Statistics and ANOVA

ME 470Spring 2012

PlanningPlanning

Product Development Process

ConceptDevelopment

ConceptDevelopment

System-LevelDesign

System-LevelDesign

DetailDesign

DetailDesign

Testing andRefinement

Testing andRefinement

ProductionRamp-Up

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

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 incomparison 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?

• 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: BoxplotBSNO

x

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.

201120102008

10

9

8

7

6

5

Year

Sta

ckedTo

tals

Individual Value Plot of StackedTotals vs Year

>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 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. 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.

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

For more on Normality and the Fat Pencil

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 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

USL (Upper Specification Limit)LSL

LCL 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

Lower Control Limit

Upper Control Limit

Are the 2 Distributions Different?X Data

Single X Multiple Xs

Y D

ata

Sin

gle

Y

Mu

ltip

le Y

s

X DataDiscrete Continuous

Y D

ata Dis

cre

te

Co

nti

nu

ou

s

One-sample t-test

Two-sample t-test

ANOVA

X DataDiscrete Continuous

Y D

ata

Dis

cre

te

Co

nti

nu

ou

s

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

Practical Applications

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

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!

>Stat>ANOVA>General Linear Model

3.01.50.0-1.5-3.0

99.9

99

90

50

10

1

0.1

Residual

Perc

ent

7.97.87.77.67.5

3.0

1.5

0.0

-1.5

-3.0

Fitted Value

Resi

dual

2.251.500.750.00-0.75-1.50-2.25-3.00

40

30

20

10

0

Residual

Fre

quency

200180160140120100806040201

3.0

1.5

0.0

-1.5

-3.0

Observation Order

Resi

dual

Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for StackedTotals

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.

Zero is contained in the interval.The years are NOT different.

Zero is NOT contained in the interval.The years are different.

Tukey Comparison

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 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 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 Number %

Brown

Yellow

Red

Orange

Green

Blue

Other

Total

Instructions for Minitab Installation

Minitab on DFS: