Announcement
Randomized (Complete) Block Designs
Gerard E. Dallal, Ph.D.

The Randomized Complete Block Designs is a bit of an odd duck. The design itself is straightforward. It's the analysis that might seem somewhat peculiar.

The design is best described in terms of the agricultural field trials that gave birth to it. When conducting field trials to compare fertilizers, plant varieties, or whatever, there is concern that some parts of a field may be more fertile than others. So, if one were comparing three fertilizers, say, it would not be a good idea to use one fertilizer here, another fertilizer over there, and the third fertilizer way out back because the effects of the fertilizers would be confounded with the natural fertility of the land.

The randomized block design goes this way.

• The field is divided into blocks and
• each block is divided into a number of units equal to the number of treatments.
• Within each block, the treatments are assigned at random so that a different treatment is applied to each unit. That is, all treatments are observed within each block. The defining feature of the Randomized (Complete) Block Design is that each block sees each treatment exactly once.*

The analysis assumes that there is no interaction between block and treatment, that is, it fits the model

Yijk = + i + j + ijk

where the s are the treatment effects and the s are the block effects.

The ANOVA table contains three lines.

 Source Degrees of Freedom Treatment a-1 Blocks b-1 Residual (a-1)(b-1)

Since there are a*b units, the total number of degrees of freedom is ab-1 and the residual degrees of freedom is

(ab-1)-(a-1)-(b-1) = (a-1)(b-1)
.

There is a better way to view randomized blocks that brings them into the mixed model framework. A Randomized (Complete) Block Design is a two-factor study in which the fixed factor TREATMENT is crossed with the random factor BLOCKS. If we fit the standard factorial model

Yijk = + i + j + (  )ij + ijk

the ANOVA table becomes

 Source Degrees of Freedom Treatment a-1 Blocks b-1 Treatment*Blocks (a-1)(b-1) Residual (n-1)ab = 0

where n is the number of observations per unit. Since n=1 (and k in the model takes on only the value 1), there are no degrees for pure error. However, in the mixed model, the proper way to test for a treatment effect is by comparing the Treatment mean square to the Interaction mean square, and we can estimate that. One might argue that the way the classic analysis works is by pooling the interaction and residual terms together and labeling them "Residual". However, since there is no pure residual, the term labeled Residual is really the Interaction, so everything works out properly!

While randomized block designs started out in agricultural field trials, they can apply to almost any field of investigation.

• In the lab, it is common for scientists to work with plates of cells divided into wells. Here, the plates are the blocks and the wells are the units. The same thing applies to gels divided into lines.
• When an experiment is replicated, each replicate can be considered a block.
• Often, the blocks are people, as in the case of a paired t test.

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*There are also Incomplete Block Designs, in which the number of units is less than the number of treatments, so that each block sees only a subset of treatments. Incomplete Block Designs are currently beyond the scope of these notes.

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