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1 Chap 17-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 17 Analysis of Variance Statistics for Business and Economics 6 th Edition

2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-2 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 one-way and two-way analysis of variance and interpret the results Conduct and interpret a Kruskal-Wallis test Analyze two-factor analysis of variance tests with more than one observation per cell

3 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-3 One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples: Average production 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

4 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-4 Hypotheses of One-Way ANOVA All population means are equal i.e., no variation in means between groups At least one population mean is different i.e., there is variation between groups Does not mean that all population means are different (some pairs may be the same)

5 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-5 One-Way ANOVA All Means are the same: The Null Hypothesis is True (No variation between groups)

6 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-6 One-Way ANOVA At least one mean is different: The Null Hypothesis is NOT true (Variation is present between groups) or (continued)

7 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-7 Variability The variability of the data is key factor to test the equality of means In each case below, the means may look different, but a large variation within groups in B makes the evidence that the means are different weak Small variation within groups A B C Group A B C Group Large variation within groups AB

8 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-8 Partitioning the Variation Total variation can be split into two parts: SST = Total Sum of Squares Total Variation = the aggregate dispersion of the individual data values across the various groups SSW = Sum of Squares Within Groups Within-Group Variation = dispersion that exists among the data values within a particular group SSG = Sum of Squares Between Groups Between-Group Variation = dispersion between the group sample means SST = SSW + SSG

9 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 17-9 Partition of Total Variation Variation due to differences between groups (SSG) Variation due to random sampling (SSW) Total Sum of Squares (SST) = +

10 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Total Sum of Squares Where: SST = Total sum of squares K = number of groups (levels or treatments) n i = number of observations in group i x ij = j th observation from group i x = overall sample mean SST = SSW + SSG

11 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Total Variation (continued)

12 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Within-Group Variation Where: SSW = Sum of squares within groups K = number of groups n i = sample size from group i X i = sample mean from group i X ij = j th observation in group i SST = SSW + SSG

13 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Within-Group Variation Summing the variation within each group and then adding over all groups Mean Square Within = SSW/degrees of freedom (continued)

14 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Within-Group Variation (continued)

15 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Between-Group Variation Where: SSG = Sum of squares between groups K = number of groups n i = sample size from group i x i = sample mean from group i x = grand mean (mean of all data values) SST = SSW + SSG

16 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Between-Group Variation Variation Due to Differences Between Groups Mean Square Between Groups = SSG/degrees of freedom (continued)

17 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Between-Group Variation (continued)

18 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Obtaining the Mean Squares

19 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap One-Way ANOVA Table Source of Variation dfSS MS (Variance) Between Groups SSGMSG = Within Groups n - KSSWMSW = Totaln - 1 SST = SSG+SSW K - 1 MSG MSW F ratio K = number of groups n = sum of the sample sizes from all groups df = degrees of freedom SSG K - 1 SSW n - K F =

20 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap One-Factor ANOVA F Test Statistic Test statistic MSG is mean squares between variances MSW is mean squares within variances Degrees of freedom df 1 = K – 1 (K = number of groups) df 2 = n – K (n = sum of sample sizes from all groups) H 0 : μ 1 = μ 2 = … = μ K H 1 : At least two population means are different

21 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Interpreting the 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 - K will typically be large Decision Rule: Reject H 0 if F > F K-1,n-K, 0 =.05 Reject H 0 Do not reject H 0 F K-1,n-K,

22 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 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 distance? Club 1 Club 2 Club

23 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap One-Factor ANOVA Example: Scatter Diagram Distance Club 1 Club 2 Club Club 1 2 3

24 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap One-Factor ANOVA Example Computations Club 1 Club 2 Club x 1 = x 2 = x 3 = x = n 1 = 5 n 2 = 5 n 3 = 5 n = 15 K = 3 SSG = 5 (249.2 – 227) (226 – 227) (205.8 – 227) 2 = SSW = (254 – 249.2) 2 + (263 – 249.2) 2 +…+ (204 – 205.8) 2 = MSG = / (3-1) = MSW = / (15-3) = 93.3

25 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap F = 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: 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 Critical Value: F 2,12,.05 = 3.89

26 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap SUMMARY GroupsCountSumAverageVariance Club Club Club ANOVA Source of Variation SSdfMSFP-valueF crit Between Groups E Within Groups Total ANOVA -- Single Factor: Excel Output EXCEL: tools | data analysis | ANOVA: single factor

27 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Kruskal-Wallis Test Use when the normality assumption for one- way ANOVA is violated Assumptions: The samples are random and independent variables have a continuous distribution the data can be ranked populations have the same variability populations have the same shape

28 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Kruskal-Wallis Test Procedure Obtain relative rankings for each value In event of tie, each of the tied values gets the average rank Sum the rankings for data from each of the K groups Compute the Kruskal-Wallis test statistic Evaluate using the chi-square distribution with K – 1 degrees of freedom

29 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Kruskal-Wallis Test Procedure The Kruskal-Wallis test statistic: (chi-square with K – 1 degrees of freedom) where: n = sum of sample sizes in all groups K = Number of samples R i = Sum of ranks in the i th group n i = Size of the i th group (continued)

30 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Decision rule Reject H 0 if W > 2 K–1, Otherwise do not reject H 0 (continued) Kruskal-Wallis Test Procedure Complete the test by comparing the calculated H value to a critical 2 value from the chi-square distribution with K – 1 degrees of freedom 2 K–1, 0 Reject H 0 Do not reject H 0

31 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Do different departments have different class sizes? Kruskal-Wallis Example Class size (Math, M) Class size (English, E) Class size (Biology, B)

32 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Do different departments have different class sizes? Kruskal-Wallis Example Class size (Math, M) Ranking Class size (English, E) Ranking Class size (Biology, B) Ranking = 44 = 56 = 20

33 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The W statistic is (continued) Kruskal-Wallis Example

34 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Since H = 6.72 >, reject H 0 (continued) Kruskal-Wallis Example Compare W = 6.72 to the critical value from the chi-square distribution for 3 – 1 = 2 degrees of freedom and =.05: There is sufficient evidence to reject that the population means are all equal

35 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way Analysis of Variance Examines the effect of Two 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 carbonation level depend on which level the line speed is set?

36 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way ANOVA Assumptions Populations are normally distributed Populations have equal variances Independent random samples are drawn (continued)

37 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Randomized Block Design Two Factors of interest: A and B K = number of groups of factor A H = number of levels of factor B (sometimes called a blocking variable) Block Group 12…K 12..H12..H x 11 x 12. x 1H x 21 x 22. x 2H ……..………..… x K1 x K2. x KH

38 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way Notation Let x ji denote the observation in the j th group and i th block Suppose that there are K groups and H blocks, for a total of n = KH observations Let the overall mean be x Denote the group sample means by Denote the block sample means by

39 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Partition of Total Variation Variation due to differences between groups (SSG) Variation due to random sampling (unexplained error) (SSE) Total Sum of Squares (SST) = + Variation due to differences between blocks (SSB) + SST = SSG + SSB + SSE The error terms are assumed to be independent, normally distributed, and have the same variance

40 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way Sums of Squares The sums of squares are Degrees of Freedom: n – 1 K – 1 H – 1 (K – 1)(K – 1)

41 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way Mean Squares The mean squares are

42 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way ANOVA: The F Test Statistic F Test for Blocks H 0 : The K population group means are all the same F Test for Groups H 0 : The H population block means are the same Reject H 0 if F > F K-1,(K-1)(H-1), Reject H 0 if F > F H-1,(K-1)(H-1),

43 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap General Two-Way Table Format Source of Variation Sum of Squares Degrees of Freedom Mean SquaresF Ratio Between groups Between blocks Error Total SSG SSB SSE SST K – 1 H – 1 (K – 1)(H – 1) n - 1

44 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap A two-way design with more than one observation per cell allows one further source of variation The interaction between groups and blocks can also be identified Let K = number of groups H = number of blocks L = number of observations per cell n = KHL = total number of observations More than One Observation per Cell

45 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap More than One Observation per Cell SST Total Variation SSG Between-group variation SSB Between-block variation SSI Variation due to interaction between groups and blocks SSE Random variation (Error) Degrees of Freedom: K – 1 H – 1 (K – 1)(H – 1) KH(L – 1) n – 1 SST = SSG + SSB + SSI + SSE (continued)

46 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Sums of Squares with Interaction Degrees of Freedom: K – 1 H – 1 (K – 1)(H – 1) KH(L – 1) n - 1

47 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way Mean Squares with Interaction The mean squares are

48 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way ANOVA: The F Test Statistic F Test for block effect F Test for interaction effect H 0 : the interaction of groups and blocks is equal to zero F Test for group effect H 0 : The K population group means are all the same H 0 : The H population block means are the same Reject H 0 if F > F K-1,KH(L-1), Reject H 0 if F > F H-1,KH(L-1), Reject H 0 if F > F (K-1)(H-1),KH(L-1),

49 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Between groups SSGK – 1 MSG = SSG / (K – 1) MSG MSE Between blocks SSBH – 1 MSB = SSB / (H – 1) MSB MSE InteractionSSI(K – 1)(H – 1) MSI = SSI / (K – 1)(H – 1) MSI MSE ErrorSSEKH(L – 1) MSE = SSE / KH(L – 1) TotalSSTn – 1

50 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Features of Two-Way ANOVA F Test Degrees of freedom always add up n-1 = KHL-1 = (K-1) + (H-1) + (K-1)(H-1) + KH(L-1) Total = groups + blocks + interaction + error The denominator of the F Test is always the same but the numerator is different The sums of squares always add up SST = SSG + SSB + SSI + SSE Total = groups + blocks + interaction + error

51 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Examples: Interaction vs. No Interaction No interaction: Block Level 1 Block Level 3 Block Level 2 Groups Block Level 1 Block Level 3 Block Level 2 Groups Mean Response Interaction is present: A B C

52 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Chapter Summary Described one-way analysis of variance The logic of Analysis of Variance Analysis of Variance assumptions F test for difference in K means Applied the Kruskal-Wallis test when the populations are not known to be normal Described two-way analysis of variance Examined effects of multiple factors Examined interaction between factors

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