STAT 512
2-Level Factorial Experiments: Basics
TWO-LEVEL FACTORIAL EXPERIMENTS (Basics)
• Treatments defined by f binary “bits”
→ 2f treatments
• For (a classical) example, fertilizer treatments defined by including:
– N (nitrogen) at level i (1 or 2)
– P (phosphorus) at level j (1 or 2)
– K (potassium) at level k (1 or 2)
*
N2
*
*
*
*
N1
*
K1
* P2
* P1
K2
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STAT 512
2-Level Factorial Experiments: Basics
• Cell Means Model:
yijkm = µijk + ijkm m = 1, ...r
• Factorial Effects Model (as we’ve been using with two factors,
“overparameterized” form with over-dots in the book):
yijkm =
µ + αi + βj + γk
+(αβ)ij + (αγ)ik + (βγ)jk
+(αβγ)ijk + ijkm
• Model matrix columns for main effects, e.g. α1 :
0
N2
0
0
0
1
N1
1
K1
1 P2
1 P1
K2
2
STAT 512
2-Level Factorial Experiments: Basics
• Two-factor interactions, e.g. (αβ)11
0
N2
0
0
0
0
N1
1
K1
0 P2
1 P1
K2
• Three-factor interactions, e.g. (αβγ)111
0
N2
0
0
0
0
N1
1
K1
0 P2
0 P1
K2
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STAT 512
2-Level Factorial Experiments: Basics
• As usual for this approach, there are more parameters than are
really needed:
– 2 first order terms for each factor ...
.
2f in all
– 4 second 
order 
terms for each pair of factors ...
f

 in all
.
4
2
– 2q qth order
 terms
 for each q-tuple of factors ...
f
q
 in all
.
2
q
• 3f of them in all, but really only need 2f parameters to model the
entire mean structure
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STAT 512
2-Level Factorial Experiments: Basics
One reparameterization that works well for 2-level factorials, is a
“full-rank” form (with no over-dots in the book):
• Replace each group of:
– 2 main effects parameters, e.g. α1 , α2
– 4 two-factor interaction parameters, e.g. (αβ)11 ...
with a single parameter as follows:
• Replace (− − −)222...2 = (− − −)
• Replace (− − −)121...2 = (− − −) × (−1)s
where s = number of 1’s in subscript
• Example:
µ121 = µ
+α1 + β2 + γ1
=µ
−α + β − γ
+(αβ)12 + (αγ)11 + (βγ)21
−(αβ) + (αγ) − (βγ)
+(αβγ)121
+(αβγ)
• Now, total of 2f parameters ... same as number of cell means
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STAT 512
2-Level Factorial Experiments: Basics
Also, this parameterization corresponds to contrasts in cell means:
• Model matrix columns for main effects, e.g. α:
+
N2
+
+
+
N1
K1
- P2
- P1
K2
• Two-factor interactions, e.g. (αβ)
+
N2
-
+
-
N1
+
K1
- P2
+ P1
K2
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STAT 512
2-Level Factorial Experiments: Basics
• Managing +1’s and -1’s ... let:
– x1 = −1 for i = 1, +1 for i = 2
– x2 = −1 for j = 1, +1 for j = 2
– x3 = −1 for k = 1, +1 for k = 2
• Then:
µijk = µ +x1 α + x2 β + x3 γ
+x1 x2 (αβ) + x1 x3 (αγ) + x2 x3 (βγ)
+x1 x2 x3 (αβγ)
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STAT 512
2-Level Factorial Experiments: Basics
8
• For a complete unreplicated 23 Factorial Design:
– y = Xθ + 

y111













y112
y121
y122
y211
y212
y221
y222

+ − − − + + +−

 
 
 
 
 
 
=
 
 
 
 
 
 
+ − − + + − −+

 
 α  

 
 β  

 
 γ  

 
  (αβ)  + 

 
  (αγ)  

 

 
  (βγ)  
+ − + − − + −+
+ − + + − − +−
+ + − − − − ++
+ + − + − + −−
+ + + − + − −−
+ + + + + + ++
µ
(αβγ)


111
112
121
122
211
212
221
222
– Columns of X are values of 1, x1 , x2 , ... x1 x2 x3 for each
experimental run.
– Basic pattern is the same for all 2f .














STAT 512
2-Level Factorial Experiments: Basics
y = Xθ + (N × 1) = (N × 2f )(2f × 1) + (N × 1)
N = 2f unreplicated, N = r2f with r units per cell
• Number of parameters is 2f because each is specified by presence
or absence of each of f symbols.
• Note odd notation ... θ contains one nuisance parameter.
– Could partition model as before:
y = 1µ + X2 φ + → H1 =
1
N J,
but since all columns of X2 have zero-sums,
H1 X2 = 0, X2|1 = X2 .
– This means “correction for the mean” or intercept is automatic
in this parameterization
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STAT 512
2-Level Factorial Experiments: Basics
• X is an orthogonal matrix:
X0 X = r2f I
θ̂ = (r2f )−1 X0 y
• If r ≥ 1:
θ̂ = 2−f M0 ȳ
where M is the model matrix for an unreplicated design and ȳ is
2f -element vector of treatment means
• Any linear combination of elements of θ is estimable (not just
those that are treatment contrasts) because M is square, of full
rank. But we generally don’t regard anything involving the first
element as meaningful, since it includes experiment-specific
effects.
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STAT 512
2-Level Factorial Experiments: Basics
Estimation of Functions of Treatment Mean: Full Model
• Let µ = (µ111... ...µ222... )0 , same size and order of elements as ȳ
• By our definition of factorial effects, µ = Mθ
• So:
0 µ = c0d
cd
Mθ
b = c0 M2−f M0 ȳ = c0 ȳ
= c0 Mθ
(because MM0 = 2f I)
• Why?
M0 M = 2f I
M is square, full rank, so 2−f M0 = M−1
M(2−f M0 ) = I
MM0 = 2f I
0 µ] = c0 E(ȳ) = c0 µ, V ar[c
0 µ] = (σ 2 /r)c0 c
d
• E[cd
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STAT 512
2-Level Factorial Experiments: Basics
12
Estimation of Treatment Means: Reduced Model
• Returning to f = 3 example, suppose we know (or are willing to
assume) that interactions involving γ don’t exist:
(αγ) = (βγ) = (αβγ) = 0
• Now:
x1







µ= 








x2
+
−
−
+
−
−
x3
−
+
x1 x2
+

+















+
−
+
−
−
+
−
+
+
−
+
+
−
−
−
+
+
−
+
−
+
+
+
−
+
+
+
+
+
+









µ
α
β
γ
(αβ)





 = M1 θ 1



STAT 512
2-Level Factorial Experiments: Basics
0
0
0 µ = c0 M
d
• Now: cd
1 θ1 = c M1 θ̂ 1 ... but this is not c ȳ
= c0 M1 (M01 M1 )−1 M01 ȳ = 2−f c0 M1 M01 ȳ
• The difference now is that M1 M01 6= 2f I
0 µ] =
• E[cd
2−f c0 M1 M01 (M1 θ 1 ) = 2−f 2f c0 M1 θ 1 = c0 µ
...if the reduced model is correct
0 µ] =
• V ar[cd
c0 M1 V ar[θ̂] M01 c
c0 M1 (M01 M1 )−1 M01 V ar[ȳ] M1 (M01 M1 )−1 M01 c
(σ 2 /r)2−2f c0 M1 M01 M1 M01 c = (σ 2 /r)2−f c0 M1 M01 c
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STAT 512
2-Level Factorial Experiments: Basics
• For full model, MM0 is 2f I
→ V ar[−] = (σ 2 /r)c0 c
– Here, let M = (M1 , M2 ), and rewrite the variance for the
FULL model:
0
2
−f 0
0 µ]
V ar[cd
=
(σ
/r)2
c
MM
c
(full)


0
M
1c 
2
−f 0
0

= (σ /r)2 (c M1 , c M2 )
M02 c
= (σ 2 /r)2−f c0 M1 M01 c + (σ 2 /r)2−f c0 M2 M02 c
0 µ]
= V ar[cd
(reduced) + something non-negative
– So, variance is no greater than, and sometimes less than, what
we would have for the full model
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STAT 512
2-Level Factorial Experiments: Basics
Examples:
• c = (0, 0, 0...1...0)0 , i.e. c0 µ is a single cell mean
– c0 M1 is a single row from M1
– c0 M1 M01 c = number of parameters included in the model, say
p1
– c0 MM0 c = 2f for full model
– reduced model gives smaller variance for estimating a single
cell mean
– (not generally interesting, since it isn’t a treatment contrast)
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STAT 512
2-Level Factorial Experiments: Basics
• c = (0, 0, 0...1.... − 1...0)0 , i.e. c0 µ is a difference (contrast)
between two cell means
– c0 M1 contains zeros, and v1 elements = ±2
– c0 M1 M01 c = 4v1
– c0 M contains zeros, and v2 elements = ±2 in v2 , v2 > v1
– c0 MM0 c = 4v2 ≥ 4v1
• c = a column from M1 (i.e. a contrast associated with a factorial
effect in the reduced model
– c0 M1 = (0, 0, 0...2f ...0)
– c0 M1 M01 c = (2f )2
– c0 MM0 c = (2f )2
– reduced model gives SAME variance for estimating an included
factorial effect
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STAT 512
2-Level Factorial Experiments: Basics
• c = a column from M2 (i.e. a contrast associated with a factorial
effect EXCLUDED from the model)
– c0 M1 = 0! ... why?
– variance for estimating something YOU SAID WAS ZERO
• Potential benefits of a reduced model are improved precision and
power associated with some treatment comparisons.
• BUT, associated risk is that estimates are biased and tests
potentially invalid if the omitted effects are actually present.
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STAT 512
2-Level Factorial Experiments: Basics
TESTING HIGHER-ORDER EFFECTS: r > 1, individual terms
• Hyp0 : (−) = 0, where “(−)” is any of the factorial effects
d = 2−f m0 ȳ
• (−)
(−)
– because (M0 M)−1 = 2−f I
– m(−) is the corresponding column from M
−2f
2
0
d = 2−2f m0 V ar[ȳ]m
• V ar[(−)]
=
2
(σ
/r)m
(−)
(−)
(−) m(−)
(last inner product is 2f )
= σ 2 /N
P
2
2
f
ˆ
2
• σ = spooled =
s
/2
ij...
ij...
q
d
• t = (−)/
s2pooled /N ... same for any (−)
compare to t[1 − α/2, 2f (r − 1)]
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STAT 512
2-Level Factorial Experiments: Basics
TESTING HIGHER-ORDER EFFECTS: r > 1, multiple terms
• Hyp0 : µ = M1 θ 1
• HypA : µ = Mθ = M1 θ 1 + M2 θ 2
• Group tests useful to address:
– inflated type-I error rates from multiple testing
– e.g. “all interactions involving γ”
• For any model, SSE =
(y − Xθ̂)0 (y − Xθ̂)
y0 (I − X(X0 X)− X0 )y
y0 y − (y0 X(X0 X)− X0 )(X(X0 X)− X0 y)
0
0
y y − (θ̂ X0 )(Xθ̂)
0
f
0
y y − r2 θ̂ θ̂
19
STAT 512
• SST =
2-Level Factorial Experiments: Basics
0
SSE(Hyp0 ) - SSE(HypA ) = y y − N θ̂ 1 θ̂ 1
0
0
f 0
f
= r2 (θ̂ θ̂ − θ̂ 1 θ̂ 1 ) = r2 θ̂ 2 θ̂ 2
0
20
0
− y y + N θ̂ θ̂
• Changing the model doesn’t change individual estimates
IN THIS CASE.
f
• F = (r2
0
θ̂ 2 θ̂ 2 /p2 )/s2pooled
compare to F [1 − α, p2 , 2f (r − 1)]
• e.g., testing (αγ) = (βγ) = 0 uses numerator mean square
2
2
d + (βγ)
d ]/2
N [(αγ)
0
STAT 512
2-Level Factorial Experiments: Basics
UNREPLICATED EXPERIMENTS: individual terms
• Unreplicated experiments are more common in factorial settings
because of the large number of possible treatments
• If full model is correct:
d ∼ indep., N ( (−) , σ 2 /N )
(−)
• For effects that are actually zero:
d ∼ i.i.d., N ( 0 , σ 2 /N )
(−)
• If most effects are actually zero (sometimes called “effect
sparsity”), a diagnostic graph for outliers can be used to identify
the (few) effects that appear to be “real”:
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STAT 512
2-Level Factorial Experiments: Basics
• For any group of effects, (−)i , i = 1...p, usually excluding µ:
d versus Φ−1 ((i − 1 )/p)
– Plot of sorted (−)
[i]
2
d [i] versus Φ−1 ( 1 + (i − 1 )/(2p))
– Plot of sorted |(−)|
2
2
(without absolute value, plot isn’t invariant to changing “high”
and “low” levels of factors)
– can get Φ−1 values from qnorm in R ...
• Reiterate: This is a useful technique only under effect sparsity,
because it relies on the few “real” effects (outliers) to be different
from the many “absent” effects.
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STAT 512
2-Level Factorial Experiments: Basics
• Example:
6
– simulated data for unreplicated 24 , with all factorial effects
zero except for α = 6, β = −4, and (αβ) = 2; and σ = 3
4
5
•A
|estimates|
3
•B
• AB
2
• ABC
1
•
0
•
• • •
• •
•
•
•
•
0.0
0.5
1.0
1.5
nonnegative normal quantiles
2.0
– so 3 “real” effects are detected, but there is also 1 false alarm
23
STAT 512
2-Level Factorial Experiments: Basics
An algorithmic alternative to normal plots – Lenth’s Method:
• Let B be the set of absolute values of estimated coefficients of
interest.
√
• Compute an initial robust estimate of σ/ N :
s0 = 1.5 × median B
• Let B ∗ be the subset of B less than 2.5 × s0
√
• Compute a refined estimate of σ/ N , “pseudo standard error”
P SE = 1.5 × median B ∗
• Treat a coefficient as significant if it is greater than t × P SE
• Lenth published a table of values of t; for α = 0.05, and B not too
small, t = 2 works reasonably well as a rule-of-thumb
24
STAT 512
2-Level Factorial Experiments: Basics
ADDITIONAL GUIDELINES FOR MODEL CONSTRUCTION
• Half-normal plots, tests, help you find a model form. BUT, what if
the model “makes no sense” – e.g. A and BCD only ...
• EFFECT HIERARCHY PRINCIPLE – If an interaction involving a
given set of factors is included in the model, all main effects and
interactions involving subsets of these factors should also be
included.
• EFFECT HEREDITY PRINCIPLE – If an interaction involving a
given set of factors is included in the model, at least one effect of
the next smallest order involving a subset of these factors should
also be included.
• For example:
25
STAT 512
2-Level Factorial Experiments: Basics
B2
→ A, B, not AB
B2
+ Heredity → OK
µ
B1
B1
A1
A2
+ Hierarchy → OK
26
STAT 512
2-Level Factorial Experiments: Basics
B1
→ A, AB, not B
µ
B2
B2
+ Heredity → OK
+ Hierarchy → add B
B1
A1
A2
27
STAT 512
2-Level Factorial Experiments: Basics
B2
B1
→ AB, not A or B
+ Heredity → add A or B
µ
+ Hierarchy → add A and B
B1
B2
A1
A2
28
STAT 512
2-Level Factorial Experiments: Basics
WHY?
• If AB is present, A and B are clearly important ... E(y) changes
when they are changed
• Given the right collection of factor levels, at least one of the two
main effects would likely be present
• Hierarchy (conservative) is generally more popular than Heredity,
except when considerable knowledge about the action of factors is
available so that informed choices can be made
Last example:
• 4 factors, ABC and AD are significant
• + Heredity → add A and AB, minimally
• + Hierarchy → add A, B, C, D; AB, AC, BC
29