Tests based only on intra-block information
Using the inter-block estimates to improve the accuracy of the intra-block estimates is relatively easy (at least for balanced designs) but it is harder to use the inter-block information to test whether the treatments have any effect on the response. Since most information about the treatments comes from the intra-block estimates,
Researchers usually use the analysis of variance test described in Section 6.6 to test whether the treatments effect the response, ignoring the inter-block information.
Inter-block information about treatment differences
If there are more blocks than treatments, the block sum of squares in an analysis of variance can be partitioned into two components, a component explained by the different treatment combinations in the blocks and a second component that is a 'residual' sum of squares at block level.
There are therefore two different sums of squares relating to differences between treatments — one at block level and another at unit level. The usual intra-block analysis compares the unit-level treatment sum of squares to the unit-level residual sum of squares. It is however also valid to compare the block-level treatment sum of squares to the block-level residual sum of squares as a second independent test for differences between the treatments.
We will not further investigate use of inter-block information to test for a treatment effect or the combination of the inter-block and intra-block test results. However we note that the split of the block-level sums of squares and associated test is closely linked to the test used in nested experiments.
Pig diets
The analysis of variance table below initially shows only the sum of squares explained by blocks and the remaining unit-level sum of squares (ignoring the nine treatments that were used in the experiment.
Click Split units to split the unit-level sum of squares into a sum of squares explained by the feeding treatments and the residual sum of squares. The F-test comparing these has a p-value of 0.0084 so we conclude from the intra-block analysis that there is very strong evidence that the treatments do affect the weight gain of the pigs. (This is the test that was used in Section 6.6.)
Now click Split block to split the between-blocks sum of squares into a component that can be explained by the different mixtures of treatments in the blocks and a 'residual' sum of squares at block level. Comparing these sums of squares in an F-test gives a p-value of 0.0400, so the block-level test only gives moderately strong evidence of a difference between the feeding treatments.
As noted in the previous page, there is much more information about the treatment differences at unit level than at block level (the intra-block parameter estimates have lower standard errors than the inter-block estimates) so we would pay more attention to the intra-block test in our conclusions.