One-way Analysis of Variance Chapter 16 One-way Analysis of Variance Fundamental Statistics for the Behavioral Sciences, 4th edition David C. Howell ©1999 Brooks/Cole Publishing Company/ITP   

Chapter 16 One-way Analysis of Variance Major Points An example The logic Calculations Unequal sample sizes Multiple comparisons Fisher’s LSD Bonferroni t Cont.

Chapter 16 One-way Analysis of Variance Major Points--cont. Assumptions of analysis of variance Magnitude of effect eta squared omega squared Review questions

Logic of the Analysis of Variance Chapter 16 One-way Analysis of Variance Logic of the Analysis of Variance Null hypothesis: H0 Population means equal m1 = m2 = m3 = m4 Alternative hypothesis: H1 Not all population means equal. Cont.

Chapter 16 One-way Analysis of Variance Logic--cont. Create a measure of variability among group means MSgroups Create a measure of variability within groups MSerror Cont.

Chapter 16 One-way Analysis of Variance Logic--cont. Form ratio of MSgroups /MSerror 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

Epinephrine and Memory Chapter 16 One-way Analysis of Variance 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 Introini-Collison, I.B., & Mcgaugh, J.L. (1986) Epinephrine modulates long-term retention of an aversively motivated discrimination. Behavioral and Neural Biology, 45, 358-365.

Chapter 16 One-way Analysis of Variance Grand mean = 3.78

Chapter 16 One-way Analysis of Variance Calculations Start with Sum of Squares (SS) We need: SStotal SSgroups SSerror Compute degrees of freedom (df ) Compute mean squares and F Cont.

Chapter 16 One-way Analysis of Variance Calculations--cont.

Degrees of Freedom (df ) Chapter 16 One-way Analysis of Variance Degrees of Freedom (df ) Number of “observations” free to vary dftotal = N - 1 Variability of N observations dfgroups = g - 1 variability of g means dferror = g (n - 1) n observations in each group = n - 1 df times g groups

Chapter 16 One-way Analysis of Variance Summary Table

Chapter 16 One-way Analysis of Variance 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. Cont.

Chapter 16 One-way Analysis of Variance Conclusions--cont. The injection of epinephrine following learning appears to consolidate that learning. High doses may have negative effect.

Chapter 16 One-way Analysis of Variance 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) Foa, E.B., Rothbaum, B.O., Riggs, D.S., & Murdock, T.B. (1991). Treatment of posttraumatic stress disorder in rape victims: A comparison between cognitive-behavioral procedures and counseling. Journal of Consulting and Clinical Psychology, 59, 715-723.

Post-Traumatic Stress Disorder Chapter 16 One-way Analysis of Variance 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 Cont.

Post-Traumatic Stress Disorder--cont. Chapter 16 One-way Analysis of Variance Post-Traumatic Stress Disorder--cont. Supportive counseling (SC) Standard counseling Waiting List Control (WL) No treatment

Chapter 16 One-way Analysis of Variance SIT = Stress Inoculation Therapy PE = Prolonged Exposure SC = Supportive Counseling WL = Waiting List Control Grand mean = 15.622

Tentative Conclusions Chapter 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance Calculations Almost the same as earlier Note differences We multiply by nj as we go along. MSerror is now a weighted average. Cont.

Chapter 16 One-way Analysis of Variance Calculations--cont.

Chapter 16 One-way Analysis of Variance Summary Table F.05(3,41) = 2.84

Chapter 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance 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

Fisher’s LSD Procedure Chapter 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance 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 Cont.

Chapter 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance Assumptions Assume: Observations normally distributed within each population Population variances are equal Homogeneity of variance or homoscedasticity Observations are independent Cont.

Chapter 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance 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 Cont.

Magnitude of Effect--cont. Chapter 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance 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 16 One-way Analysis of Variance 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? Cont.

Review Questions--cont. Chapter 16 One-way Analysis of Variance 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? Cont.

Review Questions--cont. Chapter 16 One-way Analysis of Variance 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?