Two factors
If eight runs of the experiment are possible, the experiment can be conducted with two replicates of a complete 22 factorial design — two runs for each combination of the levels of the two factors.
Since only two main effects and one 2-factor interaction are estimated, the residual sum of squares has four degrees of freedom and it is possible to test the significance of all terms.
Three factors
With eight runs of the experiment, a single replicate of a 23 factorial design can be used,
Factor | ||||
---|---|---|---|---|
Run | A | B | C | Response |
1 | -1 | -1 | -1 | y—- |
2 | -1 | -1 | +1 | y—+ |
3 | -1 | +1 | -1 | y-+- |
4 | -1 | +1 | +1 | y-++ |
5 | +1 | -1 | -1 | y+— |
6 | +1 | -1 | +1 | y+-+ |
7 | +1 | +1 | -1 | y++- |
8 | +1 | +1 | +1 | y+++ |
As in other complete factorial designs, all main effects and interactions can be estimated. For example, the main effect for A is the difference between the average response at its low and high levels,
main effect for A = (y+++ + y++- + y+-+ + y+— - y-++ - y-+- -y—+ - y—-) / 4
The signs used when combining the 8 response values for the main effects and interactions are given in the table below. Note that the interaction columns are products of the corresponding main effect columns.
Factor | 2-factor interactions | 3-factor interation | ||||||
---|---|---|---|---|---|---|---|---|
Run | A | B | C | AB | AC | BC | ABC | Response |
1 | -1 | -1 | -1 | +1 | +1 | +1 | -1 | y—- |
2 | -1 | -1 | +1 | +1 | -1 | -1 | +1 | y—+ |
3 | -1 | +1 | -1 | -1 | +1 | -1 | +1 | y-+- |
4 | -1 | +1 | +1 | -1 | -1 | +1 | -1 | y-++ |
5 | +1 | -1 | -1 | -1 | -1 | +1 | +1 | y+— |
6 | +1 | -1 | +1 | -1 | +1 | -1 | -1 | y+-+ |
7 | +1 | +1 | -1 | +1 | -1 | -1 | -1 | y++- |
8 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | y+++ |
Tests with three factors
With no replicates, there are no residual degrees of freedom if all main effects and interactions are allowed, so no tests are possible for the significance of the factors unless some interactions are negligible (assumed to be zero) so their sums of squares can be combined to form a residual sum of squares.
Popcorn
An experiment was conducted to assess how the yield of popcorn depended on the brand (ordinary or gourmet), ratio of oil to popcorn (low or high) and batch size (1/3 cup or 2/3 cup). There was only one replicate of the experiment and hence 8 batches of popcorn were tested.
If all interactions are modelled, no degrees of freedom are left for the residual sum of squares in the anova table, so significance of the effects cannot be tested.
We will now assume that there is no interaction between the factors. There is no information in the data to support this assumption, but previous experiments may help in deciding whether the assumption is warranted.
Click the checkboxes to turn off all four interactions and combine their sums of squares into the residual sum of squares. Since we now have an estimate of experimental variability — the mean residual sum of squares estimates the error variance — we can now test the significance of the main effects. From the p-values, we conclude that Size and Brand are important factors, but the Oil-popcorn ratio is not.