to vary the condition can be found in the RS table section in the plot interpretation page.
All design can be analyzed by PLS, however it is the only possibility for D-optimal designs.
The settings are done automatically.
The number of PCs tested is the maximal number of PC.
The analysis is done on the real value matrix with all X-variable weighted with the option 1/std.
The response variables are also weighted with the same option 1/std.
A full cross-validation is run to validate the model.
An uncertainty test is run on the optimal number of component so as to get the Jack-knife test results.
In addition to the usual results of PLS, some special plots are useful in DoE. They are all computed from the Jack-knife uncertainty test developed by Dr. Harald Martens.
The analysis with PLS provides some advantages:
There are some drawbacks as well: