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 | |||||||||||
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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 |
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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.
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Tyre type | Fast wear | Standard wear | Slow wear | ||||||
Tyre A |
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Tyre B |
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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.
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Tolerance group |
Control | Codeine only |
Acupuncture only |
Codeine + Acupuncture |
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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.