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The Completely Randomized Model: One-Factor Analysis of Variance

Chapter Topics. The Completely Randomized Model: One-Factor Analysis of Variance F -Test for Difference in c Means The Tukey-Kramer Procedure ANOVA Assumptions Kruksal-Wallis Rank Test for Differences in c Medians. One-Factor Analysis of Variance.

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The Completely Randomized Model: One-Factor Analysis of Variance

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  1. Chapter Topics • The Completely Randomized Model: One-Factor Analysis of Variance • F-Test for Difference in c Means • The Tukey-Kramer Procedure • ANOVA Assumptions • Kruksal-Wallis Rank Test for Differences in c • Medians

  2. One-Factor Analysis of Variance • Evaluate the Difference Among the Means of 2 or More (c) Populations • e.g., Several Types of Tires, Oven Temperature Settings • Assumptions: • Samples are Randomly and Independently Drawn (This condition must be met.) • Populations are Normally Distributed (F test is Robust to moderate departures from normality.) • Populations have Equal Variances

  3. One-Factor ANOVATest Hypothesis H0: m1 = m2 = m3 = ... = mc • All population means are equal • No treatment effect (NO variation in means among groups) H1: not all the mkare equal • At least ONE population mean is different • (Others may be the same!) • There is treatment effect • Does NOT mean that all the means are different: m1¹m2¹ ... ¹mc

  4. One-Factor ANOVA:No Treatment Effect H0: m1 = m2 = m3 = ... = mc H1: not all the mkare equal The Null Hypothesis is True m1 = m2= m3

  5. Commonly referred to as: Sum of Squares Within, or Sum of Squares Error, or Within Groups Variation Commonly referred to as: Sum of Squares Among, or Sum of Squares Between, or Sum of Squares Model, or Among Groups Variation One-Factor ANOVAPartitions of Total Variation Total Variation SST Variation Due to Treatment SSA Variation Due to Random Sampling SSW = +

  6. Among-Group Variation nj =the number of observations in group j c =the number of groups _ Xj the sample mean of group j _ _ X the overall or grand mean mimj Variation Due to Differences Among Groups.

  7. Within-Group Variation the ith observation in group j the sample mean of group j Summing the variation within each group and then adding over all groups. mj

  8. One-Way ANOVA Summary Table Degrees Sum of Source of Mean F Test Statistic Squares Variation of Square Freedom (Variance) MSA = Among SSA MSA = c - 1 MSW (Factor) SSA/(c - 1) Within SSW n - c MSW = (Error) SSW/(n - c) Total n - 1 SST = SSA+SSW

  9. One-Factor ANOVA F Test Example Machine1Machine2Machine3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40 • As production manager, you want to see if 3 filling machines have different mean filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the .05 level, is there a difference in mean filling times?

  10. Summary Table Source of Degrees of Sum of Mean Variation Freedom Squares Square (Variance) MSA MSW Among 3 - 1 = 2 47.1640 23.5820 F = = 25.60 (Machines) Within 15 - 3 = 12 11.0532 .9211 (Error) Total 15 - 1 = 14 58.2172

  11. One-Factor ANOVA ExampleSolution Test Statistic: Decision: Conclusion: • H0: m1 = m2 = m3 • H1: Not All Equal • a = .05 • df1= 2 df2 = 12 • Critical Value(s): MSA 23 5820 . = F = = 25 . 6 MSW . 9211 Reject at a = 0.05 a = 0.05 There is evidence that at least one m i differs from the rest. F 0 3.89

  12. The Tukey-Kramer Procedure • Tells Which Population Means • Are Significantly Different • e.g., m1 = m2¹m3 • Post Hoc (a posteriori) • Procedure • Done after rejection of equal means in ANOVA • Ability for Pairwise Comparisons: • Compare absolute mean differences with ‘critical range’ f(X) m m m X = 1 2 3 2 groups whose means may be significantly different.

  13. The Tukey-Kramer Procedure: Example Machine1Machine2Machine325.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40 • 1. Compute absolute mean differences: 2. Compute Critical Range: 3. Each of the absolute mean difference is greater. There is a significance difference between each pair of means. = 1.618

  14. Kruskal-Wallis Rank Test for c Medians • Extension of Wilcoxon Rank Sum Test Tests the equality of more than 2 (c) population medians • Distribution-free test procedure • Used to analyze completely randomized experimental designs

  15. Kruskal-Wallis Rank Test • Assumptions: • Independent random samples are drawn • Continuous dependent variable • Data may be ranked both within and among samples • Populations have same variability • Populations have same shape • Robust with regard to last 2 conditions • Use F Test in completely randomized designs and when the more stringent assumptions hold.

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