ENGM 720 - Lecture 06

Multiple Comparisons, 7 Tools of Ishikawa

4/29/2020 ENGM 720: Statistical Process Control 1

Assignment:

  Reading: • Chapter 4.5, Chapter 5 – 5.2, 5.4

• Finish reading • Review for Exam I • Covers material through hypothesis tests and seven tools Assignments: • Obtain the Hypothesis Test (Chart &) Tables • Access Previous Assignment Solutions & Prepare Notebook: • Download Assignment 4 & Assignment 4 Solutions 4/29/2020 ENGM 720: Statistical Process Control 2

Multiple Comparisons



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    

k

H

A

:

At least two of the means are not equal

4/29/2020 ENGM 720: Statistical Process Control 3

Multiple Comparisons

 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 4/29/2020 ENGM 720: Statistical Process Control 4

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 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.

4/29/2020 ENGM 720: Statistical Process Control 5

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

.

4/29/2020 ENGM 720: Statistical Process Control 6

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 µ 1 , µ 2 , ... , µ k , and a common variance σ 2 .

Hypothesis to be tested is:

H

H 0 A : :  1   2    

k

At least two of the means are not equal 4/29/2020 ENGM 720: Statistical Process Control 7

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 Overall Mean  Fixed Effect Statistical Model: Observed Value

y ij

 

i

 e

ij

 where e

ij

  

i

 e

ij

.

i

th Treatment Effect Error in Measurement are independent and identically distributed N(0,

σ 2

).

 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 .

4/29/2020 ENGM 720: Statistical Process Control 8

Definitions

 For a fixed effects model, if we restrict:

i k

  1 

i

 0  Then

H

0 :

H A

:  1 

i

   2 

j

   

k for at least one pair (i, j)

is equivalent to:

H

0 :

H A

:  1 

i

  2  0    

k

 0

for at least one i

4/29/2020 ENGM 720: Statistical Process Control 9

Analysis of the Fixed Effects Model

Total Mean 1 y 11 y 12 ...

y 1n T 1  y 1  2 y 21 y 22 ...

y 2n T 2  y 2 

Treatment

… … … y i i1 y i2 ...

… … … y in T i  y i  … … … … … … k y k1 y k2 ...

y kn T k  y k  T  y  4/29/2020 TM 720: Statistical Process Control 10

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

Sum of Squares Errors: The error sum of squares

Factor level 2

is a measure of the variability

within

the factor levels.

Factor level 3 X 3



X 1



X 2



Sum of Squares Treatments (SSTr)

4/29/2020 ENGM 720: Statistical Process Control 11

Analysis of the Fixed Effects Model

 P-values: The plausibility 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 errors (SSE).

BASE Larger SSTr Smaller SSTr Larger SSE Smaller SSE 4/29/2020 ENGM 720: Statistical Process Control 12

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

SST



i

 1

j n k

  1 

y ij



y

   2

k n

 

i

 1

j

 1

y ij

2 

kn y

2   

i

 1

j n k

  1

y ij

2 

y

2  

kn

Total Sum of Squares SST Treatment Sum of Squares SSTr

4/29/2020

Error Sum of Squares SSE

ENGM 720: Statistical Process Control 13

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)

SSTr



i k

  1

n



y i

 

y

   2 

i k

  1

n y i

2  

kn y

 2  

i k

  1

y i

2 

n



y

 2 

kn

measures the variability between the factor levels, and the sum of squares for error (SSE)

SSE



i

 1

j n k

  1 

y ij



y i

  2

k

 

i

 1

j n

  1

y ij

2 

i k

  1

n y i

2  

i k

  1

j n

  1

y ij

2 

i k

  1

y i

2 

k

measures the variability within the factor levels. 4/29/2020 ENGM 720: Statistical Process Control 14

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 (µ i ) are all equal depends upon the relative size of the sum of squares for treatments ( SSTr ) to the sum of squares for error ( SSE ) Definitely NOT Likely the Same 4/29/2020 Possibly Likely the Same ENGM 720: Statistical Process Control Definitely VERY Likely the Same 15

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

MSTr



SSTr k

 1 and 4/29/2020

MSE



SSE N



k

ENGM 720: Statistical Process Control 16

Analysis of the Fixed Effects Model

 F-Test for One Factor Layout: A p-value for the null hypothesis that the factor level means µ i , 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 a F k-1, N - k distribution.

4/29/2020 ENGM 720: Statistical Process Control 17

Analysis of the Fixed Effects Model

Source Treatments Degrees of Freedom k-1 Sum of Squares SSTr Error N-k SSE Total N-1 SST Mean Squares

MSTr



SSTr k

 1

MSE



SSE N



k

F-statistic

F



MSTr MSE

p-value

P

(

F k

 1 ,

N



k



F

) 4/29/2020 ENGM 720: Statistical Process Control 18

ANOVA Example

 The tensile strength of a synthetic fiber used to make cloth for 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.

4/29/2020 ENGM 720: Statistical Process Control 19

ANOVA Example

RANDOMIZATION PROCEDURE Test Sequence 1 2 3 4 5 6 7 8 9 10 ...

Run Number 8 18 10 23 17 5 14 6 15 20 ...

4/29/2020 15 1 2 3 4 5 Percentage of Cotton 20 25 30 6 7 11 12 21 22 8 9 10 13 14 15 23 24 25 35 26 27 28 29 30 Percentage of Cotton 20 30 20 35 30 15 25 20 25 30 ...

ENGM 720: Statistical Process Control 20

ANOVA Example

Percentage of Cotton Observation 1 4 5 2 3 15 7 7 15 11 9 20 12 17 12 18 18 25 14 18 18 19 19 30 19 25 22 19 23 Total Average 4/29/2020 49 77 88 108 9.8

15.4

17.6

21.6

Tensile Strength of Synthetic Fiber (lb/in 2 )

ENGM 720: Statistical Process Control 35 7 10 11 15 11 54 10.6

21

ANOVA Example

Source of Variation % Cotton

(Treatments)

Error Total Degrees of Freedom 5-1= 4 25-5= 20 25-1= 24 Sum of Squares 475.76

161.20

Mean Square 475.76 =118.94

4 161.20 =8.06

20 F 118.94 =14.8

8.06

636.96

4/29/2020 ENGM 720: Statistical Process Control 22

Critical Points for the F Distribution

Alpha = 0.05

DOF #2 (v2) 1 2 3 4 5 6 7 8 9 10 26 27 28 29 30 40 60 120 INF 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 4.84

4.75

4.67

4.60

4.54

4.49

4.45

4.41

4.38

4.35

4.32

4.30

4.28

4.26

4.24

1 2 3 4 5 6 7 8 Degrees of Freedom #1 (v1) 9 10 12 15 20 24 30 14 60 120 INF 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

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

10.13

7.71

6.61

5.99

5.59

5.32

5.12

4.96

9.55

6.94

5.79

5.14

4.74

4.46

4.26

4.10

9.28

6.59

5.41

4.76

4.35

4.07

3.86

3.71

9.12

6.39

5.19

4.53

4.12

3.84

3.63

3.48

9.01

6.26

5.05

4.39

3.97

3.69

3.48

3.33

8.94

6.16

4.95

4.28

3.87

3.58

3.37

3.22

8.89

6.09

4.88

4.21

3.79

3.50

3.29

3.14

8.85

6.04

4.82

4.15

3.73

3.44

3.23

3.07

8.81

6.00

4.77

4.10

3.68

3.39

3.18

3.02

8.79

5.96

4.74

4.06

3.64

3.35

3.14

2.98

8.74

5.91

4.68

4.00

3.57

3.28

3.07

2.91

8.70

5.86

4.62

3.94

3.51

3.22

3.01

2.85

8.66

5.80

4.56

3.87

3.44

3.15

2.94

2.77

8.64

5.77

4.53

3.84

3.41

3.12

2.90

2.74

8.62

5.75

4.50

3.81

3.38

3.08

2.86

2.70

8.71

5.87

4.64

3.96

3.53

3.24

3.03

2.86

8.57

5.69

4.43

3.74

3.30

3.01

2.79

2.62

8.55

5.66

4.40

3.70

3.27

2.97

2.75

2.58

8.53

5.63

4.37

3.67

3.23

2.93

2.71

2.54

4.23

4.21

4.20

4.18

4.17

4.08

4.00

3.92

3.84

3.98

3.89

3.81

3.74

3.68

3.63

3.59

3.55

3.52

3.49

3.47

3.44

3.42

3.40

3.39

3.37

3.35

3.34

3.33

3.32

3.23

3.15

3.07

3.00

3.59

3.49

3.41

3.34

3.29

3.24

3.20

3.16

3.13

3.10

3.07

3.05

3.03

3.01

2.99

2.98

2.96

2.95

2.93

2.92

2.84

2.76

2.68

2.61

3.36

3.26

3.18

3.11

3.06

3.01

2.96

2.93

2.90

2.87

2.84

2.82

2.80

2.78

2.76

2.74

2.73

2.71

2.70

2.69

2.61

2.53

2.45

2.37

3.20

3.11

3.03

2.96

2.90

2.85

2.81

2.77

2.74

2.71

2.68

2.66

2.64

2.62

2.60

2.59

2.57

2.56

2.55

2.53

2.45

2.37

2.29

2.21

3.09

3.00

2.92

2.85

2.79

2.74

2.70

2.66

2.63

2.60

2.57

2.55

2.53

2.51

2.49

2.47

2.46

2.45

2.43

2.42

2.34

2.25

2.18

2.10

3.01

2.91

2.83

2.76

2.71

2.66

2.61

2.58

2.54

2.51

2.49

2.46

2.44

2.42

2.40

2.39

2.37

2.36

2.35

2.33

2.25

2.17

2.09

2.01

2.95

2.85

2.77

2.70

2.64

2.59

2.55

2.51

2.48

2.45

2.42

2.40

2.37

2.36

2.34

2.32

2.31

2.29

2.28

2.27

2.18

2.10

2.02

1.94

2.90

2.80

2.71

2.65

2.59

2.54

2.49

2.46

2.42

2.39

2.37

2.34

2.32

2.30

2.28

2.27

2.25

2.24

2.22

2.21

2.12

2.04

1.96

1.88

2.85

2.75

2.67

2.60

2.54

2.49

2.45

2.41

2.38

2.35

2.32

2.30

2.27

2.25

2.24

2.22

2.20

2.19

2.18

2.16

2.08

1.99

1.91

1.83

2.79

2.69

2.60

2.53

2.48

2.42

2.38

2.34

2.31

2.28

2.25

2.23

2.20

2.18

2.16

2.15

2.13

2.12

2.10

2.09

2.00

1.92

1.83

1.75

2.72

2.62

2.53

2.46

2.40

2.35

2.31

2.27

2.23

2.20

2.18

2.15

2.13

2.11

2.09

2.07

2.06

2.04

2.03

2.01

1.92

1.84

1.75

1.67

2.65

2.54

2.46

2.39

2.33

2.28

2.23

2.19

2.16

2.12

2.10

2.07

2.05

2.03

2.01

1.99

1.97

1.96

1.94

1.93

1.84

1.75

1.66

1.57

2.61

2.51

2.42

2.35

2.29

2.24

2.19

2.15

2.11

2.08

2.05

2.03

2.01

1.98

1.96

1.95

1.93

1.91

1.90

1.89

1.79

1.70

1.61

1.52

2.57

2.47

2.38

2.31

2.25

2.19

2.15

2.11

2.07

2.04

2.01

1.98

1.96

1.94

1.92

1.90

1.88

1.87

1.85

1.84

1.74

1.65

1.55

1.46

2.74

2.64

2.55

2.48

2.42

2.37

2.33

2.29

2.26

2.22

2.20

2.17

2.15

2.13

2.11

2.09

2.08

2.06

2.05

2.04

1.95

1.86

1.78

1.69

2.49

2.38

2.30

2.22

2.16

2.11

2.06

2.02

1.98

1.95

1.92

1.89

1.86

1.84

1.82

1.80

1.79

1.77

1.75

1.74

1.64

1.53

1.43

1.32

2.45

2.34

2.25

2.18

2.11

2.06

2.01

1.97

1.93

1.90

1.87

1.84

1.81

1.79

1.77

1.75

1.73

1.71

1.70

1.68

1.58

1.47

1.35

1.22

1.69

1.67

1.65

1.64

1.62

1.51

1.39

1.26

1.03

2.41

2.30

2.21

2.13

2.07

2.01

1.96

1.92

1.88

1.84

1.81

1.78

1.76

1.73

1.71

4/29/2020 ENGM 720: Statistical Process Control 23

ANOVA Example

Anova: Single Factor SUMMARY

Groups

15 20 25 30 35

Count

5 5 5 5 5

Sum

49 77 88 108 54

Average Variance

9.8

15.4

17.6

21.6

10.8

11.2

9.8

4.3

6.8

8.2

ANOVA

Source of Variation

Between Groups Within Groups Total

SS

475.76

161.2

636.96

df

4 20

MS

118.94

8.06

F

14.757

P-value F crit

9E-06 2.866081

24 4/29/2020 ENGM 720: Statistical Process Control 24

ANOVA Example

Mean Fiber Strength

30 25 20 15 10 5 0 4/29/2020 15 20 25

Percentage Cotton

ENGM 720: Statistical Process Control 30 35 25

Ishikawa’s “Magnificent Seven” Tools

 The

Seven Tools

are: • Histogram / Stem & Leaf Diagram • • • • • • Cause & Effect (Fishbone) Diagram Defect Concentration Diagram Check Sheet Scatter (Plot) Diagram Pareto Chart Control Chart

- not covered on exam!

 The tools were not invented by Ishikawa, but were very successfully put into methodical use by him  The first six are used before starting to use the seventh • They are also reused when needed to find an assignable cause 4/29/2020 ENGM 720: Statistical Process Control 26

Ishikawa’s Tools: Histogram

 A histogram is a bar chart that takes the shape of the distribution of the data. The process for creating a histogram depends on the purpose for making the histogram.

• One purpose of a histogram is to see the shape of a distribution. To do this, we would like to have as much data as possible, and use a fine resolution.

• A second purpose of a histogram is to observe the frequency with which a class of problems occurs. The resolution is controlled by the number of problem classes. 4/29/2020 ENGM 720: Statistical Process Control 27

Histogram of Lab 01 Results

Histograms - 4 Distributions

30 15 10 25 20 5 0 2 4 6 8 10 12 14

Bins

16 18 20 22 24 26 More 4/29/2020 ENGM 720: Statistical Process Control 28 N(13,4) N(12,2) N(12,4) N(14,4) N(14,6) Count Sum

Ishikawa’s Tools: Fishbone Diagram

 Cause & Effect diagram constructed by brainstorming • Identified problem at the “head” • Connects potential causes along the spine • Sub causes are listed along the major “bones” •

Man

•

Material

•

Method

•

Machine

•

Environment

4/29/2020 ENGM 720: Statistical Process Control 29

Cause & Effect Diagram, Cont.

 The purpose of the cause and effect diagram is to obtain as many potential influencers of a process, so that the problem solving can take a more directed approach.

Skill Level Man Attention Level Dusty Environment Temperature Humidity Orifice Clogs Worn Parts Machine Method Low RPM Travel Limits Poor Conductor Poor Mixing Poor Vendor Material Bad Paint 4/29/2020 ENGM 720: Statistical Process Control 30

Ishikawa’s Tools: Defect Diagram



A defect concentration diagram graphically records the frequency of a defect with respect to product location.

•

Obtain a digital photo or multi-view part print showing all product faces.

•

Operator tallies the number and location of defects as they occur on the diagram.

4/29/2020 ENGM 720: Statistical Process Control 31

Ishikawa’s Tools: Check Sheet



Check sheets are used to collect data (values or pieces of information) in a consistent manner.

•

List each of the known / possible

•

problems Record each occurrence including time orientation.

Title Header Info: Date, Time, Location, Operator, etc.

Times of Problem Occurrence (periodic) List of Prob Types Raw Data recorded here Time of Occurrence Statistics Instructions, settings, comments, etc.

Statistics For Prob Types Overall Statistics 4/29/2020 ENGM 720: Statistical Process Control 32

Ishikawa’s Tools: Scatter Plot

 A scatter plot shows the relationship between any two variables of interest: • Plot one variable along the X-axis and the other along the Y-axis Y Y Y X X X • The presence of a relationship can be inferred or ruled out, but it cannot determine if a cause and effect relationship exists 4/29/2020 ENGM 720: Statistical Process Control 33

Ishikawa’s Tools: Pareto Chart

 80% of any problem is the result of 20% of the potential causes • Histogram categories are sorted by the magnitude of the bar • A line graph is overlaid, and depicts the cumulative proportion of defects • Quickly identifies where to focus efforts 120 100 80 60 40 20 0

Pareto Chart for Paint Defects Defect Type

120% 100% 80% 60% 40% 20% 0% 4/29/2020 ENGM 720: Statistical Process Control 34

Use of Ishikawa’s Tools



Removing special causes of variation



Preparation for:

•

hypothesis tests

•

control charts

•

process improvement Statistical Quality Control and Improvement

Time Improving Process Capability and Performance Continually Improve the System Characterize Stable Process Capability Head Off Shifts in Location, Spread Identify Special Causes - Bad (Remove) Identify Special Causes - Good (Incorporate) Reduce Variability Center the Process LSL  0 USL 4/29/2020 ENGM 720: Statistical Process Control 35

Questions & Issues

4/29/2020 ENGM 720: Statistical Process Control 36