1. Incomplete Block Designs

2. Randomized Block Design
We want to compare t treatments
Group the N = bt experimental units into b homogeneous blocks of size t.
In each block we randomly assign the t treatments to the t experimental units in each block.
The ability to detect treatment to treatment differences is dependent on the within block variability.

3. Comments
The within block variability generally increases with block size.
The larger the block size the larger the within block variability.
For a larger number of treatments, t, it may not be appropriate or feasible to require the block size, k, to be equal to the number of treatments.
If the block size, k, is less than the number of treatments (k < t)then all treatments can not appear in each block. The design is called an Incomplete Block Design.

4. Comments regarding Incomplete block designs
When two treatments appear together in the same block it is possible to estimate the difference in treatments effects.
The treatment difference is estimable.
If two treatments do not appear together in the same block it not be possible to estimate the difference in treatments effects.
The treatment difference may not be estimable.

5. Example
Consider the block design with 6 treatments and 6 blocks of size two. 1 2 2 3 1 3 4 5 5 6 4 6
The treatments differences (1 vs 2, 1 vs 3, 2 vs 3, 4 vs 5, 4 vs 6, 5 vs 6) are estimable.
If one of the treatments is in the group {1,2,3} and the other treatment is in the group {4,5,6}, the treatment difference is not estimable.

6. Definitions
Two treatments i and i* are said to be connected if there is a sequence of treatments i0 = i, i1, i2, … iM = i* such that each successive pair of treatments (ij and ij+1) appear in the same block
In this case the treatment difference is estimable.
An incomplete design is said to be connected if all treatment pairs i and i* are connected.
In this case all treatment differences are estimable.

7. Example
Consider the block design with 5 treatments and 5 blocks of size two. 1 2 2 3 1 3 4 5 1 4
This incomplete block design is connected.
All treatment differences are estimable.
Some treatment differences are estimated with a higher precision than others.

8. Definition
An incomplete design is said to be a Balanced Incomplete Block Design.
if all treatments appear in exactly r blocks.
This ensures that each treatment is estimated with the same precision
The value of l is the same for each treatment pair.
if all treatment pairs i and i* appear together in exactly l blocks.
This ensures that each treatment difference is estimated with the same precision.

9. Some Identities bk = rt r(k-1) = l (t – 1)
Let b = the number of blocks.
t = the number of treatments
k = the block size
r = the number of times a treatment appears in the experiment.
l = the number of times a pair of treatment appears together in the same block
bk = rt
Both sides of this equation are found by counting the total number of experimental units in the experiment.
r(k-1) = l (t – 1)
Both sides of this equation are found by counting the total number of experimental units that appear with a specific treatment in the experiment.

10. BIB Design A Balanced Incomplete Block Design
(b = 15, k = 4, t = 6, r = 10, l = 6)

11. An Example
A food processing company is interested in comparing the taste of six new brands (A, B, C, D, E and F) of cereal.
For this purpose:
subjects will be asked to taste and compare these cereals scoring them on a scale of
For practical reasons it is decided that each subject should be asked to taste and compare at most four of the six cereals.
For this reason it is decided to use b = 15 subjects and a balanced incomplete block design to assess the differences in taste of the six brands of cereal.

12. The design and the data is tabulated below:

13. Analysis for the Incomplete Block Design
Recall that the parameters of the design where b = 15, k = 4, t = 6, r = 10, l = 6
denotes summation over all blocks j containing treatment i.

14. Anova Table for Incomplete Block Designs
Sums of Squares
SS yij2 =
S Bj2/k =
S Qi2 =
Anova Sums of Squares
SStotal = SS yij2 –G2/bk =
SSBlocks = S Bj2/k – G2/bk =
SSTr = (S Qi2 )/(r – 1) =
SSError = SStotal - SSBlocks - SSTr =

15. Anova Table for Incomplete Block Designs