# TM 720 - Lecture 06

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```					    TM 720 - Lecture 06

Multiple Comparisons,
7 Tools of Ishikawa

12/10/2011        TM 720: Statistical Process Control   1
Assignment:
•   Chapter 4.5, 5
•   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 3 & 4 Solutions
•   E-mail instructor with name and contact information for proctor.
(Distance students, only if haven’t responded. E-mail, preferred.)

12/10/2011           TM 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:
H0 : 1  2    k
H A : At least two of the means are not equal
12/10/2011   TM 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

12/10/2011          TM 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.

12/10/2011      TM 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.
12/10/2011   TM 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 µ1, µ2, ... , µk, and
a common variance σ2.
   Hypothesis to be tested is:
H0       :   1   2     k
HA       :        At least two of the means are not equal

12/10/2011         TM 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

   Fixed Effect Statistical Model:
y ij   i  e ij     i  e ij .
where eij are independent and identically distributed N(0,σ2).

   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.

12/10/2011         TM 720: Statistical Process Control   8
Definitions
   For a fixed effects model, if we restrict:
k


i 1
i   0

   Then
H0 :  1  2    k
H A : i   j for at least one pair (i, j)
is equivalent to:
H0 :  1   2     k  0
H A :  i  0 for at least one i

12/10/2011         TM 720: Statistical Process Control   9
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           T   

Mean        y1         y2          …            yi        …     yk           y


12/10/2011                TM 720: Statistical Process Control            10
Analysis of the Fixed Effects Model

   Sum of Squares Treatments:
The sum of squares treatments
Sum of Squares Errors (SSE)
is a measure of the variability
between the factor levels.
Factor level 1

   Error Sum of Squares:
The error sum of squares      Factor level 2

is a measure of the
variability within the        Factor level 3

factor levels.
X3 X1      X2

Sum of Squares Treatments (SSTr)

12/10/2011     TM 720: Statistical Process Control                   11
Analysis of the Fixed Effects Model

   P-values:
The plausibility of          BASE
the null hypothesis
(that the factor level       Larger
SSTr
means are all equal)
depends upon the             Smaller
relative size of the         SSTr
sum of squares for
Larger
treatments (SSTr) to         SSE
the sum of squares
for error (SSE).             Smaller
SSE

12/10/2011       TM 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   y ij  y  
k     n                   k   n                     k   n  2
y 
 y ij  kny 2   y ij 
2              2                   2

i 1 j 1                i 1 j 1           i 1 j 1   kn

Total Sum of Squares
SST

Treatment Sum of Squares               Error Sum of Squares
SSTr                                 SSE

12/10/2011                 TM 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)
k                          k                    k        2
y i2 y 
SSTr   n y i   y                ny i2  kny 2           
2

i 1                     i 1                i 1 n      kn
measures the variability between the factor levels,
   and the sum of squares for error (SSE)
k

 y i2
SSE   y ij  y i     y ij   ny i2    y ij 
k      n                   k       n         k          k       n
22                   2                               i 1

i 1 j 1                  i 1 j 1         i 1        i 1 j 1             k
measures the variability within the factor levels.
12/10/2011              TM 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 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).

12/10/2011    TM 720: Statistical Process Control   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

SSTr
MSTr 
k 1
and

SSE
MSE 
N k
12/10/2011         TM 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 µi, are all equal is calculated as

p-value = P(X  F)

where the F-statistic is:
MSTr
F
MSE
and the random variable X has a Fk-1, N - k distribution.

12/10/2011      TM 720: Statistical Process Control   17
Analysis of the Fixed Effects Model

Degrees of   Sum of         Mean
Source       Freedom      Squares        Squares          F-statistic   p-value
SSTr          MSTr
Treatments       k-1          SSTr        MSTr             F          P ( Fk 1, N  k  F )
k 1          MSE
SSE
Error            N-k          SSE         MSE 
N k

Total            N-1          SST

12/10/2011            TM 720: Statistical Process Control                      18
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.

12/10/2011    TM 720: Statistical Process Control   19
Percentage of Cotton
15      20     25    30       35

ANOVA Example
1       6     11    21       26
2       7     12    22       27
3       8     13    23       28
RANDOMIZATION                              4       9     14    24       29
PROCEDURE                                  5      10     15    25       30

Test Sequence   Run Number           Percentage of Cotton
1              8                        20
2             18                        30
3             10                        20
4             23                        35
5             17                        30
6              5                        15
7             14                        25
8              6                        20
9             15                        25
10            20                        30
.
.
.             .
.
.                        .
.
.
12/10/2011          TM 720: Statistical Process Control            20
ANOVA Example
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
Tensile Strength of Synthetic Fiber (lb/in2)
12/10/2011           TM 720: Statistical Process Control          21
ANOVA Example
Source of        Degrees of        Sum of
Variation         Freedom          Squares        Mean Square       F
% Cotton                                        475.76 =118.94   118.94 =14.8
5-1= 4            475.76
(Treatments)                                      4               8.06
161.20 =8.06
Error            25-5= 20          161.20
20
Total            25-1= 24          636.96

12/10/2011            TM 720: Statistical Process Control           22
Critical Points for the F-
Distribution Alpha = 0.05
DOF #2                                                        Degrees of Freedom #1 (v1)
(v2)     1      2      3      4      5      6      7      8      9       10     12       15   20   24     30     14     60    120    INF
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

12/10/2011                           TM 720: Statistical Process Control                                         23
ANOVA Example
Anova: Single Factor

SUMMARY
Groups             Count       Sum       Average Variance
15            5          49      9.8    11.2
20            5          77    15.4      9.8
25            5          88    17.6      4.3
30            5         108    21.6      6.8
35            5          54    10.8      8.2

ANOVA
Source of Variation     SS          df        MS        F    P-value   F crit
Between Groups          475.76             4 118.94   14.757   9E-06 2.866081
Within Groups            161.2            20   8.06

Total                   636.96            24

12/10/2011              TM 720: Statistical Process Control           24
ANOVA Example
Mean Fiber Strength

30

25
Tensile Strength (lb/in^2)

20

15

10

5

0
15       20            25            30    35
Percentage Cotton

12/10/2011                               TM 720: Statistical Process Control        25
Visual Tests of Comparison
   The visual tests are similar to a quick, down & dirty
Anova:
•   Based on a Dot Plot of data – direct from the measurements
•   Similar to Box & Whisker plot, but doesn’t obscure number of
observations
•   Easier to visualize than ANOVA, and easier to compute
•   Significance approaches .05 or better, if n is large enough!

   Reference:
•   Lenth, R. V. (1994.) Design, Data, an Deduction: An Introduction to
Experimental Design. Belmont, CA; Duxbury Press. pp. 82-88.

12/10/2011          TM 720: Statistical Process Control      26
Visual Tests of Comparison
   Steps:
• Plot a dot for each observation until all treatment
data are plotted
•   Plot the data distribution for each treatment side-by-side

• Count the number of left and right stragglers
•   Left stragglers are less than the larger of the two minima
•   Right stragglers are greater than the smaller of the two maxima

• Perform the test:
•   Tukey’s Quick Test         works better for larger data sets
•   Three-Straggler Rule       works better for smaller data sets
•   Modified Quick Test        works well across the board

12/10/2011            TM 720: Statistical Process Control         27
Visual Tests of Comparison
   Tukey’s Quick Test:
•   If the total number of stragglers is 8 or more, then the locations can be
judged statistically significant at the .05 level.
• Significance approaches .035, if sample size is larger.
   Three-Straggler Rule:
•   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
• Also has significance near .035 for large sample sizes
   Modified Quick Test:
•   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 is almost exactly .05
12/10/2011            TM 720: Statistical Process Control           28
Visual Tests of Comparison
   Tukey’s Quick Test:
•   If the total number of stragglers is 8 or more, then the locations can be
judged statistically significant at the .05 level.
• Significance approaches .035, if sample size is larger.
   Three-Straggler Rule:
•   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
• Also has significance near .035 for large sample sizes
   Modified Quick Test:
•   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 is almost exactly .05
12/10/2011            TM 720: Statistical Process Control           29
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

12/10/2011          TM 720: Statistical Process Control   30
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.

12/10/2011    TM 720: Statistical Process Control   31
Histogram of Lab 01 Results
Histograms - 4 Distributions

30

25

20
N(13,4)
N(12,2)
Frequency

N(12,4)
15                                                                                               N(14,4)
N(14,6)
Count
Sum
10

5

0
2      4     6   8   10      12       14          16     18      20   22   24   26   More
Bins

12/10/2011            TM 720: Statistical Process Control                        32
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

12/10/2011      TM 720: Statistical Process Control   33
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.
Man                   Method

Skill Level
Low RPM

Attention Level
Travel Limits
Dusty

Poor Conductor
Temperature       Humidity
Poor Mixing
Orifice Clogs
Poor Vendor
Worn Parts

Machine                  Material

12/10/2011                     TM 720: Statistical Process Control            34
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.

12/10/2011       TM 720: Statistical Process Control   35
Ishikawa’s Tools: Check Sheet
Title
   Check sheets are            Header Info: Date, Time, Location, Operator, etc.
used to collect data
(values or pieces of                              Times of Occurrence (periodic)

information) in a
consistent manner.
• List each of the         Types
of                      Raw Data recorded here
Type of
Error
known / possible         Errors                                                      Statistics
problems
• Record each
Time of Occurrence Statistics         Overall
occurrence                                                                           Statistics
including time-
Instructions, settings, comments, etc.
orientation.

12/10/2011          TM 720: Statistical Process Control                         36
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
12/10/2011         TM 720: Statistical Process Control       37
Ishikawa’s Tools: Pareto Chart
 80% of any problem is                                              Pareto Chart for Paint Defects
the result of 20% of the                          120                                                                                             120%
potential causes
• Histogram categories are
100                                                                                             100%

Cumulative %
80                                                                                              80%

Frequency
sorted by the magnitude of                     60                                                                                              60%
the bar
• A line graph is overlaid,                      40

20
40%

20%
and depicts the cumulative
proportion of defects                           0                                                                                              0%

Off-Color

Thin Coat
Tacky

Thick Coat
Abrasion
Dirt/Dust

Blisters

Wrong
• Quickly identifies where to

Color
focus efforts                                                                            Defect Type

12/10/2011      TM 720: Statistical Process Control                                                                        38
Use of Ishikawa’s Tools
   Removing          Statistical Quality Control and Improvement

Improving Process Capability and Performance
special causes
of variation                                                                            Continually Improve the System

   Preparation
for:                                                            Characterize Stable Process Capability

• hypothesis
tests
• control                              Time

charts
• process                                                                    Identify Special Causes - Bad (Remove)

Identify Special Causes - Good (Incorporate)
improvement
Reduce Variability

Center the Process
LSL    0     USL

12/10/2011           TM 720: Statistical Process Control                                                     39
Questions & Issues

12/10/2011   TM 720: Statistical Process Control   40

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