1. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-1 Business Statistics: A Decision-Making Approach 8 th Edition Chapter 12 Analysis of Variance

2. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-2 Chapter Goals After completing this chapter, you should be able to: Recognize situations in which to use analysis of variance Understand different analysis of variance designs Perform a single-factor hypothesis test and interpret results (manually and with the aid of Excel) Conduct and interpret post-analysis of variance pairwise comparisons procedures Set up and perform randomized blocks analysis Analyze two-factor analysis of variance test with replications results with Excel or Minitab

3. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-3 Chapter Overview Analysis of Variance (ANOVA) F-test Tukey- Kramer test Fisher’s Least Significant Difference test One-Way ANOVA Randomized Complete Block ANOVA Two-factor ANOVA with replication

4. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-4 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 0.05 significance level, is there a difference in mean distance? Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204

5. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-5 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

6. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-6 One-Factor ANOVA Example Computations Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 x 1 = 249.2 x 2 = 226.0 x 3 = 205.8 x = 227.0 n 1 = 5 n 2 = 5 n 3 = 5 n T = 15 k = 3 SSB = 5 [ (249.2 – 227) 2 + (226 – 227) 2 + (205.8 – 227) 2 ] = 4716.4 SSW = (254 – 249.2) 2 + (263 – 249.2) 2 +…+ (204 – 205.8) 2 = 1119.6 MSB = 4716.4 / (3-1) = 2358.2 MSW = 1119.6 / (15-3) = 93.3

7. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-7 F = 25.275 One-Factor ANOVA Example Solution H 0 : μ 1 = μ 2 = μ 3 H A : μ i not all equal  = 0.05 df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence that at least one μ i differs from the rest 0  = 0.05 F 0.05 = 3.885 Reject H 0 Do not reject H 0 Critical Value: F  = 3.885

8. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-8 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-053.885 Within Groups 1119.61293.3 Total5836.014 ANOVA -- Single Factor: Excel Output EXCEL: tools | data analysis | ANOVA: single factor

9. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-9 Logic of Analysis of Variance Investigator controls one or more independent variables Called factors (or treatment variables) Each factor contains two or more levels (or categories/classifications) Observe effects on dependent variable Response to levels of independent variable Experimental design: the plan used to test hypothesis

10. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-10 Completely Randomized Design Experimental units (subjects) are assigned randomly to treatments Only one factor or independent variable With two or more treatment levels Analyzed by One-factor analysis of variance (one-way ANOVA) Called a Balanced Design if all factor levels have equal sample size

11. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-11 One-Way Analysis of Variance Evaluate the difference among the means of three or more populations Examples: ● Accident rates for 1 st, 2 nd, and 3 rd shift ● Expected mileage for five brands of tires Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn Data’s measurement level is interval or ratio

12. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-12 Hypotheses of One-Way ANOVA All population means are equal i.e., no treatment effect (no variation in means among groups) 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)

13. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-13 One-Factor ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect)

14. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-14 One-Factor ANOVA At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or (continued)

15. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-15 Partitioning the Variation Total variation can be split into two parts: SST = Total Sum of Squares (total variation) SSB = Sum of Squares Between (variation between samples) SSW = Sum of Squares Within (within each factor level) SST = SSB + SSW

16. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-16 Partitioning the Variation Total Variation (SST) = the aggregate dispersion of the individual data values across the various factor levels Within-Sample Variation (SSW) = dispersion that exists among the data values within a particular factor level Between-Sample Variation (SSB) = dispersion among the factor sample means SST = SSB + SSW (continued)

17. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-17 Partition of Total Variation Variation Due to Factor (SSB) Variation Due to Random Sampling (SSW) Total Variation (SST) Commonly referred to as:  Sum of Squares Within  Sum of Squares Error  Sum of Squares Unexplained  Within Groups Variation Commonly referred to as:  Sum of Squares Between  Sum of Squares Among  Sum of Squares Explained  Among Groups Variation = +

18. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-18 Total Sum of Squares Where: SST = Total sum of squares k = number of populations (levels or treatments) n i = sample size from population i x ij = j th measurement from population i x = grand mean (mean of all data values) SST = SSB + SSW

19. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-19 Total Variation (continued)

20. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-20 Sum of Squares Between Where: SSB = Sum of squares between k = number of populations n i = sample size from population i x i = sample mean from population i x = grand mean (mean of all data values) SST = SSB + SSW

21. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-21 Between-Group Variation Variation Due to Differences Among Groups Mean Square Between = SSB/degrees of freedom

22. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-22 Between-Group Variation (continued)

23. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-23 Sum of Squares Within Where: SSW = Sum of squares within k = number of populations n i = sample size from population i x i = sample mean from population i x ij = j th measurement from population i SST = SSB + SSW

24. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-24 Within-Group Variation Summing the variation within each group and then adding over all groups Mean Square Within = SSW/degrees of freedom

25. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-25 Within-Group Variation (continued)

26. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-26 Hartley’s F-test statistic One-Way ANOVA Table Source of Variation dfSSMS Between Samples SSBMSB = Within Samples n T - kSSWMSW = Totaln T - 1 SST = SSB+SSW k - 1 MSB MSW F ratio k = number of populations n T = sum of the sample sizes from all populations df = degrees of freedom SSB k - 1 SSW n T - k F =

27. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-27 One-Factor ANOVA F Test Statistic Test statistic MSB is mean squares between variances MSW is mean squares within variances Degrees of freedom df 1 = k – 1 (k = number of populations) df 2 = n T – k (n T = sum of sample sizes from all populations) H 0 : μ 1 = μ 2 = … = μ k H A : At least two population means are different

28. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-28 Interpreting One-Factor ANOVA F Statistic The F statistic is the ratio of the between estimate of variance and the within estimate of variance The ratio must always be positive df 1 = k -1 will typically be small df 2 = n T - k will typically be large The ratio should be close to 1 if H 0 : μ 1 = μ 2 = … = μ k is true The ratio will be larger than 1 if H 0 : μ 1 = μ 2 = … = μ k is false

29. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-29 ANOVA Steps 1.Specify parameter of interest 2.Formulate hypotheses 3.Specify the significance level,  4.Select independent, random samples Compute sample means and grand mean 5.Determine the decision rule 6.Verify the normality and equal variance assumptions have been satisfied 7.Create ANOVA table 8.Reach a decision and draw a conclusion

30. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-30 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

31. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-31 Tukey-Kramer Critical Range where: q  = Value from standardized range table with k and n T - k degrees of freedom for the desired level of  MSW = Mean Square Within n i and n j = Sample sizes from populations (levels) i and j

32. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-32 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 the q value from the table in appendix J with k and n T - k degrees of freedom for the desired level of 

33. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-33 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:

34. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-34 Tukey-Kramer in PHStat

35. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-35 Randomized Complete Block ANOVA Like One-Way ANOVA, we test for equal population means (for different factor levels, for example)......but we want to control for possible variation from a second factor (with two or more levels) Used when more than one factor may influence the value of the dependent variable, but only one is of key interest Levels of the secondary factor are called blocks

36. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-36 Assumptions Populations are normally distributed Populations have equal variances The observations within samples are independent The date measurement must be interval or ratio Application examples Testing 5 routes to a destination through 3 different cab companies to see if differences exist Determining the best training program (out of 4 choices) for various departments within a company Randomized Complete Block ANOVA (continued)

37. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-37 Partitioning the Variation Total variation can now be split into three parts: SST = Total sum of squares SSB = Sum of squares between factor levels SSBL = Sum of squares between blocks SSW = Sum of squares within levels SST = SSB + SSBL + SSW

38. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-38 Sum of Squares for Blocking Where: k = number of levels for this factor b = number of blocks x j = sample mean from the j th block x = grand mean (mean of all data values) SST = SSB + SSBL + SSW

39. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-39 Partitioning the Variation Total variation can now be split into three parts: SST and SSB are computed as they were in One-Way ANOVA SST = SSB + SSBL + SSW SSW = SST – (SSB + SSBL)

40. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-40 Mean Squares

41. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-41 Randomized Block ANOVA Table Source of Variation dfSSMS Between Samples SSBMSB Within Samples (k–1)(b-1)SSWMSW Totaln T - 1SST k - 1 MSBL MSW F ratio k = number of populationsn T = sum of the sample sizes from all populations b = number of blocksdf = degrees of freedom Between Blocks SSBLb - 1MSBL MSB MSW

42. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-42 Blocking Test Blocking test: df 1 = b - 1 df 2 = (k – 1)(b – 1) MSBL MSW F = Reject H 0 if F > F 

43. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-43 Main Factor test: df 1 = k - 1 df 2 = (k – 1)(b – 1) MSB MSW F = Reject H 0 if F > F  Main Factor Test

44. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-44 Fisher’s Least Significant Difference Test To test which population means are significantly different e.g.: μ 1 = μ 2 ≠ μ 3 Done after rejection of equal means in randomized block ANOVA design Allows pair-wise comparisons Compare absolute mean differences with critical range x  =  123

45. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-45 Fisher’s Least Significant Difference (LSD) Test where: t  /2 = Upper-tailed value from Student’s t-distribution for  /2 and (k - 1)(b - 1) degrees of freedom MSW = Mean square within from ANOVA table b = number of blocks k = number of levels of the main factor NOTE: This is a similar process as Tukey-Kramer

46. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-46 Fisher’s Least Significant Difference (LSD) Test (continued) If the absolute mean difference is greater than LSD then there is a significant difference between that pair of means at the chosen level of significance Compare:

47. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-47 Two-Factor ANOVA With 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?

48. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-48 Two-Factor ANOVA Assumptions Populations are normally distributed Populations have equal variances Independent random samples are drawn Data must be interval or ratio level (continued)

49. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-49 Two-Way ANOVA Sources of Variation Two Factors of interest: A and B a = number of levels of factor A b = number of levels of factor B n T = total number of observations in all cells

50. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-50 Two-Way ANOVA Sources of Variation SST Total Variation SS A Variation due to factor A SS B Variation due to factor B SS AB Variation due to interaction between A and B SSE Inherent variation (Error) Degrees of Freedom: a – 1 b – 1 (a – 1)(b – 1) n T – ab n T - 1 SST = SS A + SS B + SS AB + SSE (continued)

51. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-51 Two Factor ANOVA Equations Total Sum of Squares: Sum of Squares Factor A: Sum of Squares Factor B:

52. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-52 Two Factor ANOVA Equations Sum of Squares Interaction Between A and B: Sum of Squares Error: (continued)

53. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-53 Two Factor ANOVA Equations where: a = number of levels of factor A b = number of levels of factor B n’ = number of replications in each cell (continued)

54. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-54 Mean Square Calculations

55. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-55 Two-Way ANOVA: The F Test Statistic F Test for Factor B Main Effect F Test for Interaction Effect H 0 : μ A1 = μ A2 = μ A3 = H A : Not all μ Ai are equal H 0 : factors A and B do not interact to affect the mean response H A : factors A and B do interact F Test for Factor A Main Effect H 0 : μ B1 = μ B2 = μ B3 = H A : Not all μ Bi are equal Reject H 0 if F > F 

56. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-56 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Factor ASS A a – 1 MS A = SS A /(a – 1) MS A MSE Factor BSS B b – 1 MS B = SS B /(b – 1) MS B MSE AB (Interaction) SS AB (a – 1)(b – 1) MS AB = SS AB / [(a – 1)(b – 1)] MS AB MSE ErrorSSEn T – ab MSE = SSE/(n T – ab) TotalSSTn T – 1

57. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-57 Features of Two-Way ANOVA F Test Degrees of freedom always add up n T - 1 = (n T - ab) + (a - 1) + (b - 1) + (a - 1)(b - 1) Total = error + factor A + factor B + interaction The denominator of the F Test is always the same but the numerator is different The sums of squares always add up SST = SSE + SS A + SS B + SS AB Total = error + factor A + factor B + interaction

58. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-58 Interaction vs. No Interaction No interaction: 12 Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels 1 2 Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels Mean Response Interaction is present:

59. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-59 When conducting tests for a Two-Factor ANOVA: Test for interaction If present, conduct a one-way ANOVA to test the levels of one of the other factors using only one level of the other factor If NO interaction, test Factor A and Factor B Interaction vs. No Interaction (continued)

60. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-60 Chapter Summary Described one-way analysis of variance The logic of ANOVA ANOVA assumptions F test for difference in k means The Tukey-Kramer procedure for multiple comparisons Described randomized complete block designs F test Fisher’s least significant difference test for multiple comparisons Described two-way analysis of variance Examined effects of multiple factors and interaction

61. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 12-61 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.