In some experiments, a factor is varied at block level so all experimental units in a block get the same factor level. The factor is then confounded with blocks so a conventional analysis of variance cannot be used to test its effect.
In analysis of variance for a nested design, the factor effect must be removed before the block effect instead of after it. The relative sizes of the factor and block sums of squares give evidence for whether the factor affects the response.
A test for the factor effect involves the ratio of the mean sum of squares explained by the factor and the mean sum of squares between blocks (within the factor levels). This F-ratio and its p-value can be interpreted in the usual way.
The experimental units may be structured in a hierarchy of more than two levels. The effect of a factor should be assessed by comparing its explained sum of squares with the sum of squares between blocks at the level at which it is applied.
The p-value for the F-test described above is identical to the p-value that would be obtained from a simple analysis of the block means or the block totals.
Both increasing the number of blocks and the number of measurements per block improve accuracy of estimating the factor effect. The relative variability of the blocks and the values within the blocks determine the best balance when designing an experiment.
In a split plot design, one factor is varied at block level and a second factor is varied within blocks.
The sum of squares for the factor that is varied at block level should be compared to the sum of squares describing variation between blocks, but the sum of squares for the factor that is varied within blocks should be compared to the residual sum of squares.
The sum of squares for interaction between the factors should be tested against the residual sum of squares.
If the factor that is varied within blocks uses the same mixture of levels within each block, raw means for the factor levels summarise the factor effects.
The experimental units are sometimes structured in a hierarchy with more than two levels (e.g. blocks, plots and sub-plots). Factors may be varied at any level and should be tested against 'residual' variation at that level.
An example is analysed involving experimental units in a four-level hierarchy.
The effect of any factor is more accurately estimated if it is varied at the lowest level of the hierarchy of experimental units.
If treatments are varied at block level, all information about treatment differences is contained in the block means. If treatments are orthogonal to blocks, all information about treatment differences is held by differences within blocks. Other experimental designs have information at both levels.
The usual estimates of treatment effects from differences within blocks are called intra-block estimates. A second set of estimates called inter-block estimates can be found by least squares from the block means.
The inter- and intra-block estimates can be combined to give a more accurate set of parameter estimates but the improvement over the intra-block estimates is usually small.
This page combines the inter- and intra-block estimates for a balanced lattice design.
In an analysis of variance table, treatment sums of squares can be separated from the sums of squares at both block and unit level and these provide two independent tests of the effect of the treatments. The unit-level test is usually more powerful.