This document discusses the Randomized Complete Block Design (RCBD). Key points: - RCBD accounts for variation among experimental units (EU) by blocking them into groups (blocks) that are presumed to be homogenous. Treatments are randomly assigned within each block. - Blocks can be modeled as fixed or random effects, depending on whether the blocks represent all possible blocks or a sample. - There is debate around testing fixed block effects due to approximate F tests. Additivity of block and treatment effects includes lack-of-fit as error. - Missing data results in loss of orthogonality, but a single missing value can be imputed using profile least squares. Power calculations change little from

1. Randomized Complete Block
Design (RCBD)
 Block--a nuisance factor included in an
experiment to account for variation among
eu’s
 Presumably, eu’s are homogenous within
a block
 Treatments are randomly assigned to eu’s
within each block

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 The model and hypotheses
RCBD

3. Blocks in RCBDs
 Blocks can be modeled as both fixed and
random effects (Soil example)
– Block: Soil type (fixed or random?)
– Treatment: Nitrogen x Watering Regimen
– Response: IR/R reflection

4. RCBD Discussion
 There is some controversy as to whether
fixed block effects should be tested
– F test is considered at best approximate
 Additivity of the block and factor effects
– Error includes lack-of-fit
– Practical considerations
 Both block and factor could have a
factorial structure

5. Missing values in RCBD’s
 Missing values result in a loss of
orthogonality (generally)
 A single missing value can be imputed
– The missing cell (yi*j*=x) can be estimated by
profile least squares
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6. Imputation
 The error df should be reduced by one,
since x was estimated
 SAS can compute the F statistic, but the p-
value will have to be computed separately
 The method is efficient only when a couple
cells are missing

7. Alternate Imputation
approaches
 The usual Type III analysis is available,
but be careful of interpretation
 Little and Rubin use MLE and simulation-
based approaches
 PROC MI in SAS v9 implements Little and
Rubin approaches

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 Power calculations change little
– b replaces n in formulas
– The error df is (a-1)(b-1)
Power analysis