STAT 512
Factorial Treatments, Split-Plot Designs
FACTORIAL TREATMENT STRUCTURE
• A treatment is defined by a combination of factors, each set to a
specific level.
• Example:
– Response: torque required to tighten nut
– Treatments: mechanical conditions, determined by:
Factor 1: type of plating (3 kinds)
Factor 2: type of metal (2 kinds)
– 2 × 3 = 6 treatments possible
Note: In general, a design may or may not contain all of them, but
for now we’ll just talk about those that do.
• Or more generally:
– f factors
– li levels for ith factor
–
Q
l
i i
treatments possible
1
STAT 512
Factorial Treatments, Split-Plot Designs
2
• Two Factors: A 2-dimensional table of treatments
factor A
level 1
factor B
level 2
...
level 1
...
level 2
...
...
...
...
level b
...
level a
...
...
• Model: A 2-factor CRD with r units/treatment:
– yijt = µij + ijt
or
α + τij + ijt
i = 1...a, j = 1...b, t = 1...r
– Could analyze exactly as before, ignoring factors, with ab − 1
d.f. for treatments
For the “nuts” example, d.f. = 5
STAT 512
Factorial Treatments, Split-Plot Designs
3
Suppose the relationship between µ’s or τ ’s is like:
µ21
τ21
µ11
µ
τ11
µ22
µ31
or
τ
τ31
µ12
τ12
µ32
A1
τ22
A2
A3
τ32
B1
B2
STAT 512
Factorial Treatments, Split-Plot Designs
• Parallel line segments
• Changing from one plating to another results in the same change
of response mean, regardless of the metal
• Changing from one metal to another results in the same change of
response mean, regardless of the plating
• e.g., the effects associated with the two factors are additive
• e.g., picking the treatment that results in the largest/smallest
mean response is the same as picking the largest/smallest level
effect for each factor (separately).
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STAT 512
Factorial Treatments, Split-Plot Designs
µij = µ + αi + βj + (αβ)ij
• µ is the new notation for the “intercept”
• corresponds to “dotted” parameterization in Chapter 9
• αi = additive effect of the ith level of factor A
• βj = additive effect of the jth level of factor B
• (αβ)ij = non-additive (synergistic, combination-specific) effect of
the (i, j) factor combination ... exactly zero if previous plotted
parallel-line patterns hold exactly
• If we knew this sort of factorial simplicity held for our system, we
could fit a model with fewer parameters (i.e. omit (αβ)’s)
• How many:
– parameters are in the full or main-effects model?
– linearly non-redundant parameters ...?
5
STAT 512
Factorial Treatments, Split-Plot Designs
6
Organization of a model matrix:
µ
1
1
...
1
1
1
...
1
...
...
1
1
...
1
1
1
α
1
1
...
1
...
...
...
...
...
1
1
...
1
...
...
...
β
...
...
...
...
...
...
...
...
...
...
...
...
...
...
a
a−1
a−1
1
...
...
1
...
1
...
...
...
...
1
1
...
1
...
...
1
...
1
...
...
...
1
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
b
b−1
b−1
b−1
...
1
...
1
...
...
...
1
(αβ)
1
...
1
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
1
ab
ab − 1
ab − 1 − (a − 1)
ab − 1 − (a − 1) − (b − 1)
(a − 1)(b − 1)
parameters
after µ
after µ, α
after µ, α, β
lin’ly indep.
STAT 512
Factorial Treatments, Split-Plot Designs
Alternatively, we can artificially specify a unique solution to the normal
equations through constraints on “estimates” of non-estimable
functions (as is often implemented in software):
• a − (a − 1) = 1 constraint on α̂’s
P
– e.g.
i α̂i = 0
• b − (b − 1) = 1 constraint on β̂’s
P
– e.g.
j β̂j = 0
d
• ab − (a − 1)(b − 1) = a + b − 1 constraints on (αβ)’s
P d
–
(αβ) = 0 ← b
i
ij
P d
–
j (αβ)ij = 0 ← a
– one of these constraints is redundant, for example:
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STAT 512
Factorial Treatments, Split-Plot Designs
sum = 0
sum = 0
sum = 0
↓
↓
↓
d
(αβ)
1,1
d
(αβ)
1,2
d
(αβ)
1,3
d
(αβ)
1,4
← sum = 0
d
(αβ)
2,1
d
(αβ)
2,2
d
(αβ)
2,3
d
(αβ)
2,4
← sum = 0
d
(αβ)
3,1
d
(αβ)
3,2
d
(αβ)
3,3
d
(αβ)
3,4
← sum = 0
↓
sum = 0
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STAT 512
Factorial Treatments, Split-Plot Designs
CRD
• r units assigned to each treatment
• yijt = µ + αi + βj + (αβ)ij + ijt
source
df
total
abr
µ
1
corrected total
abr − 1
α
a−1
β
b−1
(αβ)
(a − 1)(b − 1)
resid(“pure error”)
ab(r − 1)
sum of squares
P
2
ijt yijt
difference
P
2
ijt (yijt − ȳ... )
P
2
i br(ȳi.. − ȳ... )
P
2
ar(ȳ
−
ȳ
)
.j.
...
j
P
2
r(ȳ
−
ȳ
−
ȳ
+
ȳ
)
ij.
i..
.j.
...
ij
P
2
ijt (yijt − ȳij. )
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STAT 512
Factorial Treatments, Split-Plot Designs
Precision and Power
• α:
0 α=
– cd
P
i ci ȳi.. ,
0 α) =
V ar(cd
σ2
br
2
i ci
P
P
– estimable linear combinations are such that i ci = 0
P
– Hyp0 : α1 = ... = αa ,
Q(α) = i br(αi − ᾱ. )2
• β: likewise using β, sums over j, and factor of ar
• (αβ): likewise using (αβ), sums over i and j, and factor of r
P
– estimable linear combinations are such that i cij = 0 for all
P
j, and j cij = 0 for all i.
10
STAT 512
Factorial Treatments, Split-Plot Designs
CBD
• a single copy of all ab treatments in each block
• ymij = µ + θm + αi + βj + (αβ)ij + mij
– θ is block parameter, m = 1...r
• one observation to each (treatment,block) combination
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STAT 512
Factorial Treatments, Split-Plot Designs
source
df
sum of squares
P
2
mij ymij
total
rab
µ
1
c.t.
rab − 1
θ
r−1
α
a−1
β
b−1
(αβ)
(a − 1)(b − 1)
difference
P
2
mij (ymij − ȳ... )
P
2
m ab(ȳm.. − ȳ... )
P
2
i rb(ȳ.i. − ȳ... )
P
2
ra(ȳ
−
ȳ
)
..j
...
j
P
2
r(ȳ
−
ȳ
−
ȳ
+
ȳ
)
.ij
.i.
..j
...
ij
resid(θ-inter’s)
(r − 1)(ab − 1)
difference
12
STAT 512
Factorial Treatments, Split-Plot Designs
13
Comparison of d.f. available for estimating σ 2 :
no assumption
about model:
“pure error”d.f.
assume
additive blocks
assume
additive blocks and
additive factors
σ
Q, variance functions
CRD
RCBD
ab(r − 1)
0
ab(r − 1)
(ab − 1)(r − 1)
ab(r − 1) + (a − 1)(b − 1)
larger?
same
(ab − 1)(r − 1) + (a − 1)(b − 1)
smaller?
same
STAT 512
Factorial Treatments, Split-Plot Designs
SMALLER BLOCKS
• Often necessary because the number of treatments is large
• Suppose a = b = block size. Here’s one way – Often undesirable; why?
(1,1)
(2,1)
(3,1)
(1,2)
(2,2)
(3,2)
(1,3)
(2,3)
(3,3)
• Another idea: Each level of each factor appears once in each block
(1,1)
(2,2)
(3,3)
(1,2)
(2,3)
(3,1)
(1,3)
(2,1)
(3,2)
• Important: BOTH designs confound/confuse “blocks” with 2 of the 8
d.f. representing “treatments”, but the second avoids confounding
blocks with main effects.
• Where have you seen second design pattern before?
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STAT 512
Factorial Treatments, Split-Plot Designs
i=1
i=2
i=3
block 1
j=1
j=2
j=3
block 2
j=2
j=3
j=1
block 3
j=3
j=1
j=2
• Information “lost” to blocks is associated with interactions, as
with LSD
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STAT 512
Factorial Treatments, Split-Plot Designs
• ANOVA for the blocked design above:
source
one rep
r reps
df
df
sum of squares
reps
r-1
blocks
2
2r
treatments
6
6
4
4
main effects
A
2
B
2
P
b(ȳi. − ȳ.. )2
i
P
2
j
a(ȳ.j − ȳ.. )
2
2
2
2
resid
diff
diff
corrected total
8
9r-1
some interactions
sum of squares
P
rb(ȳ.i. − ȳ... )2
i
P
2
j
ra(ȳ..j − ȳ... )
• Also, BIBD’s can be used with factorial treatments ... Section 9.6
– All treatment contrasts with the same c0 c are estimated with
equal BIBD-suboptimal precision ... often NOT desirable with
factorial experiments!
16
STAT 512
Factorial Treatments, Split-Plot Designs
SPLIT-PLOT DESIGNS
• Example: Animal nutrition study
– Response: characteristics of circulating blood in 5-week-old
infants
– Factor A: level of iron in pre-natal (dam) diet (a levels)
– Factor B: level of iron in infant diet (b levels)
• CRD: Randomly select abr dams, each with one infant
– VERY expensive
• How can experimental material be blocked and/or fewer dams be
required?
– e.g. several infants/dam
– a problem we’ll have to face: groups of related units (infants)
are all assigned the same level of one of the treatment factors
(dam diet) ... same problem as in the first design on slide 14.
17
STAT 512
Factorial Treatments, Split-Plot Designs
Other examples:
• Agriculture:
– “plots” = one factor (e.g. irrigation)
– “splits” another (e.g. fertilizer)
• Integrated Circuits:
– “wafer”
– “sub-wafer”
• Chemical processes:
– “Hard-to-change”(e.g. catalyst)
– “Easy-to-change” (e.g. reaction time)
18
STAT 512
Factorial Treatments, Split-Plot Designs
19
Back to animal nutrition experiment:
• Randomly select ar dams
• Randomly assign each of the a diets to r dams
1(
... )
2(
... )
...
a(
... )
1(
... )
2(
... )
...
a(
... )
...
1(
...
... )
r dams
where #(
dam.
2(
...
... )
r dams
...
...
...
a(
... )
r dams
... ) identifies the diet assigned to the associated
• Would be a CRD in factor A alone
– “noise” is between-dam variability
STAT 512
Factorial Treatments, Split-Plot Designs
20
• Will need b infants born to each dam
• Randomly assign each of the b diets to one infant from each dam
1(1 2 3 ... b)
2(1 2 3 ... b)
...
a(1 2 3 ... b)
1(1 2 3 ... b)
2(1 2 3 ... b)
...
a(1 2 3 ... b)
...
...
...
...
1(1 2 3 ... b)
2(1 2 3 ... b)
...
a(1 2 3 ... b)
r dams
r dams
...
r dams
b infants/dam
b infants/dam
...
b infants/dam
• WITHIN dams that got the same diet, would be a CBD in factor
B alone
– “noise” is between-infant, within-dam variability
– likely much smaller (than between-dam variability)
STAT 512
Factorial Treatments, Split-Plot Designs
• Will need a mixed linear model for this:
yit(i)j = µ+
whole-plot
split-plot
αi + δit +
βj + (αβ)ij + itj
• ∼ N (0, σ2 ) δ ∼ N (0, σδ2 )
• i = 1...a, levels of factor A
• t = 1...r, dams within A-level
• j = 1...b, levels of factor B, and infants within dams
21
STAT 512
whole-plot
split-plot
overall
Factorial Treatments, Split-Plot Designs
source
df
α
a−1
resid
a(r − 1)
c.t.
ar − 1
β
b−1
(αβ)
(a − 1)(b − 1)
resid
a(r − 1)(b − 1)
c.t.
arb − 1
sum of squares
P
2
rb(ȳ
−
ȳ
)
i..
...
i
P
2
it b(ȳit. − ȳi.. ) (fewer df)
P
2
it b(ȳit. − ȳ... )
P
2
j ar(ȳ..j − ȳ... )
P
2
ij r(ȳi.j − ȳi.. − ȳ..j + ȳ... )
difference (more df)
P
2
(y
−
ȳ
)
itj
...
itj
• Why MUST dams be a random effect in this design?
• E(M SEwhole ) = bσδ2 + σ2 (relatively large, test for A relatively
less powerful)
• E(M SEsplit ) = σ2 (relatively small, test for B and AB
relatively more powerful)
22
STAT 512
Factorial Treatments, Split-Plot Designs
23
Q: Why is the (αβ) interaction in the split-plot part of the analysis?
• Small example: a = b = 2, 4 whole-plots:
design
α contrast
β contrast
(αβ) contrast
i=1
j=1
j=2
−
−
−
+
+
−
i=1
j=1
j=2
−
−
−
+
+
−
i=2
j=1
j=2
+
+
−
+
−
+
i=2
j=1
j=2
+
+
−
+
−
+
• As with β, (αβ) is represented by a contrast within blocks/plots.
• (αβ) is orthogonal to both main effects, so it is orthogonal to
blocks (which are counfounded with α)
STAT 512
Factorial Treatments, Split-Plot Designs
24
Can also formulate the split-plot design as CBD for A and CBD for B:
• Sometimes can improve power/precision for A by doing this ...
• Select r “dams.2” (“grandmothers”, selection is not necessarily
random in this case)
• From the offspring of each dam.2, randomly select a females
dam.2 #1
1(
... )
2(
... )
...
a(
... )
dam.2 #2
1(
... )
2(
... )
...
a(
... )
...
dam.2 #r
...
1(
...
... )
r dams
2(
...
... )
r dams
...
...
...
a(
... )
r dams
• Assign one to each of the a diets
CBD for dam-diets, within offspring of dam.2 animals
STAT 512
Factorial Treatments, Split-Plot Designs
25
• Again, will need b infants from each dam:
dam.2 #1
1(1 2 3 ... b)
2(1 2 3 ... b)
...
a(1 2 3 ... b)
dam.2 #2
1(1 2 3 ... b)
2(1 2 3 ... b)
...
a(1 2 3 ... b)
...
...
...
...
...
dam.2 #r
1(1 2 3 ... b)
2(1 2 3 ... b)
...
a(1 2 3 ... b)
r dams
r dams
...
r dams
• Assign one to each of the b diets
CBD for infant-diets, within offspring of dams
STAT 512
Factorial Treatments, Split-Plot Designs
• Model:
ytij = µ+
whole-plot
split-plot
γt + αi + δti +
βj + (αβ)ij + tij
• new blocking structure with r − 1 df for whole-plot blocks
• estimate bσδ2 + σ2 with r − 1 fewer df; off-set by smaller σδ ?
• estimate of σ2 is not affected
26
STAT 512
whole-plot
split-plot
overall
Factorial Treatments, Split-Plot Designs
source
df
blocks
r−1
α
a−1
resid
(r − 1)(a − 1)
c.t.
ra − 1
β
b−1
(αβ)
(a − 1)(b − 1)
resid
(r − 1)a(b − 1)
c.t.
rab − 1
sum of squares
P
2
ab(ȳ
−
ȳ
)
t..
...
r
P
2
rb(ȳ
−
ȳ
)
.i.
...
i
difference
P
2
it b(ȳti. − ȳ... )
P
2
j ra(ȳ..j − ȳ... )
P
2
ij r(ȳ.ij − ȳ.i. − ȳ..j + ȳ... )
difference
P
2
(y
−
ȳ
)
tij
...
tij
27
STAT 512
Factorial Treatments, Split-Plot Designs
28
Pattern extends to more than 2 strata and/or more than 1 factor per
stratum, e.g.:
stratum
applied factors
1
A
2
B,C
3
D
degrees of freedom
units
effects
treat’s
residual
blocks
c.t.
ra
α
a−1
diff
0
ra − 1
(b − 1) + ...
diff
ra − 1
rabc − 1
(d − 1) + ...
diff
rabc − 1
rabcd − 1
bc × ra
β, γ, (βγ),
(αβ), (αγ), (αβγ)
d × rabc
δ, (αδ), (βδ), (γδ), ...