Making all block means equal

We showed earlier that it is poor to test for equal treatment means with an Analysis of Variance (anova) table that ignores the blocks. The extra variability in the response that is caused by differences between the blocks makes it far harder to detect treatment differences.

Adding or subtracting a value from each block to make all block means equal would get rid of the differences between the blocks. 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.

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. Six of these preparations were obtained from each of four fish (the blocks) and three preparations from each block were randomly chosen to get the NaCN treatment, the other three being controls.

The mean amino acid uptake from each fish (block) is shown as a coloured vertical line. (Click on any cross to see the 6 response values in the block.) Click Make all block means equal to add or subtract a constant from the values in each block in order to make the four block means equal.

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 there is strong evidence that the NaCN treatment affects amino acid uptake.

This analysis is not perfect however. The anova table still needs a further small modification that will be described on the next page.


Grazing cattle in Uganda

Five observers watched a group of 10 Zebu cattle for 88 minutes in Uganda. Each person reported the number of minutes that each animal was observed grazing.

The experiment was conducted in order to compare the five observers, but there is also considerable variation between the animals (blocks).

Again click Make all block means equal to add or subtract a constant from the values in each block in order to make the block means equal.

Again, removing the block effect does not change the treatment sum of squares but reduces the residual sum of squares considerably. The p-value becomes close to zero, giving strong evidence of a difference between observers.

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.

On the next page, we describe the correct anova table for testing equal treatment means for randomised block data.