Statistics and ANOVA

ME 470

Fall 2009

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.

How Do We Describe the World? Quiz for the day You need to install Minitab on your

computers. Sign on as localmgr >Start>Run \\tibia\Public\Course Software\Minitab Double click on Minitab R15 Install What can we say about our M&Ms?

>Stat>Basic Statistics>Display Descriptive Statistics

2004, 2005, 2006 Data

Descriptive Statistics: stackedTotal

Variable StackedYear N N* Mean SE Mean StDev MinimumstackedTotal 2004 60 0 23.467 0.188 1.455 20.000 2005 60 0 20.692 0.135 1.046 18.000 2006 90 0 21.792 0.232 2.202 19.000

Variable StackedYear Q1 Median Q3 MaximumstackedTotal 2004 23.000 23.500 24.000 27.000 2005 20.000 21.000 21.000 23.000 2006 21.000 22.000 22.000 40.000

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.

http://en.wikipedia.org/wiki/Box_plot

>Stat>Basic Statistics>Normality Test

Select StackedTotal_2004

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.

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>I-MR

Upper Control Limit

Lower Control Limit

Absolute difference between two adjacent points.

Are the 3 Distributions Different?X Data

Single X Multiple Xs

Y D

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X DataDiscrete Continuous

Y D

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One-sample t-test

Two-sample t-test

ANOVA

X DataDiscrete Continuous

Y D

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Dis

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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 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

1525

Effect of Year on M&M Production

General Linear Model: stackedTotal versus StackedYear

Factor Type Levels ValuesStackedYear fixed 3 2004, 2005, 2006

Analysis of Variance for stackedTotal, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PStackedYear 2 235.27 235.27 117.63 39.22 0.000Error 207 620.89 620.89 3.00Total 209 856.16

S = 1.73189 R-Sq = 27.48% R-Sq(adj) = 26.78%

Unusual Observations for stackedTotal

Obs stackedTotal Fit SE Fit Residual St Resid 25 27.0000 23.4667 0.2236 3.5333 2.06 R 34 20.0000 23.4667 0.2236 -3.4667 -2.02 R209 40.0000 21.7917 0.1826 18.2083 10.57 R

R denotes an observation with a large standardized residual.

This low p-value indicates that at least one year is different from the others.

>Stat>ANOVA>General Linear Model

We use the Tukey comparison to determine if the years are different. Confidence intervals that contain zero suggest no difference.

Tukey 95.0% Simultaneous Confidence IntervalsResponse Variable stackedTotalAll Pairwise Comparisons among Levels of StackedYearStackedYear = 2004 subtracted from:

StackedYear Lower Center Upper ---+---------+---------+---------+---2005 -3.522 -2.775 -2.028 (---*----)2006 -2.357 -1.675 -0.993 (----*---) ---+---------+---------+---------+--- -3.0 -1.5 0.0 1.5

StackedYear = 2005 subtracted from:

Difference SE of AdjustedStackedYear of Means Difference T-Value P-Value2006 1.100 0.2886 3.811 0.0005

Because “0.0” is not contained in the range, we concluded that 2004 is statistically different from both 2005 and 2006.

StackedYear = 2005 subtracted from:

StackedYear Lower Center Upper ---+---------+---------+---------+---2006 0.4183 1.100 1.782 (---*----) ---+---------+---------+---------+--- -3.0 -1.5 0.0 1.5

Again, because “0.0” is not in the range, we conclude that 2005 is statistically different than 2006.

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