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Two-Way Mixed Design ANOVA

Two-Way Mixed Design ANOVA. Two-Way Mixed Design ANOVA. Mixed designs involve 2 or more factors, at least one of which is within-subjects, and at least one of which is between-subjects. 3. Men. 2.5. Women. 2. Number of drinks. 1.5. 1. 0.5. 0. Mon. Tue. Wed. Thu. Fri. Sat. Sun.

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Two-Way Mixed Design ANOVA

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  1. Two-Way Mixed Design ANOVA

  2. Two-Way Mixed Design ANOVA • Mixed designs involve 2 or more factors, at least one of which is within-subjects, and at least one of which is between-subjects. PSYC 6130A, PROF. J. ELDER

  3. 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 • Source: 2004 Canadian Addiction Survey PSYC 6130A, PROF. J. ELDER

  4. 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

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

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

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

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

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

  10. Drinking Habits of Men and Women: ANOVA Summary Table PSYC 6130A, PROF. J. ELDER

  11. Step 1. State the Hypothesis • Null hypothesis has 3 parts, e.g., • Mean number of drinks is the same for men and women • Mean number of drinks is the same for each day of the week • No interactions PSYC 6130A, PROF. J. ELDER

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

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

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

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

  16. Step 5. Calculate the Test Statistics (Between Subjects) (Within Subjects) PSYC 6130A, PROF. J. ELDER

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

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

  19. Assumptions • Independent random sampling • Multivariate normal distribution • Homogeneity of variance (not a huge concern if design is balanced). • Sphericity • Homogeneity of covariance across groups (new) PSYC 6130A, PROF. J. ELDER

  20. A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 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A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A Homogeneity of Covariance Across Groups Male Female A 40.00 A How many drinks on Saturday? = 1.09 + 0.21 * fridrk How many drinks on Saturday? = 1.63 + 0.40 * fridrk R-Square = 0.04 R-Square = 0.16 30.00 A A How many drinks on Saturday? 20.00 A A A A A A A A A A A A A 10.00 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 0.00 0.00 10.00 20.00 30.00 40.00 0.00 10.00 20.00 30.00 40.00 How many drinks on Friday? How many drinks on Friday? PSYC 6130A, PROF. J. ELDER

  21. Testing Homogeneity of Covariance Across Groups in SPSS Box's Test of Equality of Covariance Matrices Box's M 5486.778 F 195.667 df1 28 df2 9E+007 Sig. .000 Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. a. Design: Intercept+gend Within Subjects Design: day PSYC 6130A, PROF. J. ELDER

  22. Homogeneity of Covariance Across Groups • Since the covariance is not homogeneous across groups (men and women), we should instead conduct 1-way ANOVAs separately for men and women. PSYC 6130A, PROF. J. ELDER

  23. Separate 1-Way Repeated Measures ANOVAs PSYC 6130A, PROF. J. ELDER

  24. 95% Confidence Interval for Mean a Difference Difference 95% Confidence Interval for Mean a a (I) day (J) day (I-J) Std. Error Sig. Lower Bound Upper Bound Difference Difference a 1 2 -1.387 * .072 .000 -1.606 -1.168 (I) day (J) day (I-J) Std. Error Sig. Lower Bound Upper Bound 1 2 -.757 * .042 .000 -.885 -.630 3 -.854 * .072 .000 -1.074 -.634 3 -.428 * .040 .000 -.550 -.305 4 .143 .055 .186 -.023 .309 4 .117 * .031 .003 .023 .211 5 .266 * .048 .000 .121 .412 5 .180 * .026 .000 .101 .258 6 .306 * .047 .000 .162 .450 6 .186 * .026 .000 .106 .266 7 .366 * .049 .000 .217 .515 7 .233 * .026 .000 .154 .312 2 1 1.387 * .072 .000 1.168 1.606 2 1 .757 * .042 .000 .630 .885 3 .533 * .079 .000 .294 .772 3 .330 * .048 .000 .185 .474 4 1.530 * .077 .000 1.295 1.765 4 .874 * .042 .000 .746 1.002 5 1.653 * .075 .000 1.426 1.881 5 .937 * .041 .000 .813 1.060 6 1.693 * .075 .000 1.465 1.922 6 .943 * .040 .000 .821 1.066 7 7 1.753 * .077 .000 1.520 1.986 .990 * .040 .000 .868 1.112 3 1 3 1 .428 * .040 .000 .305 .550 .854 * .072 .000 .634 1.074 2 -.330 * .048 .000 -.474 -.185 2 -.533 * .079 .000 -.772 -.294 4 .544 * .041 .000 .421 .668 4 .997 * .075 .000 .770 1.225 5 .607 * .039 .000 .489 .725 5 1.120 * .073 .000 .898 1.343 6 .614 * .037 .000 .501 .726 6 1.160 * .073 .000 .939 1.382 7 .660 * .038 .000 .545 .776 7 1.220 * .072 .000 1.002 1.438 4 1 -.117 * .031 .003 -.211 -.023 4 1 -.143 .055 .186 -.309 .023 2 -.874 * .042 .000 -1.002 -.746 2 -1.530 * .077 .000 -1.765 -1.295 3 -.544 * .041 .000 -.668 -.421 3 -.997 * .075 .000 -1.225 -.770 5 .063 .027 .386 -.018 .144 5 .123 .046 .162 -.017 .264 6 .069 .028 .308 -.017 .155 6 .163 * .044 .005 .029 .298 7 .116 * .026 .000 .036 .196 7 5 1 .223 * .047 .000 .080 .365 -.180 * .026 .000 -.258 -.101 2 -.937 * .041 .000 -1.060 -.813 5 1 -.266 * .048 .000 -.412 -.121 3 -.607 * .039 .000 -.725 -.489 2 -1.653 * .075 .000 -1.881 -1.426 4 -.063 .027 .386 -.144 .018 3 -1.120 * .073 .000 -1.343 -.898 6 .006 .025 1.000 -.068 .081 4 -.123 .046 .162 -.264 .017 7 .053 .025 .665 -.022 .128 6 .040 .036 1.000 -.070 .150 6 1 -.186 * .026 .000 -.266 -.106 7 .100 .039 .218 -.019 .218 2 -.943 * .040 .000 -1.066 -.821 6 1 -.306 * .047 .000 -.450 -.162 3 -.614 * .037 .000 -.726 -.501 2 -1.693 * .075 .000 -1.922 -1.465 4 -.069 .028 .308 -.155 .017 3 -1.160 * .073 .000 -1.382 -.939 5 -.006 .025 1.000 -.081 .068 4 -.163 * .044 .005 -.298 -.029 7 .047 .025 1.000 -.028 .122 5 -.040 .036 1.000 -.150 .070 7 1 -.233 * .026 .000 -.312 -.154 2 7 -.990 * .040 .000 -1.112 -.868 .060 .035 1.000 -.048 .167 3 -.660 * .038 .000 -.776 -.545 7 1 -.366 * .049 .000 -.515 -.217 4 -.116 * .026 .000 -.196 -.036 2 -1.753 * .077 .000 -1.986 -1.520 5 -.053 .025 .665 -.128 .022 3 -1.220 * .072 .000 -1.438 -1.002 6 -.047 .025 1.000 -.122 .028 4 -.223 * .047 .000 -.365 -.080 5 -.100 .039 .218 -.218 .019 6 -.060 .035 1.000 -.167 .048 Repeated Measures Post-Hoc Tests (Bonferroni) Men Women PSYC 6130A, PROF. J. ELDER

  25. End of Lecture 18

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