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 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

One-Factor ANOVA Test Hypothesis H0: m1 = m2 = m3 = ... = mc All population means are equal No treatment effect (NO variation in means among groups) H1: not all the mk are 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: m1 ¹ m2 ¹ ... ¹ mc

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

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

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 mi mj Variation Due to Differences Among Groups.

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. m j

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

One-Factor ANOVA F Test Example Machine1 Machine2 Machine3 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?

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

One-Factor ANOVA Example Solution H0: m1 = m2 = m3 H1: Not All Equal a = .05 df1= 2 df2 = 12 Critical Value(s): Test Statistic: Decision: Conclusion: 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 3.89

The Tukey-Kramer Procedure Tells Which Population Means Are Significantly Different e.g., m1 = m2 ¹ m3 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.

The Tukey-Kramer Procedure: Example 1. Compute absolute mean differences: Machine1 Machine2 Machine3 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 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

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

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.