1. Design Lecture: week3
HSTS212

2. One Factor Analysis

3. ANOVA Table

5. Example

6. Box plot

7. ANOVA

8. Randomized Block Design
HSTS312

9. Randomized Block Design
1. Experimental Units (Subjects) Are Assigned Randomly within Blocks
Blocks are Assumed Homogeneous
2. One Factor or Independent Variable of Interest
2 or More Treatment Levels or Classifications
3. One Blocking Factor
HSTS312

10. 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 to remove the effect of the nuisance factor from the analysis

11. 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

12. 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

13. 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

14. 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)

15. 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)

16. Model

17. Hypothesis

18. Randomized Block Design
Factor Levels:
(Treatments)
A, B, C, D
Experimental Units
 Treatments are randomly assigned within blocks
Block 1 A C D B
Block 2
Block 3 .
Block b
Are the mean training times the same for 3 different methods?
9 subjects
3 methods (factor levels)
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19. Randomized Block F-Test
1. Tests the Equality of 2 or More (p) Population Means
2. Variables
One Nominal Independent Variable
One Nominal Blocking Variable
One Continuous Dependent Variable
Note:
There is one dependent variable in the ANOVA model. MANOVA has more than one dependent variable.
Ask, what are nominal & interval scales?
HSTS312

20. Randomized Block F-Test Assumptions
1. Normality
Probability Distribution of each Block-Treatment combination is Normal
2. Homogeneity of Variance
Probability Distributions of all Block-Treatment combinations have Equal Variances
HSTS312

21. Manual ANOVA

22. Table

23. Example

24. Table

25. 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.

26. Balanced Incomplete Block Design (BIBD)
Notation
a-number of treatments
k-number of treatments/block
r-number of treatment reps
b-number of blocks
N=ar=bk
r(k-1)=l(a-1)

27. 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 pair of treatments appears together in the same block is
The statistical model for the BIBD is

28. Example

29. The sum of squares

30. The testing statistic for testing equality of the treatment effects:
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

31. Table

33. THE LATIN SQUARE DESIGN
used to eliminate two nuisance sources of variability
Latin square, is a square containing p rows and p columns
contains one of the p letters that corresponds to the treatments
each letter occurs once and only once in each row and column

34. 4 x 4
A B D C
B C A D
C D B A
D A C B

35. The model completely additive

37. Example

38. SPSS
A 1 1 8 C
B 1 3 4
D 1 4 6
E 1 5 4
B 2 1 7
E 2 2 2
A 2 3 9
C 2 4 8
D 2 5 2
D 3 1 1
A 3 2 7 C
E 3 4 6
B 3 5 3
C 4 1 7
D 4 2 3
E 4 3 1
B 4 4 6
A 4 5 8
E 5 1 3
B 5 2 8
D 5 3 5 A
C 5 5 8

39. ANOVA TABLE