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 Friedmans Test

• When data are non-normal, test is based on ranks
• Procedure to obtain test statistic
• Rank r treatments within each block (1smallest,
  rlargest) adjusting for ties
• Compute rank sums for treatments (Rj ) 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 Tukeys 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