Transcript TM 720 Lecture 00: Slide Format

TM 720 - Lecture 12
Introduction to Designed
Experiments
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Assignment:


Reading:
•
Chapter 12
• Start reading
Assignment:
•
None.
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What is an Experiment?

Montgomery (2001):
•
A test or series of tests in which purposeful changes
are made to the input variables of a process or system
so that we may observe or identify the reasons for
changes that may be observed in the output response.

Strategic manipulation of a system in order to
observe and understand its’ response.

Usually, sequential experiments are better:
•
•
One at a time – misses interactions
All variable combinations – too expensive
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Some Examples of Elements in
Experimentation



Purpose:
•
•
•
Characterizing
Screening
Optimizing
Strategy:
•
•
•
One-factor-at-a-time
Comprehensive
Sequential
Design:
•
•
•
•
Simple Comparison
Response Surface methods
Factorial
Fractional Factorial
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Experimental Factors

Design Factors
• Design (varied) Factors
• Constant (held-constant) Factors
• Allowed to Vary Factors

Nuisance Factors
• Controllable
• Uncontrollable
• Noise
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Identifying Factors & Ranges




Experience
Team Approach
Fishbone Diagrams
•
Four M’s and an E
• Man
• Material
• Machine
• Method
• Environment
Trial/Pilot Runs
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Controlled Experimental Observation


Control
•
•
•
Blocking
Randomization
Replication
• Replication vs. Repeated Measures
Observation
•
•
•
Main Effects
Interaction Effects
Estimation
• Location
• Variation
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7 Steps of Designed Experiments
1.
2.
3.
4.
5.
6.
7.
Statement of problem
Selection of response variable
Choice of factors, levels, and ranges
Choice of experimental design
Perform the experiment protocol
Statistical analysis of the data
Conclusions & recommendations
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Can
do in
any
order
Example: Eye Drop
Effectiveness




Purpose: Determine better of two eye drops
Blocking Variable: Patients
Why Block
•
Variation (due to patients) is great, perhaps greater
than effect of medication
To Block
•
•
Assign one medicine to an eye, and the other
medicine to the other eye
Randomized variables? (left vs. right eye,…)
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Example: Gas Mileage (Octane)




Purpose: Mileage w/ Fuel Quality
Desired Blocking Variable: Time of Day
(Traffic Load)
Why Repeated Measures, Not Replications
•
•
Can’t empty tank and replace octane immediately
Variation (due to sequence of trips) is just giving
information on measurement accuracy for traffic
To Replicate
•
•
Repeat trip conditions (time, etc.) with both octanes
Lurking variables? (summer vs. academic year,…)
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Effects: Main


A Main Effect is the difference between
responses at different levels of a Design
Factor
Example: Intelligence Drug
• Design Factors
• Drug
• Student
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Yes
85%
100%
No
75%
90%
Avg
Good
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Effects: Interactions


An Interaction is the failure of a factor to
produce the same effect on the response
at different levels
Example: Intelligence Drug
• Design Factors
• Drug
• School
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Yes
75%
100%
No
45%
99%
HSU
SDSMT
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Some Terms

Primary Factors = Design Factors - manipulated levels

Treatments = Levels

Blocking - making comparisons under homogeneous
conditions

Replications - all actions required to set the
experimental conditions are taken for each observation.

Repeated Measures - observations that cannot be
randomized in order.
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Introduction to Comparisons

Comparisons usually look for an effect that is
comparably large with respect to the variation present

Visual Inference Testing

•
•
Dot Diagrams / Barcode Plots
Applications
•
•
Small Data Sets
Subjective?
Statistical Inference Testing
•
Applications
•
•
•
Large Data Sets
More powerful to find smaller effects
Objective?
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Visual Tests of Comparison

Stragglers are:
•
•
•

Left stragglers are observations less than the larger of
the two minima
Right stragglers are observations greater than the
smaller of the two maxima
Total number of stragglers = Left stragglers + Right
stragglers
Tukey’s Quick Test
•
Tukey, J. W. (1959)
If the total number of stragglers is 8 or more, then the
locations can be judged statistically significant at the .05
level
• Significance level is about .035 for larger sample sizes
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Visual Tests of Comparison

Three-Straggler Rule
•
Lenth, R. V. (1994)
If there are at least 3 left stragglers and at least 3
right stragglers, then the locations can be judged
statistically significant at the .05 level
• Should have at least 5 observations in each set
• Significance is about .035 for larger sample sizes

Modified Quick Test
•
Lenth, R. V. (1994)
Conclude a statistical difference in location if the total
number of stragglers is 8 or more, or if there are at
least 3 stragglers at each end.
• Significance level is almost exactly .05
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Ex 1: Popcorn Brand - Method



Purpose:
•
Determine the best process for popping corn seeds
Response Variable:
•
Number of un-popped seeds (50% unbroken by flower)
Factors:
•
•
•
Brand (design, two discrete levels)
•
•
Orville Redenbacher - Regular
Jolly Time - Yellow
Method (design, three discrete levels)
•
•
•
Microwave Bowl
Hot Air
Oil Skillet
Time (constant, continuous 2:00 minutes, except as noted)
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Experiment Data
Brand
Pilot Runs
1OR 2JT
Orville Redenbacher Regular
Jolly Time - Yellow
1
2
3
4
1
2
3
4
Method
M.W
4*
5*
26
18
7
37
5
9
14
34
H.A.
64*
18*
22
26
32
38
20
22
20
17
Oil
446*
10*
10
110
86
14
9
224
4
81
* Time was 2:30 (min:sec); otherwise, time was 2:00 min
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Run Sequence
Run Order (Brand)
Mthd PR1 PR2
E1
E2
E3
E4
E5
E6
E7
E8
MW
OR
4*
JT
5*
JT
5
JT
9
JT
14
OR
26
OR
18
OR
7
JT
34
OR
37
HA
OR
64*
JT
18*
OR OR
22 26
JT
20
JT
22
JT
20
OR OR
32 38
JT
17
Oil
OR
446*
JT
10*
JT
9
OR OR OR
10 110 86
JT OR
224 14
JT
4
* Time was 2:30 (min:sec); otherwise, time was 2:00 min
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JT
81
Basic Statistical Concept

Noise results in variation = Experimental Error
•
Should be unavoidable, certainly uncontrolled, and
indicates that the measured value is a Random Variable
(abbreviated r.v.).
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Definitions


Analysis-of-variance (ANOVA) is a
statistical method used to test
hypotheses regarding more than two
sample means.
For a one-factor experiment the
hypothesis tested is:
H 0 : 1   2    n
H1 : At least two of the means are not equal
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Definitions

The strategy in an analysis of variance is to compare the variability
between sample means to the variability within sample means. If
they are the same, the null hypothesis is accepted. If the variability
between is bigger than within, the null hypothesis is rejected.
Null
Hypothesis
Alternative
Hypothesis
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Definitions

An experimental unit is the item measured during an
experiment. The errors in these measurements are
described by random variables.

It is important that the error in measurement be the
same for all treatments (random variables must be
independent and have the same distribution).

The easiest way to assure the error is the same for all
treatments is to randomly assign experimental units to
treatment conditions.
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Definitions

The variable measured in an experiment is
called the dependent variable.

The variable manipulated or changed in an
experiment is called the independent
variable.

Independent variables are also called factors,
and the sample means within a factor are
called levels or treatments.
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Definitions



Random samples of size n are selected from each of k
different populations. The k different populations are
classified on the basis of a single criterion or factor.
(one-factor and k treatments)
It is assumed that the k populations are independent
and normally distributed with means µ1, µ2, ... , µk, and
a common variance 2.
Hypothesis to be tested is
H 0 : 1   2    k
H1 : At least two of the means are not equal
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Definitions


A fixed effects model assumes that the treatments have
been specifically chosen by the experimenter, and our
conclusions apply only to the levels chosen
Fixed Effect Statistical Model:
yij  i  eij     i  eij .


where eij is a iid N(0,2).
A random effects model assumes the treatments are
random samples from a larger population, and our
conclusions apply to the larger population in general.
Because the fixed effects model assumes that the
experiment is performed in a random manner, a one-way
ANOVA with fixed effects is often called a completely
randomized design.
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Definitions

For a fixed effects model, if we restrict:
k
 i  0
i 1

Then
H 0 : 1  2    k
H A : i   j
for at least one pair (i, j)
is equivalent to:
H 0 : 1   2     k  0
H A :  i  0 for at least one i
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Analysis of the Fixed Effects
Model
Treatment
1
2
…
i
…
k
y11
y21
…
yi1
…
yk1
y12
...
y22
...
…
yi2
...
…
yk2
...
y1n
y2n
…
yin
…
ykn
Total
T1

T2

…
Ti

…
Tk
Mean
y1

y2

…
yi

…
yk
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
T

y


28
Analysis of the Fixed Effects
Model

Sum of Squares
Treatments:
The sum of squares
treatments is a measure
of the variability between
the factor levels.
Sum of Squares Errors (SSE)
Factor level 1
Factor level 2

Error Sum of Squares:
The error sum of squares
is a measure of the
variability within the
factor levels.
Factor level 3
X3 X1
X2
Sum of Squares Treatments (SSTr)
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Analysis of the Fixed Effects
Model

P-values:
The believability of
the null hypothesis
(that the factor level
means are all equal)
depends upon the
relative size of the
sum of squares for
treatments (SSTr) to
the sum of squares
for error (SSE).
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BASE
Larger
SSTr
Smaller
SSTr
Larger
SSE
Smaller
SSE
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Analysis of the Fixed Effects
Model

Sum of Squares Partition for One Factor Layout:
In a one factor layout the total variability in the data
observations is measured by the total sum of squares SST
which is defined to be
k
n

SST    yij  y
i 1 j 1
  
k
2
n
i 1 j 1
yij2
 kny2
k
n

i 1 j 1
yij2
y2

kn
Total Sum of Squares
SST
Treatment Sum of Squares
SSTr
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Error Sum of Squares
SSE
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Analysis of the Fixed Effects
Model

Sum of Squares Partition for One Factor Layout:
This can be partitioned into two components
SST = SSTr + SSE,
where the sum of squares for treatments (SSTr)
k
k
k y2
i
SSTr  n yi  y 2  nyi2  kny2 
i 1
i 1
i 1 n



y2

kn
measures the variability between the factor levels, and the sum of
squares for error (SSE)
k
k
n

SSE    yij  yi
i 1 j 1
  
2
k
n
i 1 j 1
yij2
k

i 1
nyi2
k
n
 yi2
   yij2  i 1
k
i 1 j 1
measures the variability within the factor levels.
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Analysis of the Fixed Effects
Model

Sum of Squares Partition for One Factor
Layout:
On an intuitive level, the plausibility of the null
hypothesis that the factor level means µi are all
equal depends upon the relative size of the
sum of squares for treatments (SSTr) to the
sum of squares for error (SSE).
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Analysis of the Fixed Effects
Model

F-Test for One Factor Layout:
In a one factor layout with k levels and n replications gives a
total sample size kn = N, the treatments are said to have k - 1
degrees of freedom and the error is said to have N - k
degrees of freedom. Mean squares are obtained by dividing
a sum of squares by its respective degrees of freedom so
that
SSTr
MSTr 
k 1
and
SSE
MSE 
N k
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Analysis of the Fixed Effects
Model

F-Test for One Factor Layout:
A p-value for the null hypothesis that the factor level means
µi, are all equal is calculated as
p-value = P(X  F),
where the F-statistic is
F
MSTr
MSE
and the random variable X has an Fk-1, N -k distribution.
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Analysis of the Fixed Effects
Model
Source
Degrees of
Freedom
Sum of
Squares
Mean
Squares
Treatments
k-1
SSTr
MSTr 
Error
N-k
SSE
MSE 
Total
N-1
SST
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SSTr
k 1
F-statistic
F
MSTr
MSE
p-value
P ( Fk 1, N  k  F )
SSE
N k
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ANOVA Example

The tensile strength of a synthetic fiber used to
make cloth for men’s shirts is of interest to a
manufacturer. It is suspected that strength is
affected by the percentage of cotton in the fiber.

Five levels of cotton percentage are of interest:
15%, 20%, 25%, 30%, and 35%.

Five observations are to be taken at each level of
cotton percentage and the 25 total observations are
to be run in random order.
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ANOVA Example
RANDOMIZATION
PROCEDURE
Test Sequence
1
2
3
4
5
6
7
8
9
10
...
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Run Number
8
18
10
23
17
5
14
6
15
20
...
15
1
2
3
4
5
Percentage of Cotton
20
25
30
6
11
21
7
12
22
8
13
23
9
14
24
10
15
25
Percentage of Cotton
20
30
20
35
30
15
25
20
25
30
...
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38
35
26
27
28
29
30
ANOVA Example
Tensile Strength of Synthetic Fiber (lb/in2)
Percentage of Cotton
Observation
15
20
25
30
35
1
7
12
14
19
7
2
7
17
18
25
10
3
15
12
18
22
11
4
11
18
19
19
15
5
9
18
19
23
11
Total
49
77
88
108
54
Average
9.8
15.4
17.6
21.6
10.6
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ANOVA Example
Source of
Variation
Degrees of
Freedom
Sum of
Squares
5 -1= 4
475.76
475.76/4 = 118.94 118.94/8.06
= 14.75
Error
24 -4= 20
161.20
161.20/20 = 8.06
Total
5*5 -1= 24
636.96
% Cotton
(Treatments)
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Mean Square
TM 720: Statistical Process Control
F
40
Critical Points for the F-Distribution
Alpha = 0.05
Degrees of Freedom #1 (v1)
DOF #2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
14
60
120
INF
(v2)
1
161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88 243.90 245.95 248.02 249.05 250.10 245.36 252.20 253.25 254.30
2
18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.42 19.48 19.49 19.50
3
10.13 9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
8.74
8.70
8.66
8.64
8.62
8.71
8.57
8.55
8.53
4
7.71
6.94
6.59
6.39
6.26
6.16
6.09
6.04
6.00
5.96
5.91
5.86
5.80
5.77
5.75
5.87
5.69
5.66
5.63
5
6.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
4.74
4.68
4.62
4.56
4.53
4.50
4.64
4.43
4.40
4.37
6
5.99
5.14
4.76
4.53
4.39
4.28
4.21
4.15
4.10
4.06
4.00
3.94
3.87
3.84
3.81
3.96
3.74
3.70
3.67
7
5.59
4.74
4.35
4.12
3.97
3.87
3.79
3.73
3.68
3.64
3.57
3.51
3.44
3.41
3.38
3.53
3.30
3.27
3.23
8
5.32
4.46
4.07
3.84
3.69
3.58
3.50
3.44
3.39
3.35
3.28
3.22
3.15
3.12
3.08
3.24
3.01
2.97
2.93
9
5.12
4.26
3.86
3.63
3.48
3.37
3.29
3.23
3.18
3.14
3.07
3.01
2.94
2.90
2.86
3.03
2.79
2.75
2.71
10
4.96
4.10
3.71
3.48
3.33
3.22
3.14
3.07
3.02
2.98
2.91
2.85
2.77
2.74
2.70
2.86
2.62
2.58
2.54
11
4.84
3.98
3.59
3.36
3.20
3.09
3.01
2.95
2.90
2.85
2.79
2.72
2.65
2.61
2.57
2.74
2.49
2.45
2.41
12
4.75
3.89
3.49
3.26
3.11
3.00
2.91
2.85
2.80
2.75
2.69
2.62
2.54
2.51
2.47
2.64
2.38
2.34
2.30
13
4.67
3.81
3.41
3.18
3.03
2.92
2.83
2.77
2.71
2.67
2.60
2.53
2.46
2.42
2.38
2.55
2.30
2.25
2.21
14
4.60
3.74
3.34
3.11
2.96
2.85
2.76
2.70
2.65
2.60
2.53
2.46
2.39
2.35
2.31
2.48
2.22
2.18
2.13
15
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
2.54
2.48
2.40
2.33
2.29
2.25
2.42
2.16
2.11
2.07
16
4.49
3.63
3.24
3.01
2.85
2.74
2.66
2.59
2.54
2.49
2.42
2.35
2.28
2.24
2.19
2.37
2.11
2.06
2.01
17
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
2.45
2.38
2.31
2.23
2.19
2.15
2.33
2.06
2.01
1.96
18
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
2.41
2.34
2.27
2.19
2.15
2.11
2.29
2.02
1.97
1.92
19
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
2.38
2.31
2.23
2.16
2.11
2.07
2.26
1.98
1.93
1.88
20
4.35
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
2.35
2.28
2.20
2.12
2.08
2.04
2.22
1.95
1.90
1.84
21
4.32
3.47
3.07
2.84
2.68
2.57
2.49
2.42
2.37
2.32
2.25
2.18
2.10
2.05
2.01
2.20
1.92
1.87
1.81
22
4.30
3.44
3.05
2.82
2.66
2.55
2.46
2.40
2.34
2.30
2.23
2.15
2.07
2.03
1.98
2.17
1.89
1.84
1.78
23
4.28
3.42
3.03
2.80
2.64
2.53
2.44
2.37
2.32
2.27
2.20
2.13
2.05
2.01
1.96
2.15
1.86
1.81
1.76
24
4.26
3.40
3.01
2.78
2.62
2.51
2.42
2.36
2.30
2.25
2.18
2.11
2.03
1.98
1.94
2.13
1.84
1.79
1.73
25
4.24
3.39
2.99
2.76
2.60
2.49
2.40
2.34
2.28
2.24
2.16
2.09
2.01
1.96
1.92
2.11
1.82
1.77
1.71
26
4.23
3.37
2.98
2.74
2.59
2.47
2.39
2.32
2.27
2.22
2.15
2.07
1.99
1.95
1.90
2.09
1.80
1.75
1.69
27
4.21
3.35
2.96
2.73
2.57
2.46
2.37
2.31
2.25
2.20
2.13
2.06
1.97
1.93
1.88
2.08
1.79
1.73
1.67
28
4.20
3.34
2.95
2.71
2.56
2.45
2.36
2.29
2.24
2.19
2.12
2.04
1.96
1.91
1.87
2.06
1.77
1.71
1.65
29
4.18
3.33
2.93
2.70
2.55
2.43
2.35
2.28
2.22
2.18
2.10
2.03
1.94
1.90
1.85
2.05
1.75
1.70
1.64
30
4.17
3.32
2.92
2.69
2.53
2.42
2.33
2.27
2.21
2.16
2.09
2.01
1.93
1.89
1.84
2.04
1.74
1.68
1.62
40
4.08
3.23
2.84
2.61
2.45
2.34
2.25
2.18
2.12
2.08
2.00
1.92
1.84
1.79
1.74
1.95
1.64
1.58
1.51
60
4.00
3.15
2.76
2.53
2.37
2.25
2.17
2.10
2.04
1.99
1.92
1.84
1.75
1.70
1.65
1.86
1.53
1.47
1.39
120
3.92
3.07
2.68
2.45
2.29
2.18
2.09
2.02
1.96
1.91
1.83
1.75
1.66
1.61
1.55
1.78
1.43
1.35
1.26
INF
3.84
3.00
2.61
2.37
2.21
2.10
2.01
1.94
1.88
1.83
1.75
1.67
1.57
1.52
1.46
1.69
1.32
1.22
1.03
7/16/2015
TM 720: Statistical Process Control
41
ANOVA Example
Anova: Single Factor
SUMMARY
Groups
Count
Sum
15
20
25
30
35
5
5
5
5
5
ANOVA
Source of Variation
Between Groups
Within Groups
SS
475.76
161.2
Total
636.96
7/16/2015
Average Variance
49
9.8
11.2
77
15.4
9.8
88
17.6
4.3
108
21.6
6.8
54
10.8
8.2
df
MS
4 118.94
20
8.06
F
P-value
F crit
14.757
9E-06 2.866081
24
TM 720: Statistical Process Control
42
ANOVA Example
Mean Fiber Strength
30
Tensile Strength (lb/in^2)
25
20
15
10
5
0
15
20
25
30
35
Percentage Cotton
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TM 720: Statistical Process Control
43
Assumptions of ANOVA Models

Analysis of Variance models make the following
assumptions with regard to the underlying structure
of the data:
•
•
•

The error variance is a Normal random variable with
mean equal to zero and variance equal to 2.
The error variance is the same (homogeneous) for all
conditions.
The error variance is independent from trial to trial.
Violation of these assumptions can have only minor
effects or very large effects depending on the data
set and the assumption.
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TM 720: Statistical Process Control
44

Violations in the
assumptions of ANOVA
models are most often
uncovered through
examining the
residuals:
eij  yij  yˆ ij
 yij  y i 
7/16/2015
Residuals of ACCURACY-NUMBER OF ERRORS
Residuals
14
12
10
8
6
4
2
0
-2
-4
-6
-1
0
1
2
3
4
5
6
7
Fitted Values of ACCURACY-NUMBER OF ERRORS
TM 720: Statistical Process Control
45
8
9
Normality Assumption

The normality assumption can be evaluated by comparing
residuals with values that would be expected from a Normal
distribution.

If fewer residuals are available (more typical), then normal
probability plots can be used. A good approximation to the
expected value of the kth smallest observation in a random sample
of size n is:
  k  0.375 
MSE  z 


  n  0.25  

Not much can be done to correct for violations of this assumption.
However, ANOVA’s are very robust with respect to this
assumption.
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TM 720: Statistical Process Control
46
Equal Variance Assumption

The equal variance assumption is usually checked by
plotting the residuals versus the predicted or fitted value.
Characteristic patterns that indicate unequal variance are
cone-shaped:
Residual
Residual
Fitted Value
7/16/2015
Fitted Value
TM 720: Statistical Process Control
47
Equal Variance Example
50
40
30
Residual
20
10
0
430
440
450
460
470
480
490
500
510
520
-10
-20
-30
-40
Predicted Value
7/16/2015
TM 720: Statistical Process Control
48
Factorial Experiments

Experiments are often performed to investigate the
effects of two or more independent variables on a
single dependent variable.

The simplest experimental design to accomplish this is
called the factorial or full factorial experiment. When
employing this design, each complete trial or replication
is done at every possible combination of the
independent variables.

Factors arranged a full factorial design are often said to
be crossed.
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TM 720: Statistical Process Control
49
Main Effects and Interactions
A2 40.6
51
Factorial Experiment -- No Interaction
60
50
Response
Factor A
A1
Factor B
B1
B2
20
30
Factor B
40
B1
30
B2
20
10
0
A1
A2 40.6
14
45
40
35
30
25
20
15
10
5
0
Factor B
B1
B2
A1
7/16/2015
A2
Factorial Experiment -- Interaction
Response
Factor A
A1
Factor B
B1
B2
20
30
Factor A
TM 720: Statistical Process Control
Factor A
A2
50
The Two-Factor Factorial Design
Factor A
Factor B
2
…
1
b
1
y111, y112,
…, y11n
y121, y122,
…, y12n
y1b1, y1b2,
…, y1bn
2
y211, y212,
…, y21n
y221, y222,
…, y22n
y2b1, y2b2,
…, y2bn
a
ya11, ya12,
…, ya1n
ya21, ya22,
…, ya2n
yab1, yab2,
…, yabn
7/16/2015
TM 720: Statistical Process Control
51
The Two-Factor Factorial Design

Fixed Effect Statistical Model:
yijk     i   j   ij  e ijk .

where eijk is an iid N(0,2) random variable.
Hypotheses:
H 0 :1   2     a  0
H A : at least one   0
H 0 : 1   2     b  0
H 0 :  ij  0
for all i,j
H A : at least one  ij  0
H A : at least one   0
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TM 720: Statistical Process Control
52
The Two-Factor Factorial Design
Total Sum of Squares
SSTO
SS Treatment A
SSA
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SS Treatment B
SSB
SS Interaction AB
SSAB
TM 720: Statistical Process Control
SS Error
SSE
53
The Two-Factor Factorial Design

Sum of Squares:
2
y
2
  yijk
 
abn
i 1 j 1 k 1
a
SSTO
b
n
yi2 y2
SSA  

abn
i 1 bn
a
b
SSB  
j 1
a
y2j 
y2

an abn
b
SSAB  
i 1 j 1
yij2
2
i 
y2j 
y
y2



n i 1 bn j 1 an abn
a
b
SSE  SSTO  SSA  SSB  SSAB
7/16/2015
TM 720: Statistical Process Control
54
The Two-Factor Factorial Design
ANOVA Table
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F0
FCrit
A Main
Effect
a-1
SSA
SSA
a-1
MSA
MSE
F a ,a-1,ab(n-1)
B Main
Effect
b-1
SSB
SSB
b-1
MSB
MSE
Fa ,b-1,ab(n-1)
(a-1)(b-1)
SSAB
SSAB
(b-1)(a-1)
MSAB
MSE
Fa,(a-1)(b-1),ab(n-1)
Error
ab(n-1)
SSE
SSE
ab(n-1)
Total
abn-1
SSTO
AB
Interaction
7/16/2015
TM 720: Statistical Process Control
55
Two-Factor ANOVA Example 1

There are two different driving routes from the
factory to the port, Route 1 and Route 2, and
the time of the day when the truck leaves the
factory is classified as being either in the
morning, the afternoon, or the evening.

Driving route will be considered as Factor A
with a=2 levels and period of day will be
considered as Factor B with b=3 levels.
7/16/2015
TM 720: Statistical Process Control
56
Two-Factor ANOVA Example 1
Morning
Period of Day
Afternoon
Evening
Route 1
X111=490
X121=511
X131=435
X112=553
X122=490
X132=468
X113=489
X123=489
X133=463
(X1=483.2)
X114=504
X124=492
X134=450
X115=519
X125=451
X135=444
(X11=511.0) (X12=486.6) (X13=452.0)
Route 2
X211=485
X221=456
X231=406
X212=489
X222=460
X232=422
X213=475
X223=464
X233=459
(X2=460.6)
X214=470
X224=485
X234=442
X215=459
X225=473
X235=464
(X21=475.6) (X22=467.6) (X23=438.6)
(X1=493.3) (X2=477.1) (X3=445.3) (X=471.9)
7/16/2015
TM 720: Statistical Process Control
57
Two-Factor ANOVA Example 1
Route 1
Morning
Route 2
490
553
489
504
519
511
490
489
492
451
435
468
463
450
444
Afternoon
Evening
ANOVA
Source of Variation
Period of Day
Route
Interaction
Error
SS
11925.6
3830.7
653.6
8968.8
Total
25378.7
7/16/2015
485
489
475
470
459
456
460
464
485
473
406
422
459
442
464
df
2
1
2
24
MS
F
P-value F crit
5962.8 15.9561 3.9E-05 3.4028
3830.7 10.2507 0.00383 4.2597
326.8 0.8745 0.42994 3.4028
373.7
29
TM 720: Statistical Process Control
58
Two-Factor ANOVA Example 1
Period of Day
Ev
Af
Mo
Mean
445.3 477.1 493.3
Ev
445.3
0
-31.8
-48 p=3
Af
477.1
0 -16.2 p=2
Mo
493.3
0
y Evening
445.3
7/16/2015
y Afternoon
477.1
yMorning
493.3
TM 720: Statistical Process Control
59
Two-Factor ANOVA Example 1
520
Drive Time
500
480
Route 1
Route 2
460
440
420
400
Morning
Afternoon
Evening
500
490
480
Drive Time
Drive Time
Period of Day
470
460
450
440
430
420
Morning
Afternoon
Evening
485
480
475
470
465
460
455
450
445
Route 1
Period of day
7/16/2015
Route 2
Route
TM 720: Statistical Process Control
60
Two-Factor ANOVA Example 2



An experimenter is interested in evaluating the relative
effectiveness of three drugs (Factor B) in bringing about
behavioral changes in two categories schizophrenics and
depressives, of patients (Factor A).
What is considered to be a random sample of 9 patients
belonging to Category a1 (schizophrenics) is randomly
divided into three subgroups with three patients in each
subgroup. Each subgroup is then assigned to one of the
drug conditions. An analogous procedure is followed for 9
patients belonging to Category a2 (depressives).
Criterion ratings are made of the behavior of each subject
before and after the administration of the drugs.
7/16/2015
TM 720: Statistical Process Control
61
Two-Factor ANOVA Example 2
Drug b1
Drug b2
Drug b3
Category a1
8 4 0
10 8 6
8 6 4
Category a2
14 10 6
4 2 0
15 12 9
ANOVA
Source of Variation
Category
Drugs
Interaction
Error
Total
7/16/2015
SS
18
48
144
106
316
df
1
2
2
12
MS
F
P-value
18 2.0377358 0.1789399
24 2.7169811 0.1063435
72 8.1509434 0.0058103
8.833
17
TM 720: Statistical Process Control
62
Two-Factor ANOVA Example 2
14
Ratings Difference
12
10
8
Category A1
Category A2
6
4
2
0
Drug B1
7/16/2015
Drug B2
Drug B3
TM 720: Statistical Process Control
63
Two-Factor ANOVA Example 2
14
Ratings Difference
12
10
Drug B1
Drug B2
Drug B3
8
6
4
2
0
Category A1
7/16/2015
Category A2
TM 720: Statistical Process Control
64
The Two Factor Factorial
Effects Model

The Fixed Effects Model is:
yijk     i   j   ij  e ijk

Estimated parameters for the Effects Model are:
yˆ ijk  yij
ˆ  y
ˆi  yi  y
ˆ j  y j   y
( ) ij  yij  yi  y j   y
7/16/2015
TM 720: Statistical Process Control
65
Three-Factor Models and
Beyond


yijkl     i   j   k   ij   ik    jk   ijk  e ijkl
Model:
a
b
c
n
y2
2
Sum of Squares:
SS 
y 

TO
a
SSA  
i 1
abcn
b y2
c
yi2 y2
y2 k  y2
y2
 j 

, SSB  

, SSC  

,
bcn abcn
abcn
abcn
j 1 acn
k 1 abn
a
yij2
b
SSAB  
a
c
SSAC  
i 1 k 1
b
c
SSBC  
b
c
SSABC  
i 1 j 1 k 1
a

i 1
i 1
y2 jk 
2
yijk

n

2
yi2 b y j  y2


bcn j 1 acn abcn
yi2k  a yi2 c y2 k  y2



bn i 1 bcn k 1 abn abcn
j 1 k 1
a
a
cn
i 1 j 1
7/16/2015
ijkl
i 1 j 1 k 1 l 1
an
a
b
y2 j 
j 1
acn

b
 
i 1 j 1
yij2
cn
y2 k  y2

abn abcn
c

k 1
a
c
 
i 1 k 1
2
yi2k  b c y jk 
 
bn j 1 k 1 an
2
yi2 b y j  c y2 k  y2



bcn j 1 acn k 1 abn abcn
TM 720: Statistical Process Control
66
Three-Factor Models and Beyond
Source
df
SS
MS
A
a 1
SSA
MS A
E(MS)
bcn k2
2
 
B
b 1
SSB
MS B
2
C
c 1
SSC
MSC
2
b 1
abn  k2
c 1
cn  ij
2
AB
a  1b  1
SSAB
MS AB
AC
a  1c  1
SSAC
MS AC
BC
b  1c  1
SSBC
MS BC
ABC
a  1b  1c  1
SSABC
MS ABC
Error
abcn  1
SSE
MS E
Total
abcn 1
SSTO
7/16/2015
a 1
acn  k2
2
a  1b  1
2
bn  ik
2
 
a  1c  1
2
an    jk
2
 
b  1c  1
2
n   ijk
2
 
a  1b  1c  1
2
TM 720: Statistical Process Control
F
MS A
MS E
MS B
F0 
MS E
MSC
F0 
MS E
F0 
MS AB
M SE
MS AC
F0 
MS E
F0 
F0 
MS BC
MS E
F0 
MS ABC
MS E
67
Questions & Issues
7/16/2015
TM 720: Statistical Process Control
68