ANOVA: Analysis of Variance

1. ANOVA: Analysis of Variance 1-way ANOVA Anthony J Greene

2. ANOVA • What is Analysis of Variance • The F-ratio • Used for testing hypotheses among more than two means • As with t-test, effect is measured in numerator, error variance in the denomenator • Partitioning the Variance • Different computational concerns for ANOVA • Degrees Freedom for Numerator and Denominator • No such thing as a negative value • Using Table B.4 • The Source Table • Hypothesis testing Anthony J Greene

3. M1 M3 M2 Anthony J Greene

4. ANOVA • Analysis of Variance • Hypothesis testing for more than 2 groups • For only 2 groups t2(n) = F(1,n) Anthony J Greene

5. Grp 1 Grp 2 Grp 3 Is the Effect Variability Large Compared to the Random Variability BASIC IDEA • As with the t-test, the numerator expresses the differences among the dependent measure between experimental groups, and the denominator is the error. • If the effect is enough larger than random error, we reject the null hypothesis. M1 = 1 M2 = 5 M3 = 1 Effect V = Random V Anthony J Greene

6. BASIC IDEA • If the differences accounted for by the manipulation are low (or zero) then F = 1 • If the effects are twice as large as the error, then F = 3, which generally indicates an effect. Anthony J Greene

7. Sources of Variance Anthony J Greene

8. Why Is It Called Analysis of Variance?Aren’t We Interested In Means, Not Variance? • Most statisticians do not know the answer to this question? • If we’re interested in differences among means why do an analysis of variance? • The misconception is that it compares 12 to 22. No • The comparison is between effect variance (differences in group means) to random variance. Anthony J Greene

9. Learning Under Three Temperature Conditions T is the treatment total, G is the Grand total M1 M2 M3 Anthony J Greene

10. Computing the Sums of Squares Anthony J Greene

11. Keep in mind the general formula for SS Grp 1 Grp 2 Grp 3 M1 = 1 M2 = 5 M3 = 1 How Variance is Partitioned This simply disregards group membership and computes an overall SS Variability Between and Within Groups is Included Anthony J Greene

12. Keep in mind the general formula for SS M1 = 1 M2 = 5 M3 = 1 T1 = 5 T2 = 25 T3 = 5 How Variance is Partitioned Imagine there were no individual differences at all. The SS for all scores would measure only the fact that there were group differences. Grp 1 Grp 2 Grp 3 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 Anthony J Greene

13. Keep in mind the general formula for SS How Variance is Partitioned SS computed within a column removes the mean. Thus summing the SS’s for each column computes the overall variability except for the mean differences between groups. Grp 1 Grp 2 Grp 3 1-12-12-10-10-1 0-11-13-11-1 0-1 4-53-56-53-54-5 M1 = 1 M2 = 5 M3 = 1 Anthony J Greene

14. 1-12-12-10-10-1 0-11-13-11-1 0-1 4-53-56-53-54-5 How Variance is Partitioned Grp 1 Grp 2 Grp 3 M1 = 1 M2 = 5 M3 = 1 Anthony J Greene

15. Computing Degrees Freedom • df between is k-1, where k is the number of treatment groups (for the prior example, 3, since there were 3 temperature conditions) • df within is N-k , where N is the total number of ns across groups. Recall that for a t-test with two independent groups, df was 2n-2? 2n was all the subjects N and 2 was the number of groups, k. Anthony J Greene

16. Computing Degrees Freedom Anthony J Greene

17. How Degrees Freedom Are Partitioned N-1 = (N - k) + (k - 1) N-1 = N - k + k – 1 Anthony J Greene

18. Partitioning The Sums of Squares Anthony J Greene

19. Computing An F-Ratio Anthony J Greene

20. Consult Table B-4 Take a standard normal distribution, square each value, and it looks like this Anthony J Greene

21. Table B-4 Anthony J Greene

22. Two different F-curves Anthony J Greene

23. ANOVA: Hypothesis Testing Anthony J Greene

24. Basic Properties of F-Curves Property 1: The total area under an F-curve is equal to 1. Property 2: An F-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis as it does so. Property 3: An F-curve is right skewed. Anthony J Greene

25. Finding the F-value having area 0.05 to its right Anthony J Greene

26. Assumptions for One-Way ANOVA • 1.Independent samples: The samples taken from the populations under consideration are independent of one another. • 2. Normal populations: For each population, the variable under consideration is normally distributed. • Equal standard deviations: The standard deviations of the variable under consideration are the same for all the populations. Anthony J Greene

27. Learning Under Three Temperature Conditions M1 = 1 M2 = 5 M3 = 1 Anthony J Greene

28. Is the Effect Variability Large Compared to the Random Variability M1 = 1 M2 = 5 M3 = 1 Learning Under Three Temperature Conditions Anthony J Greene

29. Learning Under Three Temperature Conditions Anthony J Greene

30. Learning Under Three Temperature Conditions Anthony J Greene

31. Learning Under Three Temperature Conditions Anthony J Greene

32. Learning Under Three Temperature Conditions Anthony J Greene

33. Learning Under Three Temperature Conditions M1 M2 M3 Anthony J Greene

34. Learning Under Three Temperature Conditions ΣX2 = 106 16936916 144 191 M1 M2 M3 Anthony J Greene

35. Learning Under Three Temperature Conditions M1 M2 M3 Anthony J Greene

36. Learning Under Three Temperature Conditions M1 M2 M3

37. Calculating the F statistic Sstotal = X2-G2/N = 46 SSbetween = SSbetween = 30 SStotal= Ssbetween + SSwithin Sswithin = 16

38. Distribution of the F-Statistic for One-Way ANOVA Suppose the variable under consideration is normally distributed on each of k populations and that the population standard deviations are equal. Then, for independent samples from the k populations, the variable has the F-distribution with df = (k – 1, n – k) if the null hypothesis of equal population means is true. Here n denotes the total number of observations. Anthony J Greene

39. ANOVA Source Table for a one-way analysis of variance Anthony J Greene

40. The one-way ANOVA test for k population means (Slide 1 of 3) Step 1 The null and alternative hypotheses are Ho: 1 = 2 = 3 = …= k Ha: Not all the means are equal Step 2 Decide On the significance level,  Step 3 The critical value of F, with df = (k - 1, N - k), where N is the total number of observations. Anthony J Greene

41. The one-way ANOVA test for k population means(Slide 2 of 3) Anthony J Greene

42. The one-way ANOVA test for k population means (Slide 3 of 3) Step 4 Obtain the three sums of squares, STT, STTR, and SSE Step 5 Construct a one-way ANOVA table: Step 6 If the value of the F-statistic falls in the rejection region, reject H0; Anthony J Greene

43. Post Hocs • H0 : 1 = 2 = 3 = …= k • Rejecting H0 means that not all means are equal. • Pairwise tests are required to determine which of the means are different. • One problem is for large k. For example with k = 7, 21 means must be compared. Post-Hoc tests are designed to reduce the likelihood of groupwise type I error. Anthony J Greene

44. Criterion for deciding whether or not to reject the null hypothesis Anthony J Greene

45. One-Way ANOVA A researcher wants to test the effects of St. John’s Wort, an over the counter, herbal anti-depressant. The measure is a scale of self-worth. The subjects are clinically depressed patients. Use α = 0.01 Anthony J Greene

46. One-Way ANOVA Compute the treatment totals, T, and the grand total, G Anthony J Greene

47. One-Way ANOVA Count n for each treatment, the total N, and k Anthony J Greene

48. One-Way ANOVA Compute the treatment means Anthony J Greene

49. One-Way ANOVA (0-1)2=1 (1-1)2=0 (3-1)2=4 (0-1)2=1 (1-1)2=0 sum Compute the treatment SSs Anthony J Greene

50. One-Way ANOVA Compute all X2s and sum them Anthony J Greene