Statistics and ANOVA

1. Statistics and ANOVA ME 470 Fall 2009

2. 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.

3. 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?

5. >Stat>Basic Statistics>Display Descriptive Statistics

6. 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 Q1 Median 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

7. Why would we care about this in design?

9. 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 . 3 5 x O N S B 2 . 3 0 Q1 2 . 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.

11. >Stat>Basic Statistics>Normality Test Select StackedTotal_2004

12. 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.

13. 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.

14. 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.

15. 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

16. 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.

17. 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.

18. The capability index is defined as: 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

19. >Stat>Control Charts>Variable Charts for Individuals>I-MR

21. Upper Control Limit Lower Control Limit Absolute difference between two adjacent points.

22. Are the 3 Distributions Different? 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

23. 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

24. 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 …

25. 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!

26. >Stat>ANOVA>General Linear Model 15 25 Effect of Year on M&M Production

29. 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.

30. >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.

31. 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.

32. 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.

33. 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.