Stat 470-6 Today: 2-way ANOVA (Section 2.3)…2.3.1 and 2.3.2; Transformation of the response.

Stat 470-6

• Today: 2-way ANOVA (Section 2.3)…2.3.1 and 2.3.2; Transformation of the response

Two-Way ANOVA

• One-way ANOVA considered impact of 1 factor with k levels (e.g. meat packaging example)

• Two-way ANOVA considers the impact of 2 factors with I and J levels respectively

• Have possible treatments for each replicate of the experiment

• If have n replicates, the the experiment has observations

Example:

• An experiment was run to understand the impact of two factors (Table speed and Wheel grit size) on the the strength of the ceramic material (bonded Si nitrate). (Jahanmir, 1996, NIST)

• Each factor has two levels (coded -1 and +1 respectively)

• The experiment was repeated 2 times

Data

Run Order Table Speed Wheel Grit Size Response2 -1 -1 691.9536 -1 -1 689.5591 +1 -1 716.9268 +1 -1 759.5814 -1 +1 701.0007 -1 +1 709.9613 +1 +1 753.3335 +1 +1 735.919

• Model:

Hypotheses

Running the Experiment

• Two-Way ANOVA Model is appropriate for experiments performed as completely randomized designs

• That is, we list the treatments (e.g., 1-8 in the ceramics example) and assign treatments to experimental units in random order

• The trials are in random order

ANOVA Table

Source of Variation

Degrees of Freedom

Sum of Squares

Mean Squares

F

Factor A I-1 Factor B J-1 A x B (I-1)(J-1) Residual (b-1)k-1) IJ(n-1) Total bk-1 Ijn-1

Return to Ceramic Data

SPEED

2.01.00.0-1.0-2.0

ST

RE

NG

TH

770

760

750

740

730

720

710

700

690

680

GRIT

2.00.0-2.0S

TR

EN

GT

H

770

760

750

740

730

720

710

700

690

680

Interaction Plot

speed

me

an

of

stre

ng

th

69

07

00

71

07

20

73

07

40

-1 1

grit

1-1

ANOVA Table

Tests of Between-Subjects Effects

Dependent Variable: STRENGTH

4010.924a 3 1336.975 4.843 .081

4144654.47 1 4144654.471 15011.920 .000

3753.505 1 3753.505 13.595 .021

222.542 1 222.542 .806 .420

34.878 1 34.878 .126 .740

1104.364 4 276.091

4149769.76 8

5115.288 7

SourceCorrected Model

Intercept

SPEED

GRIT

SPEED * GRIT

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .784 (Adjusted R Squared = .622)a.

Residuals

• Must still do residual analysis

• What would happen if the experiment was unreplicated (l =1)?

• What could we do to address this?

Multi-Way (or N-Way) ANOVA (Section 2.4)

• Can extend model to more that 2 factors

• Approach is the same

Experiment Situation

• Have N factors

• The experiment is performed as a completely randomized design

• Assumptions:

Transformations (Section 2.5)

• Often one will perform a residual analysis to verify modeling assumptions…and at least one assumption fails

• A defect that can frequently arise in non-constant variance

• This can occur, for example, when the data follow a non-normal, skewed distribution

• The F-test in ANOVA is only slightly violated

• In such cases, a variance stabalizing transformation may be applied

Transformations

• Several transformations may be attemted:

– Y*=

– Y*=

– Y*=

Transformations

• Analyze the data on the Y* scale, choosing the transformation where:

– The simplest model results,

– There are no patterns in the residuals

– One can interpret the transformation

Example

• An engineer wishes to study the impact of 4 factors on the rate of advance of a drill. Each of the 4 factors (labeled A-D) were studied at 2 levels

A B C D Y -1 -1 -1 -1 1.68 +1 -1 -1 -1 1.98 -1 +1 -1 -1 3.28 +1 +1 -1 -1 3.44 -1 -1 +1 -1 4.98 +1 -1 +1 -1 5.70 -1 +1 +1 -1 9.97 +1 +1 +1 -1 9.07 -1 -1 -1 +1 2.07 +1 -1 -1 +1 2.44 -1 +1 -1 +1 4.09 +1 +1 -1 +1 4.53 -1 -1 +1 +1 7.77 +1 -1 +1 +1 9.43 -1 +1 +1 +1 11.75 +1 +1 +1 +1 16.30

Example

• Would like to fit an N-way ANOVA to these data (main effects and 2-factor interactions only)

• Model:

Example

Tests of Between-Subjects Effects

Dependent Variable: Y

257.614a 10 25.761 25.406 .001

606.144 1 606.144 597.781 .000

3.331 1 3.331 3.285 .130

43.494 1 43.494 42.894 .001

165.508 1 165.508 163.225 .000

20.885 1 20.885 20.597 .006

9.000E-02 1 9.000E-02 .089 .778

1.416 1 1.416 1.397 .290

2.839 1 2.839 2.800 .155

9.060 1 9.060 8.935 .030

.783 1 .783 .772 .420

10.208 1 10.208 10.067 .025

5.070 5 1.014

868.829 16

262.684 15

SourceCorrected Model

Intercept

A

B

C

D

A * B

A * C

A * D

B * C

B * D

C * D

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .981 (Adjusted R Squared = .942)a.

Example

Residuals vs. Predicted

Predicted Value for Y

1614121086420

Re

sid

ua

l fo

r Y

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

Example

Tests of Between-Subjects Effects

Dependent Variable: SQRTY

9.876a 10 .988 53.540 .000

88.512 1 88.512 4798.513 .000

.103 1 .103 5.610 .064

1.735 1 1.735 94.084 .000

7.011 1 7.011 380.070 .000

.688 1 .688 37.296 .002

2.269E-04 1 2.269E-04 .012 .916

1.683E-02 1 1.683E-02 .912 .383

5.731E-02 1 5.731E-02 3.107 .138

6.818E-02 1 6.818E-02 3.696 .113

3.726E-03 1 3.726E-03 .202 .672

.192 1 .192 10.409 .023

9.223E-02 5 1.845E-02

98.480 16

9.968 15

SourceCorrected Model

Intercept

A

B

C

D

A * B

A * C

A * D

B * C

B * D

C * D

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .991 (Adjusted R Squared = .972)a.

Example

Residuals vs. Predicted

Predicted Value for SQRTY

4.03.53.02.52.01.51.0

Re

sid

ua

l fo

r S

QR

TY

1.0.9.8.7.6.5.4.3.2.1.0

-.1-.2-.3-.4-.5-.6-.7-.8-.9

-1.0