1. ETM 620 - 09U 1 Analysis of Variance (ANOVA) Suppose we want to compare more than two means? For example, suppose a manufacturer of paper used for grocery bags is concerned about the tensile strength of the paper. Product engineers believe that tensile strength is a function of the hardwood concentration and want to test several concentrations for the effect on tensile strength. If there are 2 different hardwood concentrations (say, 5% and 15%), then a z-test or t-test is appropriate: H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2

2. ETM 620 - 09U 2 Comparing > 2 Means What if there are 3 different hardwood concentrations (say, 5%, 10%, and 15%)? H 0 : μ 1 = μ 2 H 0 : μ 1 = μ 3 H 0 : μ 2 = μ 3 H 1 : μ 1 ≠ μ 2 H 1 : μ 1 ≠ μ 3 H 1 : μ 2 ≠ μ 3 How about 4 different concentrations (say, 5%, 10%, 15%, and 20%)? All of the above, PLUS H 0 : μ 1 = μ 4 H 0 : μ 2 = μ 4 H 0 : μ 3 = μ 4 H 1 : μ 1 ≠ μ 4 H 1 : μ 2 ≠ μ 4 H 1 : μ 3 ≠ μ 4 What about 5 concentrations? 10? and

3. ETM 620 - 09U 3 Comparing > 2 Means Also, suppose α = 0.05 (1 – α ) = P(accept H 0 | H 0 is true) = 0.95 4 concentrations: (0.95) 4 = 0.814 5 concentrations: _____________ 10 concentrations:_____________ Instead, use Analysis of Variance (ANOVA) treatment, factor, independent variable: that which is varied (a levels) observation, replicates, dependent variable: the result of concern (n per treatment) randomization: performing experimental runs in random order so that other factors don’t influence results.

4. ETM 620 - 09U 4 One-Way ANOVA 1. Calculate and check residuals, e ij = O i - E i plot residuals vs treatments normal probability plot 2. Perform ANOVA and determine if there is a difference in the means 3. Identify which means are different using graphical methods,Tukey’s procedure, etc.: 4. Model: y ij = μ + τ i + ε ij

5. ETM 620 - 09U 5 Our Example Six specimens were made at each of the 4 hardwood concentrations. The 24 specimens were tested in random order on a tensile test machine, with the following results: HardwoodObservations Concentration (%)123456TotalsAverages 57815119106010.00 101217131819159415.67 1514181917161810217.00 2019252223182012721.17 38315.96

6. ETM 620 - 09U 6 To determine if there is a difference … 1. Calculate sums of squares 2. Calculate degrees of freedom df treat = a – 1 = _____ df E = a(n – 1) = _____ df total = an – 1 = _____

7. ETM 620 - 09U 7 Determining the Difference 3. Mean Square, MS = SS/df MS treat = ___________ MS E = ___________ 4. Calculate F = MS treat / MS E = _____________

8. ETM 620 - 09U 8 Organizing the Results 5. Build the ANOVA table and determine significance fixed α -level  compare to F α,a-1, a(n-1) p – value  find p associated with this F with degrees of freedom a-1, a(n-1) ANOVA Source of VariationSSdfMSFP-valueF crit Treatment382.793127.619.63.6E-063.1 Error130.17206.5083 Total512.9623

9. ETM 620 - 09U 9 Conclusion? 6. Draw the picture and state your conclusion … Conclusion: Why? E(MS E ) = σ 2 always E(MS treat ) = σ 2 only if the means are equal

10. ETM 620 - 09U 10 In Excel, In Data Analysis, choose ANOVA: Single Factor, then …

11. ETM 620 - 09U 11 Gives the result …

12. ETM 620 - 09U 12 In Minitab, STAT / ANOVA / One-Way (Unstacked) … gives the following results …

13. ETM 620 - 09U 13 Note.. STAT / ANOVA / One-Way … gives the same results, but the input looks like …

14. ETM 620 - 09U 14 Checking residuals … Calculate residuals, Plot against normal score to check normality assumption,

15. ETM 620 - 09U 15 In Excel,

16. ETM 620 - 09U 16 In Minitab, Select Graphs… and Normal plot of residuals when you perform the ANOVA …

17. ETM 620 - 09U 17 Other plots can check for independence Residuals vs fitted values Residuals vs treatment means Residuals vs time Residuals vs …?

18. ETM 620 - 09U 18 Which means are different? Graphical methods Numerical methods Tukey’s test Duncan’s Multiple Range test

19. ETM 620 - 09U 19 Tukey’s test Create confidence intervals around the difference in means using the Studentized Range Statistic, q α (a,f) where a = number of treatment levels and f = degrees of freedom for error. In our example, q α (a,f) = _________ compare this value to the differences in the means …

20. ETM 620 - 09U 20 In Minitab, Select Comparisons… and Tukey’s when you perform the ANOVA.

21. ETM 620 - 09U 21

22. ETM 620 - 09U 22 Random effects model The analysis we just did is an example of a fixed effects model Set levels of a factor of interest Assumes we can identify and test at all possible levels Alternatively, the factor may have a large number of levels (too big to test them all) want to make conclusions about the whole population based on a random sampling of the possible levels this is called a random effects model same model, same analysis, same conclusions, but the underlying hypotheses are different: H 0 : σ τ 2 = 0 H 1 : σ τ 2 > 0

23. ETM 620 - 09U 23 Blocking Creating a group of one or more people, machines, processes, etc. in such a manner that the entities within the block are more similar to each other than to entities outside the block. Balanced design: each treatment appears in each block. Model: y ij = μ + τ i + β j + ε ij

24. ETM 620 - 09U 24 Example: Robins Air Force Base uses CO 2 to strip paint from F-15’s. You have been asked to design a test to determine the optimal pressure for spraying the CO 2. You realize that there are five machines that are being used in the paint stripping operation. Therefore, you have designed an experiment that uses the machines as blocking variables. You emphasized the importance of balanced design and a random order of testing. The test has been run with these results (values are minutes to strip one fighter):

25. ETM 620 - 09U 25 ANOVA: One-Way with Blocking Construct the ANOVA table

26. ETM 620 - 09U 26 Blocking Example Your turn: fill in the blanks in the following ANOVA table (from Excel): 2. Make decision and draw conclusions: ANOVA Source of VariationSSdfMSFP-valueF crit Rows89.733244.8678.4920.01054.458968 Columns77.733____________0.0553_______ Error42.26785.2833 Total209.73___