1. PSYC 3030 Review Session April 19, 2004

2. Housekeeping Exam: –April 26, 2004 (Monday) –RN 203 –Use pencil, bring calculator & eraser –Make use of your cheat sheet After the exam: –Blueberry Hill –Have a drink!

3. Outline 2-way ANOVA: theories and interpretations 3-way ANOVA: Interactions in graphs ANCOVA Repeated measures ANOVA

4. 2-way ANOVA: Data Group/ Agegrp 50607080 Mono3117349736714006 Biling2448309434863717

5. 2-way ANOVA: Computations When doing tests, use MS, not SS. Computations: –SS A = [A] – [Y] –SS B = [B] – [Y] –SS AB = [AB] – [A] – [B] + [Y] –SS Error = [ABS] – [AB] –SS Total = [ABS] – [Y]

6. 2-way ANOVA: Unequal N’s Type I SS additive, but not used in test and generally ignored with unequal N’s Type III SS not additive, but used in tests (e.g., when you test for interaction) Additive means whether the SS for each factor adds to the Model SS. In this case, Type I SS will add up to equal the model SS, but not Type III SS.

7. 2-way ANOVA: SAS output Sum of Source DF Squares Mean Square F Value Pr > F Model 7 326927.6000 46703.9429 75.28 <.0001 Error 32 19852.0000 620.3750 Corrected Total 39 346779.6000 R-Square Coeff Var Root MSE rt Mean 0.942753 3.685061 24.90733 675.9000 Source DF Type III SS Mean Square F Value Pr > F group 1 59752.9000 59752.9000 96.32 <.0001 agegrp 3 254156.0000 84718.6667 136.56 <.0001 group*agegrp 3 13018.7000 4339.5667 7.00 0.0009 You can do a max of 3 contrasts here.

8. 2-way ANOVA: Test State your hypotheses Find the F-obs Find the F-crit (remember to put dfs) Decision rule Comparison Statistical conclusion Research conclusion

9. Contrast DF Contrast SS Mean Square F Value Pr > F c1 1 9220.820000 9220.820000 14.86 0.0005 c2 1 3422.500000 3422.500000 5.52 0.0252 Contrast for Group*Agegrp

10. 2-way ANOVA: Regression Setting up the model: Y ij = μ · + τ 1 X ij1 + ….+ τ r-1 X ij(r-1) + ε ijk Y ij = X β + ε ijk

11. 2-way ANOVA: Regression NWK p. 834  full model Y ijk = μ.. + effect that you are interested in + ε ijk  reduced model Determine the composition of SS in ANOVA in regards to SS in Regression e.g., SS agegrp = SS agegrp-lin + SS agegrp-quad + SS agegrp-cubic

12. 2-way ANOVA: Regression Find SS for full and reduced models Make use of Type III SS in the ANOVA SAS output  SS in the reduced model SS in regression could be combined to become SS in ANOVA

13. 2-way ANOVA: Tests Effects Lack of fit: –SSE in ANOVA = SSPE –SSE in Reg = SSPE + SSLF –  SSLF = SSE(Reg) – SSE(ANOVA) –In regression models, SSLF = SSE – SSPE –SSE can be found in the full model, SSPE is the error terms that are beyond the degree that you are testing. –E.g., if you are testing the linear term and a df = 3 for a factor, the quadratic and the cubic terms will be the error terms

14. 2-way ANOVA: contrast Number of levels in the other factor Sample size in each cell

15. 2-way ANOVA: 1-way ANOVA How are the SS’s relating to each other? In 1-way ANOVA, the SS may or may not include SS from other factors. Hint: Use df to determine the composition of SS in 1-way and 2-way ANOVAs.

16. 3-way ANOVA: Mixed or Random Error terms A, B fixed A fixed B random A, B random AMSEMSAB BMSE MSAB ABMSE

17. 3-way ANOVA: graphs Examine graphs to look for significant effects Understand what information you can get from each plot When plots are comparing side-by- side, what is the product of overlaying one on the other?

18. C = 1

19. C = 2

20. Compare c = 1, 2 side by side

21. Average c = 1, 2

22. ANCOVA: Assumptions Random assignment to treatment Same regression slopes Covariate & treatment independent Covariate values fixed Linearity Normality Homogeneity of variance

23. ANCOVA: Data The MEANS Procedure N LANGUAGE Obs Variable Mean Std Dev ------------------------------------------------------------------------------------------------------- 1 40 TOTENON 12.18 4.65 eppvtstd 112.60 15.96 2 29 TOTENON 12.00 6.39 eppvtstd 93.97 16.83 -------------------------------------------------------------------------------------------------------

24. ANCOVA: before adjustment Sum of Source DF Squares Mean Square F Value Pr > F Model 1 0.514855 0.514855 0.02 0.8955 Error 67 1985.775000 29.638433 Corrected Total 68 1986.289855 R-Square Coeff Var Root MSE TOTENON Mean 0.000259 44.98733 5.444119 12.10145 Source DF Type III SS Mean Square F Value Pr > F LANGUAGE 1 0.51485507 0.51485507 0.02 0.8955

25. ANCOVA: example

26. ANCOVA: after adjustment Sum of Source DF Squares Mean Square F Value Pr > F Model 2 526.078865 263.039433 11.89 <.0001 Error 66 1460.210990 22.124409 Corrected Total 68 1986.289855 R-Square Coeff Var Root MSE TOTENON Mean 0.264855 38.86856 4.703659 12.10145 Source DF Type I SS Mean Square F Value Pr > F LANGUAGE 1 0.5148551 0.5148551 0.02 0.8792 eppvtstd 1 525.5640103 525.5640103 23.75 <.0001 Source DF Type III SS Mean Square F Value Pr > F LANGUAGE 1 115.6774967 115.6774967 5.23 0.0254 eppvtstd 1 525.5640103 525.5640103 23.75 <.0001

27. ANCOVA: after adjustment Standard Parameter Estimate Error t Value Pr > |t| Intercept -4.118837416 B 3.42056945 -1.20 0.2328 LANGUAGE 1 -3.021557703 B 1.32142430 -2.29 0.0254 LANGUAGE 2 0.000000000 B... eppvtstd 0.171539921 0.03519559 4.87 <.0001 NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. Least Squares Means TOTENON Standard LANGUAGE LSMEAN Error Pr > |t| 1 10.8315192 0.7931531 <.0001 2 13.8530769 0.9526099 <.0001

28. ANCOVA: Find the adjusted means

29. ANCOVA: Regression & more Set up the regression model Test parallel slopes in ANOVA and regression Compare 1-way ANOVA and 1-way ANCOVA results  Where did the error go? What’s the advantage of running ANCOVA vs. ANOVA?

30. Repeated measure: Statistical Assumptions Different error terms for B/W subj and W/in subj factor(s) Compound symmetry  homogeneity of variance If violated: p-values biased downwards (actual α > nominal α) Solution: Geiser-Greenhouse, Huyhn- Feldt estimation methods

31. Repeated measure: Designs Objective: Control for individual differences Carry-over effect might override actual treatment effect  counterbalance order of treatment Sample designs: Completely randomized btw Ss design, completely w/in Ss design, Mixed design.

32. Repeated measure: Data Are all the nonwords the same? The four group literacy study: –B/w subj. effect: GROUP –W/in subj. effect: TYPE of nonwords

33. Repeated measure: Errors Total variation Between Subjects Within Subjects GROUP Ss w/in groups TYPE TYPE x GROUP TYPE x Ss w/in groups B/w Ss error term W/in Ss error term

34. Profile plot …

35. Repeated measure: B/w subj. The GLM Procedure Repeated Measures Analysis of Variance Tests of Hypotheses for Between Subjects Effects Source DF Type III SS Mean Square F Value Pr > F LANGUAGE 3 212.224621 70.741540 6.50 0.0004 Error 128 1392.305682 10.877388 B/w Ss error term

36. Repeated measure: W/in subj The GLM Procedure Repeated Measures Analysis of Variance Univariate Tests of Hypotheses for Within Subject Effects Source DF Type III SS Mean Square F Value Pr > F type 1 75.5824517 75.5824517 40.08 <.0001 type*LANGUAGE 3 0.2314133 0.0771378 0.04 0.9889 Error(type) 128 241.3897989 1.8858578 W/in Ss error term