Math 558 Lecture #34
Split plot design Another model
Another commonly used model for the split-plot design is
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yijk =μ+τi+βj+(τβ)ij+γk+(βγ)jk+εijk i = 1, 2, ….r
j = 1, 2……a k = 1, 2…….b
In this model, (τβ)ij is still the whole-plot error, but (τγ)ik that corresponds to replicates × factor B interaction and (τβγ)ijk corresponds to replicates ×AB interaction have been pooled with εijk to form the subplot error. If we denote the variance of the subplot error term εijk by σ2 and make the same assumptions as for the previous model the expected mean squares are given below
Statistical Analysis
Model Term τi (replicates)
βj(A) (τβ)ij
γ k ( B ) βγjk(AB) εijk
σε2 + abστ2
σε2+bσ2 +rb∑β2j τβ a−1
σε2 + bσ2 (Whole plot error) τβ
σ ε2 + r a ∑ γ k2 ab−1
σε2+ r ∑ ∑(βγ)2jk (a−1)(b−1)
(Subplot error)
Note that now both the subplot treatment (B) and the AB interaction are tested against the subplot error mean square. in order to use this model we make the assumption that the interactions of replicates × B and replicates × AB are negligible.
Paper tensile strength experiment whole-plot factor: preparation method sub-plot factor: cooking temperature
prep rep 1 rep 2 rep 3
method 123123123
Temperature oF 200
30 34 29 35 41 26 37 38 33 36 42 36
28 31 31 32 36 30 40 42 32 41 40 40
31 35 32 37 40 34 41 39 39 40 44 45
Split plot designs in R Tensile strength example
TSData <- data.frame(list(
observation = 1:36,
replicate = rep(c("I","II","III"),each =12),
A = rep(rep(c("A1","A2","A3"),each = 4),3),
B = rep(c("B1","B2","B3","B4"),times = 9 ),
yield = c(30,35,37,36,34,41,38, 42,29,26,33,36,28,32,40,41, 31,36,42,40, 31,30,32,40,31,37,41,40,35,40,39,44,32,34,39,45) ))
observation replicate A B yield 1 IA1B130 2 IA1B235 3 IA1B337 4 IA1B436 5 IA2B134 6 IA2B241 7 IA2B338 8 IA2B442 9 IA3B129
10 IA3B226 11 IA3B333 12 IA3B436 13 IIA1B128 14 IIA1B232 15 IIA1B340 16 IIA1B441 17 IIA2B131 18 IIA2B236
observation replicate A B yield 19 IIA2B342 20 IIA2B440 21 IIA3B131 22 IIA3B230 23 IIA3B332 24 IIA3B440 25 IIIA1B131 26 IIIA1B237 27 IIIA1B341 28 IIIA1B440 29 IIIA2B135 30 IIIA2B240 31 IIIA2B339 32 IIIA2B444 33 IIIA3B132 34 IIIA3B234 35 IIIA3B339 36 IIIA3B445
Wrong analysis
If you don’t take into account the special design structure and fit the following model in R
aov.wrong < −aov(yield ∼ replicate + A ∗ B, data = TSData) You will get the misleading results given below summary(aov.wrong)
Df replicate 2 A 2 B 3 A:B 6
Residuals 22
Sum Sq 77.6 128.4 434.1 75.2 107.8
Mean Sq 38.78 64.19 144.69 12.53 4.90
F value 7.915 13.104 29.536 2.557
Pr(>F) 0.002572 ** 0.000179 *** 6.74e-08 *** 0.049434 *
Correct analysis taking replicates as blocks
library(agricolae)
aov.right < −aov(yield ∼ replicate + A ∗ B + Error(replicate : A), data = TSData)
replicate A Residuals
B A:B Residuals
Df Sum Sq 2 77.56 2 128.39 4 36.28 Df Sum Sq 3 434.1 6 75.2
Mean Sq 38.78 64.19 9.07 Mean Sq 144.69 12.53 3.97
F value 4.276 7.078
F value 36.427 3.154
0.1016 0.0485 *
Pr(>F) 7.45e-08 *** 0.0271 *
Correct analysis without taking replicates as blocks
aov.right1<- aov(yield A*B+Error(replicate:A), data =TSData)
Df A 2 Residuals 6 B 3 A:B 6
Residuals 18
Sum Sq 128.4 113.8 434.1 75.2 71.5
Mean Sq 64.19 18.97 144.69 12.53 3.97
F value 3.384
36.427 3.154
Pr(>F) 0.104
7.45e-08 *** 0.0271 *
Using sp.plot
TSData$A = as.factor(TSData$A)
TSData$B = as.factor(TSData$B)
attach(TSData)
modelSP<- sp.plot(block = replicate, pplot = A, splot = B,Y = yield)
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