Statistics and ANOVA

ME 470Fall 2013

Here are some interesting on-the-spot designs from the past and this class.

Fall 2009, 0.27 Cost/Height

Winner, Spring 201015” Tall8$ Cost0.533 Cost/Height

Fall 2011Height = 12Cost = 6Cost/Height = 0.5

Fall 2011Height = 24Cost = 12Cost/height = 0.5

I really enjoy the on-the-spot design.

What did you learn about the design process? There are many challenges in product development

Trade-offs Dynamics Details Time pressure Economics

Why do I love product development? Getting something to work Satisfying societal needs Team diversity Team spirit

Design is a process that requiresmaking decisions.

Planning

Product Development PhasesConceptDevelopment

System-LevelDesign

DetailDesign

Testing andRefinement

ProductionRamp-Up

Concept Development Process

Perform Economic Analysis

Benchmark Competitive Products

Build and Test Models and Prototypes

IdentifyCustomerNeeds

EstablishTargetSpecifications

GenerateProductConcepts

SelectProductConcept(s)

Set FinalSpecifications

PlanDownstreamDevelopment

MissionStatement Test

ProductConcept(s)

DevelopmentPlan

You will practice theentire concept development process with your group project

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.

Cummins asked a capstone group to investigate improvements for turbo charger lubrication sealing.

5.9L High Output Cummins Engine

Cummins Inc. was dissatisfied with the integrity of their turbocharger oil sealing capabilities.

Here are pictures of oil leakage.

Oil Leakage into Compressor Housing Oil Leakage on Impellor Plate

The students developed four prototypes for testing. After testing, they wanted to know which solution to present to Cummins. You will analyze their data to make a suggestion.

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? We will look at the results first and then you

can do the analysis on your own.

Statistics can help us examine the data and draw justified conclusions.

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?

Why would we care about this data in design?

If I am a plant manager, do I like one distribution better than another?

How do we interpret the boxplot?BS

NOx

2.45

2.40

2.35

2.30

2.25

2.20

Boxplot of BSNOx

(Q2), median

Q1

Q3

largest value excluding outliers

smallest value excluding outliersoutliers are marked as ‘*’

Values between 1.5 and 3 times away from the middle 50% of the data are outliers.

This is a density description of the data.

The Anderson-Darling normality test is 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 TestMeasures 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. Anderson-Smaller 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.

http://www.statit.com/support/quality_practice_tips/normal_probability_plot_interpre.shtml

For more on Normality and the Fat Pencil

You can use the “fat pencil” test in addition to the p-value.

Walter Shewhart

www.york.ac.uk/.../ histstat/people/welcome.htm

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.

What does the 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 depends on the spec limit and the process standard deviation.

Cp = (allowable range)/6s = (USL - LSL)/6s

USL (Upper Specification Limit)

LSL

LCL

UCL (Upper Control Limit)

http://lorien.ncl.ac.uk/ming/spc/spc9.htm

Lower Control Limit for 2008

Upper Control Limit for 2008

Minitab prints results in the Session window that lists any failures.Test Results for I Chart of StackedTotals by C4

TEST 1. One point more than 3.00 standard deviations from center line.Test Failed at points: 129

TEST 2. 9 points in a row on same side of center line.Test Failed at points: 15, 110, 111, 112, 113

TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on one side of CL).Test Failed at points: 52, 66, 119, 160, 161

TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on one side of CL).Test Failed at points: 91, 97

TEST 7. 15 points within 1 standard deviation of center line (above and below CL).Test Failed at points: 193, 194, 195, 196, 197, 198, 199, 200

This chart is extremely helpful for deciding what statistical technique to use.

X DataSingle X Multiple Xs

Y D

ata Si

ngle

Y

Mul

tiple

Ys

X DataDiscrete Continuous

Y D

ata Dis

cret

e C

ontin

uous

One-sample t-test

Two-sample t-test

ANOVA

X DataDiscrete Continuous

Y D

ata D

iscr

ete

Con

tinuo

us

Chi-Square

Simple Linear

Regression

Logistic Regression

ANOVAMultiple Linear

Regression

Multiple Logistic

Regression

Multiple Logistic

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

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

Use with one factor, varied over multiple levels Two-way ANOVA

Use with two factors, varied over multiple levels Balanced ANOVA

Use with two or more factors and equal sample sizes in each cell General Linear Model

Use anytime!

Practical Applications

Determine if our break pedal sticks more than other companies

Compare 3 different suppliers of the same component

Compare 6 combustion recipes through simulation Determine the variation in the crush force Compare 3 distributions of M&M’s And MANY more …

General Linear Model: StackedTotals versus C4

Factor Type Levels ValuesC4 fixed 3 2008, 2010, 2011

Analysis of Variance for StackedTotals, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PC4 2 6.6747 6.6747 3.3374 4.71 0.010Error 203 143.8559 143.8559 0.7086Total 205 150.5306

S = 0.841813 R-Sq = 4.43% R-Sq(adj) = 3.49%

This p value indicates that the assumption that there is no difference between years is not correct!

The null hypothesis for ANOVA is that there is no difference between years.

What are some conclusions that you can reach?

Is there a statistical difference between years?

201120102008

7.9

7.8

7.7

7.6

7.5

7.4

Year

Mea

n

Main Effects Plot for StackedTotalsFitted Means

Grouping Information Using Tukey Method and 95.0% Confidence

C4 N Mean Grouping2010 57 7.9 A2008 86 7.7 A B2011 63 7.4 B

Means that do not share a letter are significantly different.

The p value indicates that there is a difference between the years. The Tukey printout tells us which years are different.

The averages for 2010 and 2008 are not statistically different. The averages for 2008 and 2011 are not statistically different.

Command:>Stat>Basic Statistics>Display Descriptive Statistics

Why would we care about this data in design?

If I am a plant manager, do I like one distribution better than another?

This is a density description of the data.

>Stat>Basic Statistics>Normality Test

Select 2008

The Anderson-Darling normality test is 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.

Command:>Stat>Control Charts>Variable Charts for Individuals>Individuals

When doing control charts for ME470, select all tests.

It may be hard to see, but highlight the “tests” tab.

Minitab prints results in the Session window that lists any failures.Test Results for I Chart of StackedTotals by C4

TEST 1. One point more than 3.00 standard deviations from center line.Test Failed at points: 129

TEST 2. 9 points in a row on same side of center line.Test Failed at points: 15, 110, 111, 112, 113

TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on one side of CL).Test Failed at points: 52, 66, 119, 160, 161

TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on one side of CL).Test Failed at points: 91, 97

TEST 7. 15 points within 1 standard deviation of center line (above and below CL).Test Failed at points: 193, 194, 195, 196, 197, 198, 199, 200

Lower Control Limit for 2008

Upper Control Limit for 2008

Command:>Stat>ANOVA>General Linear Model

What are some conclusions that you can reach?

Is there a statistical difference between years?

201120102008

7.9

7.8

7.7

7.6

7.5

7.4

Year

Mea

n

Main Effects Plot for StackedTotalsFitted Means

General Linear Model: StackedTotals versus C4

Factor Type Levels ValuesC4 fixed 3 2008, 2010, 2011

Analysis of Variance for StackedTotals, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PC4 2 6.6747 6.6747 3.3374 4.71 0.010Error 203 143.8559 143.8559 0.7086Total 205 150.5306

S = 0.841813 R-Sq = 4.43% R-Sq(adj) = 3.49%

This p value indicates that the assumption that there is no difference between years is not correct!

The null hypothesis for ANOVA is that there is no difference between years.

Command:>Stat>ANOVA>General Linear Model

Grouping Information Using Tukey Method and 95.0% Confidence

C4 N Mean Grouping2010 57 7.9 A2008 86 7.7 A B2011 63 7.4 B

Means that do not share a letter are significantly different.

The p value indicates that there is a difference between the years. The Tukey printout tells us which years are different.

The averages for 2010 and 2008 are not statistically different. The averages for 2008 and 2011 are not statistically different.

Here is a useful reference if you feel that you need to do more reading.

http://www.StatisticalPractice.comThis recommendation is thanks to Dr. DeVasher.

You can also use the help in Minitab for more information.

Let’s look at what happened with plain M&M’s

What do you see with the boxplot?

Do we see anything that looks unusual?

General Linear Model: stackedTotal versus StackedYear

Factor Type Levels ValuesStackedYear fixed 4 2004, 2005, 2006, 2009

Analysis of Variance for stackedTotal, using Adjusted SS for TestsSource DF Seq SS Adj SS Adj MS F PStackedYear 3 1165.33 1165.33 388.44 149.39 0.000 Look at low P-value!Error 266 691.63 691.63 2.60Total 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 St Resid 25 27.0000 23.4667 0.2082 3.5333 2.21 R 34 20.0000 23.4667 0.2082 -3.4667 -2.17 R209 40.0000 21.7917 0.1700 18.2083 11.36 R215 21.0000 17.4917 0.2082 3.5083 2.19 R

R denotes an observation with a large standardized residual.

Grouping Information Using Tukey Method and 95.0% ConfidenceStackedYear N Mean Grouping2004 60 23.5 A2006 90 21.8 B2005 60 20.7 C2009 60 17.5 D

Means that do not share a letter are significantly different.Tukey 95.0% Simultaneous Confidence IntervalsResponse Variable stackedTotalAll Pairwise Comparisons among Levels of StackedYearStackedYear = 2004 subtracted from:

StackedYear Lower Center Upper -------+---------+---------+---------2005 -3.531 -2.775 -2.019 (---*---)2006 -2.365 -1.675 -0.985 (-*--)2009 -6.731 -5.975 -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 QuizName:____________ 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 Number %

Brown

Yellow

Red

Orange

Green

Blue

Other

Total

Instructions for Minitab Installation

Minitab on DFS: