Transcript Statistics - Rose

Statistics and ANOVA
ME 470
Spring 2012
Product Development Process
Planning
Concept
System-Level
Development Design
Detail
Design
Testing and
Refinement
Production
Ramp-Up
Concept Development Process
Mission
Statement
Identify
Customer
Needs
Establish
Target
Specifications
Generate
Product
Concepts
Select
Product
Concept(s)
Test
Product
Concept(s)
Perform Economic Analysis
Benchmark Competitive Products
Build and Test Models and Prototypes
Set
Final
Specifications
Plan
Downstream
Development
Development
Plan
We will use statistics to make
good design decisions!
We will categorize populations by the mean,
standard deviation, and use control charts to
determine if a process is in control.
We may be forced to run experiments to
characterize our system. We will use valid
statistical tools such as Linear Regression,
DOE, and Robust Design methods to help us
make those characterizations.
Let’s consider the Toyota problem.
What was the first clue that there was a problem?
Starting in 2003, NHSTA received information regarding reports of accelerator pedals
that were operating improperly.
How many reports causes the manufacturer to suspect a
problem?
To issue a recall NHTSA would need to prove that a substantial number of failures
attributable to the defect have occurred or is likely to occur in consumers’ use of the
vehicle or equipment and that the failures pose an unreasonable risk to motor vehicle
safety.
ODI conducted a VOQ-based assessment of UA rates on the subject Lexus in
comparison to two peer vehicles and concluded the Lexus LS400t vehicles were not
overrepresented in the VOQ database.
How might we look at two populations and decide this?
How can we use statistics to make sense of data
that we are getting?
• Quiz for the day
• What can we say about our M&Ms?
What kinds of questions can we
answer?
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What does the data look like?
What is the mean, the standard deviation?
What are the extreme points?
Is the data normal?
Is there a difference between years? Did one class get
more M&Ms than another?
• If you were packaging the M&Ms, are you doing a good
job?
• If you are the designer, what factors might cause the
variation?
>Stat>Basic Statistics>Display Descriptive Statistics
Results for 2008, 2010, 2011
(From the “Session”)
Why would we care about this in design?
Assessing Shape: Boxplot
largest value excluding outliers
Boxplot of BSNOx
2.45
Q3
2.40
(Q2), median
BSNOx
2.35
2.30
Q1
2.25
2.20
outliers are marked as ‘*’
smallest value excluding outliers
Values between 1.5 and 3 times away from the middle 50% of the data are outliers.
Individual Value Plot of StackedTotals vs Year
10
StackedTotals
9
8
7
6
5
2008
2010
Year
2011
>Stat>Basic Statistics>Normality Test
Select 2008
Anderson-Darling normality test:
Used to determine if data follow a normal distribution. If the p-value is lower than the
pre-determined level of significance, the data do not follow a normal distribution.
Anderson-Darling Normality Test
Measures the area between the fitted line (based on chosen distribution) and the
nonparametric step function (based on the plot points). The statistic is a squared
distance that is weighted more heavily in the tails of the distribution. AndersonSmaller Anderson-Darling values indicates that the distribution fits the data better.
The Anderson-Darling Normality test is defined as:
H0: The data follow a normal distribution.
Ha: The data do not follow a normal distribution.
Another quantitative measure for reporting the result of the normality test is the p-value. A
small p-value is an indication that the null hypothesis is false. (Remember: If p is low, H0
must go.)
P-values are often used in hypothesis tests, where you either reject or fail to reject a null
hypothesis. The p-value represents the probability of making a Type I error, which is
rejecting the null hypothesis when it is true. The smaller the p-value, the smaller is the
probability that you would be making a mistake by rejecting the null hypothesis.
It is customary to call the test statistic (and the data) significant when the null hypothesis H0
is rejected, so we may think of the p-value as the smallest level α at which the data are
significant.
Note that our p value is quite low, which makes us consider rejecting
the fact that the data are normal. However, in assessing the closeness
of the points to the straight line, “imagine a fat pencil lying along the
line. If all the points are covered by this imaginary pencil, a normal
distribution adequately describes the data.” Montgomery, Design
and Analysis of Experiments, 6th Edition, p. 39
If you are confused about whether or not to consider the data normal,
it is always best if you can consult a statistician. The author has
observed statisticians feeling quite happy with assuming very fat
lines are normal.
For more on Normality and the Fat Pencil
http://www.statit.com/support/quality_practice_tips/normal_probability_plot_
interpre.shtml
Walter Shewhart
Developer of Control Charts in the late 1920’s
You did Control Charts in DFM. There the emphasis was on tolerances. Here the
emphasis is on determining if a process is in control. If the process is in control, we want
to know the capability.
www.york.ac.uk/.../ histstat/people/welcome.htm
What does this data tell us about our
process?
SPC is a continuous improvement tool which minimizes tampering or
unnecessary adjustments (which increase variability) by distinguishing
between special cause and common cause sources of variation
Control Charts have two basic uses:
Give evidence whether a process is operating in a state of statistical
control and to highlight the presence of special causes of variation so
that corrective action can take place.
Maintain the state of statistical control by extending the statistical
limits as a basis for real time decisions.
If a process is in a state of statistical control, then capability studies my
be undertaken. (But not before!! If a process is not in a state of
statistical control, you must bring it under control.)
SPC applies to design activities in that we use data from manufacturing to
predict the capability of a manufacturing system. Knowing the
capability of the manufacturing system plays a crucial role in selecting
the concepts.
Voice of the Process
Control limits are not spec limits.
Control limits define the amount of fluctuation that a
process with only common cause variation will have.
Control limits are calculated from the process data.
Any fluctuations within the limits are simply due to
the common cause variation of the process.
Anything outside of the limits would indicate a
special cause (or change) in the process has
occurred.
Control limits are the voice of the process.
The capability index is defined as:
Cp = (allowable range)/6s = (USL - LSL)/6s
LSL
LCL
USL (Upper Specification Limit)
UCL (Upper Control Limit)
http://lorien.ncl.ac.uk/ming/spc/spc9.htm
>Stat>Control Charts>Variable Charts for Individuals>Individuals
Select all tests
These test failures fall into the category of “special cause variations”, statistically
unlikely events that are worth looking into as possible problems
Upper Control
Limit
Lower Control
Limit
Are the 2 Distributions Different?
X Data
Single X
Multiple Xs
Y Data
Logistic
Regression
One-sample ttest
Two-sample ttest
Simple
Linear
Regression
Discrete
Discrete X Data Continuous
Continuous
Continuous
Y Data
Discrete
Single Y
Chi-Square
ANOVA
Multiple Ys
Y Data
Discrete X Data Continuous
Multiple
Logistic
Regression
Multiple
Logistic
Regression
ANOVA
Multiple
Linear
Regression
When to use ANOVA

The use of ANOVA is appropriate when
 Dependent variable is continuous
 Independent variable is discrete, i.e. categorical
 Independent variable has 2 or more levels under study
 Interested in the mean value
 There is one independent variable or more

We will first consider just one independent variable
Practical Applications

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Compare 3 different suppliers of the same
component
Compare 4 test cells
Compare 2 performance calibrations
Compare 6 combustion recipes through simulation
Compare our brake failure rate with other companies
Compare 3 distributions of M&M’s
And MANY more …
ANOVA Analysis of Variance


Used to determine the effects of categorical independent
variables on the average response of a continuous variable
Choices in MINITAB
 One-way ANOVA


Two-way ANOVA


Use with two factors, varied over multiple levels
Balanced ANOVA


Use with one factor, varied over multiple levels
Use with two or more factors and equal sample sizes in each cell
General Linear Model

Use anytime!
>Stat>ANOVA>General Linear Model
Residual Plots for StackedTotals
Normal Probability Plot
Versus Fits
99.9
3.0
1.5
90
Residual
Percent
99
50
10
1
0.1
0.0
-1.5
-3.0
-3.0
-1.5
0.0
Residual
1.5
3.0
7.5
7.6
7.7
Fitted Value
Histogram
Versus Order
1.5
30
Residual
Frequency
7.9
3.0
40
20
10
0
7.8
0.0
-1.5
-3.0
-3.00 -2.25 -1.50 -0.75 0.00
Residual
0.75
1.50
2.25
1
20
40
60
80
100 120 140 160 180 200
Observation Order
This p value indicates
that the assumption that
the years are different is
correct
>Stat>ANOVA>General Linear Model ---Select Comparisons
We use the Tukey comparison to determine if the years are different.
Confidence intervals that contain zero suggest no difference.
Tukey Comparison
Zero is contained in the interval.
The years are NOT different.
Zero is NOT contained in the interval.
The years are different.
Let’s look at what happened with plain
M&M’s
What do you see with the boxplot?
What do you see with the boxplot?
Do we see anything that looks unusual?
General Linear Model: stackedTotal versus StackedYear
Factor
StackedYear
Type Levels
fixed
4
Values
2004, 2005, 2006, 2009
Analysis of Variance for stackedTotal, using Adjusted SS for Tests
Source
DF
Seq SS Adj SS Adj MS
F
P
StackedYear 3 1165.33 1165.33 388.44 149.39 0.000 Look at low P-value!
Error
266 691.63 691.63 2.60
Total
269 1856.96
S = 1.61249 R-Sq = 62.75% R-Sq(adj) = 62.33%
Unusual Observations for stackedTotal
Obs stackedTotal Fit
SE Fit Residual
25
27.0000 23.4667 0.2082 3.5333
34
20.0000 23.4667 0.2082 -3.4667
209
40.0000 21.7917 0.1700 18.2083
215
21.0000 17.4917 0.2082 3.5083
St Resid
2.21 R
-2.17 R
11.36 R
2.19 R
R denotes an observation with a large standardized residual.
Grouping Information Using Tukey Method and 95.0% Confidence
StackedYear N Mean Grouping
2004
60 23.5 A
2006
90 21.8 B
2005
60 20.7 C
2009
60 17.5
D
Means that do not share a letter are significantly different.
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable stackedTotal
All Pairwise Comparisons among Levels of StackedYear
StackedYear = 2004 subtracted from:
StackedYear
2005
2006
2009
Lower Center
-3.531 -2.775
-2.365 -1.675
-6.731 -5.975
Upper -------+---------+---------+---------2.019
(---*---)
-0.985
(-*--)
-5.219 (--*--)
-------+---------+---------+---------5.0
-2.5
0.0
Zero is not contained in the intervals. Each year is
statistically different. (2004 got the most!)
StackedYear = 2005 subtracted from:
StackedYear Lower Center Upper -------+---------+---------+--------2006
0.410 1.100 1.790
(-*--)
2009
-3.956 -3.200 -2.444
(--*--)
-------+---------+---------+---------5.0
-2.5
0.0
StackedYear = 2006 subtracted from:
StackedYear Lower Center Upper -------+---------+---------+--------2009
-4.990 -4.300 -3.610
(--*--)
-------+---------+---------+---------5.0
-2.5
0.0
Implications for design
• Is there a difference in production
performance between the plain and peanut
M&Ms?
Individual Quiz
Name:____________
Section No:__________
CM:_______
You will be given a bag of M&M’s. Do NOT eat the M&M’s.
Count the number of M&M’s in your bag. Record the number of each color,
and the overall total. You may approximate if you get a piece of an M&M.
When finished, you may eat the M&M’s. Note: You are not required to eat the
M&M’s.
Color
Brown
Yellow
Red
Orange
Green
Blue
Other
Total
Number
%
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