Randomized Complete Block and Repeated Measures (Each Subject Receives Each Treatment) Designs

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