1. Chapter 10: Analysis of Variance: Comparing More Than Two Means

2. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 2 Where We’ve Been Presented methods for estimating and testing hypotheses about a single population mean Presented methods for comparing two population means

3. Where We’re Going Discuss the critical elements in the design of a sampling experiment Investigate completely randomized, randomized block, and factorial designs Show how to analyze data using a technique called analysis of variance Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 3

4. 4 10.1: Elements of a Designed Experiment Elements of Designed Experiments Factors (possible impacting the response variable) Quantitative Factors (Numerical) Qualitative Factors (Non-numerical) Response Variable (or dependent variable)

5. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 5 10.1: Elements of a Designed Experiment Elements of Designed Experiments Factors (possible impacting the response variable) Quantitative Factors (Numerical) Qualitative Factors (Non-numerical) Response Variable (or dependent variable) Factor Levels are the values of the factors

6. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 6 10.1: Elements of a Designed Experiment Treatments are the factor-level combinations  In the example above, a variety of different GPA – Hours Studied combinations could occur within each subset (Yes or No) of the Study Group factor

7. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 7 10.1: Elements of a Designed Experiment An experimental unit is the object on which the response and factors are observed or measured  In the example above, an individual student would be the experimental unit

8. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 8 10.1: Elements of a Designed Experiment In a designed experiment the analyst controls the treatments and the selection of experimental units to each treatment In an observational experiment the analyst observes the treatment and response on a sample of experimental units

9. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 9 10.1: Elements of a Designed Experiment In a designed experiment the analyst controls the treatments and the selection of experimental units to each treatment In an observational experiment the analyst observes the treatment and response on a sample of experimental units The method by which the experimental units are selected determines the type of experiment

10. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 10 10.1: Elements of a Designed Experiment Population of Experimental Units Sample of Experimental Units Apply factor-level combinations Treatment 1 Sample Treatment 2 Sample Treatment 3 Sample … Treatment k Sample

11. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 11 10.2: The Completely Randomized Design The completely randomized design is a design in which treatments are randomly assigned to the experimental units or in which independent random samples of experimental units are selected for each treatment.

12. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 12 10.2: The Completely Randomized Design Randomly assign observations to treatments Completely Randomized Design Randomly assign treatments to observations

13. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 13 10.2: The Completely Randomized Design Very often the object is to determine whether the varying treatments result in different means: H 0 : µ 1 = µ 2 = µ 3 = µ 4 = ··· = µ k H a : At least two of the k treatment means differ

14. Testing the equity of the means involves comparing the variability among the different treatments as well as within the treatments, adjusted for degrees of freedom. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 14 10.2: The Completely Randomized Design

15. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 15 10.2: The Completely Randomized Design Adjusting for degrees of freedom produces comparable measures of variability

16. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 16 10.2: The Completely Randomized Design

17. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 17 10.2: The Completely Randomized Design The ratio of the variability among the treatment means to that within the treatment means is an F -statistic: with k-1 numerator and n-k denominator degrees of freedom.

18. 10.2: The Completely Randomized Design If F*  1, the difference between the treatment means may be attributable to sampling error. If F* > 1 (significantly), there is support for the alternative hypothesis that the treatments themselves produce different results. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 18

19. 10.2: The Completely Randomized Design Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 19

20. 10.2: The Completely Randomized Design Conditions required for a Valid ANOVA F-Test: Completely Randomized Design 1. The samples are randomly selected in an independent manner from the k treatment populations. 2. All k sampled populations have distributions that are approximately normal. 3. The k population variances are equal. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 20

21. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 21 10.2: The Completely Randomized Design The USGA compares the driving distance of four brands of golf balls.  H 0 : µ 1 = µ 2 = µ 3 = µ 4  H a : The mean distances differ for at least two of the brands   =.10  Test Statistic: F = MST/MSE  Rejection region: F > 2.25 = F.10 with v 1 = 3 and v 2 = 36

22. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 22 10.2: The Completely Randomized Design

23. The USGA compares the driving distance of four brands of golf balls.  H 0 : µ 1 = µ 2 = µ 3 = µ 4  H a : The mean distances differ for at least two of the brands   =.10Test Statistic: F = MST/MSE  Rejection region: F > 2.25 = F.10 with v 1 = 3 and v 2 = 36 SourceDegrees of Freedom Sum of Squares Mean Square Fp-value Brands32,794.39931.4643.99.000 Error36762.3021.18 Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 23

24. 10.2: The Completely Randomized Design The USGA compares the driving distance of four brands of golf balls.  H 0 : µ 1 = µ 2 = µ 3 = µ 4  H a : The mean distances differ for at least two of the brands   =.10Test Statistic: F = MST/MSE  Rejection region: F > 2.25 = F.10 with v 1 = 3 and v 2 = 36 SourceDegrees of Freedom Sum of Squares Mean Square Fp-value Brands32,794.39931.4643.99.000 Error36762.3021.18 Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 24 Since the calculated F > 2.25, we reject the null hypothesis.

25. 10.2: The Completely Randomized Design Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 25 If the conditions for ANOVA are not met, a nonparametric procedure is recommended (see Chapter 14). If the null hypothesis is not rejected, that is not conclusive proof that the treatment means are all equal.

26. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 26 10.3: Multiple Comparisons of Means Suppose the ANOVA F-test indicates differences in the means. To determine the differences, we would compare the differences of the means. With k treatment means, there are c = k(k – 1)/2 pairs of means to be compared, and each would have a significance level smaller than the overall .

27. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 27 10.3: Multiple Comparisons of Means To retain the overall confidence level, various techniques are available for pair wise comparisons:  Tukey – treatment sample sizes are equal  Bonferroni - treatment sample sizes may be unequal  Scheffé – general procedure for all linear combinations of treatment means

28. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 28 10.3: Multiple Comparisons of Means Let’s go back to the four brands of golf balls in the previous example:  Rank the treatment means with an overall 95% level of confidence using Tukey’s procedure.  Estimate the highest ranked golf ball's mean driving distance.

29. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 29 10.3: Multiple Comparisons of Means Brand Comparison95% Confidence Interval µ A - µ B -15.82 < µ A - µ B < - 4.74 µ A - µ C -24.71 < µ A - µ C < -13.63 µ A - µ D -4.08 < µ A - µ D < 7.00 µ B - µ C -14.43 < µ B - µ C < -3.35 µ B - µ D 6.2 < µ B - µ D < 17.28 µ C - µ D 15.09 < µ C - µ D < 26.17 Pair wise Comparisons for Four Golf ball Brands Based on a SAS ANOVA report (see pages 506-7) Brand C outperforms each of the other brands.

30. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 30 10.3: Multiple Comparisons of Means To construct a confidence interval on Brand C, we can use the descriptive statistics from the ANOVA and a straightforward one- sample t-based confidence interval (see section 7.3):

31. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 31 10.4: The Randomized Block Design The randomized block design:  Blocks (matched sets of experimental units) are formed. Each of the b blocks has k experimental units, one for each treatment.  One experimental unit from each block is randomly assigned to each treatment, for a total of n = bk responses.

32. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 32 10.4: The Randomized Block Design To test the equity of the means, we use the ratio MST/MSE ~ F

33. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 33 10.4: The Randomized Block Design

34. Conditions required for a valid ANOVA F – Test  The b blocks are randomly selected and all k treatments are applied (in random order) to each block.  The distribution of observations corresponding to all bk block-treatment combinations are approximately normal.  The bk block-treatment distributions have equal variances. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 34

35. 10.4: The Randomized Block Design Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 35 Total Sum of Squares SS(Total) df=n-1 Sum of Squares for Treatments SST df=k-1 Sum of Squares for Treatments SST df=k-1 Sum of Squares for Error SSE df=n-k Sum of Squares for Blocks SSB df=b-1 Sum of Squares for Error SSE df=n-b-k+1=(b-1)(k-1) Completely Randomized Design Randomized Block Design

36. 10.4: The Randomized Block Design Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 36 Suppose the golf balls analyzed above are analyzed again using ten real golfers instead of a machine.  Each golfer is a block  Each brand is a treatment assigned in random order to each golfer The ten drives for each brand produce the following means: Brand ABrand BBrand CBrand D 227 yards233.2 yards245.3 yards220.7 yards

37. 10.4: The Randomized Block Design Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 37 SourcedfSSMSFp Treatment (Brand)32,298.71,099.654.31.000 Block (Golfer)912,073.91,341.5 Error27546.620.2 Total3915,919.2 ANOVA Table for the Golf Ball Tests

38. 10.4: The Randomized Block Design Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 38 Equity of Means 95% Confidence Intervals for the Golf Balls’ Distance µ A - µ B µ A - µ C µ A - µ D µ B - µ C µ B - µ D µ C - µ D (-11.9,--.4)(-24.0, -2.6)(.6, 12.0)(-17.9, -6.4)(6.7, 18.2)(18.9, 30.3) None of the confidence intervals contain zero, so we can be 95% certain all of the brand means differ.

39. 10.4: The Randomized Block Design Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 39 SourcedfSSMSFp Treatment (Brand)32,298.71,099.654.31.000 Block (Golfer)912,073.91,341.5 Error27546.620.2 Total3915,919.2 ANOVA Table for the Golf Ball Tests To test for block mean differences, use the ratio of MSB to MEE

40. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 40 10.5: Factorial Experiments A complete factorial experiment is a factorial experiment in which every factor-level combination is utilized. That is, the number of treatments in the experiment equals the total number of factor-level combinations.

41. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 41 10.5: Factorial Experiments Say two factors, A and B, are involved. Results could vary among observations because … Factor A (only) matters Factors A and B both matter and they interact Factor B (only) matters Factors A and B both matter, independently

42. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 42 10.5: Factorial Experiments

43. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 43 Total Sum of Squares SS(Total) df=n-1 Sum of Squares for Treatments SST df=ab-1 Main effect sum of squares Factor A SS(A) df=a-1 Main effect sum of squares Factor B SS(B) df = b-1 Interaction sum of squares Factors A and B SS(AB) df = (a-1)(b-1) Sum of Squares for Error SSE df=n-ab Sum of Squares for Error SSE df= n - ab Stage 1Stage 2 10.5: Factorial Experiments

44. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 44 10.5: Factorial Experiments Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

45. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 45 10.5: Factorial Experiments Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

46. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 46 10.5: Factorial Experiments Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

47. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 47 10.5: Factorial Experiments Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

48. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 48 10.5: Factorial Experiments Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment  Conditions Required: Response distribution for each factor-level combination is normal. Response variance is constant for all treatments. Random and independent samples of experimental units are associated with each treatment.

49. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 49 10.5: Factorial Experiments The four brands of golf balls are tested again, this time with a driver and a 5 iron. Each brand-club combination (eight in all) is assigned randomly to four experimental units in a sequence of swings by Iron Byron.  Are the treatment means equal?  Do the factors “brand“ and “club” interact?

50. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 50 10.5: Factorial Experiments SourcedfSSMSF Model733659.814808.54140.35 Brand132,092.1132,093.11936.75 Club3800.74266.917.79 Interaction3765.96255.327.45 Error24822.2434.26 Total3134,482.05 TABLE 10.13: ANOVA Summary Table for Example 10.10

51. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 51 10.5: Factorial Experiments SourcedfSSMSF Model733659.814808.54140.35 Brand132,092.1132,093.11936.75 Club3800.74266.917.79 Interaction3765.96255.327.45 Error24822.2434.26 Total3134,482.05 TABLE 10.13: ANOVA Summary Table for Example 10.10 Reject the null hypothesis

52. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 52 10.5: Factorial Experiments SourcedfSSMSF Model733659.814808.54140.35 Brand132,092.1132,093.11936.75 Club3800.74266.917.79 Interaction3765.96255.327.45 Error24822.2434.26 Total3134,482.05 TABLE 10.13: ANOVA Summary Table for Example 10.10

53. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 53 10.5: Factorial Experiments SourcedfSSMSF Model733659.814808.54140.35 Brand132,092.1132,093.11936.75 Club3800.74266.917.79 Interaction3765.96255.327.45 Error24822.2434.26 Total3134,482.05 TABLE 10.13: ANOVA Summary Table for Example 10.10 Reject the null hypothesis

54. Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance 54 10.5: Factorial Experiments Further analysis (see text) suggests that, although the factor “Club” clearly has an impact on distance, the results for “Brand “ are more ambiguous: Brand B hit with a 5 iron outdistances the others, but not when hit with the driver.