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Chapter 4   Blocks

4.1   Introduction to blocks

4.1.1   Variability of experimental units

In most experiments, the experimental units that are available differ from each other and this variability results in less accurate estimates of the effect of the treatment.

4.1.2   Blocks of similar experimental units

There is often enough information about differences between the experimental units before the experiment begins to allow them to be grouped into similar blocks of units.

4.1.3   Intentional variation in experimental units

Experiments are sometimes designed to have variable experimental units to give confidence that the conclusions will hold for a wide range of different situations.

4.1.4   Blocks and replicates

Some authors use the term 'replicate' to denote a separate repeat of the whole experiment. Such replicates should be treated as blocks.

4.2   Paired data

4.2.1   Randomised blocks with pairs

Groups of similar experimental units are called blocks. A randomised block design consists of a completely randomised design within each block. The special case of blocks of size two (pairs) and a single factor with two levels is particularly easy to analyse.

4.2.2   Matched pairs

Experimental units may be grouped into pairs that are 'similar' using any available information. In a randomised block design, one experimental unit in each pair is randomly given each factor level. The resulting estimate of the factor effect is more accurate than in a completely randomised experiment.

4.2.3   Analysis of differences

With paired data, the differences between the two measurements in each pair hold all information about the effect of the factor. Their average provides an estimate of the factor's effect.

4.2.4   Inference about differences

Standard univariate inference methods can be used to test whether the factor affects the response and to find a confidence interval for the factor effect.

4.3   Randomised blocks

4.3.1   Randomised block design

If the experimental units have been grouped into similar blocks, a randomised block design randomly allocates treatments separately within each block.

4.3.2   Randomised block examples

This page shows a few data sets from randomised block experiments.

4.3.3   Comparison of designs

Randomised block designs result in more accurate estimates of the effects of the factors of interest than in a completely randomised experiment with the same experimental units.

4.3.4   Confidence intervals for differences

If there is a control treatment and a single replicate in each block, treatments can be compared to the control with CIs of the form described in the previous section. They are narrower than the corresponding CIs that take no account of the blocks.

4.3.5   Removing block effects

The variation between blocks can be removed by adding/subtracting a value to each block to make all block means equal. This reduces the residual (unexplained) sum of squares in a simple anova table.

4.4   Anova for randomised blocks

4.4.1   Model for randomised blocks

The simplest model for a single factor in a randomised block design is very similar to the model without interaction for designs with two factors.

4.4.2   Explained sums of squares

Adding terms to the model for blocks and for the factor reduce the residual sum of squares. Each reduction is the sum of squared differences between the fitted values from two models.

4.4.3   Removing the block effect

The explained sums of squares for blocks is the sums of squared differences between the block means and overall mean. The treatment sum of squares is similarly the sum of squared differences between the treatment and overall means.

4.4.4   Removing block and factor effects

Adding a term for blocks to the model can be thought of as adjusting the values to make all block means equal. The residuals from the full model can be thought of as a further adjustment to make all treatment means equal.

4.4.5   Anova table and examples

An anova table shows these sums of squares and associated degrees of freedom. The F-ratio for treatments in the table is the basis of a test for equal treatment means. Several examples are given.

4.4.6   Model assumptions and trandformation

The anova table test for equal treatment means is based on a model that makes assumptions about the data. When the unexplained response variation is not constant, a transformation of the response may fit the model better.

4.4.7   Confidence intervals for treatment means

If it is concluded that all treatment means are not equal, confidence intervals can be used to report the differences.

4.5   Interaction between blocks and factor

4.5.1   Varying factor effect

If the blocks are very different from each other, the factor may have a different effect within different blocks. This is called interaction between the block and factor.

4.5.2   Interaction sums of squares

Interaction between the effect of blocks and the factor is modeled with an interaction term in the same way as the interaction between two factors. It reduces the residual sum of squares by an amount equal to the sum of squared differences between the fitted values for the models with and without the term.

4.5.3   Anova test for interaction

The explained and residual sums of squares can be presented in an analysis of variance table. The interaction sum of squares is used to test for whether the effect of the factor is the same in all blocks.

4.6   Two or more factors

4.6.1   Randomised blocks and treatments

In many randomised block experiments, the treatments are combinations of levels of two or more separate factors. Each such treatment should have the same number of replicates be randomly applied to experimental units within each block.

4.6.2   Analysis of variance

The treatment sum of squares in the earlier analysis of variance table for randomised block designs can be spllit into explained sums of squares for the separate factors and their interaction.