Transcript Randomized Complete Block Design (RCBD)

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

RCBD

The model and hypotheses


Yij     i   j  ij   ij ,
2
ij   ij iid N (0,  )
H o : i  0
RCBD

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
RCBD

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
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
x
ay'i*. by'. j*  y'..
a  1b  1
Imputation
The error df should be reduced by one,
since x was estimated
 SAS can compute the F statistic, but the pvalue will have to be computed separately
 The method is efficient only when a couple
cells are missing

Imputation
The usual Type III analysis is available,
but be careful of interpretation
 Little and Rubin use MLE and simulationbased approaches
 PROC MI in SAS v9 implements Little and
Rubin approaches

Power analysis

Power calculations change little
– b replaces n in formulas
For Ho : L  0, use  
bL2
 2  ci2
For H o :   0, use  
b 
– The error df is (a-1)(b-1)
2
2
i