Outline Experiments with Two Factors (14.1) Experiments with Three - - PDF document
2/15/2007 219323 Probability and Statistics for Software Statistics for Software and Knowledge Engineers Lecture 12: Multifactor Experimental Multifactor Experimental Design and Analysis Monchai Sopitkamon, Ph.D. Outline Experiments
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Lecture 12: Multifactor Experimental Multifactor Experimental Design and Analysis
Monchai Sopitkamon, Ph.D.
Experiments with Two Factors (14.1) Experiments with Three or More Factors
(14.2)
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Two-Factor Experimental Designs (14.1.1) Models for Two-Factor Experiments (14 1 2) Models for Two-Factor Experiments (14.1.2) Analysis of Variance Table (14.1.3) Pairwise Comparisons of the Factor Level
Means (14.1.4)
ANOVA in Chapter 11 concerns relationship
between a response variable of interest and between a response variable of interest and various levels of a single factor of interest
Objectives:
– To simultaneously investigate the relationship between a response variable and various levels
– To extend the ANOVA technique to analyze the To extend the ANOVA technique to analyze the structure of the relationship between the response variable and the two factors of interest – To assess any interaction between the two factors where the response variable in influenced
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Investigate how a response variable
depends on two factors of interest depends on two factors of interest
Ex: washing quality (response var) of a
detergent may depend on two factors: water temperature and detergent type.
Ex: a student’s grade (response var) may
depend on two factors: in-class t ti d dili concentration and diligence.
A tw o factor experim ent w ith a × b experim ental configurations
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Consider an experiment w/ two factors, A
and B. If factor A has a levels and factor B and B. If factor A has a levels and factor B has b levels, then there are ab experimental configurations or cells.
In a complete balanced experiment, n
replicate measurements are taken at each configuration, resulting in a total sample size
The data observation xijk represents the
value of the kth measurement obtained at the ith level of factor A and the jth level of factor B.
Data observations and sam ple averages for a tw o factor experim ent
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Data set for the car body assem bly line exam ple
An observation xijk for the kth observation of
level i of factor A and level j of factor B can j be written as: xijk = µij + εijk where µij are unknown parameters of interest in a two-factor experiment or cell means, and εijk is the error term with N(0,σ2)
Consider three aspects of the relationship
p p between the expected value of the response variable and the two factors of interest.
– the interaction effect between factors A and B – the main effect of factor A – the main effect of factor B
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Cell m eans for a tw o factor experim ent
I nterpretation of m odels for tw o factor experim ents
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I nterpretation of m odels for tw o factor experim ents
I nterpretation of m odels for tw o factor experim ents
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I nterpretation of m odels for tw o factor experim ents
I nterpretation of m odels for tw o factor experim ents
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The sum of squares decom position for a tw o factor experim ent
Total Sum of Squares (SST) – a measure of
the total variability in the data set the total variability in the data set
= = = = = =
a i b j n k ijk a i b j n k ijk
1 1 1 2 ... 2 2 1 1 1 ...
where
1 1 1 a i b j n k ijk
1 1 1 ... T i j k
= = =
ijk
k-th observation in level i.of factor A and level j of factor B nT: total number of observations: abn
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The sum of squares decom position for a tw o factor experim ent
Sum of Squares for Factor A (SSA) – a
measure of the variability due to factor A. measure of the variability due to factor A.
2 ... 1 2 1 2 ...
a i i a i i
= ⋅ ⋅ = ⋅ ⋅
where l h l l
b n
sample averages at each level
number of levels of factor B
: n
number of observations at level i of factor A and level j of factor B
1 1
= = ⋅ ⋅ = j k ijk i
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The sum of squares decom position for a tw o factor experim ent
Sum of Squares for Factor B (SSB) – a
measure of the variability due to factor B. measure of the variability due to factor B. where
2 ... 1 2 1 2 ...
b j j b j j
= ⋅ ⋅ = ⋅ ⋅
a n
l
number of levels of factor A
1 1
= = ⋅ ⋅ = i k ijk j
sample averages at each level of factor B.
number of observations at level i of factor A and level j of factor B
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The sum of squares decom position for a tw o factor experim ent
Sum of Squares for Interaction (SSAB) – a
measure of the variability due to interaction measure of the variability due to interaction effect between factors A and B.
= = ⋅ ⋅ ⋅ ⋅ ⋅
a i b j j i ij
1 1 2 ...
where
1
= ⋅ = n k ijk ij
Cell averages at level i of factor A and level j of factor B.
number of observations at level i of factor A and level j of factor B
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The sum of squares decom position for a tw o factor experim ent
Sum of Squares for Error (SSE) – a
measure of the variability within each of the measure of the variability within each of the ab experimental configuration.
= = ⋅ = = = = = =
a i b j ij a i b j n k ijk a i b j n k ij ijk
1 1 2 1 1 1 2 1 1 1 2 .
where
1
= ⋅ = n k ijk ij
Cell averages at level i of factor A and level j of factor B.
number of observations at level i of factor A and level j of factor B
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Analysis of variance table for a tw o factor experim ent Reject H0 if F-statistic > F-critical (from
Table IV or Excel’s FINV function)
chapter)
Ex.9 pg.654: Car Body Assembly Line
The analysis of variance table for the car body assem bly line exam ple
No machine main effect (accept H0) There is solder main effect (reject H0) No interaction effect between machine and
solder (accept H0)
the car body assem bly line exam ple
Excel sheet
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Ex.74 pg.655: Company Transportation Costs
The analysis of variance table for the com pany transportation costs exam ple
There is route main effect (reject H0) There is period main effect (reject H0) No interaction effect between the route taken
and the period of the day (accept H0)
the com pany transportation costs exam ple
Excel sheet
A plot of the cell sam ple averages for the com pany transportation transportation costs exam ple
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If H0 is accepted for interaction effect but H0
is rejected for one or more factor, we’d like is rejected for one or more factor, we d like to be able to tell which samples are different and by how much.
Factor A: perform multiple comparisons to
compare all levels simultaneously
– By computing the differences μI – μj for 1 ≤ i < j ≤ a among all a(a 1)/2 pairs of factor level ≤ a among all a(a – 1)/2 pairs of factor level means
Compute confidence intervals
n ab a j i j i ) 1 ( , , − ⋅ ⋅ ⋅ ⋅
α
If the CI for the difference μI – μj contains 0,
then factor levels i and j of factor A are not then factor levels i and j of factor A are not significantly different.
If the CI for the difference μI – μj does not
contains 0, then factor levels i and j for factor A are significantly different.
The CI indicates by how much the factor
l l h t b diff t level means are shown to be different.
If H0 is accepted for factor A, then all CIs will
contain 0, which means no significant difference between different levels of factor A
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If there is interaction effect, then these CIs
are not interpretable since the different are not interpretable since the different effects of factor A depend on the level of factor B.
If factor A has two levels (a = 2), constructed
CI for μ1 – μ2 is equivalent to the CI constructed from a two-sample t-procedure w/ pooled variance estimate as in Chapter w/ pooled variance estimate as in Chapter 9.3.2.
Factor B: perform multiple comparisons to
compare all levels simultaneously compare all levels simultaneously
– By computing the differences μI – μj for 1 ≤ i < j ≤ b among all b(b – 1)/2 pairs of factor level means
Compute confidence intervals
n ab b j i j i ) 1 ( , , − ⋅ ⋅ ⋅ ⋅
α
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– No interaction effect between machines and No interaction effect between machines and solder formulations, but solder main effect existed. – Perform pairwise comparisons of b = 4 levels of solder formulations. q0.05,4, 24 = 3.90 (From Table V)
) 1 ( , ,
−
n ab b α
Excel sheet
CIs of Diff lower upper μ1 − μ2
2.365 1 3 1 182 1 313
significantly different than one
μ1 − μ3
1.313 μ1 − μ4
μ2 − μ3
0.197 μ2 − μ4
μ3 − μ4
significantly different than one another
welding strength than any other 3 solder types
Com parisons of the solders for the car body assem bly line exam ple
Excel sheet
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Costs Costs
– No interaction effect between route and period, but route and period main effects existed. – Perform pairwise comparisons of both factors A (route, a = 2) and B (period, b = 3 levels). Consider Factor A: Route (a = 2) Consider Factor A: Route (a 2) q0.05,2, 24 = 2.92 (From Table V)
) 1 ( , ,
−
n ab a α
Excel sheet
CIs of Diff lower upper
mins longer than route 2, and takes as much as 37 mins longer on average. μ1 − μ2 8.0 37.2 Since no interaction effect between route and period of day, this result holds regardless of time of day the truck leaves the factory.
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Costs Costs
Consider Factor B: Period of time (b = 3) Let level 1 be morning, level 2 be afternoon, level 3 be evening q = 3 53 (From Table V) q0.05,3, 24 = 3.53 (From Table V)
) 1 ( , ,
−
n ab b α
Excel sheet
CIs of Diff lower upper μ1 μ2 5 4 37 8
driving time between morning
μ1 − μ2
37.8 μ1 − μ3 26.4 69.6 μ2 − μ3 10.2 53.4
than in the morning and afternoon.
and between 26 and 69 mins compared to the morning period driving time between morning and afternoon periods. and between 26 and 69 mins compared to the morning period. Excel sheet
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Three-Factor Experiments (14.2.1) 2k Experiments (14 2 2) 2 Experiments (14.2.2)
Investigate how a response variable
depends on three or more factors of interest. depends on three or more factors of interest.
This is similar analysis to that for the two-
factor one, except that there are more interaction terms to be considered.
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The sum of squares decom position for three factor experim ent
Analysis of variance table for a three factor experim ent
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If there are k factors that effect a response
variable, each factor having the number of g levels as l1,…, lk, the total number of experiments required is
l1 x l2 x ⋅ ⋅ ⋅ x lk
If there are n replicates for each experiment,
the total sample size required is
nT = l1 x l2 x ⋅ ⋅ ⋅ x lk x n
f
This sample size can become large for large
number of k factors.
Screening experiments are therefore
necessary where each factor is considered at only 2 levels, or li = 2, 1 ≤ i ≤ k
So the total number of experiments with k
factors, each factor having 2 levels, are now d d t reduced to 2 x 2 x 2 x ⋅ ⋅ ⋅ x 2
This is known as 2k experimental design, and
the total sample size is
2k
k times
nT = 2k x n Where n is the number of replicates at each experimental configuration.
Use the typical model where the data
together with an error term.
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The cell means are
– the sum of the k individual factor main effects + – all the possible two-way interaction effects + – higher order interaction effects
There are r-way interaction effects (the
number of ways that a subset of r factors can b t k f th k f t ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ r k be taken from the k factors.
Construct the ANOVA table as usual with all
rows having 1 degree of freedom, except for the error’s degrees of freedom, which are 2k(n – 1).
The total degrees of freedom are 2kn – 1
Data set for chem ical yields exam ple
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the chem ical yields exam ple
The sam ple averages for the chem ical yields exam ple