1. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-1 Chapter 10 Analysis of Variance Statistics for Managers Using Microsoft ® Excel 4 th Edition

2. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-2 Chapter Goals After completing this chapter, you should be able to: Recognize situations in which to use analysis of variance (ANOVA) Understand different analysis of variance designs Evaluate assumptions of the model Perform a single-factor ANOVA and interpret the results Conduct and interpret a Tukey-Kramer post-analysis to determine which means are different Analyze two-factor analysis of variance tests Conduct and interpret a Tukey-Kramer post-analysis procedure to determine which factors are different

3. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-3 General ANOVA Analysis Investigator controls one or more independent variables Called factors or treatment variables One factor contains three or more levels or groups or categories/classifications Other factors contains two or more levels or groups or categories/classifications Experimental design: the plan used to test the hypothesis

4. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-4 One-Factor ANOVA Also known as Completely Randomized Design and One-way ANOVA Experimental units (subjects) are assigned randomly to treatments Subjects are assumed homogeneous Only one factor or independent variable With three or more treatment levels Analyzed by one-factor analysis of variance

5. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-5 One-Factor Analysis of Variance Evaluates the difference among the means of three or more groups Examples: Accident rates for 1 st, 2 nd, and 3 rd shift Expected mileage for five brands of tires Assumptions Populations are normally distributed (test with Box plot or Normal Probability Plot) Populations have equal variances (use Levene’s Test for Homogeneity of Variance) Samples are randomly and independently drawn

6. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-6 Why Analysis of Variance? We could compare the means in pairs using a t test for difference of means Each t test contains Type 1 error The total Type 1 error with k pairs of means is 1- (1 -  ) k If there are 5 means and you use  =.05 Must perform 10 comparisons Type I error is 1 – (.95) 10 =.40 40% of the time you will reject the null hypothesis of equal means in favor of the alternative even when the null is true!

7. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-7 Hypotheses of One-Factor ANOVA All population means are equal i.e., no treatment effect At least one population mean is different i.e., there is a treatment effect Does not mean that all population means are different (some pairs may be the same)

8. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-8 One-Factor ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect)

9. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-9 One-Factor ANOVA At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or (continued)

10. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-10 One-Factor ANOVA F Test Example You want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the.05 significance level, is there a difference in mean driving distance? Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204

11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-11 One-Factor ANOVA Example: Scatter Diagram 270 260 250 240 230 220 210 200 190 Distance Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 Club 1 2 3

12. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-12 SUMMARY GroupsCountSumAverageVariance Club 151246249.2108.2 Club 25113022677.5 Club 351029205.894.2 ANOVA Source of Variation SSdfMSFP-valueF crit Between Groups 4716.422358.225.2754.99E-053.89 Within Groups 1119.61293.3 Total5836.014 ANOVA -- Single Factor: Excel Output EXCEL: Tools | Data Analysis | ANOVA: Single Factor

13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-13 F = 25.275 One-Factor ANOVA Example Solution H 0 : μ 1 = μ 2 = μ 3 H 1 : μ i not all equal  =.05 df 1 = 2 df 2 = 12 Test Statistic: p-value: 4.99E-05 Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence that at least one μ i differs from the rest 0  =.05 Reject H 0 Do not reject H 0

14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-14 The Tukey-Kramer Procedure Tells which population means are significantly different e.g.: μ 1 = μ 2  μ 3 Done after rejection of equal means in ANOVA Allows pair-wise comparisons Compare absolute mean differences with critical range x μ 1 = μ 2 μ 3

15. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-15 The Tukey-Kramer Procedure: Example 1. Compute absolute mean differences: Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 2. Find a Q U value from the table in appendix E.9 with c = 3 (across the table) and n – c = 15 – 3 = 12 degrees of freedom (down the table) for the desired level of  (  =.05 used here): Get MSW from the ANOVA output.

16. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-16 The Tukey-Kramer Procedure: Example 5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance. 3. Compute Critical Range: 4. Compare: (continued) PhStat does all the calculations for you but you must input the Q value

17. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-17 Tukey-Kramer in PHStat

18. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-18 Two-Factor ANOVA (With or Without Replication) Examines the effect of Two or more factors of interest on the dependent variable e.g.: Percent carbonation and line speed on soft drink bottling process Interaction between the different levels of these two factors e.g.: Does the effect of one particular percentage of carbonation depend on which level the line speed is set?

19. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-19 Assumptions Populations are normally distributed Populations have equal variances Independent random samples are drawn (continued) Two-Factor ANOVA (With or Without Replication)

20. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-20 Two-Factor ANOVA Summary Table Without Replication Source of Variation Degrees of Freedom Sum of Square s Mean Squares F Statistic p-value Rows Factor A r – 1SSAMSA MSA/ MSE f (F A ) Columns Factor B c – 1SSBMSB MSB/ MSE f (F B ) Error (r-1)(c-1)SSEMSE Total r  c–1 SST

21. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-21 Two-Factor ANOVA Summary Table With Replication Source of Variation Degrees of Freedom Sum of Squares Mean Squares F Statistic p-value Sample Factor A (Row) r – 1SSA MSA = SSA/(r – 1) MSA/ MSE f (F A ) Columns Factor B c – 1SSB MSB = SSB/(c – 1) MSB/ MSE f (F B ) Interaction (AB) (r – 1)(c – 1)SSAB MSAB = SSAB/ [(r – 1)(c – 1)] MSAB/ MSE f (F A&B ) Within Error r  c  n ’ – 1) SSE MSE = SSE/[r  c  n ’ – 1)] Total r  c  n ’ – 1 SST

22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-22 Two-Factor ANOVA: The F Test Statistic F Test for Factor B Effect F Test for Interaction Effect H 0 : μ 1A = μ 2A = μ 3A = H 1 : Not all means are equal H 0 : Interaction of A and B is equal to zero H 1 : Interaction of A and B is not zero F Test for Factor A Effect H 0 : μ 1B = μ 2B = μ 3B = H 1 : Not all means are equal Only for Two-Factor with replication

23. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-23 As production manager, you want to see if 3 filling machines have different mean filling times when used with 5 types of boxes. At the.05 level, is there a difference in machines, in boxes? Box Machine1 Machine2 Machine3 1 25.40 23.40 20.00 2 26.31 21.80 22.20 3 24.10 23.50 19.75 4 23.74 22.75 20.60 5 25.10 21.60 20.40 Two-Factor ANOVA Without Replication

24. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-24 Summary Table Source of Variation Degrees of Freedom Sum of Squares Mean Square.6309 Rows ( Boxes ) 5 - 1 = 42.6507.6627 Columns ( Machines ) 3 - 1 = 247.164 1.0503 Total(3 · 5)-1 = 1458.2172 P-ValueF.6543 Error 8.4025 23.58222.4525.0005 (5-1)(3-1) = 8

25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-25 Two-Factor ANOVA With Replication As production manager, you want to see if 3 filling machines have different mean filling times when used with 5 types of boxes. At the.05 level, is there a difference in machines, in boxes? Is there an interaction? Box Machine1 Machine2 Machine3 1 25.40 23.40 20.00 26.40 24.40 21.00 2 26.31 21.80 22.20 25.90 23.00 22.00 3 24.10 23.50 19.75 24.40 22.40 19.00 4 23.74 22.75 20.60 25.40 23.40 20.00 5 25.10 21.60 20.40 26.20 22.90 21.90

26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-26 Summary Table Source of Variation Degrees of Freedom Sum of Squares Mean Square 3.6868 Sample ( Boxes ) 5 - 1 = 47.47141.8678 Columns ( Machines ) 3 - 1 = 2106.298.5066 Total3 · 5 ·2 -1 = 29131.0720 P-ValueF.0277 Within (Error) 7.5994 53.149104.9081.52E-09 (5-1)(3-1) = 8 Interaction 5·3·(2-1)=15 9.7032 1.21292.3941.0690

27. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-27 The Tukey-Kramer Procedure: With Replication Factor A 1.No Macro - compute by formula only 2.MSW (Within) from ANOVA Printout 3.Q from Table E.9 in the Book page 860 alpha =.05 or.01 r is the number of levels of factor A (across table) rc(n’-1) (down the table) c is the number of levels of factor B n’ is the number of replications

28. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-28 The Tukey-Kramer Procedure: With Replication Factor B 1.No Macro compute by formula only 2.MSW (Within) from ANOVA Printout 3.Q from Table E.9 in the Book page 860 alpha =.05 or.01 c is the number of levels of factor B (across table) rc(n’-1) (down the table) r is the number of levels of factor A n’ is the number of replications

29. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 10-29 Chapter Summary Described one-factor analysis of variance ANOVA assumptions ANOVA test for difference in c means The Tukey-Kramer procedure for multiple comparisons Described two-factor analysis of variance Examined effects of multiple factors Examined interaction between factors for the model with and without replicated observations The Tukey-Kramer procedure for multiple comparisons for both factor A and factor B for the model with replication