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**11 Analysis of Variance Chapter Overview of ANOVA One-Factor ANOVA**

Multiple Comparisons Tests for Homogeneity of Variances McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. 1

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Overview of ANOVA Analysis of variance (ANOVA) is a comparison of means. ANOVA allows you to compare more than two means simultaneously. Proper experimental design efficiently uses limited data to draw the strongest possible inferences. 2

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**Overview of ANOVA 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). 3

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**Overview of ANOVA 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: H0: m1 = m2 = m3 = m4 H1: Not all the means are equal The test uses the F distribution. If we cannot reject H0, we conclude that observations within each treatment have a common mean m. 4

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**Overview of ANOVA ANOVA Assumptions**

Analysis 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. 5

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**One-Factor ANOVA (Completely Randomized Design)**

One-Factor ANOVA as a Linear Model An equivalent way to express the one-factor model is to say that treatment j came from a population with a common mean (m) plus a treatment effect (Aj) plus random error (eij): yij = m + Aj + eij 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. 6

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**One-Factor ANOVA (Completely Randomized Design)**

One-Factor ANOVA as a Linear Model A fixed effects model only looks at what happens to the response for particular levels of the factor. H0: A1 = A2 = … = Ac = 0 H1: Not all Aj are zero If the H0 is true, then the ANOVA model collapses to yij = m + eij 7

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**One-Factor ANOVA (Completely Randomized Design)**

Steps in Performing One-Factor ANOVA Step 1: State the hypotheses H0: m1 = m2 = … = mc H1: 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) For the given a, obtain the critical value from Appendix F or Excel. 8

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**One-Factor ANOVA (Completely Randomized Design)**

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

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**One-Factor ANOVA (Completely Randomized Design)**

Steps in Performing One-Factor ANOVA Step 4: Make the Decision Reject H0 if the critical value F exceeds the test statistic Fa or if the p-value given by Excel is < a. MegaStat and MINITAB can also be used to perform the ANOVA. 10

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**Multiple Comparison Tests**

Tukey’s Test 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. For c groups, there are c(c – 1) distinct pairs of means to be compared. 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. 11

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**Multiple Comparison Tests**

Tukey’s Test Named for statistician John Wilder Tukey (1915 – 2000) 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. The hypotheses are: H0: mj = mk H1: mj ≠ mk 12

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**Tests for Homogeneity of Variances**

ANOVA Assumptions 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 Hartley’s Fmax Test. 13

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**Tests for Homogeneity of Variances**

Hartley’s Fmax Test Named for H.O. Hartley (1912 – 1980). The hypotheses are H0: s12 = s22 = … = sc2 H1: Not all the sj2 are equal The test statistic is the ratio of the largest sample variance to the smallest sample variance Fmax = smax2 smin2

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**Tests for Homogeneity of Variances**

Levene’s Test Levene’s test is a more robust alternative to Hartley’s Fmax test. 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. A computer program (e.g., MINITAB) is needed to perform this test. 15

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