Statistics for Managers using Microsoft Excel 6th Edition Chapter 11 Analysis of Variance Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-1 Learning Objectives In this chapter, you learn: The basic concepts of experimental design How to use one-way analysis of variance to test for differences among the means of several populations (also referred to as groups in this chapter) How to use two-way analysis of variance and interpret the interaction effect How to perform multiple comparisons in a one-way analysis of variance and a two-way analysis of variance Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall

11-2 Chapter Overview DCOVA Analysis of Variance (ANOVA) One-Way ANOVA F-test TukeyKramer Multiple Comparisons Levene Test For Homogeneity of Variance Randomized Block Design (On Line Topic) Tukey Multiple Comparisons Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall Two-Way ANOVA Interaction Effects Tukey Multiple Comparisons

11-3 General ANOVA Setting DCOVA Investigator controls one or more factors of interest Each factor contains two or more levels Levels can be numerical or categorical Different levels produce different groups Think of each group as a sample from a different population Observe effects on the dependent variable Are the groups the same? Experimental design: the plan used to collect the data Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-4 Completely Randomized Design DCOVA Experimental units (subjects) are assigned randomly to groups

Only one factor or independent variable Subjects are assumed homogeneous With two or more levels Analyzed by one-factor analysis of variance (ANOVA) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-5 One-Way Analysis of Variance DCOVA Evaluate the difference among the means of three or more groups Examples: Accident rates for 1st, 2nd, and 3rd shift Expected mileage for five brands of tires Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn

Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-6 Hypotheses of One-Way ANOVA DCOVA H0 : 1 2 3 c All population means are equal i.e., no factor effect (no variation in means among groups) H1 : Not all of the population means are the same At least one population mean is different i.e., there is a factor effect Does not mean that all population means are

different (some pairs may be the same) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-7 One-Way ANOVA DCOVA H0 : 1 2 3 c H1 : Not all j are the same The Null Hypothesis is True All Means are the same: (No Factor Effect) 1 2 3 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-8 One-Way ANOVA H0 : 1 2 3 c DCOVA (continued) H1 : Not all j are the same The Null Hypothesis is NOT true At least one of the means is different (Factor Effect is present) or

1 2 3 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 1 2 3 11-9 Partitioning the Variation DCOVA Total variation can be split into two parts: SST = SSA + SSW SST = Total Sum of Squares (Total variation) SSA = Sum of Squares Among Groups (Among-group variation) SSW = Sum of Squares Within Groups (Within-group variation) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-10 Partitioning the Variation (continued) SST = SSA + SSW DCOVA Total Variation = the aggregate variation of the individual

data values across the various factor levels (SST) Among-Group Variation = variation among the factor sample means (SSA) Within-Group Variation = variation that exists among the data values within a particular factor level (SSW) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-11 Partition of Total Variation DCOVA Total Variation (SST) = Variation Due to Factor (SSA) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall + Variation Due to Random Error (SSW) 11-12 Total Sum of Squares DCOVA

SST = SSA + SSW c nj SST ( Xij X) Where: 2 j 1 i 1 SST = Total sum of squares c = number of groups or levels nj = number of observations in group j Xij = ith observation from group j X = grand mean (mean of all data values) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-13 Total Variation DCOVA (continued) 2 2 SST ( X 11 X ) ( X 12 X ) ( X cn X ) 2

c Response, X X Group 1 Group 2 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall Group 3 11-14 Among-Group Variation DCOVA SST = SSA + SSW c SSA n j ( X j X)2 Where: j1 SSA = Sum of squares among groups c = number of groups nj = sample size from group j Xj = sample mean from group j X = grand mean (mean of all data values) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall

11-15 Among-Group Variation (continued) DCOVA c SSA n j ( X j X)2 j1 Variation Due to Differences Among Groups SSA MSA c 1 Mean Square Among = SSA/degrees of freedom i j Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-16 Among-Group Variation DCOVA

(continued) 2 2 SSA n1 (X1 X) n 2 (X 2 X) n c (X c X) 2 Response, X X3 X1 Group 1 Group 2 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall X2 X Group 3 11-17 Within-Group Variation DCOVA SST = SSA + SSW c

SSW j1 nj ( Xij X j ) 2 i1 Where: SSW = Sum of squares within groups c = number of groups nj = sample size from group j Xj = sample mean from group j Xij = ith observation in group j Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-18 Within-Group Variation (continued) c SSW j1 nj

DCOVA ( Xij X j )2 i1 Summing the variation within each group and then adding over all groups SSW MSW n c Mean Square Within = SSW/degrees of freedom j Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-19 Within-Group Variation DCOVA (continued) 2 2

SSW (X11 X1 ) (X12 X 2 ) (X cn c X c ) 2 Response, X X1 Group 1 Group 2 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall X2 X3 Group 3 11-20 Obtaining the Mean Squares DCOVA The Mean Squares are obtained by dividing the various sum of squares by their associated degrees of freedom SSA MSA c1 Mean Square Among (d.f. = c-1) SSW

MSW n c Mean Square Within (d.f. = n-c) SST MST n 1 Mean Square Total (d.f. = n-1) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-21 One-Way ANOVA Table Source of Variation Degrees of Freedom Sum Of Squares Among Groups c-1 Within

Groups n-c SSW n1 SST Total SSA DCOVA Mean Square (Variance) F SSA MSA = c-1 SSW MSW = n-c FSTAT = MSA MSW c = number of groups

n = sum of the sample sizes from all groups df = degrees of freedom Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-22 One-Way ANOVA F Test Statistic DCOVA H0: 1= 2 = = c H1: At least two population means are different Test statistic MSA FSTAT MSW MSA is mean squares among groups MSW is mean squares within groups Degrees of freedom df1 = c 1 (c = number of groups)

df2 = n c (n = sum of sample sizes from all populations) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-23 Interpreting One-Way ANOVA F Statistic DCOVA The F statistic is the ratio of the among estimate of variance and the within estimate of variance The ratio must always be positive df1 = c -1 will typically be small df2 = n - c will typically be large Decision Rule: Reject H if F 0 STAT > F, otherwise do not reject H0 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall

0 Do not reject H0 Reject H0 F 11-24 One-Way 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? Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall Club 1 254 263 241 237 251

Club 2 234 218 235 227 216 DCOVA Club 3 200 222 197 206 204 11-25 One-Way ANOVA Example: Scatter Plot DCOVA Club 1 254 263 241 237 251 Club 2 234 218 235 227

216 Club 3 200 222 197 206 204 Distance 270 260 250 240 230 220 210 x1 249.2 x 2 226.0 x 3 205.8 x 227.0 X2

200 190 1 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall X1 2 Club X X3 3 11-26 One-Way ANOVA Example Computations DCOVA Club 1 254 263 241 237 251 Club 2 234

218 235 227 216 Club 3 200 222 197 206 204 X1 = 249.2 n1 = 5 X2 = 226.0 n2 = 5 X3 = 205.8 n3 = 5 X = 227.0 n = 15 c=3 SSA = 5 (249.2 227)2 + 5 (226 227)2 + 5 (205.8 227)2 = 4716.4 SSW = (254 249.2)2 + (263 249.2)2 ++ (204 205.8)2 = 1119.6 MSA = 4716.4 / (3-1) = 2358.2 MSW = 1119.6 / (15-3) = 93.3

Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall F STAT 2358.2 93.3 25.275 11-27 One-Way ANOVA Example Solution DCOVA Test Statistic: H 0: 1 = 2 = 3 H1: j not all equal MSA 2358.2 FSTAT 25.275 MSW 93.3 = 0.05 df1= 2 df2 = 12 Decision: Reject H0 at = 0.05 F = 3.89

Conclusion: = .05 There is evidence that at least one j differs Reject H F = 3.89 FSTAT = 25.275 from the rest Critical Value: 0 Do not reject H0 0 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-28 One-Way ANOVA Excel Output DCOVA SUMMARY Groups Count Sum

Average Variance Club 1 5 1246 249.2 108.2 Club 2 5 1130 226 77.5 Club 3 5 1029 205.8 94.2

ANOVA Source of Variation SS df MS Between Groups 4716.4 2 2358.2 Within Groups 1119.6 12 93.3 Total 5836.0

14 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall F 25.275 P-value 4.99E-05 F crit 3.89 11-29 The Tukey-Kramer Procedure DCOVA Tells which population means are significantly different e.g.: 1 = 2 3 Done after rejection of equal means in ANOVA Allows paired comparisons Compare absolute mean differences with critical

range 1= 2 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 3 x 11-30 Tukey-Kramer Critical Range DCOVA Critical Range Q MSW 1 1 2 n j n j' where: Q = Upper Tail Critical Value from Studentized Range Distribution with c and n - c degrees of freedom (see appendix E.10 table) MSW = Mean Square Within nj and nj = Sample sizes from groups j and j Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-31 The Tukey-Kramer Procedure: Example

DCOVA Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 1. Compute absolute mean differences: Club 3 200 222 197 206 204 x1 x 2 249.2 226.0 23.2 x1 x 3 249.2 205.8 43.4 x 2 x 3 226.0 205.8 20.2 2. Find the Q value from the table in appendix E.10 with c = 3 and (n c) = (15 3) = 12 degrees of freedom:

Q 3.77 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-32 The Tukey-Kramer Procedure: Example (continued) DCOVA 3. Compute Critical Range: Critical Range Q MSW 1 1 93.3 1 1 3.77 16.285 2 n j n j' 2 5 5 4. Compare: 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. Thus, with 95% confidence we can conclude

that the mean distance for club 1 is greater than club 2 and 3, and club 2 is greater than club 3. Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall x1 x 2 23.2 x1 x 3 43.4 x 2 x 3 20.2 11-33 ANOVA Assumptions Randomness and Independence Select random samples from the c groups (or randomly assign the levels) Normality DCOVA The sample values for each group are from a normal population Homogeneity of Variance

All populations sampled from have the same variance Can be tested with Levenes Test Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-34 ANOVA Assumptions Levenes Test DCOVA Tests the assumption that the variances of each population are equal. First, define the null and alternative hypotheses: H0: 21 = 22 = =2c

H1: Not all 2j are equal Second, compute the absolute value of the difference between each value and the median of each group. Third, perform a one-way ANOVA on these absolute differences. Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-35 Levene Homogeneity Of Variance Test Example DCOVA H0: 21 = 22 = 23 H1: Not all 2j are equal Calculate Medians Club 1 Club 2 Calculate Absolute Differences Club 3 Club 1 Club 2 Club 3

237 216 197 14 11 7 241 218 200 10 9 4 251 227 204 Median 0

0 0 254 234 206 3 7 2 263 235 222 12 8 18 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-36 Levene Homogeneity Of Variance

Test Example (continued) DCOVA Anova: Single Factor SUMMARY Groups Count Sum Average Variance Club 1 5 39 7.8 36.2 Club 2 5 35 7 17.5 Club 3

5 31 6.2 50.2 F Pvalue Source of Variation Between Groups Within Groups Total SS df 6.4 2 415.6 12 422

14 MS 3.2 0.092 34.6 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall F crit 0.912 3.885 Since the p-value is greater than 0.05 there is insufficient evidence of a difference in the variances 11-37 Factorial Design: Two-Way ANOVA DCOVA 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? Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-38 Two-Way ANOVA (continued) DCOVA Assumptions Populations are normally distributed

Populations have equal variances Independent random samples are drawn Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-39 Two-Way ANOVA Sources of Variation DCOVA Two Factors of interest: A and B r = number of levels of factor A c = number of levels of factor B n = number of replications for each cell n = total number of observations in all cells n = (r)(c)(n) Xijk = value of the kth observation of level i of factor A and level j of factor B Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-40 Two-Way ANOVA Sources of Variation SST = SSA + SSB + SSAB + SSE

SSA Factor A Variation SSB SST Total Variation Factor B Variation SSAB n-1 DCOVA (continued) Degrees of Freedom: r1 c1 Variation due to interaction between A and B (r 1)(c 1) SSE rc(n 1) Random variation (Error) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall

11-41 Two-Way ANOVA Equations DCOVA Total Variation: r c n SST ( Xijk X) 2 i1 j1 k 1 Factor A Variation: r 2 SSA cn ( Xi.. X) i 1 Factor B Variation: c SSB rn ( X. j. X)2 j1

Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-42 Two-Way ANOVA Equations (continued) DCOVA Interaction Variation: r c SSAB n ( Xij. Xi.. X.j. X)2 i 1 j1 Sum of Squares Error: r c n SSE ( Xijk Xij. ) 2 i 1 j1 k 1 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall

11-43 Two-Way ANOVA Equations r where: X c Xi.. (continued) n X DCOVA ijk i1 j 1 k 1 rcn n X c

Grand Mean ijk j1 k 1 Mean of ith level of factor A (i 1, 2, ..., r) cn r n X X. j. i1 k 1 rn ijk Mean of jth level of factor B (j 1, 2, ..., c) Xijk Xij. Mean of cell ij k 1 n n Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall

r = number of levels of factor A c = number of levels of factor B n = number of replications in each cell 11-44 Mean Square Calculations SSA MSA Mean square factor A r 1 DCOVA SSB MSB Mean square factor B c 1 SSAB MSAB Mean square interaction (r 1)(c 1) SSE MSE Mean square error rc(n' 1) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-45 Two-Way ANOVA: The F Test Statistics F Test for Factor A Effect H0: 1..= 2.. = 3..= = r.. H1: Not all i.. are equal

DCOVA F STAT MSA MSE Reject H0 if FSTAT > F F Test for Factor B Effect H0: .1. = .2. = .3.= = .c. H1: Not all .j. are equal F STAT H0: the interaction of A and B is equal to zero Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall Reject H0 if FSTAT > F F Test for Interaction Effect H1: interaction of A and B is not F STAT zero

MSB MSE MSAB MSE Reject H0 if FSTAT > F 11-46 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Factor A SSA r1 Factor B SSB

c1 AB (Interaction) SSAB (r 1)(c 1) Error SSE rc(n 1) Total SST n1 Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall Mean Squares MSA = SSA /(r 1) MSB = SSB /(c 1) MSAB

= SSAB / (r 1)(c 1) DCOVA F MSA MSE MSB MSE MSAB MSE MSE = SSE/rc(n 1) 11-47 Features of Two-Way ANOVA F Test DCOVA Degrees of freedom always add up n-1 = rc(n-1) + (r-1) + (c-1) + (r-1)(c-1) Total = error + factor A + factor B + interaction The denominators of the F Test are always the

same but the numerators are different The sums of squares always add up SST = SSE + SSA + SSB + SSAB Total = error + factor A + factor B + interaction Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-48 Examples: Interaction vs. No Interaction No interaction: line segments are parallel Factor B Level 3 Factor B Level 2 Factor A Levels Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall Mean Response Factor B Level 1

Mean Response DCOVA Interaction is present: some line segments not parallel Factor B Level 1 Factor B Level 2 Factor B Level 3 Factor A Levels 11-49 Multiple Comparisons: The Tukey Procedure DCOVA Unless there is a significant interaction, you can determine the levels that are significantly different using the Tukey procedure Consider all absolute mean differences and

compare to the calculated critical range Example: Absolute differences for factor A, assuming three levels: X1.. X 2.. X1.. X3.. X 2.. X3.. Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-50 Multiple Comparisons: The Tukey Procedure DCOVA Critical Range for Factor A: Critical Range Q MSE c n' (where Q is from Table E.10 with r and rc(n1) d.f.) Critical Range for Factor B:

Critical Range Q MSE r n' (where Q is from Table E.10 with c and rc(n1) d.f.) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-51 Chapter Summary Described one-way analysis of variance The logic of ANOVA ANOVA assumptions F test for difference in c means The Tukey-Kramer procedure for multiple comparisons The Levene test for homogeneity of variance Described two-way analysis of variance

Examined effects of multiple factors Examined interaction between factors Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-52 Statistics for Managers using Microsoft Excel 6th Edition Online Topic The Randomized Block Design Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-53 Learning Objective To learn the basic structure and use of a randomized block design Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-54 The Randomized Block Design DCOVA

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) Levels of the secondary factor are called blocks Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-55 Partitioning the Variation DCOVA Total variation can now be split into three parts: SST = SSA + SSBL + SSE SST = Total variation SSA = Among-Group variation SSBL = Among-Block variation SSE = Random variation Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-56

Sum of Squares for Blocks DCOVA SST = SSA + SSBL + SSE r SSBL c ( Xi. X) 2 i1 Where: c = number of groups r = number of blocks Xi. = mean of all values in block i X = grand mean (mean of all data values) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-57 Partitioning the Variation DCOVA Total variation can now be split into three parts: SST = SSA + SSBL + SSE SST and SSA are

computed as they were in One-Way ANOVA Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall SSE = SST (SSA + SSBL) 11-58 Mean Squares DCOVA SSBL MSBL Mean square blocking r 1 MSA Mean square among groups SSA c 1 SSE MSE Mean square error (r 1)(c 1) Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-59 Randomized Block ANOVA Table DCOVA Source of Variation

SS df MS Among Blocks SSBL r-1 MSBL Among Groups SSA c-1 MSA Error SSE (r1)(c-1) MSE SST

rc - 1 Total F MSBL MSE MSA MSE c = number of populations rc = total number of observations r = number of blocks df = degrees of freedom Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-60 Testing For Factor Effect DCOVA H 0 : .1 .2 .3 .c H1 : Not all population means are equal MSA FSTAT = MSE Main Factor test: df1 = c 1

df2 = (r 1)(c 1) Reject H0 if FSTAT > F Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-61 Test For Block Effect DCOVA H 0 : 1. 2. 3. ... r. H1 : Not all block means are equal MSBL FSTAT = MSE Blocking test: df1 = r 1 df2 = (r 1)(c 1) Reject H0 if FSTAT > F Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-62 Topic Summary

Examined the basic structure and use of a randomized block design Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-63 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. Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 11-64