Tests based on block totals or means
In the previous page, we developed an analysis of variance table to test the effect of a factor whose levels are allocated at block level. A much simpler analysis gives the same results.
If the block means or block totals are analysed with a simple analysis of variance, the p-value for testing the effect of the factor is identical to the p-value from an analysis of the individual measurements.
This result may make you question the point of the analysis developed in the first half of this section. However it provides a basis for analysis of a more common type of experiment (a split plot experiment) that is the subject of the next section.
Combability of hair
The table below shows the totals and means of the five combability measurements from the 16 hair swatches.
Swatch | ||||||||
---|---|---|---|---|---|---|---|---|
Formulation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
A | 183 133 190 153 173 |
173 173 115 198 150 |
80 75 68 70 58 |
115 125 120 125 148 |
145 113 98 138 140 |
73 123 100 75 115 |
123 138 100 253 93 |
38 55 35 38 53 |
Total | 832 | 809 | 351 | 633 | 634 | 486 | 707 | 219 |
Mean | 166.4 | 161.8 | 70.2 | 126.6 | 126.8 | 97.2 | 141.4 | 43.8 |
9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
B | 255 110 195 93 213 |
118 200 145 155 108 |
133 155 145 240 230 |
150 130 110 185 105 |
65 60 105 90 100 |
95 65 45 65 60 |
145 175 125 145 180 |
60 55 60 45 65 |
Total | 866 | 726 | 903 | 680 | 420 | 330 | 770 | 285 |
Mean | 173.2 | 145.2 | 180.6 | 136.0 | 84.0 | 66.0 | 154.0 | 57.0 |
The diagram below shows analysis of variance tables for the individual measurements, the swatch totals and the swatch means.
Source | Ssq | df | MSq | F | p-value |
---|---|---|---|---|---|
Formulation | 5,968 | 1 | 5,968 | 0.11 | 0.7424 |
Residual | 743,493 | 14 | 53,107 | ||
Total | 749,460 | 15 |
Observe that the p-value for testing for a difference between the two formulations is identical for all three analyses.