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Chapter 13 Analysis of Variance (ANOVA). PSY295-001 Spring 2003. Major Points. An example Requirements The logic Calculations Unequal sample sizes. Cont. Major Points--cont. Multiple comparisons Fisher’s LSD Bonferroni t Assumptions of analysis of variance Magnitude of effect
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Chapter 13 Analysis of Variance (ANOVA) PSY295-001 Spring 2003
Major Points • An example • Requirements • The logic • Calculations • Unequal sample sizes Chapter 13 One-way Analysis of Variance Cont.
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
Example • Daily caffeine intake (DV) • Year in school (IV) • four groups • 6 pairs of comparison Chapter 13 One-way Analysis of Variance
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
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
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
The data Chapter 13 One-way Analysis of Variance
The steps • Measure total (or overall) variability • Two components • between treatments variability • within treatments variability Chapter 13 One-way Analysis of Variance
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
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
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.
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
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
Grand mean = 3.78 Chapter 13 One-way Analysis of Variance
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.
Calculations--cont. Chapter 13 One-way Analysis of Variance
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
Summary Table Chapter 13 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. Chapter 13 One-way Analysis of Variance Cont.
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
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
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.
Post-Traumatic Stress Disorder--cont. • Supportive counseling (SC) • Standard counseling • Waiting List Control (WL) • No treatment Chapter 13 One-way Analysis of Variance
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
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
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.
Calculations--cont. Chapter 13 One-way Analysis of Variance
Summary Table F.05(3,41) = 2.84 Chapter 13 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 13 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 13 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 Chapter 13 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 13 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 Chapter 13 One-way Analysis of Variance Cont.
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
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.
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
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.
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
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
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.
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.
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