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Chapter 17

Chapter 17. Comparing Multiple Population Means: One-factor ANOVA. What if we have more than 2 conditions/groups?. Interest - the effects of 3 drugs on depression - Prozac, Zoloft, and Elavil

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Chapter 17

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  1. Chapter 17 Comparing Multiple Population Means: One-factor ANOVA

  2. What if we have more than 2 conditions/groups? • Interest - the effects of 3 drugs on depression - Prozac, Zoloft, and Elavil • Select 24 people with depression, randomly assign (blindly) to one of four conditions: 1) Prozac, 2) Zoloft, 3) Elavil, and 4) Placebo • After 1 month of drug therapy, we measure depression

  3. Research Design and Data Prozac Zoloft Elavil Placebo 10 14 19 21 8 12 15 27 15 18 14 20 12 16 16 23 9 13 18 15 6 17 20 22

  4. Multiple t-tests? • Differences between drugs? Prozac vs. Zoloft Prozac vs. Elavil Prozac vs. Placebo Zoloft vs. Elavil Zoloft vs Placebo Elavil vs. Placebo • 6 separate t-tests

  5. Probability Theory (Revisited) • The probability of making a correct decision when the null is false is 1 - α(generally .95) • Each test is independent • The probability of making the correct decision across all 6 tests is the product of those probabilities or, (.95)(.95)(.95)(.95)(.95)(.95) = .735

  6. Type 1 error & multiple t-tests • Thus, the probability of a type 1 error is not α, but 1 - (1 - α)C, where C is the number of comparisons • Or, in the present case 1 - .735 = .265

  7. t statistic as a ratio Hmmm… obtained difference t = ———————————————— difference expected by chance (“error”) Easy – Pool Variance

  8. Differences in the t test • M1 – M2 or MD • Can we subtract multiple means from one another? M1 – M2 – M3 – M4= ???? M4– M1– M2– M3= ???? Is there another statistic that tells us how much things differ from one another?

  9. What statistic describes how scores differ from one another? • Variance • How do a set a means differ from one another? Answer – variance between means/groups

  10. t statistic as a ratio obtained difference t = ———————————————— difference expected by chance (“error”) variance between means/groups t = ———————————————— pooled variance

  11. F statistic between-groups variance estimate F = —————————————— within-groups variance estimate Mean-square Treatment (MST or MSB) s2B F = ———————————————— = — Mean-square Error (MSE or MSW)s2W

  12. ANOVA • Analysis of Variance, or ANOVA, allows us to compare multiple group means, without compromising α • And, even though an ANOVA uses variances and the F statistic, it helps test hypotheses about means

  13. F statistic • Between-groups variance (MST or MSB) is based on the variability between the groups • Within-groups variance (MSE or MSW) is a measure of the variability within the groups • if there is no difference between these 2 measure of variability (due to no differences between groups), F will be close to 1 • if there is greater variability between-groups (due to differences between groups), F will be greater than 1

  14. Between-groups variance (MST, MSB or s2B) k groups where Miis the mean of the ith group, and MGis the grand mean (the mean of all scores)

  15. Within-groups variance(MSE, MSW, or s2W) k groups

  16. SST (Sums of Squares Total) • The sums of squares total can be used either as a check, or to calculate SSW

  17. An ANOVA Table • The results of an ANOVA are often presented in a table: Source SS df MS F Between Within Total

  18. An ANOVA Table • The results of an ANOVA are often presented in a table: Source SS df MS F Between 180 2 90.0 36.00 Within 30 12 2.5 Total 210 14

  19. Procedure for Completing an ANOVA • 1. Arrange Data by Group • 2. Compute for each group (k groups): Σx Σx2 M SS(x) n

  20. Procedure for Completing an ANOVA • 3. Compute the grand mean ( MG), by adding all the scores and dividing by N MG= Σx/N • 4. Compute SSB = Σni( Mi- MG)2 • 5. Compute SSW SSW = SS(x1) + SS(x2) + ···+ SS(xk) • 6. Compute SST = Σx2- (Σx)2/N

  21. Procedure for Completing an ANOVA • 7. Compute df dfB = k - 1 dfW = N - k dfT = N -1 • 8. Fill in ANOVA table • 9. Compute MS (SS/df) • 10. Compute F = MSB/MSW

  22. 1. ANOVA Calculations Prozac Zoloft Elavil Placebo 10 14 19 21 8 12 15 27 15 18 14 20 12 16 16 23 9 13 18 15 6 17 20 22

  23. 2. ANOVA Calculations Prozac (Group 1) 10 ΣX1= 50 8 ΣX12= 650 15 M1= 10 12 SS(X1) = 50 9 n1 = 6 6

  24. 2. ANOVA Calculations Zoloft (Group 2) 14 ΣX2= 90 12 ΣX22= 1378 18 M2= 15 16 SS(X2) = 28 13 n2 = 6 17

  25. 2. ANOVA Calculations Elavil (Group 3) 19 ΣX3= 102 15 ΣX32= 1762 14 M3= 17 16 SS(X3) = 28 18 n3 = 6 20

  26. 2. ANOVA Calculations Placebo (Group 4) 21 ΣX4= 128 27 ΣX42= 2808 20 M4= 21.33 23 SS(X4) = 77.33 15 n4 = 6 22

  27. 3. ANOVA Calculations MG = Σx/N = (ΣX1+ΣX2 + ΣX3 + ΣX4)/ (n1+n2+n3+n4) = (60+90+102+128)/(6+6+6+6) = 380/24 = 15.83

  28. 4. ANOVA Calculations SSB = Σni ( Mi- XG)2 = 6(10 - 15.83)2 + 6(15 - 15.83)2 + 6(17 - 15.83)2 + 6(21.33 - 15.83)2 = 6(34.03) + 6(.69) + 6(1.36) + 6(30.25) = 204.18 + 4.14 + 8.16 + 181.5 = 398.00

  29. 5. ANOVA Calculations SSW = SS(X1) + SS(X2) + ···+ SS(Xk) = 50 + 28 + 28 + 77.33 = 183.33

  30. 6. ANOVA Calculations SST = ΣX2- (ΣX)2/N = (650 + 1378 + 1762 + 2808) - (60 + 90 + 102 + 128)2/24 = 6598 - 144400/24 = 6598 - 6016.67 = 581.33

  31. Check SST = SSB + SSW 581.33 = 398 + 183.33 581.33 = 581.33

  32. 7. ANOVA Calculations dfB = k -1 = 4 -1 = 3 dfW = N - k = 24 - 4 = 20 dfT = N - 1 = 23

  33. 8. ANOVA Calculations Source SS df MS F Between 398.00 3 Within 183.33 20 Total 581.33 23

  34. 8. ANOVA Calculations Source SS df MS F Between 398.00 3 132.67 Within 183.33 20 9.17 Total 581.33 23

  35. 8. ANOVA Calculations Source SS df MS F Between 398.00 3 132.67 14.47 Within 183.33 20 9.17 Total 581.33 23

  36. 8. ANOVA Calculations Source SS df MS F Between 398.00 3 132.67 14.47 Within 183.33 20 9.17 Total 581.33 23

  37. Hypothesis test of Anti-depressants • 1. State and Check Assumptions • About the population • Normally distributed? - don’t know • Homogeneity of variance – we’ll check • About the sample • Independent Random sample? – yes • Independent samples • About the sample • Interval level

  38. Hypothesis test of Anti-depressants • 2. Hypotheses HO: μProzac= μZoloft= μElavil= μPlacebo HA : the null is wrong

  39. That’s an Odd HA • You might think that the alternative hypothesis should look like this: HA: μProzac≠ μZoloft≠ μElavil≠ μPlacebo Accepting this alternative indicates that all of the means are unequal, which is not what ANOVA determines

  40. What does ANOVA determine? • That at least one of the means is different than at least one other mean • Since, that is a difficult statement to write, we say • “the null is wrong”

  41. Hypothesis test of Anti-depressants • 3. Choose test statistic • 4groups • independent samples One-factor ANOVA

  42. Hypothesis test of Anti-depressants • 4. Set Significance Level α= .05 Critical Value Non-directional Hypothesis with dfB= k – 1 and dfW= N – k dfB = 3 and dfW= 21 From Table D Fcrit= 3.07, so we reject HO if F≥ 3.07

  43. Hypothesis test of Anti-depressants • 5. Compute Statistic Source SS df MS F Between 398.00 3 132.67 14.47 Within 183.33 20 9.17 Total 581.33 23

  44. Hypothesis test of Anti-depressants • 6. Draw Conclusions • because our F falls within the rejection region, we reject the HO, and • conclude that at least one medicine is better than at least one other medicine in treating depression

  45. Violations of Assumptions • As with t-tests, ANOVA is fairly ROBUST to violations of normality and homogeneity of variance, but • IF there are severe violations of these assumptions, • Use a Kruskal-Wallis H test (a non-parametric alternative)

  46. Procedure for completing a Kruskal-Wallis H • 1. Arrange data in columns, 1 group per column, skipping columns between groups • 2. Rank all the scores, assigning the lowest rank (1) to the lowest score (put ranks in the column next to the raw scores) • 3. Sum the ranks in each column (ΣTj) • 4. Square the sum of the ranks of each column (ΣTj)2

  47. Procedure for completing a Kruskal-Wallis H test • 5. Compute SSB • 6. Compute H

  48. Procedure for completing a Kruskal-Wallis H test • 6. Compute df = k - 1 • 7. H is distributed as a χ2 • Look up critical value in χ2(chi-square) table with appropriate df

  49. Dependent Samples(more than 2 conditions) • Experiments are often conducted comparing more than 2 conditions • ANOVA • Kruskal-Wallis H • Samples are often related - “dependent samples” (within-subjects, repeated measures, etc.)

  50. Dependent Samples ANOVA SS(T) = SS(B) + SS(Bl) + SS(E) Calculate SS(T), SS(B), and SS(Bl) SS(E) = SS(T) - SS(B) - SS(Bl)

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