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## Statistically Quality Design

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### Title: FULL FACTORIAL DESIGNED EXPERIMENT Author: Jrmark Last modified by: NCKU Created Date: 7/3/2002 8:09:14 AM Document presentation format: – PowerPoint PPT presentation

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Title: Statistically Quality Design

1
Statistically Quality Design
• Topics
• Four perspectives in quality improvement
• Review DOE topics and terminologies
• Implementation plan and procedure for
experimental design
• Full factorial design and Yates algorithm
• Full factorial design example improving wave
solder process at TDY company
• Concepts and examples for conducting Block and
Latin square designs

2
Four Perspectives in Quality Improvement
• Downstream Perspective Customers Quality
• such as fuel consumption, noise, failure rates,
pollution, etc.
• Midstream Perspective Manufacturing Quality
(spec. drawings)
• important for production or trading.
• Upstream Quality of Design (robustness of
objective function)
• good for design development after product
planning
• Origin Quality of Technology (robustness of
technology )
• good for technology development prior to product
planning
• functionality of generic function
• example Hooks Law for spring
• Quality Engineering
• ????????????????????????????/??

3
Characteristics of Technology Development
• Technology Readiness (???) ???????????

• ????????????
• Flexibility (???)????????????????
• Reproducibility (???) R D Manufacturing
Market

4
Statistically Quality Design
• Topics
• Four perspectives in quality improvement
• Review DOE topics and terminologies
• Implementation plan and procedure for
experimental design
• Full factorial design and Yates algorithm
• Full factorial design example improving wave
solder process at TDY company
• Concepts and examples for conducting Block and
Latin square designs

5
Design of Experiments (Topics)
• CRD
• completely randomized design
• Full Factorial
• all possible combinations of factors and levels
• Fractional Factorial
• assume some interaction will not occur, a factor
is assigned
• Latin Square
• each level of each factor appears only once with
each level of two other factors
• Yates Notation Algorithm
• On-line V.S. Off-line Quality Control

6
Design of Experiments (Topics cont.)
• System Design
• the selection of materials, parts, equipment, and
process parameters
• Parameter Design
• study the effect of noise factor in DOE
• Tolerance Design
• the specification of appropriate tolerance,
product and process parameters
• Signal to Noise Ratio (S/N Ratio)
• a comparison of the influence of control factors
(signal) to that of noise factors
• Orthogonal Array
• a design where correlation between factors is
zero

7
Experimental Design Terminology
• ANOVA
• Analysis of Variance
• Experimental Unit
• largest collection of experiment material
• Treatment
• what is done to the experiment materials
• Sampling Unit
• a part of experimental unit

8
Experimental Design Terminology (cont.)
• Experimental/ Sampling Errors
• a measure of variation
• Randomization
• a system of using random number
• Replication
• number of times a specific combination of factor
level is run during an experiment
• Factor
• an input to a process produces an effect.
controllable factors vs. noise factors
• Level
• a setting or value of a factor
• Run
• number of trials for each condition of an
experiment

9
Experimental Design Terminology (cont.)
• Quality Characteristic
• the response variable (output)
• Interaction
• the combination of two factors generates a result
that is different from individual factor.
• main effect vs. interaction effect
• DOF
• independent piece of information
• Resolution
• number of letters in the shortest length in
defining relation
• the lower the number, the more saturated the
design is

10
ANOVA (?????)
• The method of analyzing data collected by CRD /
RCBD
• ANOVA equation
• ANOVA Table

gt SST SStrt
SSE
source of variation d.f. SS MS F
Between Trt. a -1
Error n a
Total N - 1
11
ANOVA
• Example
• ???????????????????,???????????????,????5??????(Ha
rdwood Concentration),5,10,15,20?25,?????5???
?(obs.,tensile strength),??????

Obs.
?? 1 2 3 4 5
5 7 7 15 11 9 49 9.8
10 12 17 12 18 18 77 15.4
15 14 18 18 19 19 88 17.6
20 19 25 22 19 23 108 21.6
25 7 10 11 15 11 54 10.8
12
ANOVA (cont.)
source of variation d.f. SS MS F
Between Trt. 5 -1 475.56 118.89 14.75062
Error 25 5 161.20 8.06
Total 25 - 1 636.96
13
Fixed Effect vs. Random Effect
• Fixed Effect
• ?????????,???????????(chosen in a nonrandom
manner/ a small hand-selected factor level)
• Radom Effect
• ????30??????,?????????,???????????30????????(study
the source of variability/ the variation
associated with a factor)

Fixed Effect Radom Effect
Effect A
Effect B
Effect AB
14
Statistically Quality Design
• Topics
• Four perspectives in quality improvement
• Review DOE topics and terminologies
• Implementation plan and procedure for
experimental design
• Full factorial design and Yates algorithm
• Full factorial design example improving wave
solder process at TDY company
• Concepts and examples for conducting Block and
Latin square designs

15
Implementation Plan and Procedure for
Experimental Design
Basically, a twelve-steps (procedure)
approach for conducting any experimental design
can be divided into the following three stages
Stage 1 (?????????)
Define the problems and state the objective of the experiment Select quality characteristic (response) and input variables (factors) Determine the desired number of runs and replications Consider the randomization of runs during the selection of the best design type
16
Stage 2 (?????????)
Conduct the experiment and record the data Analyze the data using analyze of mean, analysis of variance Use Yates algorithm and normal probability plot to determine the significant main and interaction effects
Stage 3 (?????????????)
Develop a fitted model using regression analysis Draw conclusion and make prediction Perform confirmatory tests Assess results and make decision
17
Steps for Experimental Design
• Statement of the problem_________________________
________
• (During this step you should estimate your
current level of quality by way of Cpk, dpm, or
total loss. This estimate will be compared with
improvements found after Step 7.)
• Objective of the experiment______________________
_________
• Star Date_____________ End Date_____________
• Select quality characteristics
• (also known as responses, dependent variables,
or output variables). These characteristics
should be related to customer needs and
expectations.

Response Type Anticipated Range How will you measure the response?
1
2
3
18
Steps for Experimental Design (cont.)
• Select factors
• (also know as parameters or input variables)
which are anticipated to have an effect on
response.
• Determine the number of resources to be used in
the experiment
• (Consider the desired number, the cost per
resource, time per experimental trial, and
maximum allowable number of resources.)
• Which design types and analysis strategies are
appropriate?
• (Discuss advantage and disadvantages of
each.)

Factor Type Controllable or Noise Range of Interest Levels Anticipated Interactions With How Measured
1
2
3
19
Steps for Experimental Design (cont.)
• Select the best design type and analysis strategy
to suit your needs
• Can all the runs be randomized?___________________
_______
• Which factors are most difficult to
randomize?________________
• Conduct the experiment and record the data
• (Monitor both of these events for accuracy)
• Analyze the data, draw conclusions, mark
predictions, and do confirmatory tests
• Assess results and make decisions
• (Evaluate your new state of quality and
compare with the quality level prior to the
improvement effort. If necessary, conduct more
experimentation.)

20
Statistically Quality Design
• Topics
• Four perspectives in quality improvement
• Review DOE topics and terminologies
• Implementation plan and procedure for
experimental design
• Full factorial design and Yates algorithm
• Full factorial design example improving wave
solder process at TDY company
• Concepts and examples for conducting Block and
Latin square designs

21
Full Factorial Experimental Design
Principles
22
Random Number Table
• There are 400 digits in this random number table,
3 appears 41 times.

23
3 Factors, 2 Levels
• Four dimensional visibility with

test combinations in a full factorial matrix
24
Label The Cells
8 Test Combinations
A-
A
(1) a
b ab
c ac
bc abc
B-
C-
B
B-
C
B
25
Yates Notation
Test Combinations
Cell A B AB C AC BC ABC
(1) - - -
- a - - - -
b - - -
- ab
- - - - c - -
- - ac
- - - -
bc - - -
- abc

26
Yates Notation
Test Combinations
27
Yates Work Session
Y yield strength , PSI
A, B and C are concentrations of 3 separate
elements
A-
A
58 56 36 39
51 53 34 32
53 48 54 59
49 49 55 61
B-
C-
B
B-
C
B
Determine the size of each contrast using Yates
algorithm
What combination of elements will give the
highest yield strength?
28
The Yates Algorithm
two variables A, B
number of variables, n 2 number of columns, n
2
For top ½ of each column
29
Yates Work Session
30
Yates Worksheet, 3 Variables
y
y
Cell
2
3
RANK
1

(1)
a
b
ab
c
ac
bc
abc
TOTAL
31
Analysis of Variance for a AB Factorial
Experiment
ANOVA of factorial experiment The total sum of
squares can be partitioned into Total SS
SS(A) SS(B) SS(AB) SSE
ANOVA Table For AXB Factorial Experiment ANOVA Table For AXB Factorial Experiment ANOVA Table For AXB Factorial Experiment ANOVA Table For AXB Factorial Experiment
Source d.f. SS MS
Factor A Factor B Interaction AB Error (a-1) (b-1) (a-1)(b-1) (n-ab) SS(A) SS(B) SS(AB) SSE SS(A)/(a-1) SS(B)/(b-1) SS(AB)/((a-1)(b-1) SSE/(n-ab)
Total (n-1) Total SS
n rab r number of times each factorial
treatment combination appears in the experiment
32
AB Factorial Experiment (Cont.)
Test each null hypothesis
33
Example A B Factorial Experiment
• The evaluation of a flame retardant was conducted
at two different laboratories on three different
materials with the following results

Materials Materials Materials
Laboratory 1 2 3
1 2 4.1 , 3.9 4.3 2.7 , 3.1 2.6 3.1 , 2.8 3.3 1.9 , 2.2 2.3 3.5 , 3.2 3.6 2.7 , 2.3 2.5
34
Example AB Factorial Experiment (Cont.)
Total For Calculating Sums of Squares Total For Calculating Sums of Squares Total For Calculating Sums of Squares Total For Calculating Sums of Squares Total For Calculating Sums of Squares
Material (B) Material (B) Material (B) Material (B) Material (B)
Laboratory 1 2 3 Total (A)
1 2 12.3 8.4 9.2 6.4 10.3 7.5 31.8 22.3
Total (B) 20.7 15.6 17.8 54.1
There are n rab (3)(2)(3) 18 observation
35
Total SS
SS(A)
SS(B)
SS(AB)
SSE
36
Example AB Factorial Experiment (Cont.)
ANOVA Table
Source d.f. SS MS F
Laboratory (A) Material (B) Interaction (AB) Error 1 2 2 12 5.0139 2.1811 .1344 .6000 5.0139 1.0906 0.672 0.0500 100.28 21.81 1.34
Total 17 7.9294
37
• Testing hypothesis to confirm interaction exists
or not

Since
the interaction is not significant
The null hypothesis is not rejected
No differences among interaction
• Since

the null hypothesis is rejected.
The laboratory and material are important.
38
Main Effect Larger Than Interaction
39
Interaction Larger Than Main Effect
40
Two-way ANOVA
• Open the two_way.mtw worksheet

41
Stat ? ANOVA ? Two-way Analysis of Variance
response
Materials
Materials
Enter OK
42
ANOVA Table
P-value lt 0.05
1. The materials and laboratory are significant
(important). 2. The interaction is not
significant.
43
Main Effects Plot
Stat ? ANOVA ? Main Effects Plot
44
Interactions Plots
Stat ? ANOVA ? Interactions Plot
45
Statistically Quality Design
• Topics
• Four perspectives in quality improvement
• Review DOE topics and terminologies
• Implementation plan and procedure for
experimental design
• Full factorial design and Yates algorithm
• Full factorial design example improving wave
solder process at TDY company
• Concepts and examples for conducting Block and
Latin square designs

46
Full Factorial Experiment Example Improving Wave
Solder Process at Teledyne
• Objective
• To determine the effect of flux type and lead
length on the DFDAU wave soldering (WS) defects
• Planted steps for statistically designed
experiment
• (1) Select output variables, 2 factors, 2 levels
and 8 Runs
• (2) Randomize the sequence of runs and labels 8
DFDAU boards
• (3) Select two touchup operators to check the WS
defects consistency
• (4) Iso-plot the major WS defects for top/rear
sides to compare one operator against another
• (5) Analyze the data using ANOVA table with
interactions or using Yates' algorithm
• (6) Plot/interpret the results and draw the
conclusions

47
Wave Soldering Process Flow Chart
48
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49
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50
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51
Statistically Designed Experiment
Run No. Flux Type Lead
Length Label Y1
New(OA) Trimmed Leads ab Y2
Old(RMA) Std. Lead
Length (1) Y3
Old(RMA) Std. Lead Length (1) Y4
New(OA) Std. Lead
Length b Y5 Old(RMA)
Trimmed Leads a Y6
New(OA) Trimmed Leads
ab Y7 New(OA)
Std. Lead Length b Y8
Old(RMA) Trimmed Leads a
New Flux Alpha 857 Old Flux RMA
Std. Lead Length (IC connect. point not trimmed
IC??????????) Trimm
ed Leads ( about .045)
52
Yates Algorithm
Notations
ANOVA Table
22 4 Combinations
Contrasts Cell A B AB (1)
- - a -
- b - - ab

Std. Leads Trimmed Leads
A-
A
Old Flux
(1) a
b ab
B-
New Flux
B
53
Operator 1
Std. Leads
Trimmed Leads
12 10.5 9 9 7.5 6
7 8 9 8 7 6
Avg. Operator
Old Flux
Std. Leads
Trm. Leads
New Flux
10.5 10 9.5 7.5 7.25 7
8 7.5 7 7 5.75 4.5
Old Flux
Operator 2
11 9.5 8 9 7 5
7 7 7 7 4.5 2
New Flux
Old Flux
New Flux
54
ANOVA Analysis For Two Factorial Experiment
Two way ANOVA for flux type and lead length
55
Interaction Plot for Flux Type and Lead Length
56
2 Factor Full Factorial Experiment Summary and
Conclusions
• Summary of Findings
• ISPLOT reveals that 2 operators
• were fairly consistent in calling
• out VIA defects
• VIA defects consist of 77
• vs. 93 of total defects, 1 vs. 2
• Only the rear VIA defects are
• considered for output measures
• Defect level 2941 ppm
• (Trimmed leads)
• Defect level 3959 ppm
• (Std. lead length)
• Conclusions
• The ANOVA/Yatess Analysis indicates lead length
to be the most
• significant factor
• Interaction between flux and lead
• length proven to be the least
• significant factor
• 26 improvement can be expected
• if using the Trimmed Leads
• 23 improvement can be expected
• if using the OA Flux
• Optimal range for the board temperature needs to
be further studied

57
Statistically Quality Design
• Topics
• Four perspectives in quality improvement
• Review DOE topics and terminologies
• Implementation plan and procedure for
experimental design
• Full factorial design and Yates algorithm
• Full factorial design example improving wave
solder process at TDY company
• Concepts and examples for conducting Block and
Latin square designs

58
Latin Square (???? )
Operators
I
II
III
Processes
A B C
B C A
C A B
1
2
3
Model
Operators I, II,
III Processes 1, 2,
3 Material Source A, B, C
59
Greco-Latin Square
I
II
III

1
2
3
Operators I, II,
III Processes 1, 2,
3 Material 1 Source A, B,
C Material 2 Source
60
Latin Square Design
• By using a Latin square design, three sources of
variation, A, B and C, can be investigated
simultaneously, providing there is no interaction
between the three factors and also that each of
them has the same number of levels r.

For example, suppose each factor has four
levels denoted by
If factor A is associated
with the rows of the table and B with the columns
of the table then each levels of factor C must
appear once in each row and once in each column.
In order to achieve this a systematic cyclic
pattern, it can be set down for the Cs as shown
in the table. To randomize the design, the
allocation of the As and Bs to the rows and
columns is then carried out at random.
61

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62
Latin Square Models
Latin Square Model
The as, ßs, ?s and es are mutually
independent.
Analysis of variance for a Latin Square design
The total sum of squares is divided into four
component parts, one for each source of variation
and one for the residual.
Here Yi is the sum of over the r observations in
which factor A is at level i, with similar
interpretation for Y.j. and Y..k and Y is the
sum of all the r2 observations.
63
The analysis and test statistics are summarized
in the following ANOVA table.
ANOVA Table for Latin Square
Source d.f S.S
M.S F Factor A
r-1 SSA Factor B
r-1 SSB Factor C r-1
SSC Residual r2 - 3r
2 SSE Total r2 - 1
SST
64
Example
The following 4 X 4 Latin Square in which the
effects of three factors, farm, type of
fertilizer applied, and method of application
(C1, C2, C3 or C4) on the yield crop are being
investigated.
Fertilizer Fertilizer Fertilizer Fertilizer

Farm
Farm
Farm
Farm
65
To ease the calculations, the data can be coded
by subtracting 33 from each observation. Then the
row and column totals and the totals for each
method of application are calculated. (??33,
????ANOVA??)
Fertilizer Fertilizer Fertilizer Fertilizer Fertilizer
1 2 3 4 Total 1 2 3 4 Total 1 2 3 4 Total 1 2 3 4 Total 1 2 3 4 Total
1 Farm 2 3 4 2
1 Farm 2 3 4 8
1 Farm 2 3 4 4
1 Farm 2 3 4 -2
Total 4 2 10 -4 12 4 2 10 -4 12 4 2 10 -4 12 4 2 10 -4 12 4 2 10 -4 12
Method Total
Method Total -4 14 2 10 12 -4 14 2 10 12 -4 14 2 10 12 -4 14 2 10 12 -4 14 2 10 12
66
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67
The calculations necessary for testing the
significant of the three factors are summarized
in the following ANOVA table.
Source d.f S.S
M.S F Farm 3
13 4.33
13.0 Fertilizer 3
25 8.33 25.0 Method
3 45 15.00
45.0 Residual 6
2 0.333 Total
15 85
Since the critical value are F0.99(3, 6) 9.78
and F0.999(3, 6) 23.70, the farm effect is
significant at 1 level. The type of fertilizer
used and the method of application are both
significant at the 0.1 level.
68
Latin Square Models
• Open the Latin Square.mtw worksheet

69
Stat ? ANOVA ? General Linear Model
70
P-value lt 0.05
The farm effect, the type of fertilizer used and
the method of application are significant at a
0.05
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