1. Randomized Complete Block and Repeated Measures (Each Subject Receives Each Treatment) Designs
KNNL – Chapters 21,27.1-2

2. Block Designs
Prior to treatment assignment to experimental units, we may have information on unit characteristics
When possible, we will create “blocks” of homogeneous units, based on the characteristics
Within each block, we randomize the treatments to the experimental units
Complete Block Designs have block size = number of treatments (or an integer multiple)
Block Designs allow the removal of block to block variation, for more powerful tests
When Subjects are blocking variable, use Repeated Measures Designs, with adjustments made to Block Analysis (in many cases, the analysis is done the same)

3. Randomized Block Design – Model & Estimates

4. Analysis of Variance

5. RBD -- Non-Normal Data Friedman’s Test
When data are non-normal, test is based on ranks
Procedure to obtain test statistic:
Rank r treatments within each block (1=smallest, r=largest) adjusting for ties
Compute rank sums for treatments (R•j ) across blocks
H0: The r populations are identical
HA: Differences exist among the r group means

6. Checking Model Assumptions
Strip plots of residuals versus blocks (equal variance among blocks – all blocks received all treatments)
Plots of residuals versus fitted values (and treatments – equal variances)
Plot of residuals versus time order (in many lab experiments, blocks are days – independent errors)
Block-treatment interactions – Tukey’s test for additivity

7. Comparing Treatment Effects (All Pairs)

8. Extensions of RCBD Can have more than one blocking variable
Gender/Age among Human Subjects
Region/Size among cities
Observer/Day among Reviewers (Note: Observers are really subjects, same individual)
Can have more than one replicate per block, but prefer to have equal treatment exposure per block
Can have factorial structures run in blocks (usual breakdown of treatment SS). Problems with many treatments (non-homogeneous blocks).
Main Effects
Interaction Effects

9. Relative Efficiency
Measures the ratio of the experimental error variance for the Completely Randomized Design (sr2) to that for the Randomized Block Design (sb2)
Computed from the Mean Squares for Blocks and Error
Represents how many observations would be needed per treatment in CRD to have comparable precision in estimating means (standard errors) as the RBD

10. Repeated Measures Design
Subjects (people, cities, supermarkets, etc) are selected at random, and assigned to receive each treatment (in random order)
Unlike block effects, which were treated as fixed, subject effects are random variables (since the subjects were selected at random)
Measurements on subjects are correlated, however conditional on a subject being selected, they are independent (no carry-over effects or order effects)
The analysis is conducted in a similar manner to Randomized Complete Block Design

11. Repeated Measures Design – Model

12. Repeated Measures Design – ANOVA

13. Comparing Treatment Effects (All Pairs)

14. Within-Subject Variance-Covariance Matrix
Common Assumptions for the Repeated Measures ANOVA
Variances of measurements for each treatment are equal: s12 = ... = sr2
Covariances of measurements for each pair treatments are the same
Note: These will not hold exactly for sample data, should give a feel if reasonable