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   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.