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: H0: 1 = 2 H1: 1 2 EGR 252 - Ch.13 1 Comparing > 2 Means What if there are 3 different hardwood concentrations (say, 5%, 10%, and 15%)? H0: 1 = 2 H0: 1 = 3 H0: 2 = 3 and and H1: 1 2 H1: 1 3 H1: 2 3 How about 4 different concentrations (say, 5%, 10%, 15%, and 20%)? All of the above, PLUS H0: 1 = 4 H0: 2 = 4

H 0: 3 = 4 and and H1: 1 4 H1: 2 4 H1: 3 4 What about 5 concentrations? 10? EGR 252 - Ch.13 2 Comparing > 2 Means Also, suppose = 0.05 (1 ) = P(accept H0 | H0 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 dont influence results. EGR 252 - Ch.13 3 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: Hardwood Observations Concentration (%) EGR 252 - Ch.13 1 2 3 4 5 6 Totals Averages 5

7 8 15 11 9 10 60 10.00 10 12 17 13 18 19 15 94

15.67 15 14 18 19 17 16 18 102 17.00 20 19 25 22 23 18 20

127 21.17 383 15.96 4 To determine if there is a difference 1. Calculate sums of squares SStotal y 2 y __________ ______ N i 1 j 1 SStreat y i2 y 2 __________ ________ n N i 1 a n

2 ij a SSE SStotal SStreat __________ _______ 2. Calculate degrees of freedom dftreat = a 1 = _____ dfE = a(n 1) = _____ dftotal = an 1 = _____ EGR 252 - Ch.13 5 Determining the Difference 3. Mean Square, MS = SS/df MStreat = ___________ MSE = ___________ 4. Calculate F = MStreat / MSE = _____________ EGR 252 - Ch.13 6 Organizing the Results 5. Build the ANOVA table and determine significance ANOVA Source of Variation

SS df Treatment 382.79 3 Error 130.17 20 Total 512.96 23 MS F 127.6 19.6 P-value 3.6E-06 F crit 3.1

6.5083 fixed -level level compare to F,a-1, a(n-1) p value find p associated with this F with degrees of freedom a-1, a(n-1) EGR 252 - Ch.13 7 Conclusion? 6. Draw the picture and state your conclusion Conclusion: Why? E(MSE) = 2 always E(MStreat) = 2 only if the means are equal EGR 252 - Ch.13 8 Which means are different? Graphical methods Numerical methods Tukeys test Duncans Multiple Range test more on these next time EGR 252 - Ch.13

9 Recall: One-Way ANOVA 1. Calculate and check residuals, eij = Oi - Ei 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 Tukeys procedure: ( y r y s ) q( , k, ) MSE 1/ n 4. Model: yij = + i + ij EGR 252 - Ch.13 10 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: yij = + i + j + ij EGR 252 - Ch.13 11 Example: Robins Air Force Base uses CO2 to strip paint from F-15s. You have been asked to design a test to determine the optimal pressure for spraying the CO2. 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): EGR 252 - Ch.13 12 ANOVA: One-Way with Blocking 1. Construct the ANOVA table k Where, b SST ( y ij y )2 i 1 j 1 k

SSA b ( y i y )2 i 1 b SSB k ( y j y )2 j 1 EGR 252 - Ch.13 SSE SST SSA SSB 13 Blocking Example Your turn: fill in the blanks in the following ANOVA table (from Excel): ANOVA Source of Variation SS df MS P-value F crit 44.867 8.492 0.0105

4.458968 0.0553 _______ Rows 89.733 2 Columns 77.733 ___ _____ Error 42.267 8 5.2833 Total 209.73 ___

F ____ 2. Make decision and draw conclusions: EGR 252 - Ch.13 14 Two-Way ANOVA Blocking is used to keep extraneous factors from masking the effects of the treatments you are interested in studying. A two-way ANOVA is used when you are interested in determining the effect of two treatments. Model: yijk = + i + j + ( )ijk + ij EGR 252 - Ch.13 15 Two-Way ANOVA w/ Replication Your fame as an experimental design expert grows. You have been called in as a consultant to help the Pratt and Whitney plant in Columbus determine the best method of applying the reflective stripe that is used to guide the Automated Guided Vehicles (AGVs) along their path. There are two ways of applying the stripe (paint and coated adhesive tape) and three types of flooring (linoleum and two types of concrete) in the facilities using

the AGVs. You have set up two identical test tracks on each type of flooring and applied the stripe using the two methods under study. You run 3 replications in random order and count the number of tracking errors per 1000 ft of track. The results are as follows: EGR 252 - Ch.13 16 Two-Way ANOVA Example Analysis is the same as with blocking, except we are now concerned with interaction effects EGR 252 - Ch.13 17 Two-Way ANOVA EGR 252 - Ch.13 18 Your Turn Fill in the blanks ANOVA Source of Variation Sample SS df

MS F F crit 0.14748 4.74722 0.4356 1 4.48 2 2.24 12.33 0.00123 3.88529 0.9644 ___ 0.4822

_____ 0.11104 3.88529 Within 2.18 ___ 0.1817 Total 8.06 17 Columns Interaction _____ 2.3976 P-value What does this mean? EGR 252 - Ch.13 19 What if Interaction Effects

are Significant? For example, suppose a new test was run using different types of paint and adhesive, with the following results: Linoleum Concrete I Concrete II Paint 10.7 10.8 12.2 10.9 11.1 12.3 11.3 10.7 12.5 Adhesive 11.2 11.9 10.9 11.6 12.2 11.6 10.9 11.7 11.9 ANOVA Source of Variation SS Sample 0.109

Columns 1.96 Interaction 2.831 df MS P-value F crit 1 0.1089 1.071 0.3211 4.7472 2 0.98 9.639 0.0032 3.8853 2 1.4156 13.92 0.0007 3.8853 Within 1.22

12 0.1017 Total 6.12 17 EGR 252 - Ch.13 F 20 Understanding Interaction Effects Graphical methods: graph means vs factors identify where the effect will change the result for one factor based on the value of the other. Interaction Tracking Errors 12.5 12 Paint 11.5 Adhesive 11

10.5 0 1 2 3 4 Floor Type EGR 252 - Ch.13 21