5.5. Advanced topics
5.5.9. An EDA approach to experimental design
18.104.22.168. Cumulative residual standard deviation plot
|Models for 2k and 2k-p designs||Given that we have a statistic to measure the quality of a model, any model, we move to the question of how to construct reasonable models for fitting data from 2k and 2k-p designs.|
|Initial simple model||
The simplest such proposed model is
|Next-step model||The logical next-step proposed model will consist of the above additive constant plus some term that will improve the predicted values the most. This will equivalently reduce the residuals the most and thus reduce the residual standard deviation the most.|
|Using the most important effects||
As it turns out, it is a mathematical fact that the factor or
interaction that has the largest estimated effect
In the previous steps in our analysis, we developed a ranked list of factors and interactions. We thus have a ready-made ordering of the terms that could be added, one at a time, to the model. This ranked list of effects is precisely what we need to cumulatively build more complicated, but better fitting, models.
|Step through the ranked list of factors||
Our procedure will thus be to step through, one by one, the ranked list
of effects, cumulatively augmenting our current model by the next term
in the list, and then compute (for all n design points) the
predicted values, residuals, and residual standard deviation. We
continue this one-term-at-a-time augmentation until the predicted
values are acceptably close to the observed responses Y (and
hence the residuals and residual standard deviation become acceptably
close to zero).
Starting with the simple average, each cumulative model in this iteration process will have its own associated residual standard deviation. In practice, the iteration continues until the residual standard deviations become sufficiently small.
|Cumulative residual standard deviation plot||
The cumulative residual standard deviation plot is a graphical summary
of the above model-building process. On the horizontal axis is a
series of terms (starting with the average, and continuing on with
various main effects and interactions). After the average, the
ordering of terms on the horizontal axis is identical to the ordering
of terms based on the half-normal probability
plot ranking based on effect magnitude.
On the vertical axis is the corresponding residual standard deviation that results when the cumulative model has its coefficients fitted via least squares, and then has its predicted values, residuals, and residual standard deviations computed. The first residual standard deviation (on the far left of the cumulative residual standard deviation plot) is that which results from the model consisting of