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ANOVA in Psychology: Understanding Analysis of Variance in One-way Designs

Explore the logic, calculations, assumptions, and effects of ANOVA in psychology, with examples and data interpretation. Learn about Fisher’s LSD, Bonferroni t, and the magnitude of effect in research analysis.

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ANOVA in Psychology: Understanding Analysis of Variance in One-way Designs

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  1. Chapter 13 Analysis of Variance (ANOVA) PSY295-001 Spring 2003

  2. Major Points • An example • Requirements • The logic • Calculations • Unequal sample sizes Chapter 13 One-way Analysis of Variance Cont.

  3. Major Points--cont. • Multiple comparisons • Fisher’s LSD • Bonferroni t • Assumptions of analysis of variance • Magnitude of effect • eta squared • omega squared • Review questions Chapter 13 One-way Analysis of Variance

  4. Example • Daily caffeine intake (DV) • Year in school (IV) • four groups • 6 pairs of comparison Chapter 13 One-way Analysis of Variance

  5. Requirements • The DV must be quantitative (Interval or Ratio) • The IV must be between subjects (so each subject is in one and only one treatment group) • The IV must have 2 or more levels. Chapter 13 One-way Analysis of Variance

  6. The F-ratio • For the t-test we had • t = mean differences between samples standard error of the mean • for the F-ratio we use, • F = variance (differences) between means variance (differences) from sampling error Chapter 13 One-way Analysis of Variance

  7. Another silly example • Alcohol consumption and driving performance (number of trashed cones) • 3 levels of the IV: 1oz, 3oz, 5oz of ETOH • N=15, n1=5, n2=5, n3=5 • What is the null hypothesis? • What is the alternative? Chapter 13 One-way Analysis of Variance

  8. The data Chapter 13 One-way Analysis of Variance

  9. The steps • Measure total (or overall) variability • Two components • between treatments variability • within treatments variability Chapter 13 One-way Analysis of Variance

  10. The steps (continued) • Between treatments variability comprised of: • treatment effects • individual differences • experimental error • Within treatments variability comprised of: • individual differences • experimental error Chapter 13 One-way Analysis of Variance

  11. The steps (continued) • Production of the F-ratio • F =Variance between treatment group means Variance within treatment groups OR • F =Treatment Effects+Individual Differences+Error Individual Differences+Error Chapter 13 One-way Analysis of Variance

  12. Logic of the Analysis of Variance • Null hypothesis: H0 Population means equal • m1 = m2 = m3 = m4 • Alternative hypothesis: H1 • Not all population means equal. Chapter 13 One-way Analysis of Variance Cont.

  13. F-ratio • Ratio approximately 1 if null true • Ratio significantly larger than 1 if null false • “approximately 1” can actually be as high as 2 or 3, but not much higher Chapter 13 One-way Analysis of Variance

  14. Epinephrine and Memory • Based on Introini-Collison & McGaugh (1986) • Trained mice to go left on Y maze • Injected with 0, .1, .3, or 1.0 mg/kg epinephrine • Next day trained to go right in same Y maze • dep. Var. = # trials to learn reversal • More trials indicates better retention of Day 1 • Reflects epinephrine’s effect on memory Chapter 13 One-way Analysis of Variance

  15. Grand mean = 3.78 Chapter 13 One-way Analysis of Variance

  16. Calculations • Start with Sum of Squares (SS) • We need: • SS total • SS within or sometimes called SS error • SS between or sometimes called SS groups • Compute degrees of freedom (df ) • Compute mean squares and F Chapter 13 One-way Analysis of Variance Cont.

  17. Calculations--cont. Chapter 13 One-way Analysis of Variance

  18. Degrees of Freedom (df ) • Number of “observations” free to vary • dftotal = N - 1 • Variability of N observations • dfgroups = k - 1 • variability of g means • dferror = k(n - 1) • n observations in each group = n - 1 df • times k groups Chapter 13 One-way Analysis of Variance

  19. Summary Table Chapter 13 One-way Analysis of Variance

  20. Conclusions • The F for groups is significant. • We would obtain an F of this size, when H0 true, less than one time out of 1000. • The difference in group means cannot be explained by random error. • The number of trials to learn reversal depends on level of epinephrine. Chapter 13 One-way Analysis of Variance Cont.

  21. Conclusions--cont. • The injection of epinephrine following learning appears to consolidate that learning. • High doses may have negative effect. Chapter 13 One-way Analysis of Variance

  22. Unequal Sample Sizes • With one-way, no particular problem • Multiply mean deviations by appropriate ni as you go • The problem is more complex with more complex designs, as shown in next chapter. • Example from Foa, Rothbaum, Riggs, & Murdock (1991) Chapter 13 One-way Analysis of Variance

  23. Post-Traumatic Stress Disorder • Four treatment groups given psychotherapy • Stress Inoculation Therapy (SIT) • Standard techniques for handling stress • Prolonged exposure (PE) • Reviewed the event repeatedly in their mind Chapter 13 One-way Analysis of Variance Cont.

  24. Post-Traumatic Stress Disorder--cont. • Supportive counseling (SC) • Standard counseling • Waiting List Control (WL) • No treatment Chapter 13 One-way Analysis of Variance

  25. SIT = Stress Inoculation Therapy PE = Prolonged Exposure SC = Supportive Counseling WL = Waiting List Control Grand mean = 15.622 Chapter 13 One-way Analysis of Variance

  26. Tentative Conclusions • Fewer symptoms with SIT and PE than with other two • Also considerable variability within treatment groups • Is variability among means just a reflection of variability of individuals? Chapter 13 One-way Analysis of Variance

  27. Calculations • Almost the same as earlier • Note differences • We multiply by nj as we go along. • MSerror is now a weighted average. Chapter 13 One-way Analysis of Variance Cont.

  28. Calculations--cont. Chapter 13 One-way Analysis of Variance

  29. Summary Table F.05(3,41) = 2.84 Chapter 13 One-way Analysis of Variance

  30. Conclusions • F is significant at a = .05 • The population means are not all equal • Some therapies lead to greater improvement than others. • SIT appears to be most effective. Chapter 13 One-way Analysis of Variance

  31. Multiple Comparisons • Significant F only shows that not all groups are equal • We want to know what groups are different. • Such procedures are designed to control familywise error rate. • Familywise error rate defined • Contrast with per comparison error rate Chapter 13 One-way Analysis of Variance

  32. More on Error Rates • Most tests reduce significance level (a) for each t test. • The more tests we run the more likely we are to make Type I error. • Good reason to hold down number of tests Chapter 13 One-way Analysis of Variance

  33. Fisher’s LSD Procedure • Requires significant overall F, or no tests • Run standard t tests between pairs of groups. • Often we replace s 2j or pooled estimate with MSerror from overall analysis • It is really just a pooled error term, but with more degrees of freedom--pooled across all treatment groups. Chapter 13 One-way Analysis of Variance

  34. Bonferroni t Test • Run t tests between pairs of groups, as usual • Hold down number of t tests • Reject if t exceeds critical value in Bonferroni table • Works by using a more strict value of a for each comparison Chapter 13 One-way Analysis of Variance Cont.

  35. Bonferroni t--cont. • Critical value of a for each test set at .05/c, where c = number of tests run • Assuming familywise a = .05 • e. g. with 3 tests, each t must be significant at .05/3 = .0167 level. • With computer printout, just make sure calculated probability < .05/c • Necessary table is in the book Chapter 13 One-way Analysis of Variance

  36. Assumptions • Assume: • Observations normally distributed within each population • Population variances are equal • Homogeneity of variance or homoscedasticity • Observations are independent Chapter 13 One-way Analysis of Variance Cont.

  37. Assumptions--cont. • Analysis of variance is generally robust to first two • A robust test is one that is not greatly affected by violations of assumptions. Chapter 13 One-way Analysis of Variance

  38. Magnitude of Effect • Eta squared (h2) • Easy to calculate • Somewhat biased on the high side • Formula • See slide #33 • Percent of variation in the data that can be attributed to treatment differences Chapter 13 One-way Analysis of Variance Cont.

  39. Magnitude of Effect--cont. • Omega squared (w2) • Much less biased than h2 • Not as intuitive • We adjust both numerator and denominator with MSerror • Formula on next slide Chapter 13 One-way Analysis of Variance

  40. h2 and w2 for Foa, et al. • h2 = .18: 18% of variability in symptoms can be accounted for by treatment • w2 = .12: This is a less biased estimate, and note that it is 33% smaller. Chapter 13 One-way Analysis of Variance

  41. Review Questions • Why is it called the analysis of “variance” and not the analysis of “means?” • What do we compare to create our test? • Why would large values of F lead to rejection of H0? • What do we do differently with unequal n ’s? Chapter 13 One-way Analysis of Variance Cont.

  42. Review Questions--cont. • What is the per comparison error rate? • Why would the familywise error rate generally be larger than .05 unless it is controlled? • Most instructors hate Fisher’s LSD. Can you guess why? • Why should they not hate it? Chapter 13 One-way Analysis of Variance Cont.

  43. Review Questions--cont. • How does the Bonferroni test work? • What assumptions does the analysis of variance require? • What does “robust test” mean? • What do we mean by “magnitude of effect?” • What do you know if h2 is .60? Chapter 13 One-way Analysis of Variance

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