PSYC 3030 Review Session April 19, 2004

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!

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

2-way ANOVA: Data Group/ Agegrp Mono Biling

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]

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.

2-way ANOVA: SAS output Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE rt Mean Source DF Type III SS Mean Square F Value Pr > F group <.0001 agegrp <.0001 group*agegrp You can do a max of 3 contrasts here.

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

Contrast DF Contrast SS Mean Square F Value Pr > F c c Contrast for Group*Agegrp

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

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

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

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

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

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.

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

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?

C = 1

C = 2

Compare c = 1, 2 side by side

Average c = 1, 2

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

ANCOVA: Data The MEANS Procedure N LANGUAGE Obs Variable Mean Std Dev TOTENON eppvtstd TOTENON eppvtstd

ANCOVA: before adjustment Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE TOTENON Mean Source DF Type III SS Mean Square F Value Pr > F LANGUAGE

ANCOVA: example

ANCOVA: after adjustment Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE TOTENON Mean Source DF Type I SS Mean Square F Value Pr > F LANGUAGE eppvtstd <.0001 Source DF Type III SS Mean Square F Value Pr > F LANGUAGE eppvtstd <.0001

ANCOVA: after adjustment Standard Parameter Estimate Error t Value Pr > |t| Intercept B LANGUAGE B LANGUAGE B... eppvtstd <.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| < <.0001

ANCOVA: Find the adjusted means

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?

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

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.

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

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

Profile plot …

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 Error B/w Ss error term

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 <.0001 type*LANGUAGE Error(type) W/in Ss error term