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Randomized complete block design
Recall 1-way AOV: ìcompletely randomized design with single factorî. There are (^8) Texperimental units (book is now using notation R ), randomly assigned among> treatments; objective of study is to obtain inferences about the treatment means. (^) " ,. (^) # , ..., .>. Reparameterization will be helpful for extending the AOV model to more complicated designs:
! (^) " œ. (^) "'. ! (^) # œ. (^) #'. ã ! (^) > œ. (^) >'.
The! 3 's are the treatment effects ; here .is the overall mean (the mean of the. 3 's) and! (^) " ! (^) # â * !> œ0.
The statistical model becomes
] 34 œ. *! 3 *% 34
where % 34 μ normal 0a ß 5 #b, or equivalently,
] 34 μ normal ˆ^. *! 3 , 5 #‰
The RCBD design incorporates an additional categorical variable, called a blocking variable or block , to account for possible heterogeneity in experimental circumstances. For instance, a typical block in an agricultural experiment is a fieldó fields differ substantially in soil quality, etc., and the same experimental treatment might produce different means in different fields.
Formally, the design is as follows: within each of ,blocks, assign 1 experimental unit at random to each of > treatments. Thus, all treatments appear within each block, and each block-treatment combination receives 1 experimental unit, which produces the observed response C 34 :
block treatment 1 2 , 1 C (^) "" C (^) "# â C (^) ", 'C"† 2 C (^) #" C (^) ## â C (^) #, 'C#† ã ã ã
C (^) >" C (^) ># â C (^) >, 'C>†
mean 'C (^) †" 'C (^) †# 'C†,
The statistical model is
] 34 œ. *! 3 * " 4 *% 34
where D! 3 3 œ 0, D" 4 4 œ 0, and % 34 μ normal 0,a 5 #b. Here
" 4 is the effect of the 4 th block on the mean of ] 34.
The AOV table for the RCBD design:
Source SS df MS J
block SS blocka b , ' " SS block^ a,'^1 b^ MS blockMS erroraa^ bb
trt SS trta b > ' " SS trt>'^ a^1 b^ MS errorMS trta a^ bb
error SS errora b a , ' "ba > ' "b (^) a ,'SS error 1 a^ ba >'b 1 b
total SS totala b ,> ' 1
Hypothesis tests
Block effect
H :! "" œ "# œ â œ ", œ 0 H :a Á
Test statistic is 0 œ MS blockMS error^ aa^ bb using
F a , ' ", a , ' "ba > ' "bbdistribution.
Treatment effect
H :! !" œ !# œ â œ !>œ 0 H :a Á
Test statistic is 0 œ (^) MS errorMS trta a^ bb using F a> ' " , a, ' " ba> ' " bb
distribution.
RCBD remarks
- Missing observation(s) produces unbalanced design; block and treatment are no longer orthogonal (& hand- calculating formulas are no longer available). Parameters can still be estimated, and effects tested, in PROC GLM. Use type III sums of squares:
MODEL Y=BLOCK TRT / SS3;
Type III SS tests the effect last, after the other variables are entered in the model.
- If blocks have been randomly chosed from some defined population of blocks (block might be a person in a study, say, or a plot of land picked from farms in southern Idaho), one might consider a model in which
" 4 μ normal 0,ˆ^5 "#‰
This is called a random effects model (or, more precisely, a mixed effects model) & the experiment is called a random blocks experiment (rather than randomized block). We will look at a few of these types of models toward the end of the course. The ordinary RCBD model is a fixed effects model.
