stat 470-6 today: 2-way anova (section 2.3)…2.3.1 and 2.3.2; transformation of the response
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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