Statistics for Managers using Microsoft Excel 6 th

Statistics for Managers using Microsoft Excel 6 th Edition Chapter 11 Analysis of Variance Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -1

Learning Objectives In this chapter, you learn: n n 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 Tukey. Kramer 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 n n n DCOVA Investigator controls one or more factors of interest n Each factor contains two or more levels n Levels can be numerical or categorical n Different levels produce different groups n Think of each group as a sample from a different population Observe effects on the dependent variable n 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 n Experimental units (subjects) are assigned randomly to groups n n Only one factor or independent variable n n 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 n Evaluate the difference among the means of three or more groups Examples: Accident rates for 1 st, 2 nd, and 3 rd shift Expected mileage for five brands of tires n Assumptions n Populations are normally distributed n Populations have equal variances n Samples are randomly and independently drawn Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -6

Hypotheses of One-Way ANOVA DCOVA n n n n All population means are equal i. e. , no factor effect (no variation in means among groups) n At least one population mean is different n i. e. , there is a factor effect n 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 The Null Hypothesis is True All Means are the same: (No Factor Effect) Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -8

One-Way ANOVA DCOVA (continued) The Null Hypothesis is NOT true At least one of the means is different (Factor Effect is present) or Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -9

Partitioning the Variation n 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 Where: 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) Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -14

Among-Group Variation DCOVA SST = SSA + SSW Where: 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 Variation Due to Differences Among Groups Mean Square Among = SSA/degrees of freedom Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -16

Among-Group Variation DCOVA (continued) Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -17

Within-Group Variation DCOVA SST = SSA + SSW 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) DCOVA Summing the variation within each group and then adding over all groups Mean Square Within = SSW/degrees of freedom Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -19

Within-Group Variation DCOVA (continued) Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 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 Mean Square Among (d. f. = c-1) Mean Square Within (d. f. = n-c) 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 Total n– 1 SST 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 H 0: μ 1= μ 2 = … = μc H 1: At least two population means are different n Test statistic MSA is mean squares among groups MSW is mean squares within groups n Degrees of freedom n n df 1 = c – 1 (c = number of groups) df 2 = 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 n The F statistic is the ratio of the among estimate of variance and the within estimate of variance n n n The ratio must always be positive df 1 = c -1 will typically be small df 2 = n - c will typically be large Decision Rule: n Reject H 0 if FSTAT > Fα, otherwise do not reject H 0 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 0 Do not reject H 0 Reject H 0 Fα 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 DCOVA Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 11 -25

One-Way ANOVA Example: Scatter Plot DCOVA Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 Distance 270 260 250 240 230 220 210 200 190 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall • • • • • 1 2 3 Club 11 -26

One-Way ANOVA Example Computations DCOVA 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 n 1 = 5 X 2 = 226. 0 n 2 = 5 X 3 = 205. 8 n 3 = 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 11 -27

One-Way ANOVA Example Solution DCOVA Test Statistic: H 0: μ 1 = μ 2 = μ 3 H 1: μj not all equal = 0. 05 df 1= 2 df 2 = 12 Decision: Reject H 0 at = 0. 05 Critical Value: Fα = 3. 89 =. 05 0 Do not reject H 0 Reject H 0 Fα = 3. 89 FSTAT Conclusion: There is evidence that at least one μj differs = 25. 275 from the rest 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 P-value 25. 275 F crit 4. 99 E-05 3. 89 11 -29

The Tukey-Kramer Procedure DCOVA n Tells which population means are significantly different n n n e. g. : μ 1 = μ 2 μ 3 Done after rejection of equal means in ANOVA Allows paired comparisons n Compare absolute mean differences with critical range μ 1 = μ 2 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall μ 3 x 11 -30

Tukey-Kramer Critical Range DCOVA where: Qα = 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 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 1. Compute absolute mean differences: 2. Find the Qα value from the table in appendix E. 10 with c = 3 and (n – c) = (15 – 3) = 12 degrees of freedom: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -32

The Tukey-Kramer Procedure: Example (continued) DCOVA 3. Compute Critical Range: 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 11 -33

ANOVA Assumptions n Randomness and Independence n n Select random samples from the c groups (or randomly assign the levels) Normality n n DCOVA The sample values for each group are from a normal population Homogeneity of Variance n n All populations sampled from have the same variance Can be tested with Levene’s Test Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -34

ANOVA Assumptions Levene’s Test n n Tests the assumption that the variances of each population are equal. First, define the null and alternative hypotheses: n n DCOVA H 0: σ21 = σ22 = …=σ2 c H 1: Not all σ2 j 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 H 0: σ21 = σ22 = σ23 H 1: Not all σ2 j 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 MS 3. 2 0. 092 F crit 0. 912 3. 885 34. 6 14 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 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 n DCOVA Examines the effect of n Two factors of interest on the dependent variable n n e. g. , Percent carbonation and line speed on soft drink bottling process Interaction between the different levels of these two factors n 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 n Assumptions n Populations are normally distributed n Populations have equal variances n 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: r – 1 c – 1 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: Factor A Variation: Factor B Variation: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -42

Two-Way ANOVA Equations (continued) DCOVA Interaction Variation: Sum of Squares Error: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -43

Two-Way ANOVA Equations (continued) where: DCOVA r = number of levels of factor A c = number of levels of factor B n’ = number of replications in each cell Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -44

Mean Square Calculations DCOVA Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -45

Two-Way ANOVA: The F Test Statistics H 0: μ 1. . = μ 2. . = μ 3. . = • • = µr. . H 1: Not all μi. . are equal H 0: μ. 1. = μ. 2. = μ. 3. = • • = µ. c. H 1: Not all μ. j. are equal H 0: the interaction of A and B is equal to zero H 1: interaction of A and B is not zero Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall DCOVA F Test for Factor A Effect Reject H 0 if FSTAT > Fα F Test for Factor B Effect Reject H 0 if FSTAT > Fα F Test for Interaction Effect Reject H 0 if FSTAT > Fα 11 -46

Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Factor A SSA r– 1 Factor B SSB c– 1 AB (Interaction) SSAB (r – 1)(c – 1) Error SSE rc(n’ – 1) Total SST n– 1 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 = SSE/rc(n’ – 1) 11 -47

Features of Two-Way ANOVA F Test DCOVA n n n Degrees of freedom always add up n n-1 = rc(n’-1) + (r-1) + (c-1) + (r-1)(c-1) n 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 n SST = SSE + SSA + SSB + SSAB n 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 n n 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 n n n 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: Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -50

Multiple Comparisons: The Tukey Procedure n DCOVA Critical Range for Factor A: (where Qα is from Table E. 10 with r and rc(n’– 1) d. f. ) n Critical Range for Factor B: (where Qα is from Table E. 10 with c and rc(n’– 1) d. f. ) Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -51

Chapter Summary n Described one-way analysis of variance n n n The logic of 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 n n 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 6 th Edition Online Topic The Randomized Block Design Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -53

Learning Objective n 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 n n n 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 n 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 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 n 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 Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall DCOVA 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 (r– 1)(c-1) MSE SST rc - 1 Total c = number of populations r = number of blocks F MSBL MSE MSA MSE rc = total number of observations df = degrees of freedom Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -60

Testing For Factor Effect MSA FSTAT = MSE n DCOVA Main Factor test: df 1 = c – 1 df 2 = (r – 1)(c – 1) Reject H 0 if FSTAT > Fα Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -61

Test For Block Effect MSBL FSTAT = MSE n Blocking test: DCOVA df 1 = r – 1 df 2 = (r – 1)(c – 1) Reject H 0 if FSTAT > Fα Copyright © 2011 Pearson Education, Inc. publishing as Prentice Hall 11 -62

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

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