This document provides an overview of analysis of variance (ANOVA) techniques. It discusses one-way ANOVA, which evaluates differences between three or more population means. Key aspects covered include partitioning total variation into between- and within-group components, assumptions of normality and equal variances, and using the F-test to test for differences. Randomized block ANOVA and two-factor ANOVA are also introduced as extensions to control for additional variables. Post-hoc tests like Tukey and Fisher's LSD are described for determining specific mean differences.

1. Analysis of Variance

3. Chapter Overview Analysis of Variance (ANOVA) F-test F-test Tukey- Kramer test Fisher’s Least Significant Difference test One-Way ANOVA Randomized Complete Block ANOVA Two-factor ANOVA with replication

8. One-Factor ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect)

9. One-Factor ANOVA At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or (continued)

11. Partitioning the Variation Total Variation = the aggregate dispersion of the individual data values across the various factor levels (SST) Within-Sample Variation = dispersion that exists among the data values within a particular factor level (SSW) Between-Sample Variation = dispersion among the factor sample means (SSB) SST = SSB + SSW (continued)

13. Total Sum of Squares Where: SST = Total sum of squares k = number of populations (levels or treatments) n i = sample size from population i x ij = j th measurement from population i x = grand mean (mean of all data values) SST = SSB + SSW

14. Total Variation (continued)

15. Sum of Squares Between Where: SSB = Sum of squares between k = number of populations n i = sample size from population i x i = sample mean from population i x = grand mean (mean of all data values) SST = SSB + SSW

16. Between-Group Variation Variation Due to Differences Among Groups Mean Square Between = SSB/degrees of freedom

17. Between-Group Variation (continued)

18. Sum of Squares Within Where: SSW = Sum of squares within k = number of populations n i = sample size from population i x i = sample mean from population i x ij = j th measurement from population i SST = SSB + SSW

19. Within-Group Variation Summing the variation within each group and then adding over all groups Mean Square Within = SSW/degrees of freedom

20. Within-Group Variation (continued)

21. One-Way ANOVA Table Source of Variation df SS MS Between Samples SSB MSB = Within Samples N - k SSW MSW = Total N - 1 SST = SSB+SSW k - 1 MSB MSW F ratio k = number of populations N = sum of the sample sizes from all populations df = degrees of freedom SSB k - 1 SSW N - k F =

25. One-Factor ANOVA Example: Scatter Diagram • • • • • 270 260 250 240 230 220 210 200 190 • • • • • • • • • • Distance Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 Club 1 2 3

26. One-Factor ANOVA Example Computations Club 1 Club 2 Club 3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 x 1 = 249.2 x 2 = 226.0 x 3 = 205.8 x = 227.0 n 1 = 5 n 2 = 5 n 3 = 5 N = 15 k = 3 SSB = 5 [ (249.2 – 227) 2 + (226 – 227) 2 + (205.8 – 227) 2 ] = 4716.4 SSW = (254 – 249.2) 2 + (263 – 249.2) 2 +…+ (204 – 205.8) 2 = 1119.6 MSB = 4716.4 / (3-1) = 2358.2 MSW = 1119.6 / (15-3) = 93.3

28. ANOVA -- Single Factor: Excel Output EXCEL: tools | data analysis | ANOVA: single factor SUMMARY Groups Count Sum Average Variance Club 1 5 1246 249.2 108.2 Club 2 5 1130 226 77.5 Club 3 5 1029 205.8 94.2 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 4716.4 2 2358.2 25.275 4.99E-05 3.885 Within Groups 1119.6 12 93.3 Total 5836.0 14

30. Tukey-Kramer Critical Range where: q  = Value from standardized range table with k and N - k degrees of freedom for the desired level of  MSW = Mean Square Within n i and n j = Sample sizes from populations (levels) i and j

32. The Tukey-Kramer Procedure: Example 5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance. 3. Compute Critical Range: 4. Compare:

33. Tukey-Kramer in PHStat

36. Sum of Squares for Blocking Where: k = number of levels for this factor b = number of blocks x j = sample mean from the j th block x = grand mean (mean of all data values) SST = SSB + SSBL + SSW

38. Mean Squares

39. Randomized Block ANOVA Table Source of Variation df SS MS Between Samples SSB MSB Within Samples (k–1)(b-1) SSW MSW Total N - 1 SST k - 1 MSBL MSW F ratio k = number of populations N = sum of the sample sizes from all populations b = number of blocks df = degrees of freedom Between Blocks SSBL b - 1 MSBL MSB MSW

43. Fisher’s Least Significant Difference (LSD) Test where: t  /2 = Upper-tailed value from Student’s t-distribution for  /2 and (k -1)(n - 1) degrees of freedom MSW = Mean square within from ANOVA table b = number of blocks k = number of levels of the main factor

44. Fisher’s Least Significant Difference (LSD) Test (continued) If the absolute mean difference is greater than LSD then there is a significant difference between that pair of means at the chosen level of significance. Compare:

47. Two-Way ANOVA Sources of Variation Two Factors of interest: A and B a = number of levels of factor A b = number of levels of factor B N = total number of observations in all cells

48. Two-Way ANOVA Sources of Variation SST Total Variation SS A Variation due to factor A SS B Variation due to factor B SS AB Variation due to interaction between A and B SSE Inherent variation (Error) Degrees of Freedom: a – 1 b – 1 (a – 1)(b – 1) N – ab N - 1 SST = SS A + SS B + SS AB + SSE (continued)

49. Two Factor ANOVA Equations Total Sum of Squares: Sum of Squares Factor A: Sum of Squares Factor B:

50. Two Factor ANOVA Equations Sum of Squares Interaction Between A and B: Sum of Squares Error: (continued)

51. Two Factor ANOVA Equations where: a = number of levels of factor A b = number of levels of factor B n’ = number of replications in each cell (continued)

52. Mean Square Calculations

53. Two-Way ANOVA: The F Test Statistic F Test for Factor B Main Effect F Test for Interaction Effect H 0 : μ A1 = μ A2 = μ A3 = • • • H A : Not all μ Ai are equal H 0 : factors A and B do not interact to affect the mean response H A : factors A and B do interact F Test for Factor A Main Effect H 0 : μ B1 = μ B2 = μ B3 = • • • H A : Not all μ Bi are equal Reject H 0 if F > F  Reject H 0 if F > F  Reject H 0 if F > F 

54. Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Factor A SS A a – 1 MS A = SS A /(a – 1) MS A MSE Factor B SS B b – 1 MS B = SS B /(b – 1) MS B MSE AB (Interaction) SS AB (a – 1)(b – 1) MS AB = SS AB / [(a – 1)(b – 1)] MS AB MSE Error SSE N – ab MSE = SSE/(N – ab) Total SST N – 1