Two-Way ANOVA

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