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

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)

Randomized Block Design – Model & Estimates

Analysis of Variance

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  H 0 : The r populations are identical  H A : Differences exist among the r group means

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

Comparing Treatment Effects (All Pairs)

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

Relative Efficiency Measures the ratio of the experimental error variance for the Completely Randomized Design (  r 2 ) to that for the Randomized Block Design (  b 2 ) 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

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

Repeated Measures Design – Model

Repeated Measures Design – ANOVA

Comparing Treatment Effects (All Pairs)

Within-Subject Variance-Covariance Matrix Common Assumptions for the Repeated Measures ANOVA Variances of measurements for each treatment are equal:     r  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