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Analysis of Variance Analysis of Variance Chapter1111 Overview of ANOVA One-Factor ANOVA Multiple Comparisons Tests for Homogeneity of Variances Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

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Overview of ANOVA 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. 11-2

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The Goal: Explaining Variation The Goal: Explaining Variation ANOVA seeks to identify sources of variation in a numerical dependent variable Y (the response variable).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).Variation in Y about its mean is explained by one or more categorical independent variables (the factors) or is unexplained (random error). Overview of ANOVA 11-3

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

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ANOVA Assumptions ANOVA Assumptions Analysis of Variance assumes that theAnalysis of Variance assumes that the - observations on Y are independent, - observations on Y are independent, - populations being sampled are normal, - populations being sampled are normal, - populations being sampled have equal variances. - 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. Overview of ANOVA 11-5

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One-Factor ANOVA (Completely Randomized Design) 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 11-6

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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 zeroA 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 = + ijIf the H 0 is true, then the ANOVA model collapses to y ij = + ij One-Factor ANOVA as a Linear Model One-Factor ANOVA as a Linear Model One-Factor ANOVA (Completely Randomized Design) 11-7

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Step 1: State the hypotheses H 0 : 1 = 2 = … = c H 1 : Not all the means are equalStep 1: State the hypotheses H 0 : 1 = 2 = … = c H 1 : Not all the means are equal Step 2: State the decision rule The degrees of freedom for the critical value F are Numerator: d.f. = c – 1 (between treatments) Denominator: d.f. = n – c (within treatments)Step 2: State the decision rule The degrees of freedom for the critical value F are Numerator: d.f. = c – 1 (between treatments) Denominator: d.f. = n – c (within treatments) For the given , obtain the critical value from Appendix F or Excel. Steps in Performing One-Factor ANOVA Steps in Performing One-Factor ANOVA One-Factor ANOVA (Completely Randomized Design) 11-8

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Step 3: Perform the Calculations Use Excel to perform the calculations. For example, here are the results of an ANOVA:Step 3: Perform the Calculations Use Excel to perform the calculations. For example, here are the results of an ANOVA: Steps in Performing One-Factor ANOVA Steps in Performing One-Factor ANOVA Figure 11.9 One-Factor ANOVA (Completely Randomized Design) 11-9

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Step 4: Make the Decision Reject H 0 if the critical value F exceeds the test statistic F or if the p-value given by Excel is < .Step 4: Make the Decision Reject H 0 if the critical value F exceeds the test statistic F or if the p-value given by Excel is < . MegaStat and MINITAB can also be used to perform the ANOVA.MegaStat and MINITAB can also be used to perform the ANOVA. Steps in Performing One-Factor ANOVA Steps in Performing One-Factor ANOVA One-Factor ANOVA (Completely Randomized Design) 11-10

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Multiple Comparison Tests Multiple Comparison Tests 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. Tukey’s studentized range test (or HSD for “honestly significant difference” test) is a multiple comparison test that has good power and is widely used.Tukey’s studentized range test (or HSD for “honestly significant difference” test) is a multiple comparison test that has good power and is widely used. Tukey’s Test Tukey’s Test 11-11

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Named for statistician John Wilder Tukey (1915 – 2000)Named for statistician John Wilder Tukey (1915 – 2000) This test is not available in Excel’s Tools > Data Analysis but is available in MegaStat.This test is not available in Excel’s Tools > Data Analysis but is available in MegaStat. Tukey’s is a two-tailed test for equality of paired means from c groups compared simultaneously.Tukey’s is a two-tailed test for equality of paired means from c groups compared simultaneously. The hypotheses are:The hypotheses are: H 0 : j = k H 1 : j ≠ k Tukey’s Test Tukey’s Test Multiple Comparison Tests Multiple Comparison Tests 11-12

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Tests for Homogeneity of Variances Tests for Homogeneity of Variances ANOVA assumes that observations on the response variable are from normally distributed populations that have the same variance.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.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 Hartley’s F max Test.Test this assumption of homogeneous variances, using Hartley’s F max Test. ANOVA Assumptions ANOVA Assumptions 11-13

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Named for H.O. Hartley (1912 – 1980).Named for H.O. Hartley (1912 – 1980). The hypotheses areThe hypotheses are H 0 : 1 2 = 2 2 = … = c 2 H 1 : Not all the j 2 are equal The test statistic is the ratio of the largest sample variance to the smallest sample varianceThe test statistic is the ratio of the largest sample variance to the smallest sample variance Hartley’s F max Test Hartley’s F max Test F max = s max 2 s min 2 Tests for Homogeneity of Variances Tests for Homogeneity of Variances 11-14

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Levene’s test is a more robust alternative to Hartley’s F max test.Levene’s test is a more robust alternative to Hartley’s F max test. Levene’s test does not assume a normal distribution.Levene’s 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. Levene’s Test Levene’s Test Tests for Homogeneity of Variances Tests for Homogeneity of Variances 11-15

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