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McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4 th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh

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11-2 Analysis of Variance Chapter Contents 11.1 Overview of ANOVA 11.2 One-Factor ANOVA (Completely Randomized Model) 11.3 Multiple Comparisons 11.4 Tests for Homogeneity of Variances 11.5 Two-Factor ANOVA without Replication (Randomized Block Model) 11.6 Two-Factor ANOVA with Replication (Full Factorial Model) 11.7 Higher Order ANOVA Models (Optional) Chapter 11

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11-3 Chapter Learning Objectives LO11-1: Use basic ANOVA terminology correctly. LO11-2: Recognize from data format when one-factor ANOVA is appropriate. LO11-3: Interpret sums of squares and calculations in an ANOVA table. LO11-4: Use Excel or other software for ANOVA calculations. LO11-5: Use a table or Excel to find critical values for the F distribution. LO11-6: Explain the assumptions of ANOVA and why they are important. LO11-7: Understand and perform Tukey's test for paired means. LO11-8: Use Hartley's test for equal variances in c treatment groups. LO11-9: Recognize from data format when two-factor ANOVA is needed. LO11-10: Interpret main effects and interaction effects in two-factor ANOVA. LO11-11: Recognize the need for experimental design and GLM (optional). Chapter 11 Analysis of Variance

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11-4 Analysis of variance (ANOVA) is a comparison of means.Analysis of variance (ANOVA) is a comparison of means. ANOVA allows you to compare more than two means simultaneously.ANOVA allows you to compare more than two means simultaneously. Proper experimental design efficiently uses limited data to draw the strongest possible inferences.Proper experimental design efficiently uses limited data to draw the strongest possible inferences Overview of ANOVA Chapter 11 The Goal: Explaining Variation ANOVA seeks to identify sources of variation in a numerical dependent variable Y (the response variable). Variation in Y about its mean is explained by one or more categorical independent variables (the factors) or is unexplained (random error). LO11-1 LO11-1: Use basic ANOVA terminology correctly.

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11-5 The Goal: Explaining Variation The Goal: Explaining Variation Each possible value of a factor or combination of factors is a treatment. We test to see if each factor has a significant effect on Y using (for example) the hypotheses: H 0 : 1 = 2 = 3 = 4 (e.g. mean defect rates are the same for all four plants) H 1 : Not all the means are equal The test uses the F distribution. If we cannot reject H 0, we conclude that observations within each treatment have a common mean. Chapter Overview of ANOVA LO11-1

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11-6 The Goal: Explaining Variation The Goal: Explaining Variation Figure 11.3 Chapter Overview of ANOVA LO11-1

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11-7 ANOVA Assumptions ANOVA Assumptions Analysis of Variance assumes that theAnalysis of Variance assumes that the - observations on Y are independent, - populations being sampled are normal, - populations being sampled have equal variances. ANOVA is somewhat robust to departures from normality and equal variance assumptions.ANOVA is somewhat robust to departures from normality and equal variance assumptions. Chapter 11 ANOVA Calculations ANOVA Calculations Software (e.g., Excel, MegaStat, MINITAB, SPSS) is used to analyze data. Large samples increase the power of the test, but power also depends on the degree of variation in Y. Lowest power would be in a small sample with high variation in Y Overview of ANOVA LO11-6 LO11-6: Explain the assumptions of ANOVA and why they are important.

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11-8 An equivalent way to express the one-factor model is to say that treatment j came from a population with a common mean ( ) plus a treatment effect (A j ) plus random error ( ij ):An equivalent way to express the one-factor model is to say that treatment j came from a population with a common mean ( ) plus a treatment effect (A j ) plus random error ( ij ): y ij = + A j + ij j = 1, 2, …, c and i = 1, 2, …, n Random error is assumed to be normally distributed with zero mean and the same variance for all treatments.Random error is assumed to be normally distributed with zero mean and the same variance for all treatments. One-Factor ANOVA as a Linear Model One-Factor ANOVA as a Linear Model Chapter One-Factor ANOVA (Completely Randomized Model) A fixed effects model only looks at what happens to the response for particular levels of the factor. H 0 : A 1 = A 2 = … = A c = 0 H 1 : Not all A j are zero If the H 0 is true, then the ANOVA model collapses to y ij = + ij One can use Excels one-factor ANOVA menu using Data Analysis to analyze data.One can use Excels one-factor ANOVA menu using Data Analysis to analyze data. LO11-2 LO11-2: Recognize from data format when one-factor ANOVA is appropriate.

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11-9 Partitioned Sum of Squares Partitioned Sum of Squares Table 11.2 Chapter 11 Use Appendix F or Excel to obtain the critical value of F for a given For ANOVA, the F test is a right-tailed test. LO One-Factor ANOVA (Completely Randomized Model) LO11-3: Interpret sums of squares and calculations in an ANOVA table.

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11-10 Decision Rule for an F-test Decision Rule for an F-test Chapter One-Factor ANOVA (Completely Randomized Model) LO11-5 LO11-5: Use a table or Excel to find critical values for the F distribution LO11-5: Use a table or Excel to find critical values for the F distribution.

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Multiple Comparisons After rejecting the hypothesis of equal mean, we naturally want to know: Which means differ significantly?After rejecting the hypothesis of equal mean, we naturally want to know: Which means differ significantly? In order to maintain the desired overall probability of type I error, a simultaneous confidence interval for the difference of means must be obtained.In order to maintain the desired overall probability of type I error, a simultaneous confidence interval for the difference of means must be obtained. For c groups, there are c(c – 1) distinct pairs of means to be compared.For c groups, there are c(c – 1) distinct pairs of means to be compared. These types of comparisons are called Multiple Comparison Tests.These types of comparisons are called Multiple Comparison Tests. Tukeys studentized range test (or HSD for honestly significant difference test) is a multiple comparison test that has good power and is widely used.Tukeys studentized range test (or HSD for honestly significant difference test) is a multiple comparison test that has good power and is widely used. Named for statistician John Wilder Tukey (1915 – 2000)Named for statistician John Wilder Tukey (1915 – 2000) This test is not available in Excels Tools > Data Analysis but is available in MegaStat and MinitabThis test is not available in Excels Tools > Data Analysis but is available in MegaStat and Minitab Tukeys Test Tukeys Test Chapter 11LO11-7 LO11-7: Understand and perform Tukey's test for paired means.

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Tests for Homogeneity of Variances ANOVA assumes that observations on the response variable are from normally distributed populations that have the same variance. The one-factor ANOVA test is only slightly affected by inequality of variance when group sizes are equal. Test this assumption of homogeneous variances, using Hartleys F max Test. ANOVA Assumptions ANOVA Assumptions Chapter 11 The hypotheses are The test statistic is the ratio of the largest sample variance to the smallest sample variance. LO11-8 LO11-8: Use Hartley's test for equal variances in c treatment groups.

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11-13 The decision rule is: Hartleys Test Hartleys Test Chapter Tests for Homogeneity of Variances LO11-8

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11-14 Levenes test is a more robust alternative to Hartleys F test.Levenes test is a more robust alternative to Hartleys F test. Levenes test does not assume a normal distribution.Levenes test does not assume a normal distribution. It is based on the distances of the observations from their sample medians rather than their sample means.It is based on the distances of the observations from their sample medians rather than their sample means. A computer program (e.g., MINITAB) is needed to perform this test.A computer program (e.g., MINITAB) is needed to perform this test. Levenes Test Levenes Test Chapter Tests for Homogeneity of Variances LO11-6 LO11-6: Explain the assumptions of ANOVA and why they are important.

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