2. Incomplete Block Design & Balanced Incomplete Block Designs

3. What is RCBD ?
Every treatment lies in every block and with one condition of homogeneity a b c d A B C D

4. Concept of Incomplete block design
Now considered that your block is large enough and lost its homogeneity condition and we cannot apply all treatments in each block
In some situations this is not possible because
(physical) block size is too small
too expensive
not advisable (think of rater having to rate 7 champagne brands)

6. Concept of balanced Incomplete block design
We call an incomplete block design balanced (BIBD) if all pairs of treatments occur together in the same block equally often a b c d e f A B C D E F

7. There are two types of analysis in the incomplete block designs
intrablock analysis
interblock analysis
Intrablock analysis:
In intrablock analysis, the treatment effects are estimated after eliminating the block effects and then the analysis and the test of significance of treatment effects are conducted further. If the blocking factor is not marked, then the intrablock analysis is sufficient enough to provide reliable, correct and valid statistical inferences.
Interblock analysis:
There is a possibility that the blocking factor is important and the block totals may carry some important information about the treatment effects. In such situations, one would like to utilize the information on block effects (instead of removing it as in the intrablock analysis) in estimating the treatment effects to conduct the analysis of design. This is achieved through the interblock analysis of an incomplete block design by considering the block effects to be random.

8. The statistical model for the BIBD
Where is the ith observation in the jth block
is the overall mean
is the effect of the ith treatment,
is the effect of the jth block,
is the NID (0, ) random error component.

9. Total sum of square Blocks sum of square Sum of square of treat.
Qi is the adjusted total for the ith treatment, which is computed as

10. with nij = 1 if treatment i appears in block j and nij = 0 otherwise

11. ANOVA Table

12. Example
suppose that a chemical engineer thinks that the time of reaction for a chemical process is a function of the type of catalyst employed. Four catalysts are currently being investigated. The experimental procedure consists of selecting a batch of raw material, loading the pilot plant, applying each catalyst in a separate run of the pilot plant, and observing the reaction time. Because variations in the batches of raw material may affect the performance of the catalysts, the engineer decides to use batches of raw material as blocks. However, each batch is only large enough to permit three catalysts to be run. Therefore, a randomized incomplete block design must be used. The balanced incomplete block design for this experiment

13. Statistical Analysis of the BIBD

14. there are a treatments and b blocks
there are a treatments and b blocks. In addition, we assume that each block contains k treatments, that each treatment occurs r times in the design and that there are
N = ar = bk
total observations. Furthermore, the number of times each pair of treatments appears in the same block is

15. This is a BIBD with a = 4, b = 4, k = 3, r = 3, = 2, and N = 12
This is a BIBD with a = 4, b = 4, k = 3, r = 3, = 2, and N = 12. The analysis of this data is as follows. The total sum of squares is

16. SUM SQUARE OF BLOCKS
SUM SQUARE OF TREATMENT