1 / 33

(One-Way) Repeated Measures ANOVA

(One-Way) Repeated Measures ANOVA. One-Way Repeated Measures ANOVA. Generalization of repeated-measures t-test to independent variable with more than 2 levels. Each subject has a score for each level of the independent variable. May be used for repeated or matched designs. 500 ms. 500 ms.

anastasia-hero
Download Presentation

(One-Way) Repeated Measures ANOVA

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. (One-Way) Repeated Measures ANOVA

  2. One-Way Repeated Measures ANOVA • Generalization of repeated-measures t-test to independent variable with more than 2 levels. • Each subject has a score for each level of the independent variable. • May be used for repeated or matched designs. PSYC 6130A, PROF. J. ELDER

  3. 500 ms 500 ms 200 ms Until Response Example: Visual Grating Detection in Noise PSYC 6130A, PROF. J. ELDER

  4. 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 Repeated Measures ANOVA Example: Grating Detection Spatial frequency = 0.5 c/deg Signal-to-noise ratio (SNR) at threshold PSYC 6130A, PROF. J. ELDER

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

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

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

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

  9. 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 Repeated Measures ANOVA Example: Grating Detection Spatial frequency = 0.5 c/deg Signal-to-noise ratio (SNR) at threshold PSYC 6130A, PROF. J. ELDER

  10. Step 1. State the Hypothesis • Same as for 1-way independent ANOVA: PSYC 6130A, PROF. J. ELDER

  11. Step 2. Select Statistical Test and Significance Level • As usual PSYC 6130A, PROF. J. ELDER

  12. 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 Step 3. Select Samples and Collect Data • Ideally, randomly sample • More probably, random assignment PSYC 6130A, PROF. J. ELDER

  13. Step 4. Find Region of Rejection PSYC 6130A, PROF. J. ELDER

  14. Step 5. Calculate the Test Statistic PSYC 6130A, PROF. J. ELDER

  15. Step 5. Calculate the Test Statistic PSYC 6130A, PROF. J. ELDER

  16. Step 6. Make the Statistical Decisions PSYC 6130A, PROF. J. ELDER

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

  18. 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

  19. Homogeneity of Variance • Homogeneity of Variance is the property that the variance in the dependent variable is the same at each level of the independent variable. • In the context of RM ANOVA, this means that the variance between subjects is the same at each level of the independent variable. • Since RM ANOVA designs are balanced by default, homogeneity of variance is not a critical issue. PSYC 6130A, PROF. J. ELDER

  20. Homogeneity of Variance PSYC 6130A, PROF. J. ELDER

  21. 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

  22. 4 4 4 3 3 3 2 2 2 1 1 1 i3 i3 i2 0 X X X 0 0 -1 -1 -1 -2 -2 -2 -3 -4 -3 -3 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 X X X i1 i1 i2 Sphericity Does Not Hold PSYC 6130A, PROF. J. ELDER

  23. 4 4 4 3 3 3 2 2 2 1 1 1 i3 i3 i2 0 0 0 X X X -1 -1 -1 -2 -2 -2 -3 -3 -3 -4 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 X X X i1 i1 i2 Sphericity Does Hold PSYC 6130A, PROF. J. ELDER

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. End of Lecture 17

  30. Multivariate Approach to Repeated Measures • Based on forming difference scores for each pair of levels of the independent variable. • e.g., for our 3-level example, there are 3 pairs • Each pair of difference scores is treated as a different dependent variable in a MANOVA. • Sphericity does not need to be assumed. • When all assumptions of the repeated measures ANOVA are met, ANOVA is usually more powerful than MANOVA (especially for small samples). • Thus multivariate approach should be considered only if there is doubt about sphericity assumption. • When sphericity does not apply, MANOVA can be much more powerful for large samples. PSYC 6130A, PROF. J. ELDER

  31. 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

  32. Reporting the Result • One-way repeated measures ANOVA reveals a significant effect of noise contrast on the signal-to-noise ratio at threshold (F[2,4]=14.4, p=.044, e =0.618). Post-hoc pairwise Bonferroni-corrected comparisons reveal that signal-to-noise ratio at threshold was higher at 4.8% noise contrast than at 14.3% noise contrast (p=.037). No other significant pairwise differences were found (p>.05). PSYC 6130A, PROF. J. ELDER

  33. Varieties of Repeated-Measures and Randomized-Blocks Designs • Simultaneous RM Design • e.g., subject rates different aspects of stimulus on comparable rating scale. • Successive RM Design • Here counterbalancing becomes important • RM Over Time • Track a dependent variable over time (e.g., learning effects) • Not likely to satisfy sphericity (scores taken closer in time will have higher covariance). • RM with Quantitative Levels • Think about regression first. • Randomized Blocks • Matched design useful if you cannot avoid serious carryover effects. • Natural blocks • Blocks of subjects are naturally occuring (e.g., children in same family) PSYC 6130A, PROF. J. ELDER

More Related