Alternative approaches

The different combinations of levels for the factors used in an experiment are called treatments.

It is sometimes convenient to consider all different treatments to be a single factor, even when there is structure to the treatments — e.g. when they are formed from levels of 2 or more distinct variables that are controlled by the experimenter.

The analysis of variance tables that arise from these two approaches (separate factors or a single combined factor) are closely related.

Tyre wear

The table below shows the results of an experiment involving six combinations of tyre and type of wear test.

Weight loss of tyre
  Fast-wear
test on
tyre A  
  Standard
test on
tyre A
  Slow-wear
test on
tyre A
  Fast-wear
test on
tyre B
  Standard
test on
tyre B
  Slow-wear
test on
tyre B
151
101
119
156
136
107
124
140
118
95
130
111
117
102
133
100
109
95
126
113
115
97
109
92

These six treatments can be treated as a single factor with six levels, or the table could be restructured as shown below as two factors, one with two levels and the other with three levels.

  Test type
Tyre type   Fast wear     Standard wear   Slow wear
Tyre A
151
101
119
156
136
107
124
140
118
95
130
111
Tyre B
117
102
133
100
109
95
126
113
115
97
109
92

The analysis of variance table below initially uses all treatments in a single factor.

Click Split treatments to show the conventional break-up of the explained sum of squares for Tyre type, test type and their interaction.

Note that the sums of squares (and degrees of freedom) for the two factors and their interaction add to the explained sum of squares for the treatments as a single factor with six levels.


Treatments in a randomised block design

The same alternatives arise in experiments with blocks. The treatments are again the different combinations of factor levels set by the experimenter and we may either consider all treatments as a single 'super-factor' in the experiment or separate it into main effects and interactions for the individual variables that have been controlled.

Acupuncture and Codeine for dental pain relief

An anaesthetist conducted an experiment to assess the effects of codeine and acupuncture for relieving dental pain. The experiment used 32 subjects who were grouped into blocks of 4 according to an initial assessment of their tolerance to pain.

The four treatment combinations of (codeine or a sugar capsule) and (active or inactive acupuncture points) were randomly given to the four subjects in each block. Pain relief scores were recorded from each subject two hours after dental treatment. The experiment was double blind since neither the subjects nor the person assessing pain relief knew which treatment had been adminstered.

  Pain relief score
Tolerance
group
  Control   Codeine
only
Acupuncture
only
Codeine +
Acupuncture
1
2
3
4
5
6
7
8
0.0
0.3
0.4
0.4
0.6
0.9
1.0
1.2
0.6
0.7
0.8
0.9
1.5
1.6
1.7
1.6
0.5
0.6
0.8
0.7
1.0
1.4
1.8
1.7
1.2
1.3
1.6
1.5
1.9
2.3
2.1
2.4

Have codeine or acupuncture affected changed the mean pain relief score? If so, by how much?

Click Split treatments to expand the treatment sum of squares (and degrees of freedom) into main effects for the two factors Use of codeine and Use of acupuncture and their interaction.

From the split sum of squares, we can test whether there is interaction between the effects of codeine and acupuncture. The p-value is 0.0923 so we would conclude that there is very little evidence of interaction.

The p-values for the two main effects are both reported as "0.0000" so it is almost certain that both codeine and acupuncture increase the mean pain relief score.