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Statistics and ANOVA

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Statistics and ANOVA

ME 470

Fall 2009

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.

- 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?

2004, 2005, 2006 Data

Descriptive Statistics: stackedTotal

Variable StackedYear N N* Mean SE Mean StDev Minimum

stackedTotal 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 Q1Median Q3 Maximum

stackedTotal 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?

largest value excluding outliers

B

o

x

p

l

o

t

o

f

B

S

N

O

x

Q3

2

.

4

5

2

.

4

0

(Q2), median

2

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3

5

x

O

N

S

B

2

.

3

0

Q1

2

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2

5

2

.

2

0

outliers are marked as ‘*’

smallest value excluding outliers

Assessing Shape: Boxplot

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

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

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 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. 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.

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

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.

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.

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

LSL

USL (Upper Specification Limit)

LCL

UCL (Upper Control Limit)

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

Upper Control Limit

Lower Control Limit

Absolute difference between two adjacent points.

X Data

Single X

Multiple Xs

X Data

X Data

Discrete

Continuous

Discrete

Continuous

Discrete

Logistic Regression

Multiple Logistic Regression

Multiple Logistic Regression

Chi-Square

Discrete

Y Data

Y Data

Single Y

Continuous

One-sample t-test

Two-sample t-test

ANOVA

Y Data

Simple Linear Regression

Multiple Linear Regression

Continuous

ANOVA

Multiple Ys

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!

- One-way ANOVA

>Stat>ANOVA>General Linear Model

15

25

Effect of Year on M&M Production

General Linear Model: stackedTotal versus StackedYear

Factor Type Levels Values

StackedYear fixed 3 2004, 2005, 2006

Analysis of Variance for stackedTotal, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

StackedYear 2 235.27 235.27 117.63 39.22 0.000

Error 207 620.89 620.89 3.00

Total 209 856.16

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

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

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 R

209 40.0000 21.7917 0.1826 18.2083 10.57 R

R denotes an observation with a large standardized residual.

>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 Intervals

Response Variable stackedTotal

All Pairwise Comparisons among Levels of StackedYear

StackedYear = 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 Adjusted

StackedYear of Means Difference T-Value P-Value

2006 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.