Replicating the whole design
The best way to estimate the error variance is with a complete replicate of the whole factorial design. A complete replicate also results in more accurate estimates of the effects of the factors. However a second replicate doubles the number of experimental runs and can be prohibitively expensive.
Replicating a single treatment
An alternative way to estimate the error variance is with r runs of the experiment at a single combination of factor levels (e.g. the low level for all factors). The variance of these repeat observations provides an estimate of the error variance and can be used in the 'Residual' row of the analysis of variance with (r - 1) degrees of freedom.
However these extra values destroy the orthogonality of the design — in the analysis of variance table, the sums of squares for the main effects and interactions will depend on the order in which they are added to the model. This is therefore a poor solution to the problem of estimating the error variance.
Replicating the 'centre point' of the design
If all factors are numerical in a 2k factorial experiment, a better solution is with several runs of the experiment at a treatment combination with the middle value for all factors. These are called centre points in the design. If the factor levels are coded as -1 and +1 in the factorial runs of the experiment, this corresponds to making r runs of the experiment with all factors at value 0.
Again the response variance at the r centre points estimates the error variance and can be used in the residual row of the analysis of variance table with r - 1 degrees of freedom. By replicating the centre point, the other factors remain orthogonal, making interpretation of the results easier.
At least 6 replicates of the centre point are recommended to get a reasonably accurate estimate of the error variance from the residual sum of squares.
Properties
Factorial designs with added centre points have several important properties:
The final bullet point means that the design could be considered wasteful, so the number of centre points is usually limited to 5 or 6 — possibly even as low as 3 if runs are very expensive.
Sugar reduction study
We now examine how the addition of centre points would alter the analysis of the sugar reduction data.
Factor | Low level | High level | Centre value |
---|---|---|---|
Saltiness | 0.10 g% | 0.20 g% | 0.15 g% |
Acidity | 0.00 g% | 0.05 g% | 0.025 g% |
Thickness | 3.00 g% | 4.00 g% | 3.50 g% |
Hotness | 0.05 g% | 0.15 g% | 0.10 g% |
To illustrate the method, we have added four response values at the centre point of the design.
Saltiness | Acidity | Thickness | Hotness | Sweetness |
---|---|---|---|---|
+1 +1 +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 |
+1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 +1 -1 -1 -1 -1 0 0 0 0 |
+1 +1 -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 0 0 0 0 |
+1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 0 0 0 0 |
7.60 7.92 7.72 7.27 8.09 7.87 7.52 6.89 8.02 8.34 8.20 8.21 8.14 7.82 8.04 7.00 7.60 7.70 7.80 7.90 |
The diagram below initially shows the full model with all interactions but without the centre points. Notice that the 4-factor interaction cannot be tested.
Click Use centre points to add the four centre points to the analysis. These centre points give a residual sum of squares with 3 degrees of freedom, allowing all interactions to be tested.
There is also a single sum of squares and degree of freedom for nonlinearity. Since the corresponding p-value is large, we can conclude that there is no evidence of nonlinearity.