1

Recall: One-Way ANOVA

1. Calculate and check residuals, eij = Oi - Ei

– plot residuals vs treatments– normal probability plot

2. Perform ANOVA and determine if there is a difference in the means

3. Identify which means are different using Tukey’s procedure:

4. Model: yij = μ + αi + εij

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2

Blocking

• Creating a group of one or more people, machines, processes, etc. in such a manner that the entities within the block are more similar to each other than to entities outside the block.

• Balanced design: each treatment appears in each block.

• Model: yij = μ + α i + βj + εij

3

Example:Robins Air Force Base uses CO2 to strip paint from F-15’s. You have been asked to design a test to determine the optimal pressure for spraying the CO2. You realize that there are five machines that are being used in the paint stripping operation. Therefore, you have designed an experiment that uses the machines as blocking variables. You emphasized the importance of balanced design and a random order of testing. The test has been run with these results (values are minutes to strip one fighter):

4

ANOVA: One-Way with Blocking

1. Construct the ANOVA table

Where,

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5

Blocking Example

Your turn: fill in the blanks in the following ANOVA table (from Excel):

2. Make decision and draw conclusions:

ANOVA

Source of Variation SS df MS F P-value F crit

Rows 89.733 2 44.867 8.492 0.0105 4.458968

Columns 77.733 ___ _____ ____ 0.0553 _______

Error 42.267 8 5.2833

Total 209.73 ___        

6

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: yijk = μ + α i + βj + (α β)ijk + εij

7

Two-Way ANOVA w/ Replication• Your fame as an experimental design expert grows. You

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:

8

Two-Way ANOVA Example

• Analysis is the same as with blocking, except we are now concerned with interaction effects

9

Two-Way ANOVA

10

Your Turn

• Fill in the blanks …

• What does this mean?

ANOVA

Source of Variation SS df MS F P-value F crit

Sample 0.4356 1 _____ 2.3976 0.14748 4.74722

Columns 4.48 2 2.24 12.33 0.00123 3.88529

Interaction 0.9644 ___ 0.4822 _____ 0.11104 3.88529

Within 2.18 ___ 0.1817

Total 8.06 17        

11

What if Interaction Effects are Significant?

• For example, suppose a new test was run using different types of paint and adhesive, with the following results:

Linoleum Concrete I Concrete IIPaint 10.7 10.8 12.2

10.9 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

ANOVA

Source of Variation SS df MS F P-value F crit

Sample 0.109 1 0.1089 1.071 0.3211 4.7472

Columns 1.96 2 0.98 9.639 0.0032 3.8853

Interaction 2.831 2 1.4156 13.92 0.0007 3.8853

Within 1.22 12 0.1017

Total 6.12 17        

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Understanding Interaction Effects• Graphical methods:

– graph means vs factors– identify where the effect will change the result for one factor

based on the value of the other.

Interaction

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11

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12

12.5

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Floor Type

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Paint

Adhesive