Adjusting data to make all block means equal
The blocks of similar experimental units in a randomised block design improve accuracy because they can be used to explain some of the observed variability in the response. It is often found that some blocks have consistently higher response values than others.
One approach that takes account of the block-to-block variation is to adjust the values in each block (adding or subtracting a constant) to 'correct' for these block differences and make all blocks have the same mean response.
After 'correcting' for the block differences in this way, we could ignore the existence of blocks and use the simpler analysis of variance for a completely randomised design to test for differences between the treatments. Although the resulting anova table is not completely correct for testing whether the treatment means are equal, its sums of squares are the basis for the test that will be described in the next section.
Amino acid uptake by fish
To examine how NaCN (sodium cyanide) affected the uptake of an amino acid by intestinal preparations from a species of fish, a randomised block experiment was conducted. Only six preparations could be obtained from each fish, so four fish were used in order to obtain enough experimental units. The fish are the four blocks in the experiment and three preparations from each were randomly chosen to get the NaCN treatment, the other three being controls.
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Treatment | 1 | 2 | 3 | 4 | ||||
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Without NaCN |
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The coloured vertical lines on the diagram below show the mean uptakes for the different fish — clearly some fish have much higher amino acid uptake than others. (Click on any cross to see the 6 response values from that fish.)
The analysis of variance table is the one that is appropriate for a completely randomised design ignoring the existence of blocks. Observe that:
If the blocks are ignored, there is only moderately strong evidence that the treatments differ (p-value = 0.0209).
Click Make all block means equal to 'correct' the data for differences between the blocks. Observe that:
Because the residual sum of squares decreases so much, the F-ratio becomes large and the p-value becomes close to zero. From this anova table,
We would now conclude that it is almost certain that the NaCN treatment affects amino acid uptake.
The treatment and residual sums of squares shown above are the basis for testing whether the treatment means are all equal. However:
The analysis described on this page is not completely correct — the residual degrees of freedom are too high.
In the next section, we describe the correct anova table for testing whether the treatment means are equal in data from a randomised block experiment.