Design and Analysis of Experiments (3)
Analysis of Variance (AVOVA)
Kyung-Ho Park
Experiments with a Single-Factor :
The Analysis of Variance
Factor : Simple (temperature)
Level : 4 (140, 150, 160, 170℃)
Treatment
temperature
Hardness (kg/cm2)
140℃
Y11
Y12
Y13
Y14
150℃
Y21
Y21
Y21
Y21
160℃
Y31
Y32
Y33
Y34
170℃
Y41
Y42
Y43
Y44
Table. Typical Data for a Single Factor Experiment
yij    ti   ij , i  1,2,  , a,
j  1,2,  , n
yij: overall mean, μ: ith treatment effect, ε: a random error compound
yij  i   ij ,  ij  N (0,  2 )
 Hypotheses
H 0 : t1  t 2    t a  0
H1 : not H 0
Treatment
temperature
Hardness (kg/cm2)
yi
140℃
7
35.8
-0.175
1
36.9
+0.925
14
35.2
-0.775
2
36.0
+0.025
35.975
-3.375
150℃
10
40.2
-0.15
11
40.4
+0.05
5
39.6
-0.75
9
41.2
+0.85
40.350
1.000
160℃
13
42.0
-0.05
4
42.8
+0.75
3
41.5
-0.55
12
41.9
-0.15
42.050
2.7000
170℃
15
39.8
+0.775
8
38.8
-0.225
6
38.5
-0.525
16
39.0
-0.025
39.025
-0.325
y=39.350
SST  SSTreatment  SS E
a
n
SST   ( yij  y ) total sum of squares
i 1 j 1
a
SSTreatment  n ( yi  y ) 2
treatment of squares
i 1
a
n
SS E   ( yij  yi ) 2
i 1 j 1
error of squares
ANOVA Table
Source
Of Variation
Sum of
Squares
Degree of
Freedom
Treatments
SSTreatment
a-1
Error
SSE
a(n-1)
Total
SST
an-1
Mean Square
F0
MSTreatments
MSTreatments/ MSE
=SSTreatment/(a-1)
MSE
=SSE/[a(n-1)]
P
0.05
Factor : Simple (temperature)
Level : 4 (140, 150, 160, 170℃)
Randomization
Order of test
1. Calc > Random data >
Sample from Columns
2. Sample : 16
3. Enter: No, Temp
4. Store sample in: No, Temp
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Temp
140
140
140
140
150
150
150
150
160
160
160
160
170
170
170
170
No
2
4
11
10
7
15
1
14
8
5
6
12
9
3
13
16
Temp
140
140
160
160
150
170
140
170
150
150
150
160
160
140
170
170
Ex.3-1. Hardness of metal with quenching temperature
[Step 1]: Input Data
Order
No
Temp
Hardness
1
2
140
36.9
2
4
140
36.0
3
11
160
41.5
4
10
160
42.8
5
7
150
39.6
6
15
170
38.5
7
1
140
35.8
8
14
170
38.8
9
8
150
41.2
10
5
150
40.2
11
6
150
40.4
12
12
160
41.9
13
9
160
42.0
14
3
140
35.2
15
13
170
39.8
16
16
170
39.0
Ex.3-1. Hardness of metal with quenching temperature
[Step 2]: Stat > ANOVA > One-way
1. response : hardness
2. Factor : temp
• Graph > click
1. individual value plot
2. Boxplots of data
3. Residual plots > Four in one
Boxplot of Hardness by Temp
43
43
42
42
41
41
40
40
Hardness
Hardness
Individual Value Plot of Hardness vs Temp
39
38
39
38
37
37
36
36
35
35
140
150
160
Temp
Individual Value Plot
170
140
150
160
Temp
Boxplot
170
Residual Plots for Hardness
Residuals Versus the Fitted Values
99
1.0
90
0.5
Residual
Percent
Normal Probability Plot of the Residuals
50
10
1
-1.0
-0.5
0.0
Residual
0.5
-0.5
-1.0
1.0
Histogram of the Residuals
4
1.0
3
0.5
2
1
0
-0.75 -0.50 -0.25 0.00
0.25
Residual
0.50
0.75
36.0
37.5
39.0
Fitted Value
40.5
42.0
Residuals Versus the Order of the Data
Residual
Frequency
0.0
1.00
0.0
-0.5
-1.0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Observation Order
O rd er
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Tem p
2
4
11
10
7
15
1
14
8
5
6
12
9
3
13
16
140
140
160
160
150
170
140
170
150
150
150
160
160
140
170
170
H ard ness R E S I1
F ITS 1
36.9
0.925
35.975
36
0.025
35.975
41.5
-0.55
42.05
42.8
0.75
42.05
39.6
-0.75
40.35
38.5
-0.525
39.025
35.8
-0.175
35.975
38.8
-0.225
39.025
41.2
0.85
40.35
40.2
-0.15
40.35
40.4
0.05
40.35
41.9
-0.15
42.05
42
-0.05
42.05
35.2
-0.775
35.975
39.8
0.775
39.025
39
-0.025
39.025
[Step 2]: Results
One-way ANOVA: Hardness versus Temp
Source
DF
SS
Temp
3
79.145
Error
12
4.615
Total
15
83.760
S = 0.6201
Level
---140
150
160
170
N
4
4
4
4
MS
26.382
0.385
R-Sq = 94.49%
F
P
68.60 0.000
R-Sq(adj) = 93.11%
Individual 95% CIs For Mean Based on
Pooled StDev
Mean StDev ----+---------+---------+---------+-
35.975
40.350
42.050
39.025
0.704 (---*--)
0.661
(---*--)
0.545
(--*---)
0.556
(--*---)
----+---------+---------+---------+----36.0
38.0
40.0
42.0
Pooled StDev = 0.620
Regression Model
[Step 4]: Stat > Regression > Fitted Line Plot
1. response : hardness
2. Factor : temp
3. Type of Regression Model : Quadratic
• Graph > click
1. Standardized
2. Residual plots > Four in one
3. Residuals versus the variables : temp
• Options >click
1. Display confidence interval
2. Display prediction interval
3. Confidence level : 95.0
Fitted Line Plot
Hardness = - 419.6 + 5.844 Temp
- 0.01850 Temp**2
Regression
95% CI
95% PI
43
42
S
R-Sq
R-Sq(adj)
40
39
38
37
36
35
34
140
145
150
155
Temp
160
165
170
Residuals Versus Temp
(response is Hardness)
2
Standardized Residual
Hardness
41
1
0
-1
-2
140
145
150
155
Temp
160
165
170
0.647807
93.5%
92.5%
Residual Plots for Hardness
Normal Probability Plot of the Residuals
Standardized Residual
99
90
Percent
Residuals Versus the Fitted Values
50
10
1
-2
-1
0
1
Standardized Residual
2
2
1
0
-1
-2
35.0
Histogram of the Residuals
Standardized Residual
Frequency
2
1
-2
-1
0
1
Standardized Residual
42.5
Residuals Versus the Order of the Data
3
0
37.5
40.0
Fitted Value
2
2
1
0
-1
-2
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Observation Order
Polynomial Regression Analysis: Hardness versus Temp
The regression equation is
Hardness = - 419.6 + 5.844 Temp - 0.01850 Temp**2
S = 0.647807
R-Sq = 93.5%
R-Sq(adj) = 92.5%
Analysis of Variance
Source
Regression
Error
Total
DF
2
13
15
SS
MS
F
P
78.3045 39.1522 93.30 0.000
5.4555 0.4197
83.7600
Sequential Analysis of Variance
Source
DF
SS
F
P
Linear
1 23.5445 5.47 0.035
Quadratic 1 54.7600 130.49 0.000
Experiments with Two Factors :
The Analysis of Variance
Factors : A, B
 No replication test
 No information on interaction between factors
 y = f(A, B)
 Replication test
 show the effect of interaction
 y = f(A, B, AB)
Tests of the effect of reaction time and pressure on the strength
of product
time
pressure
100 psi
110 psi
120 psi
130 psi
100 min
105 min
110 min
305
322
320
302
325
322
335
350
342
337
348
344
366
326
338
364
324
336
372
330
348
374
330
348
[Step 1]: Input Data
tim e
P ressure S trength
100
100
305
100
100
302
100
110
335
100
110
337
100
120
366
100
120
364
100
130
372
100
130
374
105
100
322
105
100
325
105
110
350
105
110
348
105
120
326
105
120
324
105
130
330
105
130
330
110
100
320
110
100
322
110
110
342
110
110
344
110
120
338
110
120
336
110
130
348
110
130
348
[Step 2]: Stat > ANOVA > Two - Way
1. Response : strength
2. Row factor: time √ Display means
3. Column factor : pressure √ Display means
4. Confidence level: 95.0
• Graph > click
1. √ Individual value plot
2. Residual plots: √ four in one
Two-way ANOVA: Strength versus time, Pressure
Source
DF
time
2
Pressure
3
Interaction 6
Error
12
Total
23
S = 1.443
SS
629.08
4059.33
3565.92
25.00
8279.33
R-Sq = 99.70%
MS
F
P
314.54 150.98 0.000
1353.11 649.49 0.000
594.32 285.27 0.000
2.08
R-Sq(adj) = 99.42%
Individual 95% CIs For Mean Based on
Pooled StDev
time
Mean ---+---------+---------+---------+-----100 344.375
(--*--)
105 331.875 (--*-)
110 337.250
(--*--)
---+---------+---------+---------+-----332.0
336.0
340.0
344.0
Individual 95% CIs For Mean Based on
Pooled StDev
Pressure
Mean -----+---------+---------+---------+---100
316.000 (*)
110
342.667
(-*)
120
342.333
(*-)
130
350.333
(*-)
-----+---------+---------+---------+---320
330
340
350
Individual Value Plot of Strength vs time, Pressure
380
370
Strength
360
350
340
330
320
310
300
Pressure
time
100 110 120 130
100
100 110 120 130
105
100 110 120 130
110
Residual Plots for Strength
Normal Probability Plot of the Residuals
Residuals Versus the Fitted Values
99
1
Residual
Percent
90
50
10
1
0
-1
-2
-1
0
Residual
1
2
300
Histogram of the Residuals
320
340
Fitted Value
360
380
Residuals Versus the Order of the Data
1
6
Residual
Frequency
8
4
2
0
0
-1
-1.6
-0.8
0.0
Residual
0.8
1.6
2
4
6
8 10 12 14 16 18 20 22 24
Observation Order
[Step 2]: Stat > ANOVA > Analysis of Means
1. Response : strength
2. √ Normal
- Factor 1: time
- Factor 2: pressure
Two-Way ANOM for Strength by time, Pressure
Alpha = 0.05
Interaction Effects
Effect
20
2.54
0
-2.54
0
-20
Pressure 100
time 100
110
120
130
100
105
110
120
Main Effects for time
110
120
130
Main Effects for Pressure
344
340
338.95
337.83
336.72
335
Mean
Mean
100
110
356
345
330
130
339.28
337.83
336.38
332
320
100
105
time
110
100
110
120
Pressure
130
[Step 3]: Stat > ANOVA > Interval plot (Interval of mean)
1. √ Multiple Y’s (with group)
2. Graph variable: Strength
3. Categorical variable : C2-C2
Interval Plot of Strength vs time, Pressure
95% CI for the Mean
400
Strength
375
350
325
300
Pressure
time
100 110 120 130
100
100 110 120 130
105
100 110 120 130
110
[Step 3]: Stat > ANOVA > Main effect plot
1. Response : strength
2. Factors : time, pressure
Main Effects Plot (data means) for Strength
time
Pressure
350
Mean of Strength
345
340
335
330
325
320
315
100
105
110
100
110
120
130
[Step 3]: Stat > ANOVA > Interactions effect
1. Response : strength
2. Factors : time, pressure
Interaction Plot (data means) for Strength
380
time
100
105
110
370
360
Mean
350
340
330
320
310
300
100
110
120
Pressure
130