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Two-Way ANOVA. Blocking is used to keep extraneous factors from masking the effects of the treatments you are interested in studying. A two-way ANOVA is used when you are interested in determining the effect of two treatments. Model: y ijk = μ + τ i + β j + ( τ β ) ijk + ε ij. - PowerPoint PPT Presentation
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ETM 620 - 09U1
Two-Way ANOVABlocking is used to keep extraneous factors
from masking the effects of the treatments you are interested in studying.
A two-way ANOVA is used when you are interested in determining the effect of two treatments.
Model: yijk = μ + τ i + βj + (τ β)ijk + εij
ETM 620 - 09U2
Two-Way ANOVA w/ ReplicationYou have been called in as a consultant to help
the Pratt and Whitney plant in Columbus determine the best method of applying the reflective stripe that is used to guide the Automated Guided Vehicles (AGVs) along their path. There are two ways of applying the stripe (paint and coated adhesive tape) and three types of flooring (linoleum and two types of concrete) in the facilities using the AGVs. You have set up two identical “test tracks” on each type of flooring and applied the stripe using the two methods under study. You run 3 replications in random order and count the number of tracking errors per 1000 ft of track. The results are as follows:
ETM 620 - 09U3
Two-Way ANOVA Example
Analysis is the same as with blocking, except we are now concerned with interaction effects
ETM 620 - 09U4
Two-Way ANOVA
ETM 620 - 09U5
Your TurnComplete the ANOVA in Minitab and fill in the
blanks …
What does this mean?
ANOVA
Source df SS MS F P-value
Stripe 10.435
6 _____ 2.40 0.147
Flooring 2 4.48 2.24 12.33 0.001
Interaction ___0.964
4 0.4822 _____ 0.111
Error ___ 2.18 0.1817
Total 17 8.06
ETM 620 - 09U6
What about interaction effects?For example, suppose a new test was run
using different types of paint and adhesive, with the following results:Linoleum Concrete I Concrete II
Paint 10.7 10.8 12.210.9 11.1 12.311.3 10.7 12.5
Adhesive 11.2 11.9 10.911.6 12.2 11.610.9 11.7 11.9
Source DF SS MS F
P-valu
e
Stripe 10.1088
90.1088
9 1.070.32
1
Flooring 2 1.96 0.98 9.640.00
3
Interaction 2
2.83111
1.41556
13.92
0.001
Error 12 1.220.1016
7
Total 17 6.12
ETM 620 - 09U7
Understanding interaction effectsGraphical methods:
graph means vs factorsidentify where the effect will change the result for one
factor based on the value of the other.
Interaction
10.5
11
11.5
12
12.5
0 1 2 3 4
Floor Type
Tra
ckin
g E
rro
rs
Paint
Adhesive
ETM 620 - 09U8
General factorial experiments (> 3 factors)
Example: 3-factor experiment. The model is:
Use Minitab:Balanced ANOVA (if the experiment is balianced,
i.e.,equal number of observations at each treatment combination.)
General Linear Model (GLM) if the experiment is unbalanced.
Allows for random effects and mixed models.
ijklijkjkikijkjiijkly )()()()(
ETM 620 - 09U9
An exampleComparison of head-up and head-down displays
Manual flying mode2 display formats 3 levels of ceiling & visibility – 5000 ft and 10 miles,
200 ft and ½ statute mile, and 0 ft and 1200 ft runway visual range
2 levels of wind direction and velocity – 090 degrees at 10 knots and 135 degrees at 21 knots
Use Balanced ANOVA with visibility and wind as random effects.“Model” is input as follows: Display| Wind| VisibilityTo discuss – use “restricted” form of the model or
not?
ETM 620 - 09U10
Minitab Output …Using balanced ANOVA and restricted model.
Source DF SS MS F P
Display 1 3274327
4 **
Wind 1 420.4 420 0.57 0.53
Visibility 2 234 117 0.16 0.86
Display*Wind 1 174 174 0.82 0.46
Display*Visibility 2 65.07 32.5 0.15 0.87
Wind*Visibility 2 1466 733 8.58 0
Display*Wind*Visibility 2 422.6 211 2.47 0.09
Error 108 9223 85.4
Total 1191527
8
** Denominator of F-test is zero.
ETM 620 - 09U11
Graphical evaluation of interactions …
X-T
rack
Dev
VisibilityWind
50002000135901359013590
30
25
20
15
10
Interval Plot of X-Track Dev vs Visibility, Wind95% CI for the Mean
X-T
rack
Dev
VisibilityWind
Display
50002000135901359013590
HUHDHUHDHUHDHUHDHUHDHUHD
40
30
20
10
0
Interval Plot of X-Track Dev vs Visibility, Wind, Display95% CI for the Mean
ETM 620 - 09U12
2k factorialsFactorial experiments with multiple factors in
which each factor can have several levels can get expensive in terms of number of trials required, especially if replication is desired.
If we choose not to replicate the experiment (i.e., only 1 observation per combination of factor levels), we lose the ability to evaluate higher level interactions.
Can get more “bang for the buck” with careful selection of two levels of each factor … i.e., a 2k factorial design.
ETM 620 - 09U13
Example: 22 factorial Look at the effect of oven temperature and
reaction time on the yield (in percent) of a process …
Oven Temp. Reaction Time 110° 50 min. 130° 70 min.
Take 2 observations at each combination with the following result:
Observation
s
Temp R.T.Interacti
on #1 #2 SUM
(1) -1 -1 1 55.5 54.5 110
a 1 -1 -1 60.2 61121.
2
b -1 1 -1 64.5 63.9128.
4
ab 1 1 1 67.7 68.7136.
4
ETM 620 - 09U14
To determine the effect of Temperature, we find the midpoint between the average “high” temp yield and the average “low” temp yield, or
Similarly, the effect of Time is …
And the interaction effect is …
4.1364.1282.121110)2(2
12
4.1281102
4.1362.12121