1. Chapter 4 Randomized Blocks, Latin Squares, and Related Designs

2. 4.1 The Randomized Complete Block Design
Nuisance factor: a design factor that probably has an effect on the response, but we are not interested in that factor.
If the nuisance variable is known and controllable, we use blocking
If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance (see Chapter 14) to remove the effect of the nuisance factor from the analysis

3. If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment
Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

4. We wish to determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester
Assignment of the tips to an experimental unit; that is, a test coupon
Structure of a completely randomized experiment
The test coupons are a source of nuisance variability
Alternatively, the experimenter may want to test the tips across coupons of various hardness levels
The need for blocking

5. To conduct this experiment as a RCBD, assign all 4 tips to each coupon
Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips
Variability between blocks can be large, variability within a block should be relatively small
In general, a block is a specific level of the nuisance factor
A complete replicate of the basic experiment is conducted in each block
A block represents a restriction on randomization
All runs within a block are randomized

6. Suppose that we use b = 4 blocks:
Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)

7. Statistical Analysis of the RCBD
Suppose that there are a treatments (factor levels) and b blocks
A statistical model (effects model) for the RCBD is
 is an overall mean, i is the effect of the ith treatment, and j is the effect of the jth block
ij ~ NID(0,2)

8. Means model for the RCBD
The relevant (fixed effects) hypotheses are
An equivalent way for the above hypothesis
Notations:

9. ANOVA partitioning of total variability:

10. SST = SSTreatment + SSBlocks + SSE
Total N = ab observations, SST has N – 1 degrees of freedom.
a treatments and b blocks, SSTreatment and SSBlocks have a – 1 and b – 1 degrees of freedom.
SSE has ab – 1 – (a – 1) – (b – 1) = (a – 1)(b – 1) degrees of freedom.
From Theorem 3.1, SSTreatment /2, SSBlocks / 2 and SSE / 2 are independently chi-square distributions.

11. The expected values of mean squares:
For testing the equality of treatment means,

12. The ANOVA table
Another computing formulas:

13. Example 4.1
4.1.2 Model Adequacy Checking
Residual Analysis
Residual:
Basic residual plots indicate that normality, constant variance assumptions are satisfied
No obvious problems with randomization

16. Can also plot residuals versus the type of tip (residuals by factor) and versus the blocks. Also plot residuals v.s. the fitted values. Figure 4.5 and 4.6 in Page 137
These plots provide more information about the constant variance assumption, possible outliers
4.1.3 Some Other Aspects of the Randomized Complete Block Design
The model for RCBD is complete additive.

17. Interactions?
For example:
The treatments and blocks are random.
Choice of sample size:
Number of blocks , the number of replicates and the number of error degrees of freedom 

18. Estimating miss values:
Approximate analysis: estimate the missing values and then do ANOVA.
Assume the missing value is x. Minimize SSE to find x
Table 4.8
Exact analysis

19. 4.1.4 Estimating Model Parameters and the General Regression Significance Test
The linear statistical model
The normal equations

20. Under the constraints,
the solution is
and the fitted values,
The sum of squares for fitting the full model:
The error sum of squares

21. The sum of squares due to treatments:

22. 4.2 The Latin Square Design
RCBD removes a known and controllable nuisance variable.
Example: the effects of five different formulations of a rocket propellant used in aircrew escape systems on the observed burning rate.
Remove two nuisance factors: batches of raw material and operators
Latin square design: rows and columns are orthogonal to treatments.

23. The Latin square design is used to eliminate two nuisance sources, and allows blocking in two directions (rows and columns)
Usually Latin Square is a p  p squares, and each cell contains one of the p letters that corresponds to the treatments, and each letter occurs once and only once in each row and column.
See Page 145

24. The statistical (effects) model is
yijk is the observation in the ith row and kth column for the jth treatment,  is the overall mean, i is the ith row effect, j is the jth treatment effect, k is the kth column effect and ijk is the random error.
This model is completely additive.
Only two of three subscripts are needed to denote a particular observation.

25. Sum of squares:
SST = SSRows + SSColumns + SSTreatments + SSE
The degrees of freedom:
p2 – 1 = p – 1 + p – 1 + p – 1 + (p – 2)(p – 1)
The appropriate statistic for testing for no differences in treatment means is
ANOVA table (Table 4-10) (Page 146)
Example 4.3

26. The residuals
Table 4.13
If one observation is missing,
Replication of Latin Squares:
Three different cases
See Table 4.14, 4.15 and 4.16
Crossover design: Pages 150 and 151

27. 4.3 The Graeco-Latin Square Design
Two Latin Squares
One is Greek letter and the other is Latin letter.
Two Latin Squares are orthogonal
Table 4.18
Block in three directions
Four factors (row, column, Latin letter and Greek letter)
Each factor has p levels. Total p2 runs

28. The statistical model:
yijkl is the observation in the ith row and lth column for Latin letter j, and Greek letter k
 is the overall mean, i is the ith row effect, j is the effect of Latin letter treatment j , k is the effect of Greek letter treatment k, l is the effect of column l.
ANOVA table (Table 4.19)
Under H0, the testing statistic is Fp-1,(p-3)(p-1) distribution.
Example 4.4

29. 4.4 Balance Incomplete Block Designs
May not run all the treatment combinations in each block.
Randomized incomplete block design (BIBD)
Any two treatments appear together an equal number of times.
There are a treatments and each block can hold exactly k (k < a) treatments.
For example: A chemical process is a function of the type of catalyst employed. See Table 4.22

30. 4.4.1 Statistical Analysis of the BIBD
a treatments and b blocks. Each block contains k treatments, and each treatment occurs r times. There are N = ar = bk total observations. The number of times each pairs of treatments appears in the same block is
The statistical model for the BIBD is

31. The sum of squares

32. The degree of freedom:
Treatments(adjusted): a – 1
Error: N – a – b – 1
The testing statistic for testing equality of the treatment effects:
ANOVA table (see Table 4.23)
Example 4.5

33. 4.4.2 Least Squares Estimation of the Parameters
The least squares normal equations:
Under the constrains,
we have

34. For the treatment effects,