Statistics for Business and Economics 7th Edition Chapter 15 Analysis of Variance Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-1 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-2 15.2  One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples: Average production for 1st, 2nd, and 3rd shifts Expected mileage for five brands of tires  Assumptions  Populations are normally distributed  Populations have equal variances  Samples are randomly and independently drawn Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-3 Hypotheses of One-Way ANOVA   H0 : μ1 = μ2 = μ3 =  = μK  All population means are equal  i.e., no variation in means between groups H1 : μi ≠ μ j for at least one i, j pair  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) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-4 One-Way ANOVA H0 : μ1 = μ2 = μ3 =  = μK H1 : Not all μi are the same All Means are the same: The Null Hypothesis is True (No variation between groups) μ1 = μ2 = μ3 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-5 One-Way ANOVA (continued) H0 : μ1 = μ2 = μ3 =  = μK H1 : Not all μi are the same At least one mean is different: The Null Hypothesis is NOT true (Variation is present between groups) or μ1 = μ2 ≠ μ3 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall μ1 ≠ μ2 ≠ μ3 Ch 15-6 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 A B A B Group C Small variation within groups Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall A B Group C Large variation within groups Ch 15-7 Partitioning the Variation  Total variation can be split into two parts: SST = SSW + SSG 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-8 Partition of Total Variation Total Sum of Squares (SST) = Variation due to random sampling (SSW) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall + Variation due to differences between groups (SSG) Ch 15-9 Total Sum of Squares SST = SSW + SSG K ni SST = ∑∑ (x ij − x) i=1 j=1 Where: SST = Total sum of squares K = number of groups (levels or treatments) ni = number of observations in group i xij = jth observation from group i x = overall sample mean Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-10 Two-Way Notation     Let xji denote the observation in the jth group and ith 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 x j• (j = 1,2, ,K)  Denote the block sample means by x •i (i = 1,2, ,H) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-43 Partition of Total Variation  SST = SSG + SSB + SSE Total Sum of Squares (SST) = Variation due to differences between groups (SSG) + Variation due to differences between blocks (SSB) + The error terms are assumed to be independent, normally distributed, and have the same variance Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Variation due to random sampling (unexplained error) (SSE) Ch 15-44 Two-Way Sums of Squares  The sums of squares are Total : Degrees of Freedom: K H SST = ∑∑ (x ji − x)2 n–1 j=1 i =1 Between - Groups : K SSG = H∑ (x j• − x)2 K–1 j=1 Between - Blocks : H SSB = K ∑ (x •i − x)2 H–1 i =1 Error : K H SSE = ∑∑ (x ji − x j• − x •i + x)2 (K – 1)(K – 1) j =1 i =1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-45 Two-Way Mean Squares  The mean squares are SST MST = n −1 SST MSG = K −1 SST MSB = H −1 SSE MSE = (K − 1)(H − 1) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-46 Two-Way ANOVA: The F Test Statistic H0: The K population group means are all the same H0: The H population block means are the same Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall F Test for Groups MSG F= MSE Reject H0 if F > FK-1,(K-1)(H-1),α F Test for Blocks MSB F= MSE Reject H0 if F > FH-1,(K-1)(H-1),α Ch 15-47 General Two-Way Table Format Source of Variation Between groups Sum of Squares Degrees of Freedom SSG K–1 Between blocks SSB H–1 Error SSE (K – 1)(H – 1) Total SST n-1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Mean Squares MSG = MSB = MSE = SSG K −1 SSB H −1 F Ratio MSG MSE MSB MSE SSE (K − 1)(H − 1) Ch 15-48 More than One Observation per Cell 15.5  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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-49 More than One Observation per Cell SST = SSG + SSB + SSI + SSE SSG Between-group variation SST Total Variation SSB Between-block variation SSI n–1 (continued) Degrees of Freedom: K–1 H–1 Variation due to interaction between groups and blocks (K – 1)(H – 1) SSE KH(L – 1) Random variation (Error) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-50 Sums of Squares with Interaction Degrees of Freedom: Total : SST = ∑∑∑ (x jil − x)2 j Between - groups : i l K SSG = HL∑ (x j•• − x)2 j=1 Between - blocks : n-1 K–1 H SSB = KL ∑ (x •i• − x)2 H–1 i=1 Interaction : K H SSI = L ∑∑ (x ji• − x j•• − x •i• + x)2 j=1 i=1 Error : SSE = ∑∑∑ (x jil − x ji• )2 i Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall j l (K – 1)(H – 1) KH(L – 1) Ch 15-51 Two-Way Mean Squares with Interaction  The mean squares are MST = SST n −1 MSG = SST K −1 MSB = SST H −1 MSI = SSI (K - 1)(H − 1) SSE MSE = KH(L − 1) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-52 Two-Way ANOVA: The F Test Statistic H0: The K population group means are all the same H0: The H population block means are the same H0: the interaction of groups and blocks is equal to zero Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall F Test for group effect MSG F= MSE Reject H0 if F > FK-1,KH(L-1),α F Test for block effect MSB F= MSE Reject H0 if F > FH-1,KH(L-1),α F Test for interaction effect MSI F= MSE Reject H0 if F > F(K-1)(H-1),KH(L-1),α Ch 15-53 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Between groups SSG K–1 MSG = SSG / (K – 1) MSG MSE MSB MSE MSI MSE Between blocks SSB H–1 MSB = SSB / (H – 1) Interaction SSI (K – 1)(H – 1) MSI = SSI / (K – 1)(H – 1) Error SSE KH(L – 1) MSE = SSE / KH(L – 1) Total SST n–1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-54 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-55 Examples: Interaction vs No Interaction   Interaction is present: No interaction: Block Level Block Level A B Groups C Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Mean Response Mean Response Block Level Block Level Block Level Block Level A B Groups C Ch 15-56 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 15-57 ... Hall Ch 15- 2 15. 2  One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples: Average production for 1st, 2nd, and 3rd shifts Expected mileage for five.. .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... as Prentice Hall Ch 15- 30 Multiple Supgroups: Example x1 = 249.2 n1 = x2 = 226.0 n2 = x3 = 205.8 n3 = Sp 93.3 MSD(k) = q = 3.77 = 9.387 n 15 (where q = 3.77 is from Table 13 for α = 05 and 12 df)