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Final Review

Final Review. Main Topics for Final Exam. Ch. 12 One-Way Independent ANOVA Ch. 13 Multiple Comparisons Ch. 14 Two-Way ANOVA Ch. 15 Repeated-Measures ANOVA Ch. 16 Two-Way Mixed Design ANOVA. Ch. 12 One-Way Independent ANOVA. Example. Sums of Squares Approach. Summary Table (SPSS).

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Final Review

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  1. Final Review

  2. Main Topics for Final Exam • Ch. 12 One-Way Independent ANOVA • Ch. 13 Multiple Comparisons • Ch. 14 Two-Way ANOVA • Ch. 15 Repeated-Measures ANOVA • Ch. 16 Two-Way Mixed Design ANOVA PSYC 6130A, PROF. J. ELDER

  3. Ch. 12 One-Way Independent ANOVA

  4. Example PSYC 6130A, PROF. J. ELDER

  5. Sums of Squares Approach PSYC 6130A, PROF. J. ELDER

  6. Summary Table (SPSS) PSYC 6130A, PROF. J. ELDER

  7. Interpreting the F Ratio PSYC 6130A, PROF. J. ELDER

  8. Effect Size and Proportion of Variance Accounted For PSYC 6130A, PROF. J. ELDER

  9. (Approxiately) Unbiased Effect Size PSYC 6130A, PROF. J. ELDER

  10. Assumptions • Independent random sampling • Normal distributions • Homogeneity of variance PSYC 6130A, PROF. J. ELDER

  11. Homogeneity of Variance: Levene’s Test • SPSS reports an F-statistic for Levene’s test • Allows the homogeneity of variance for two or more variables to be tested. PSYC 6130A, PROF. J. ELDER

  12. Ch. 13: Multiple Comparisons

  13. Final Recommendations for Post-Hoc Comparisons • If comparisons are strictly simple: • For k=3 (3 treatment groups): • For k>3 • If sample sizes are equal or nearly equal and variances appear to be homogeneous • Use Dunnett’s test when you are comparing multiple treatment group means with a control group mean. • Use Tukey’s HSD when you have to do your calculations by hand. • Otherwise use REGWQ. • If sample sizes are very different or variances appear unequal • Bonferroni (preferably sequential) with separate variances and • One of the methods offered by SPSS (e.g., Games-Howell SPSS) • If comparisons include complex comparisons: • Use Scheffe’s test PSYC 6130A, PROF. J. ELDER

  14. Recommendations for Planned Comparisons • If k=3, comparisons are strictly simple: • Use Fisher’s LSD • If sample sizes are very different or variances appear unequal, use separate variances tests • Otherwise • Use Bonferroni (preferably sequential) PSYC 6130A, PROF. J. ELDER

  15. Ch. 14: Two-Way (Independent) ANOVA

  16. Balanced Design Factor B Factor A PSYC 6130A, PROF. J. ELDER

  17. Types of Effects PSYC 6130A, PROF. J. ELDER

  18. Degrees of Freedom Tree PSYC 6130A, PROF. J. ELDER

  19. Step 5. Calculate the Test Statistics PSYC 6130A, PROF. J. ELDER

  20. Step 5. Calculate the Test Statistics PSYC 6130A, PROF. J. ELDER

  21. SPSS Output PSYC 6130A, PROF. J. ELDER

  22. Assumptions of Two-Way Independent ANOVA • Same as for One-Way • If balanced, don’t have to worry about homogeneity of variance. PSYC 6130A, PROF. J. ELDER

  23. Simple Effects • A main effect is an effect of one factor measured by collapsing (pooling) over all other factors. • A simple effect is an effect of one factor measured by fixing all other factors. • Although we found significant main effects, given the significant interaction, these main effects do not necessarily imply similarly significant simple effects. PSYC 6130A, PROF. J. ELDER

  24. Test of Homogeneity of Variances Signal to Noise at Threshold Levene Multiple Comparisons Statistic df1 df2 Sig. Dependent Variable: Signal to Noise at Threshold 12.229 2 57 .000 LSD Mean 95% Confidence Interval (I) Noise Contrast (J) Noise Contrast Difference (Michelson units) (Michelson units) (I-J) Std. Error Sig. Lower Bound Upper Bound .043 .148 .010120 * .004492 .028 .00112 .01912 .500 .004800 .004492 .290 -.00420 .01380 .148 .043 -.010120 * .004492 .028 -.01912 -.00112 .500 -.005320 .004492 .241 -.01432 .00368 .500 .043 -.004800 .004492 .290 -.01380 .00420 .148 .005320 .004492 .241 -.00368 .01432 *. The mean difference is significant at the .05 level. Planned or Posthoc Pairwise Comparisons • Since there are 3 levels of noise, we can consider using Fisher’s LSD. • However, since variances do not appear homogeneous, we should not use an LSD based on pooling the variance over all 3 conditions. PSYC 6130A, PROF. J. ELDER

  25. Multiple Comparisons Dependent Variable: Signal to Noise at Threshold Games-Howell Mean 95% Confidence Interval (I) Noise Contrast (J) Noise Contrast Difference (Michelson units) (Michelson units) (I-J) Std. Error Sig. Lower Bound Upper Bound .043 .148 .010120 * .003787 .030 .00085 .01939 .500 .004800 .004573 .552 -.00648 .01608 .148 .043 -.010120 * .003787 .030 -.01939 -.00085 .500 -.005320 .005028 .546 -.01762 .00698 .500 .043 -.004800 .004573 .552 -.01608 .00648 .148 .005320 .005028 .546 -.00698 .01762 *. The mean difference is significant at the .05 level. Planned or Posthoc Pairwise Comparisons • Alternative when variances appear heterogeneous: • Compute Fisher’s LSD by hand, calculating standard error separately for each test (not difficult) • One of the unequal variance post-hoc tests offered by SPSS PSYC 6130A, PROF. J. ELDER

  26. Planned or Posthoc Pairwise Comparisons • It is also possible to test differences between cell means. Note that in this design, there are 15 possible pairwise cell comparisons. • It doesn’t make that much sense to compare 2 cells that are not in the same row or column (i.e. that differ in both factors). • It is more likely that you would follow a significant simple effect test with a set of pairwise comparisons within a factor while holding the other factor constant. There are 9 such comparisons possible here. • For example, within a spatial frequency condition, what noise conditions differ significantly? • This defines a total of 6 pairwise comparisons (2 families of 3 comparisons each). PSYC 6130A, PROF. J. ELDER

  27. Interaction Comparisons • If significant interactions are found in a design that is 2x3 or larger, it may be of interest to test the significance of smaller (e.g., 2x2) interactions. • These can be tested by ignoring specific subsets of the data for each test (e.g., by using the SPSS Select Cases function). PSYC 6130A, PROF. J. ELDER

  28. Ch. 15: (One-Way) Repeated Measures ANOVA

  29. Group Noise Total .04 .15 .50 Mean Subject 1 .08480 .06830 .06540 .07283 2 .08290 .06090 .07610 .07330 3 .08880 .06440 .07120 .07480 Group Total .08550 .06453 .07090 .07364 Mean Example: Grating Detection Spatial frequency = 0.5 c/deg PSYC 6130A, PROF. J. ELDER

  30. Example Grating Detection PSYC 6130A, PROF. J. ELDER

  31. Sum of Squares Analysis PSYC 6130A, PROF. J. ELDER

  32. Degrees of Freedom Tree PSYC 6130A, PROF. J. ELDER

  33. Test Statistic PSYC 6130A, PROF. J. ELDER

  34. SPSS Output PSYC 6130A, PROF. J. ELDER

  35. Assumptions • Independent random sampling • Multivariate normal distribution • Homogeneity of variance (not a huge concern, since there is the same number of observations at each treatment level). • Sphericity (new). PSYC 6130A, PROF. J. ELDER

  36. Sphericity • Sphericity is the property that the degree of interaction (covariance) between any two different levels of the independent variable is the same. • Sphericity is critical for RM ANOVA because the error term is the average of the pairwise interactions. • Violations generally lead to inflated F statistics (and hence inflated Type I error). PSYC 6130A, PROF. J. ELDER

  37. 0.078 0.078 0.068 0.076 0.076 0.074 0.074 0.066 0.072 0.072 Noise = .50 Noise = .50 Noise = .15 0.064 0.07 0.07 0.068 0.068 0.062 0.066 0.066 0.064 0.064 0.06 0.082 0.084 0.086 0.088 0.09 0.06 0.062 0.064 0.066 0.068 0.07 0.082 0.084 0.086 0.088 0.09 Noise = .04 Noise = .14 Noise = .04 Sphericity • Does sphericity appear to hold? • Do these graphs suggest that the RM design will yield a large increase in statistical power? PSYC 6130A, PROF. J. ELDER

  38. Testing Sphericity • Mauchly (1940) test: provided automatically by SPSS • Test has low power (for small samples, likely to accept sphericity assumption when it is false). PSYC 6130A, PROF. J. ELDER

  39. Alternative: Assume the Worst! (Total Lack of Sphericity) • Conservative Geisser-Greenhouse F Test (1958) • Provides a means for calculating a correct critical F value under the assumption of a complete lack of sphericity (lower bound): PSYC 6130A, PROF. J. ELDER

  40. Estimating Sphericity • What if your F statistic falls between the 2 critical values (assuming sphericity or assuming total lack of sphericity)? • Solution: estimate sphericity, and use estimate to adjust critical value. • Two different methods for calculating e : • Greenhouse and Geisser (1959) • Huynh and Feldt (1976) – less conservative PSYC 6130A, PROF. J. ELDER

  41. Mauchly's Test of Sphericity Measure: MEASURE_1 a Epsilon Approx. Greenhous Within Subjects Effect Mauchly's W Chi-Square df Sig. e-Geisser Huynh-Feldt Lower-bound noise .383 .960 2 .619 .618 1.000 .500 Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. SPSS Output PSYC 6130A, PROF. J. ELDER

  42. Pairwise Comparisons Measure: MEASURE_1 95% Confidence Interval for Mean a Difference Difference a (I) noise (J) noise (I-J) Std. Error Sig. Lower Bound Upper Bound 1 2 .021 * .002 .037 .003 .039 3 .015 .004 .197 -.015 .045 2 1 -.021 * .002 .037 -.039 -.003 3 -.006 .005 1.000 -.046 .034 3 1 -.015 .004 .197 -.045 .015 2 .006 .005 1.000 -.034 .046 Based on estimated marginal means *. The mean difference is significant at the .05 level. a. Adjustment for multiple comparisons: Bonferroni. Post-Hoc Comparisons • If very confident about sphericity, use standard methods (e.g., Fisher’s LSD, Tukey’s HSD), with MSresid as error term. • Otherwise, use conservative approach: Bonferroni test. • Error term calculated separately for each comparison, using only the data from the two levels. • This means that sphericity need not be assumed. PSYC 6130A, PROF. J. ELDER

  43. Ch. 16 Two-Way Mixed Design ANOVA

  44. 3 Men 2.5 Women 2 Number of drinks 1.5 1 0.5 0 Mon Tue Wed Thu Fri Sat Sun Day Example: Drinking Habits of Men and Women PSYC 6130A, PROF. J. ELDER

  45. Degrees of Freedom Analysis PSYC 6130A, PROF. J. ELDER

  46. Multi-Subscript Notation PSYC 6130A, PROF. J. ELDER

  47. Notation Factor B Subjects Factor A PSYC 6130A, PROF. J. ELDER

  48. Sum of Squares Analysis PSYC 6130A, PROF. J. ELDER

  49. Forming the F Statistics (Between Subjects) (Within Subjects) PSYC 6130A, PROF. J. ELDER

  50. SPSS Output PSYC 6130A, PROF. J. ELDER

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