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Split-Plot Designs
STA 6208 Course Notes
Spring 2009
Classical Example:
Consider an agricultural experiment to determine effects of I (^) Factor A: level of irrigation I (^) Factor B: variety of seed
Levels of B can easily be applied separately to small plots of land. Levels of A cannot (since sprinklers irrigate a large area).
Compromise — create two types of EUs: I (^) whole plots to which levels of A are randomized I (^) split plots (or subplots) that subdivide whole plots, and to which levels of B are randomized, with whole plots as blocks A is the whole-plot factor, and B is the split-plot factor.
1
Another Example: Concrete Pillars
I (^) Factor A: composition of mix I (^) Factor B: reinforcement technique
Composition is a property of a batch of concrete mix, but reinforcement can be applied separately to each pillar —
I (^) Whole plots: batches of mix I (^) Split plots: individual pillars (several from each batch)
Notation:
a = # levels of A ( ≥ 2) b = # levels of B ( ≥ 2) n = # whole plots per level of A ( ≥ 2)
We will assume that
split plots per whole plot = b
Randomization
- Randomize levels of A to whole plots (as in a CRD, or a RCB design, or ...)
- Independently, randomize levels of B to split plots, using whole plots as blocks (as in a RCB design)
Example: a = 2, b = 3, n = 4, CRD on A
A 1 A 2 A 2 A 2 A 1 A 2 A 1 A 1
B 2
B 1
B 3
B 2
B 1
B 3
B 1
B 2
B 3
B 2
B 1
B 3
B 2
B 1
B 3
B 1
B 3
B 2
B 3
B 2
B 1
B 2
B 3
B 1
Note: The a levels of factor A are assigned to the an whole plots as in a balanced CRD.
Example: a = 2, b = 3, n = 4, RCB design on A
A 1 A 2 A 2 A 1 A 1 A 2 A 1 A 2
B 3
B 2
B 1
B 1
B 3
B 2
B 3
B 1
B 2
B 2
B 1
B 3
B 3
B 1
B 2
B 2
B 1
B 3
B 1
B 2
B 3
B 3
B 1
B 2
Block 1 Block 2 Block 3 Block 4
Note: When a RCB design is used for factor A, there are n blocks, each containing a whole plots.
Whatever design is used for A, factors A and B are always crossed. For the analysis, we will also assume that they are both fixed. (But A or B or both could be random.)
4
Analysis of a Split Plot Design
CRD on Whole-Plot Factor
Let yijk be the response from the split plot that receives level j of B, within the kth^ whole plot that receives level i of A. Model equation: yijk = μ + αi + ηk(i) + βj + αβij + ijk
i = 1,... , a j = 1,... , b k = 1,... , n
μ = overall mean
αi , βj = main effects αβij = interaction effect (with the usual sum-to-zero constraints)
ηk(i) = whole-plot error ∼ i.i.d. N(0, σ η^2 ) ijk = split-plot error ∼ i.i.d. N(0, σ^2 )
indep.
5
Note:
I (^) Responses from the same whole plot are correlated. I (^) Whole-plot effects do not “interact” with factor B — whole plots are blocks for B
Hasse diagram:
M (^11)
vvv
vvv
vvv GGG
GGG
GGG
A aa− 1
SSS S
SSSSS SSSS SSSS
S B^
b b− 1
whole-plot error
GGG GG
AB ab (a−1)(b−1)
xxx
xxx
xx
split-plot error +^3 (SPE) abna(b−1)(n−1)
ANOVA and Expected Mean Squares:
Source df MS E(MS )
A a − 1 MSA σ^2 + b σ^2 η + nb Q(α) WPE a (n − 1) MSWPE σ^2 + b σ^2 η
B b − 1 MSB σ^2 + na Q(β) AB (a − 1)(b − 1) MSAB σ^2 + n Q(αβ) SPE a (b − 1)(n − 1) MSSPE σ^2
Degrees of freedom are the same as in the Hasse diagram, and formulas for sums of squares can be surmised from the corresponding positions of their terms in the Hasse diagram.
[
SAS ©R Example (Oats Variety and Manuring)
]
Note: In SAS ©R, the whole-plot error term is designated as if it were a “Block × A interaction” — a description that is consistent with the position it occupies in the Hasse diagram.
(If a CRD on factor A were used instead, the whole-plot error term would be designated as “whole plot nested within A.”)
12
Contrast Inference
Assuming no AB interaction, consider main effect contrasts:
Factor A Contrasts
i
wi αi , where
i
wi = 0
Unbiased estimate:
i wi^ y^ i••
Can show V
i wi^ y^ i••
σ^2 + b σ^2 η
E(MSWPE )
i w^ 2 i /nb.
A (1 − α)100% CI:
∑
i
wi y (^) i•• ± tα/ 2 , df (^) WPE
MSWPE
i
w (^) i^2
nb
13
Factor B Contrasts
j
wj βj , where
j
wj = 0
Unbiased estimate:
j wj^ y^ • j•
Can show V
j wj^ y^ • j•
= σ^2
j w^
2 j /na.
A (1 − α)100% CI:
∑
j
wj y (^) • j• ± tα/ 2 , df (^) SPE
MSSPE
j
w (^) j^2
na
If AB interaction is present, must consider more general contrasts. Letting
μij = μ + αi + βj + αβij
the general form of a contrast is ∑
i
j
wij μij , where
i
j
wij = 0,
which has unbiased estimate ∑
i
j
wij y (^) ij•
(when the whole-plot factor A has a CRD or a RCB design). The variance of this estimate is generally complicated.
However, there are relatively uncomplicated formulas for inference about the simple effects:
Simple Effects of B (within level i of A)
μij − μij′ , where j 6 = j′
Unbiased estimate: y (^) ij• − y (^) ij′•
Can show V(y (^) ij• − y (^) ij′•) = 2 σ^2 /n.
A (1 − α)100% CI:
y (^) ij• − y (^) ij′• ± tα/ 2 , df (^) SPE
2 MSSPE / n
16
Simple Effects of A (within level j of B)
μij − μi′j , where i 6 = i′
The unbiased estimate y (^) ij• − y (^) i′j• has
V(y (^) ij• − y (^) i′j•) = 2
(b − 1) σ^2 ︸ ︷︷ ︸ (b−1) E(MSSPE )
- σ^2 + b σ^2 η ︸ ︷︷ ︸ E(MSWPE )
nb.
A (Satterthwaite) approximate (1 − α)100% CI:
y (^) ij• − y (^) i′j• ± tα/ 2 , ν
(b − 1) MSSPE + MSWPE
nb
where ν =
(b − 1) MSSPE + MSWPE
(b − 1)^2 MS^2 SPE df (^) SPE
MS^2 WPE
df (^) WPE 17
Generalizations
Many extensions of the basic design and analysis are possible:
I (^) Can have multiple whole-plot or split-plot factors (with interactions among factors, but not between factors and blocking criteria) — see textbook
I (^) Can have more than one level of plot splitting (e.g. split-split plots — Sec. 16.4)
I (^) Can incorporate whole-plot or split-plot covariates (Sec. 17.4)
