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Psychology 10

Psychology 10. Analysis of Psychological Data April 16, 2014. The plan for today. Another example of ANOVA Asking the really interesting question: Which means are different? Tukey’s HSD. Effect size for ANOVA. Another ANOVA example (made up data).

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Psychology 10

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  1. Psychology 10 Analysis of Psychological Data April 16, 2014

  2. The plan for today • Another example of ANOVA • Asking the really interesting question: Which means are different? • Tukey’s HSD. • Effect size for ANOVA.

  3. Another ANOVA example(made up data) • A researcher was interested in comparing four different treatments for depression. • Dependent variable: the Beck Depression Inventory (higher scores mean more depression). • Treatments: Placebo, Cognitive Restructuring, Assertiveness Training, and Exercise/nutrition counseling.

  4. Depression example (cont.) • 10 observations in each group, so n = 10 for each group, and N = 40. • The grand total, G, was 713.

  5. Depression example (cont.) • For the placebo group, T was 228 and SX2 was 5370. • For the cognitive group, T was 116 and SX2 was 1474. • For the assertiveness group, T was 160 and SX2 was 2690. • For the exercise group, T was 209 and SX2 was 4513.

  6. Depression example (cont.) • Let’s keep track of that information in an ANOVA table.

  7. Depression example (cont.) Source SS df MS F ----------------------------------------------------------- Between 762.875 Within ----------------------------------- Total

  8. Depression example (cont.) • Now let’s think about the between-groups df. • We have 4 groups, so df = 4 – 1 = 3. • We can add that to our table and compute the mean square.

  9. Depression example (cont.) Source SS df MS F ----------------------------------------------------------- Between 762.875 3 254.292 Within ----------------------------------- Total

  10. Depression example (cont.) • Next, we’ll consider the SS within each group. • Placebo: 5370 – 2282/10 = 171.6. • Cognitive: 1474 – 1162/10 = 128.4. • Assertiveness: 2690 – 1602/10 = 130.0. • Exercise: 4513 – 2092/10 = 144.9. • SSW = 171.6 + 128.4 + 130.0 + 144.9 = 574.9.

  11. Depression example (cont.) Source SS df MS F ----------------------------------------------------------- Between 762.875 3 254.292 Within 574.90 ----------------------------------- Total

  12. Depression example (cont.) • What is the df within groups? We have N = 40, and we have 4 groups. 40 – 4 = 36. • Another way to think about it: We have 4 groups, each with 9 df. 4 × 9 = 36. • So we can add the df and MS to our ANOVA table.

  13. Depression example (cont.) Source SS df MS F ----------------------------------------------------------- Between 762.875 3 254.292 Within 574.900 36 15.969 ----------------------------------- Total

  14. Depression example (cont.) • At this point, we could go ahead and calculate the F statistic. • However, it’s a good idea to check our work first. • The overall SX2 = 14047. • SSTotal = 14047 – 7132 / 40 = 1337.775. • Check: 762.875 + 574.900 = 1337.775. • Let’s finish the table.

  15. Depression example (cont.) Source SS df MS F ----------------------------------------------------------- Between 762.875 3 254.292 15.92 Within 574.900 36 15.969 ----------------------------------- Total 1337.775 39

  16. Depression example (cont.) • We obtained an F with 3 and 36 df that had the value 15.92. • What decision do we make? • From the table, the critical value of F for an alpha level of .05 is 2.86. • We reject the null hypothesis and conclude that there is evidence that the population BDI means differ by treatment.

  17. Assumptions • Independence between groups. • Independence within groups. • Equal variances in all of the populations. • Normality in each population.

  18. Checking the assumptions • The groups were created by random assignment, so we know we have independence between groups. • We know little about how the data were generated, so we are not in a position to evaluate independence within groups.

  19. Assumptions (cont.) • The sums of squares within the four groups were 171.6, 128.4, 130, and 144.9. • These would result in estimated standard deviations of 4.37, 3.78, 3.80, and 4.01. • Those are very similar estimates, so it is plausible that each population has the same variance.

  20. Assumptions (cont.) • Normality for each population could be assessed by looking graphically at each group. • However, it is very difficult to evaluate normality with a sample as small as 10.

  21. Addressing the real question • So far, all we know is that there are differences somewhere among the means. • We want to know specifically which means differ. • Two ways to approach this: • A priori procedures; • Post hoc procedures.

  22. A priori procedures • A priori procedures allow us to test specific hypotheses that were defined before we ever observed data. • They tend to be more powerful than post hoc procedures, and thus should be used when possible. • Unfortunately, they are more complicated and your book doesn’t cover them, so we won’t either.

  23. Post hoc procedures • Post hoc procedures allow us to investigate which means are different from which after we have rejected the omnibus null hypothesis. • We will consider Tukey’s HSD.

  24. Tukey’s HSD • The idea behind the HSD is that if we sort our means into ascending sequence and start by testing the largest difference, then the next largest, etc., we will eventually know that none of the rest of the differences will be significant. • The Studentized range distribution (q) is an adjusted form of the t distribution that accounts for that issue.

  25. Tukey’s HSD (cont.) • Procedure: • Look up q in the table of the Studentized range statistic (Table B.5 in the book). • Means that are separated by at least one HSD are significantly different.

  26. Tukey’s HSD, Depression example • In the depression example, we had 4 groups and 36 df within groups. • From Table B.5, q for a .05 alpha level is 3.85 (using 30 df). • The MSWwas 15.969 and n was 10. • HSD = 3.85 √(15.969 / 10) = 4.87. • Any means that differ by as much as 4.87 are significantly different.

  27. Depression example (cont.) • Our totals were 228, 116, 160, and 209, so the means are 22.8 for the placebo group, 11.6 for the cognitive group, 16.0 for the assertiveness group, and 20.9 for the exercise group.

  28. Depression example, (cont.) • 22.8 – 11.6 = 11.2 ( > HSD), so placebo and cognitive are significantly different. • 22.8 – 16.0 = 6.8 ( > HSD), so placebo and assertiveness are significantly different. • 22.8 – 20.9 = 1.9 ( < HSD), so placebo and exercise are not significantly different.

  29. Effect size • Sometimes it is desirable to have a measure of the global effect in an ANOVA. • One (somewhat flawed) such measure is “eta squared” (h2). • In the depression example, h2 = 762.875 / 1337.775 = .57.

  30. Example

  31. Useful formulas

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