Business Statistics: A Decision-Making Approach 6th Edition Chapter 11 Analysis of Variance Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, Inc Chap 11-1 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 single-factor hypothesis test and interpret results Conduct and interpret post-analysis of variance pairwise comparisons procedures Set up and perform randomized blocks analysis Analyze two-factor analysis of variance test with replications results Business Statistics: A Decision- Chap 11-2 Chapter Overview Analysis of Variance (ANOVA) One-Way ANOVA Randomized Complete Block ANOVA Two-factor ANOVA with replication F-test TukeyKramer test Business Statistics: A Decision- F-test Fisher’s Least Significant Difference test Chap 11-3 General ANOVA Setting Investigator controls one or more independent variables Called factors (or treatment variables) Each factor contains two or more levels (or categories/classifications) Observe effects on dependent variable Response to levels of independent variable Experimental design: the plan used to test hypothesis Business Statistics: A Decision- Chap 11-4 One-Way Analysis of Variance Evaluate the difference among the means of three or more populations 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 Business Statistics: A Decision- Chap 11-5 Completely Randomized Design Experimental units (subjects) are assigned randomly to treatments Only one factor or independent variable Analyzed by With two or more treatment levels One-factor analysis of variance (one-way ANOVA) Called a Balanced Design if all factor levels have equal sample size Business Statistics: A Decision- Chap 11-6 Hypotheses of One-Way ANOVA H0 : μ1 = μ2 = μ3 = = μk All population means are equal i.e., no treatment effect (no variation in means among groups) HA : Not all of the population means are the same At least one population mean is different i.e., there is a treatment effect Does not mean that all population means are different (some pairs may be the same) Business Statistics: A Decision- Chap 11-7 One-Factor ANOVA H0 : μ1 = μ2 = μ3 = = μk HA : Not all μi are the same All Means are the same: The Null Hypothesis is True (No Treatment Effect) μ1 = μ2 = μ3 Business Statistics: A Decision- Chap 11-8 One-Factor ANOVA H0 : μ1 = μ2 = μ3 = = μk (continue d) HA : Not all μi are the same At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or μ1 = μ2 ≠ μ3 Business Statistics: A Decision- μ1 ≠ μ2 ≠ μ3 Chap 11-9 Partitioning the Variation Total variation can be split into two parts: SST = SSB + SSW SST = Total Sum of Squares SSB = Sum of Squares Between SSW = Sum of Squares Within Business Statistics: A Decision- Chap 11-10 Fisher’s Least Significant Difference (LSD) Test LSD = t α/2 MSW b where: tα/2 = Upper-tailed value from Student’s t-distribution for α/2 and (k -1)(n - 1) degrees of freedom MSW = Mean square within from ANOVA table b = number of blocks k = number of levels of the main factor Business Statistics: A Decision- Chap 11-43 Fisher’s Least Significant Difference (LSD) Test (continued) LSD = t α/2 Is x i − x j > LSD ? MSW b Compare: If the absolute mean difference is greater than LSD then there is a significant difference between that pair of means at the chosen level of significance Business Statistics: A Decision- x1 − x x1 − x x2 − x3 etc Chap 11-44 Two-Way ANOVA Examines the effect of Two or more 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 percentage of carbonation depend on which level the line speed is set? Business Statistics: A Decision- Chap 11-45 Two-Way ANOVA (continued) Assumptions Populations are normally distributed Populations have equal variances Independent random samples are drawn Business Statistics: A Decision- Chap 11-46 Two-Way ANOVA Sources of Variation Two Factors of interest: A and B a = number of levels of factor A b = number of levels of factor B N = total number of observations in all cells Business Statistics: A Decision- Chap 11-47 Two-Way ANOVA Sources of Variation SST = SSA + SSB + SSAB + SSE SSA Variation due to factor A SST Total Variation SSB Variation due to factor B SSAB N-1 Variation due to interaction between A and B SSE Inherent variation (Error) Business Statistics: A Decision- (continued) Degrees of Freedom: a–1 b–1 (a – 1)(b – 1) N – ab Chap 11-48 Two Factor ANOVA Equations Total Sum of Squares: a n′ b SST = ∑∑∑ ( x ijk − x ) i=1 j=1 k =1 Sum of Squares Factor A: a ′ SS A = bn ∑ ( x i − x ) i=1 Sum of Squares Factor B: b SSB = an′∑ ( x j − x )2 j=1 Business Statistics: A Decision- Chap 11-49 Two Factor ANOVA Equations (continued) Sum of Squares Interaction Between A and B: SS a AB b = n′∑∑ ( x ij − x i − x j + x )2 i=1 j =1 Sum of Squares Error: a b n′ SSE = ∑∑∑ ( x ijk − x ij ) i=1 j =1 k =1 Business Statistics: A Decision- Chap 11-50 Two Factor ANOVA Equations a where: b xi = x= j=1 k =1 n′ ∑∑∑ x i =1 j=1 k =1 abn′ n′ ∑∑ x b (continued) ijk = Grand Mean ijk bn′ = Mean of each level of factor A a xj = n′ ∑∑ x i=1 k =1 an′ ijk = Mean of each level of factor B x ijk x ij = ∑ = Mean of each cell k =1 n′ n′ Business Statistics: A Decision- a = number of levels of factor A b = number of levels of factor B n’ = number of replications in each cell Chap 11-51 Mean Square Calculations SS A MS A = Mean square factor A = a −1 SSB MSB = Mean square factor B = b −1 MS AB SS AB = Mean square interaction = (a − 1)(b − 1) SSE MSE = Mean square error = N − ab Business Statistics: A Decision- Chap 11-52 Two-Way ANOVA: The F Test Statistic H0: μA1 = μA2 = μA3 = • • • HA: Not all μAi are equal H0: μB1 = μB2 = μB3 = • • • HA: Not all μBi are equal H0: factors A and B not interact to affect the mean response HA: factors A and B interact Business Statistics: A Decision- F Test for Factor A Main Effect MS A F= MSE Reject H0 if F > Fα F Test for Factor B Main Effect MSB F= MSE Reject H0 if F > Fα F Test for Interaction Effect MS AB F= MSE Reject H0 if F > Fα Chap 11-53 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Factor A SSA Degrees of Freedom a–1 Factor B SSB b–1 AB (Interaction) SSAB (a – 1)(b – 1) Error SSE N – ab Total SST N–1 Business Statistics: A Decision- Mean Squares F Statistic MSA MSA MSE = SSA /(a – 1) MSB = SSB /(b – 1) MSAB = SSAB / [(a – 1)(b – 1)] MSB MSE MSAB MSE MSE = SSE/(N – ab) Chap 11-54 Features of Two-Way ANOVA F Test Degrees of freedom always add up N-1 = (N-ab) + (a-1) + (b-1) + (a-1)(b-1) Total = error + factor A + factor B + interaction The denominator of the F Test is always the same but the numerator is different The sums of squares always add up SST = SSE + SSA + SSB + SSAB Total = error + factor A + factor B + interaction Business Statistics: A Decision- Chap 11-55 Examples: Interaction vs No Interaction No interaction: Interaction is present: Factor B Level Factor B Level Factor A Levels Business Statistics: A Decision- Mean Response Factor B Level Mean Response Factor B Level Factor B Level Factor B Level Factor A Levels Chap 11-56 Chapter Summary Described one-way analysis of variance Described randomized complete block designs The logic of ANOVA ANOVA assumptions F test for difference in k means The Tukey-Kramer procedure for multiple comparisons F test Fisher’s least significant difference test for multiple comparisons Described two-way analysis of variance Examined effects of multiple factors and interaction Business Statistics: A Decision- Chap 11-57 ... of all data values) Business Statistics: A DecisionChap 11-13 Total Variation (continued) SST = ( x11 − x )2 + ( x12 − x )2 + + ( x knk − x )2 Response, X X Group Group Business Statistics:... freedom µi µj Business Statistics: A Decision- Chap 11-16 Between-Group Variation (continued) SSB = n1 ( x1 − x ) + n ( x − x ) + + nk ( x k − x ) 2 Response, X X3 X1 Group Group Business Statistics:... of freedom µi Business Statistics: A Decision- Chap 11-19 Within-Group Variation (continued) SSW = ( x11 − x1 ) + ( x12 − x ) + + ( x knk − x k ) 2 Response, X X1 Group Group Business Statistics: