1. Lecture 14 Analysis of Variance Experimental Designs (Chapter 15.3)
Randomized Block (Two-Way) Analysis of Variance
Announcement: Extra office hours, today after class and Monday, 9:00-10:20

2. 15.3 Analysis of Variance Experimental Designs
Several elements may distinguish between one experimental design and another:
The number of factors (1-way, 2-way, 3-way,… ANOVA).
The number of factor levels.
Independent samples vs. randomized blocks
Fixed vs. random effects
These concepts will be explained in this lecture.

3. Number of factors, levels
Example: 15.1, modified
Methods of marketing: price, convenience, quality => first factor with 3 levels
Medium: advertise on TV vs. in newspapers => second factor with 2 levels
This is a factorial experiment with two “crossed factors” if all 6 possibilities are sampled or experimented with.
It will be analyzed with a “2-way ANOVA”. (The book got this term wrong.)

4. One - way ANOVA Single factor Two - way ANOVA Two factors
Response
Response
Treatment 3 (level 1)
Treatment 2 (level 2)
Treatment 1 (level 3)
Level 3
Level2
Factor A
Level 1
Level2
Level 1
Factor B

5. Randomized blocks
This is something between 1-way and 2-way ANOVA: a generalization of matched pairs when there are more than 2 levels.
Groups of matched observations are collected in blocks, in order to remove the effects of unwanted variability. => We improve the chances of detecting the variability of interest.
Blocks are like a second factor => 2-way ANOVA is used for analysis
Ideally, assignment to levels within blocks is randomized, to permit causal inference.

6. Randomized blocks (cont.)
Example: expand 13.03
Starting salaries of marketing and finance MBAs: add accounting MBAs to the investigation.
If 3 independent samples of each specialty are collected (samples possibly of different sizes), we have a 1-way ANOVA situation with 3 levels.
If GPA brackets are formed, and if one samples MBAs per bracket, one from each specialty, then one has a blocked design. (Note: the 3 samples will be of equal size due to blocking.)
Randomization is not possible here: one can’t assign each student to a specialty
=> No causal inference.

7. Models of fixed and random effects
Fixed effects
If all possible levels of a factor are included in our analysis or the levels are chosen in a nonrandom way, we have a fixed effect ANOVA.
The conclusion of a fixed effect ANOVA applies only to the levels studied.
Random effects
If the levels included in our analysis represent a random sample of all the possible levels, we have a random-effect ANOVA.
The conclusion of the random-effect ANOVA applies to all the levels (not only those studied).

8. Models of fixed and random effects (cont.)
Fixed and random effects - examples
Fixed effects - The advertisement Example (15.1): All the levels of the marketing strategies considered were included. Inferences don’t apply to other possible strategies such as emphasizing nutritional value.
Random effects - To determine if there is a difference in the production rate of 50 machines in a large factory, four machines are randomly selected and the number of units each produces per day for 10 days is recorded.

9. 15.4 Randomized Blocks Analysis of Variance
The purpose of designing a randomized block experiment is to reduce the within-treatments variation, thus increasing the relative amount of between treatment variation.
This helps in detecting differences between the treatment means more easily.

10. Examples of Randomized Block Designs
Factor
Response
Units
Block
Varieties of Corn
Yield
Plots of Land
Adjoining plots
Blood pressure Drugs
Blood pressure
Patient
Same age, sex, overall condition
Number of breaks
Worker productivity
Worker
Shifts

11. Randomized Blocks
Block all the observations with some commonality across treatments
Treatment 4
Treatment 3
Treatment 2
Treatment 1
Block3
Block2
Block 1

12. Randomized Blocks
Block all the observations with some commonality across treatments

13. Partitioning the total variability
The sum of square total is partitioned into three sources of variation
Treatments
Blocks
Within samples (Error)
Recall. For the independent samples design we have:
SS(Total) = SST + SSE
SS(Total) = SST + SSB + SSE
Sum of square for treatments
Sum of square for blocks
Sum of square for error

14. Sums of Squares Decomposition
= observation in ith block, jth treatment
= mean of ith block
= mean of jth treatment

15. Calculating the sums of squares
Formulas for the calculation of the sums of squares
SSB=
SST =

16. Calculating the sums of squares
Formulas for the calculation of the sums of squares
SSB=
SST =

17. Mean Squares
To perform hypothesis tests for treatments and blocks we need
Mean square for treatments
Mean square for blocks
Mean square for error

18. Test statistics for the randomized block design ANOVA
Test statistic for treatments
Test statistic for blocks
df-T: k df-B: b df-E: n-k-b+1

19. The F test rejection regions
Testing the mean responses for treatments
F > Fa,k-1,n-k-b+1
Testing the mean response for blocks
F> Fa,b-1,n-k-b+1

20. Randomized Blocks ANOVA - Example
Are there differences in the effectiveness of cholesterol reduction drugs?
To answer this question the following experiment was organized:
25 groups of men with high cholesterol were matched by age and weight. Each group consisted of 4 men.
Each person in a group received a different drug.
The cholesterol level reduction in two months was recorded.
Can we infer from the data in Xm15-02 that there are differences in mean cholesterol reduction among the four drugs?

21. Randomized Blocks ANOVA - Example
Solution
Each drug can be considered a treatment.
Each 4 records (per group) can be blocked, because they are matched by age and weight.
This procedure eliminates the variability in cholesterol reduction related to different combinations of age and weight.
This helps detect differences in the mean cholesterol reduction attributed to the different drugs.

22. Randomized Blocks ANOVA - Example
Conclusion: At 5% significance level there is sufficient evidence
to infer that the mean “cholesterol reduction” gained by at least
two drugs are different.
Treatments
Blocks
b-1
K-1
MST / MSE
MSB / MSE

23. Required Conditions for Test
The sample from each block in each population is a simple random sample from the block in that population
There are conditions that are similar to the populations being normal and having equal variance but they are more complicated (the book’s description is wrong). We shall discuss this more when we cover regression. For now, you should just look for outliers.

24. Criteria for Blocking
Goal is to find criteria for blocking that significantly affect the response variable
Effect of teaching methods on student test scores.
Good blocking variable: GPA
Bad blocking variable: Hair color
Ideal design of experiment is often to make each subject a block and apply the entire set of treatments to each subject (e.g., give different drugs to each subject) but not always physically possible.

25. Test of whether blocking is effective
We can test for whether blocking is effective by testing whether the means of different blocks are the same.
We now consider the blocks to be “treatments” and look at
Under the null hypothesis that the mean in each block is the same, F has an F distribution with (b-1,n-k-b+1) dof

26. Practice Problems
15.38, 15.40