Variations of One Factor ANOVA Designs - Agricultural Statistical Data Analysis Using Stata - page 85

These designs are considered balanced because pairs of treatments
occur an equal number of times in the experiment. In addition, each
treatment is replicated the same number of times throughout the
experiment. Incomplete block designs are usually employed because
of some limitation in space or equipment to test all the treatments
within each block.
Balanced incomplete block (BIB) designs will have k experimental
units, b blocks, t treatments, r replications, and λ blocks in which
pairs of treatments occur. The total number of experimental units,
N, = rt = bk . In addition, these designs will have k < t . Finally, λ = r ( k -
1)/( t -1) in these experiments.
Load the dataset Broccoli Germination.dta. This is a dataset of
broccoli root growth as a function of temperature. There were only
three growth chambers ( k = 3) available to control the temperature
and the experiment called for four germination temperatures ( t = 4).
The experiment was set up as a BIB design with four temperature
treatments of 10, 20, 30, and 40°C. Root growth was measured as
mm•h -1 . Four separate runs ( b = 4) were conducted in such a fashion
so that each temperature occurred three times ( r = 3) in the experi-
ment and was paired with each other temperature within a run twice
(λ = 2).

1 2 3 1 2 3

Run 1 10°C 20°C 40°C Run 2 30°C 20°C 10°C

Chamber Chamber

1 2 3 1 2 3

Run 3 40°C 10°C 30°C Run 4 40°C 20°C 30°C

If you look at the dataset in the Data Editor window you will notice
that there are missing data indicating the design is incomplete. It is,
however, balanced because each treatment occurs the same number
of times and is paired with other temperatures the same number of
times. To meet these requirements, particularly as the number of
treatments increase, can become quite complex and it is best to con-
sult a textbook or statistician before proceeding.