Math 558 Lecture #33
Split Plot Design Linear Model
The linear model for the split-plot design is
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yijk = μ+τi +βj +(τβ)ij +γk +(τγ)ik + (βγ)jk + (τβγ)ijk + εijk
i = 1, 2, ….r
j = 1, 2……a k = 1, 2…….b
Linear Model
τi corresponds to the replicates
βj corresponds to factor A
(τβ)ij corresponds to replicates × factor A. Which is whole plot error.
γk corresponds to subplot factor B
(τγ)ik corresponds to replicates × factor B interaction (βγ)jk corresponds to factor A × factor B interaction (τβγ)ijk corresponds to (replicates ×AB) subplot error
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
This is a typical data layout for a split plot design.
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: Important Features
randomization restriction (different from CRD and RCBD)
split-plot can be considered as two superimposed CR designs or RCB designs
A: whole-plot factor(a levels); B: sub-plot factor (b levels), r replicates (can be considered blocks)
RCB Design for factor A with a (factor A levels) in r number of blocks.
RCB Design for factor B with b(factor B levels) in ra
Statistical Analysis
SS =ab∑(y ̄ −y ̄)2,df=r−1 r ii…..
SS =rb∑(y ̄−y ̄)2,df=a−1 A j.j….
SS =b∑(y ̄−y ̄−y ̄+y ̄)2,df=(r−1)(a−1) rA i,j ij. i.. .j. …
SS =ar∑(y ̄ −y ̄)2df=(b−1) B k..k…
SS =a∑(y ̄−y ̄−y ̄+y ̄)2,df=(r−1)(b−1) rB i,k i.k i.. ..k …
SS =r∑(y ̄−y ̄−y ̄+y ̄)2df=(a−1)(b−1) AB j,k .jk .j. ..k …
SS =∑ (y ̄ −y ̄ −y ̄ −y ̄ +y ̄ +y ̄ +y ̄ +y ̄)2, rAB i,j,k ijk ij. i.k .jk i.. .j. ..k …
df=(r-1)(a-1)(b-1).
Statistical Analysis
Model Term
σ2 + abστ2
σ2+abσ2 +rb∑β2j τβ a−1
σ2 + abσ2 τβ
σ 2 + a σ 2 + r a ∑ γ k2 τγ b−1
σ2 + aσ2 τγ
σ2 + σ2 + r ∑ ∑(βγ)2jk τβγ (a−1)(b−1)
σ2 + σ2 τβγ
not estimable
(τβγ)ijk ε(ijk)h
Statistical Analysis
Note that factor A effects will be estimated using the whole plots and factor B and the A*B interaction effects will be estimated using the split plots.The difference in the size of the whole plot and split plots will effect the precision of the effect estimates. Therefore, in the statistical analysis of split-plot designs, we must take into account the presence of two different sizes of experimental units.
The statistical testing is carried out as follows
The main factor (A) in the whole plot is tested against the whole-plot error
sub-plot treatment (B) is tested against the replicates × sub-plot treatment interaction.
The AB interaction is tested against the subplot error.
Statistical Analysis Hypothesis tests
1. Test for factor A Whole plot analysis:
H0 :βj =0 forallj Ha :βj ̸=0 forsomej
FA = MSA MSEwhole plots
Note that under null hypothesis FA = 1 2. Sub-plot analysis:
Test for AB interaction
H0 :(βγ)jk =0 forallj,k Ha :(βγ)jk ̸=0 forsomej,k
FAB = MSAB MSEABR
Hypotheses tests
3. Sub-plot analysis: Test for factor B
H0 :(γ)k =0 forallk Ha :(γ)jk ̸=0 forsomek
FB = MSB MSEBR
Tensile strength ANOVA
sources of variation Replicates Prep method A WPE(R×A) Temperature B Rep × B AB SPE(Rep × AB) Total
sum of squares 77.5 128.39 36.28 434.08 20.67 75.17 50.83 822.97
Degrees of freedom 2
Mean square 38.78 64.20 9.07 144.69 3.45 12.53 4.24
F P-value 7.08 0.05
41.94 < 0.01 2.96 0.05
Discussion
We have two different error terms for the experiment. We can see that the subplot error (4.24) is less than the whole-plot error (9.07). The reason is that the subplots are generally more homogeneous than the whole plots. Because the subplot treatments are compared with greater precision, it is preferable to assign the treatment we are most interested in to the subplots, if possible.
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