ST 380
Probability and Statistics for the Physical Sciences
Factors
The characteristics of measurements made under different conditions
are affected by various factors.
A textile engineer identifies the dye on a fiber by dissolving it in an
organic solvent; the amount of the dye extracted depends on:
the temperature of the solvent;
the length of time the fiber is left in the solvent.
The factors are temperature and time; the levels that are used might
be 20◦ C or 30◦ C, and 15, 20, or 25 minutes.
Combining these factor levels creates 6 possible treatments.
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Multifactor Analysis of Variance
Introduction
ST 380
Probability and Statistics for the Physical Sciences
Two Factors
Example 11.7
The response X is thermal conductivity of asphalt mix (W/(m◦ K)).
The factors are:
Asphalt binder grade: PG58, PG64, or PG70.
Coarse aggregate content: 38%, 41%, or 44%;
In R
asphalt <- read.table("Data/Example-11-07.txt", header = TRUE)
boxplot(Cond ~ AsphGr, asphalt)
boxplot(Cond ~ AggCont, asph)
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Probability and Statistics for the Physical Sciences
The boxplots show a strong effect of AggCont, and a possible effect
of AsphGr.
To quantify these impressions, we need to test appropriate null
hypotheses.
Because both factors may affect the response, the hypotheses must
be set up carefully.
The hypotheses are defined in the context of a statistical model.
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ST 380
Probability and Statistics for the Physical Sciences
Notation
Write Xi,j,k for:
the k th response (k = 1 or 2)
when AsphGr is at level i (i = 1, 2, or 3)
and AggCont is at level j (j = 1, 2, or 3)
The Additive Model
We assume that
µi,j = E (Xi,j,k ) = µ + αi + βj ,
k = 1, 2,
for parameters µ, α1 , α2 , α3 , β1 , β2 , and β3 .
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ST 380
Probability and Statistics for the Physical Sciences
Estimability
The model is over-parametrized as it stands.
If a constant c is added to µ and subtracted from each of the α’s (or
from each of the β’s), the sum remains the same.
That is, different sets of parameter values produce the same values
for E (Xi,j,k ), so we cannot estimate the parameters.
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ST 380
Probability and Statistics for the Physical Sciences
Constraints
We can eliminate the nonuniqueness by imposing constraints on the
α’s and β’s.
One possibility, used in the book, is:
I
X
i=1
αi =
J
X
βj = 0.
j=1
Another approach, used in all software, is based on choosing a
reference level of each factor.
The parameter associated with the reference level is set to zero,
which also eliminates the nonuniqueness.
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ST 380
Probability and Statistics for the Physical Sciences
In R, the reference level defaults to the first level, while in SAS (and
JMP?) the default is the last level:
in R: α1 = β1 = 0
in SAS: αI = βJ = 0.
Note that, in the R convention,
µ1,1 = E (X1,1,k ) = µ + α1 + β1 = µ.
That is, in this “reference level” approach, µ is actually the expected
response for the treatment in which both factors are at their
respective reference levels.
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Probability and Statistics for the Physical Sciences
Hypotheses
The level of the binder grade, AsphGr, has no effect on E (X ) if
αi = 0,
i = 1, 2, . . . , I .
We test this as a null hypothesis against the alternative that some of
the α’s are nonzero.
As in the single-factor case, the usual test statistic is a ratio of mean
squares, and is F -distributed under the null hypothesis.
A similar statistic tests the null hypothesis that AggCont has no
effect:
βj = 0, j = 1, 2, . . . , J.
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Probability and Statistics for the Physical Sciences
In R
asphaltAov <- aov(Cond ~ AsphGr + factor(AggCont), asphalt)
summary(asphaltAov)
Output
Df
Sum Sq
AsphGr
2 0.002089
factor(AggCont) 2 0.008297
Residuals
13 0.000991
--Signif. codes: 0 *** 0.001
Mean Sq F value
Pr(>F)
0.001045
13.7 0.00063 ***
0.004149
54.4 4.83e-07 ***
0.000076
** 0.01 * 0.05 . 0.1
1
Both factors have very significant effects, especially AggCont.
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Multifactor Analysis of Variance
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ST 380
Probability and Statistics for the Physical Sciences
Pairwise Comparisons
Knowing that AsphGr has a significant effect on conductivity, the
next question is what kind of effect:
TukeyHSD(asphaltAov, "AsphGr")
Output
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = Cond ~ AsphGr + factor(AggCont), data = asphalt)
$AsphGr
diff
lwr
upr
p adj
PG64-PG58 0.01166667 -0.001645642 0.024978975 0.0892142
PG70-PG58 -0.01466667 -0.027978975 -0.001354358 0.0306046
PG70-PG64 -0.02633333 -0.039645642 -0.013021025 0.0004494
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Probability and Statistics for the Physical Sciences
Binder grade PG70 gives significantly lower conductivity than the
other grades, but PG58 and PG64 are not significantly different.
TukeyHSD(asphaltAov, "factor(AggCont)") shows that all three
levels of AggCont give significantly different conductivities.
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Probability and Statistics for the Physical Sciences
Parameter Estimates
When there is only one factor, pairwise comparisons are the most
common inferences.
We can also estimate the parameters themselves, which will be
important when more factors are involved:
asphaltLm <- lm(Cond ~ AsphGr + factor(AggCont), asphalt)
summary(asphaltLm)
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Probability and Statistics for the Physical Sciences
Output
Call:
lm(formula = Cond ~ AsphGr + factor(AggCont), data = asphalt)
Residuals:
Min
1Q
Median
-0.011333 -0.004583 -0.001167
3Q
0.003583
Max
0.015333
Coefficients:
(Intercept)
AsphGrPG64
AsphGrPG70
factor(AggCont)41
factor(AggCont)44
--Signif. codes: 0
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Estimate Std. Error t value Pr(>|t|)
0.841000
0.004602 182.730 < 2e-16
0.011667
0.005042
2.314 0.03766
-0.014667
0.005042 -2.909 0.01219
-0.017333
0.005042 -3.438 0.00441
-0.051667
0.005042 -10.248 1.35e-07
*** 0.001 ** 0.01 * 0.05 . 0.1
Multifactor Analysis of Variance
Two Factors
1
***
*
*
**
***
ST 380
Probability and Statistics for the Physical Sciences
Output, continued
Residual standard error: 0.008732 on 13 degrees of freedom
Multiple R-squared: 0.9129, Adjusted R-squared: 0.8861
F-statistic: 34.05 on 4 and 13 DF, p-value: 8.953e-07
Interpretation
The “Coefficients” are the estimated parameters:
(Intercept)
µ̂
AsphGrPG64
AsphGrPG70
α̂2
α̂3
factor(AggCont)41 β̂2
factor(AggCont)44 β̂3
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Probability and Statistics for the Physical Sciences
Notes
AsphGrPG58 and factor(AggCont)38 are not in the output,
because the corresponding parameters α1 and β1 are constrained to
be zero.
Recall that the intercept µ is the expected response for this
combination of factor levels.
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