ST 516
Experimental Statistics for Engineers II
Blocked Designs
Recall the paired comparison design: two treatments applied to the
same experimental material.
E.g. hardness testing
Treatment: two types of pointed tip pressed into a sample of metal
with known force;
Pairs: both types tested on the same samples.
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
E.g. shear strength of girders
Treatment: two methods;
Pairs: both methods tested on the same girders.
Statistical model is
yi,j = µi + βj + i,j ,
i = 1, 2;
j = 1, 2, . . . , n.
or, in effects form,
yi,j = µ + τi + βj + i,j ,
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i = 1, 2;
Blocked Designs
j = 1, 2, . . . , n.
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
More Than Two Treatments
If feasible, apply all treatments to each unit.
Treatments are assigned randomly:
time order of runs;
subsamples of metal coupon.
This is the randomized complete block design (RCBD).
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
Statistical Analysis
Same statistical model as for paired comparisons:
yi,j = µ + τi + βj + i,j ,
i = 1, 2, . . . , a;
j = 1, 2, . . . , b.
Here:
yi,j = response for i th treatment in the j th block
µ = overall mean
τi = i th treatment effect
βj = j th block effect
i,j = random error.
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
Same general questions as for the completely random design (CRD):
Are there any differences among treatments?
H0 : τ1 = τ2 = · · · = τa = 0.
If so, how to describe them:
regression modeling for a quantitative factor;
pairwise comparisons and other contrasts for a qualitative factor.
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
In addition: Was blocking necessary?
H0 : β1 = β2 = · · · = βb = 0.
But note: because treatments are randomized only within blocks,
testing this hypothesis has problems:
F -statistic may not be F -distributed under the null hypothesis;
Corresponding P-value may not be valid.
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
Also note: blocks are often viewed as a random sample from a
population of possible blocks.
That would mean that the β’s are random variables, not constants.
If Var(βj ) = σβ2 , the null hypothesis becomes H0 : σβ2 = 0.
H0 implies not only that β1 = β2 = · · · = βb = 0, but in addition
that all future β’s will also be zero.
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
Estimating the parameters
P
P
Under the natural constraints
τi = βj = 0:
E (ȳ·· ) = µ
E (ȳi· ) = µ + τi
E (ȳ·j ) = µ + βj
So the obvious estimates are:
µ̂ = ȳ··
τ̂i = ȳi· − ȳ··
β̂j = ȳ·j − ȳ··
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
Analysis of Variance
Sums of squares:
SSTotal =
SSTreatments =
SSBlocks =
SSError =
a X
b
X
(yi,j − µ̂)2
i=1 j=1
a
X
b
τ̂i2
i=1
b
X
a
β̂j2
j=1
a X
b X
yi,j − µ̂ − τ̂i − β̂j
2
i=1 j=1
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
The ANOVA identity:
SSTotal
df = N − 1
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= SSTreatments + SSBlocks +
a−1
b−1
Blocked Designs
SSError
N −a−b+1
= (a − 1)(b − 1)
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
Example: yield (%) of a manufacturing process; factors are Extrusion
Pressure and Batch (of resin, the raw material in the process). Data
file: graft.txt
R commands
graft <-read.table("data/graft.txt", header = TRUE);
graftLong <- reshape(graft,
varying = c("Batch1", "Batch2", "Batch3",
"Batch4", "Batch5", "Batch6"),
v.names = "Yield", timevar = "Batch",
direction = "long")
# boxplots:
par(mfcol = c(1, 2))
plot(Yield ~ factor(Batch) + factor(Pressure), data = graftLong)
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
R command for ANOVA
summary(aov(Yield ~ factor(Batch) + factor(Pressure),
data = graftLong))
Output
Df Sum Sq Mean Sq F value
Pr(>F)
factor(Batch)
5 192.252 38.450 5.2487 0.005532 **
factor(Pressure) 3 178.171 59.390 8.1071 0.001916 **
Residuals
15 109.886
7.326
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
1
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
R command for effects
summary(lm(Yield ~ factor(Batch) + factor(Pressure),
data = graftLong))
Call:
lm(formula = Yield ~ factor(Batch) + factor(Pressure), data = graftLong)
Residuals:
Min
1Q Median
-3.5708 -1.3333 -0.3167
3Q
1.1417
Max
4.1792
Coefficients:
(Intercept)
factor(Batch)2
factor(Batch)3
factor(Batch)4
factor(Batch)5
factor(Batch)6
factor(Pressure)8700
factor(Pressure)8900
factor(Pressure)9100
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Estimate Std. Error t value Pr(>|t|)
90.721
1.657 54.735 < 2e-16 ***
2.050
1.914
1.071 0.301043
3.300
1.914
1.724 0.105201
2.850
1.914
1.489 0.157175
-2.375
1.914 -1.241 0.233684
6.750
1.914
3.527 0.003050 **
-1.133
1.563 -0.725 0.479457
-3.900
1.563 -2.496 0.024713 *
-7.050
1.563 -4.512 0.000414 ***
Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
R command for (Tukey) pairwise comparisons
TukeyHSD(aov(Yield ~ factor(Batch) + factor(Pressure),
data = graftLong),
"factor(Pressure)")
Output
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = Yield ~ factor(Batch) + factor(Pressure), data = graftLong)
$‘factor(Pressure)‘
diff
lwr
upr
p adj
8700-8500 -1.133333 -5.637161 3.370495 0.8854831
8900-8500 -3.900000 -8.403828 0.603828 0.1013084
9100-8500 -7.050000 -11.553828 -2.546172 0.0020883
8900-8700 -2.766667 -7.270495 1.737161 0.3245644
9100-8700 -5.916667 -10.420495 -1.412839 0.0086667
9100-8900 -3.150000 -7.653828 1.353828 0.2257674
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Blocked Designs
Randomized Complete Block Design
ST 516
Experimental Statistics for Engineers II
Random Effects
If Block (Batch) is a random effect with variance σβ2 , then the
expected mean square is
E (MSBlocks ) = σ 2 + aσβ2 .
So we estimate σβ2 by
σ̂β2 =
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MSBlocks − MSResiduals
38.450 − 7.326
=
= 7.781.
a
4
Blocked Designs
Randomized Complete Block Design