6:1 LAB RESULTS -WITHIN-S ANOVA

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1 6:1 LAB RESULTS -WITHIN-S ANOVA T1/T2/T3/T4. SStotal =(1-12) (18-12) 2 = = SSpill + SSsubj + SSpxs df = 9-1 = 8 P1 P2 P3 Ms Ms-Mg SSsubj= 3x( ) = df =3-1= 2 Mp Mp - Mg SSpill = = 3x( ) df=3-1=2 Predicted from Main Effects Row Means Deviations Col Means = Mg SSpxs = 42.0 = 1x( ) df = (3-1)(3-1) = 4 With Main Effects removed Row Means Deviations Col Means = Mg SSpxs = 42.0 = 1x( ) *C1. Edited output. DATA LIST FREE / p1 p2 p3. BEGIN DATA END DATA. MANOVA p1 p2 p3 /WSF = pill(3) /PRINT = CELL. Mean Std. Dev. N Variable.. p Variable.. p Variable.. p Tests of Between-Subjects Effects. WITHIN CELLS CONSTANT Tests involving 'PILL' Within-Subject Effect. WITHIN CELLS PILL *C2. Using menus. Edited Output (also gives default polynomial contrasts). GLM p1 p2 p3 /WSF = pill(3) POLYNOMIAL. Tests of Within-Subjects Effects pill Sphericity Error(pill) Sphericity Tests of Within-Subjects Contrasts (default polynomial contrasts omitted) Tests of Between-Subjects Effects Intercept Error

2 T5. Partition Pill Effect. P1 P2 P3 Mj L SS (nj = 3) p1v p2v Sum = = SS pill OR Subj 1 Subj 2 Subj 3 P1 P2 P3 P1 P2 P3 P1 P2 P3 y L SS (nj = 1) Pill (Main Effect) p1v p2v Sum = T6. Partitioning SS Interaction. Subj 1 Subj 2 Subj 3 P1 P2 P3 P1 P2 P3 P1 P2 P3 y L SS Pill p1v p2v Subj (arbitrary contrasts) s12v s1v Pill x Subj p1xs p1xs p1xs df = 2 p2xs p2xs P2xS df = 2 *C3a. Partitioning using MANOVA and /PRINT = SIGN(UNIV)command. * Coefficients for t-test differ from preceding; normalized. MANOVA p1 p2 p3 /WSF = pill(3) /PRINT = SIGN(UNIV) /CONTR(pill) = HELMERT. Tests of Between-Subjects Effects. WITHIN CELLS CONSTANT Tests involving 'PILL' Within-Subject Effect. Univariate F-tests with (1,2) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F T T WITHIN CELLS PILL Estimates for T2 --- Individual univariate.9500 confidence intervals Estimates for T3 --- Individual univariate.9500 confidence intervals

3 *C3b Partitioning using MANOVA and /WSD command. MANOVA p1 p2 p3 /WSF = pill(3) /CONTR(pill) = HELMERT /WSD pill(1) pill(2). Tests of Between-Subjects Effects. WITHIN+RESIDUAL CONSTANT Estimates for T1 --- Individual univariate.9500 confidence intervals CONSTANT Tests involving 'PILL(1)' Within-Subject Effect. WITHIN+RESIDUAL PILL(1) Estimates for T2 --- Individual univariate.9500 confidence intervals PILL(1) Tests involving 'PILL(2)' Within-Subject Effect. WITHIN+RESIDUAL PILL(2) Estimates for T3 --- Individual univariate.9500 confidence intervals PILL(2) *C3c. Partitioning using GLM and HELMERT on menus (note SSs do not add up). GLM p1 p2 p3 /WSF = pill(3) HELMERT. Tests of Within-Subjects Effects pill Sphericity Error(pill) Sphericity Tests of Within-Subjects Contrasts Source pill Type III Sum of df Mean Square F Sig. pill Level 1 vs. Later Level 2 vs. Level Error(pill) Level 1 vs. Later Level 2 vs. Level Tests of Between-Subjects Effects Intercept Error

4 *C4. Partitioning using GLM and normalized coefficients (SSs add up). GLM p1 p2 p3 /WSF = pill(3) SPEC( ). Tests of Within-Subjects Effects pill Sphericity Error(pill) Sphericity Tests of Within-Subjects Contrasts Source pill Type III Sum of df Mean Square F Sig. pill L L Error(pill) L L Tests of Between-Subjects Effects Intercept Error *Partition WSF using MMATRIX and normalized coeff s; = SQRT(6) = SQRT(2). GLM p1 p2 p3 /WSFACTOR = pill 3 /MMATRIX p1-2/ p2 1/ p3 1/2.4495; p1 0 p2-1/ p3 1/ Custom Hypothesis Tests T1 T2 L1 Contrast Estimate Std. Error Sig Univariate Test Results Source Transformed Sum of df Mean Square F Sig. Variable Contrast T T Error T T *C5. Partitioning by computing Normalized Contrast Scores. COMP n1v23 = *p *p *p3. COMP n2v3 = *p *p3. MANOVA n1v23 /PRINT = CELL. SS Num =3x( ) 2 SS Error = Mean Std. Dev. N Variable.. n1v WITHIN CELLS CONSTANT MANOVA n2v3 /PRINT = CELL. Mean Std. Dev. N Variable.. n2v WITHIN CELLS CONSTANT LIST p1 p2 p3 n1v23 n2v3. p1 p2 p3 n1v23 n2v

5 *C6 - Post Hoc Tests for Within-S Design. * - first analysis repeats earlier contrasts, one is pairwise comparison. MANOVA p1 p2 p3 /WSF = pill(3) /PRINT = SIGN(UNIV) /CONTR(pill) = SPEC( ). Tests involving 'PILL' Within-Subject Effect. Univariate F-tests with (1,2) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F T T <<pairwise Estimates for T3 --- Individual univariate.9500 confidence intervals PILL GLM p1 p2 p3 /WSFACTOR = pill 3 /EMMEANS = TABLES(pill) COMPARE ADJ(LSD). Estimated Marginal Means Estimates pill Mean Std. 95% Confidence Interval Error Lower Bound Upper Bound Pairwise Comparisons (I) (J) Mean Difference Std. Sig.(a) 95% Confidence Interval for pill pill (I-J) Error Difference(a) Lower Bound Upper Bound t = 8/2.646 = <<<p same as above *Pairwise comparisons are essentially paired t-tests; compare ps and ts. TTEST PAIR p1 p2 p3. Mean N Std. Deviation Std. Error Mean Pair p p Pair p p Pair p p Paired Samples Correlations N Correlation Sig. Pair 1 p1 & p Pair 2 p1 & p Pair 3 p2 & p Paired Samples Test Paired Differences t df Sig. (2-tail Mean Std. Std. Error 95% Confidence Interval Deviation Mean of the Difference Lower Upper Pair 1 p1 - p Pair 2 p1 - p Pair 3 p2 - p

6 *Bonferroni adjustment - ps = 3 x p from LSD or paired t-tests. GLM p1 p2 p3 /WSFACTOR = pill 3 /EMMEANS = TABLES(pill) COMPARE ADJ(BONF). Pairwise Comparisons (I) (J) Mean Difference Std. Sig.(a) 95% Confidence Interval for pill pill (I-J) Error Difference(a) Lower Bound Upper Bound *Other Post Hoc Tests (SNK, TUKEY) - not in lab. To do other Post Hoc tests, use TTEST PAIR to get pairwise ts and multiply by SQRT(2) to obtain qs. Use Tables or SPSS algorithms to obtain critical values to determine significance or use SPSS algorithms to determine p values. Note: Some statisticians disagree with use of q statistics for Within-S factors. *Using SPSS algorithms to determine p values. DATA LIST FREE / comp str t. BEGIN DATA END DATA. COMPUTE qobs = SQRT(2)*t. *Compute p values for qobs: LSD, SNK, TUKEY, BONF (df = 2). COMP plsd = 1 - CDF.SRANGE(qobs, 2, 2). COMP psnk = 1 - CDF.SRANGE(qobs, str, 2). COMP ptuk = 1 - CDF.SRANGE(qobs, 3, 2). COMP pbon = 3*(1 - CDF.SRANGE(qobs, 2, 2)). LIST. comp str t qobs plsd psnk ptuk pbon *Determining critical values with SPSS. *Read str. DATA LIST FREE / str. BEGIN DATA 2 3 END DATA. FORMAT str (f2.0). *Compute critical values for LSD (t and q), SNK, TUKEY, BONF (df = 2). COMP tlsd = IDF.T(1 -.05/2, 2). COMP qlsd = IDF.SRANGE(1-.05, 2, 2). COMP qsnk = IDF.SRANGE(1-.05, str, 2). COMP qtuk = IDF.SRANGE(1-.05, 3, 2). COMP qbonf = IDF.SRANGE(1 - (.05/3), 2, 2). LIST. str tlsd qlsd qsnk qtuk qbonf

7 *C7. Within-S Analysis (and Between-S Analysis) with data in Between-S Format. DATA LIST FREE / subj pill mood. BEGIN DATA END DATA. *Correct Within-S ANOVA specifying Pill and Subj main effects; Residual = PxS interaction. MANOVA mood BY subj(1 3) pill(1 3) /DESIGN pill subj. RESIDUAL PILL SUBJ (Model) (Total) GLM mood BY pill subj /DESIGN pill subj. Corrected Model (a) Intercept pill subj Error Total Corrected a R Squared =.863 (Adjusted R Squared =.725) *Full Factorial: shows that error is Pill X Subj interaction. MANOVA mood BY pill(1 3) subj(1 3). RESIDUAL pill subj pill BY subj (Model) (Total) R-Squared = *INCORRECT Between-S ANOVA: pill effect NS despite df = 2, 6 (vs 2, 4 for WS ANOVA). MANOVA mood BY pill(1 3). WITHIN CELLS pill vs..040 for WS (Model) (Total)

8 *Using Between-S Format to Obtain P x S Interaction Deviations (compare to T1/T2/T3). MANOVA mood BY subj(1 3) pill(1 3) /PMEAN = TABLE(subj*pill) /DESIGN pill subj. RESIDUAL PILL SUBJ (Model) (Total) Adjusted and Estimated Means CELL Obs. Mean Adj. Mean Est. Mean Raw Resid. Std. Resid *Using GLM to obtain PxS deviations and calculate SS PxS. GLM mood BY pill subj /DESIGN pill subj /SAVE PRED(pmain). GLM mood BY pill subj /SAVE PRED(pcell). COMPUTE inter = pcell - pmain. COMPUTE inter2 = inter**2. FORMAT pmain pcell inter inter2 (F5.2). DESCR inter2 /STAT = SUM. N Sum inter LIST. subj pill mood pmain pcell inter inter *Partitioning Error Term in Between-S Format; equivalent to Within-S partitioning. MANOVA mood by subj(1 3) pill(1 3) /CONTR(pill) = HELMERT /DESIGN subj pill(1) VS 1 pill(2) VS 2 pill(1) BY subj = 1 pill(2) BY subj = 2. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * W A R N I N G * Too few degrees of freedom in RESIDUAL * * * error term for the following test(s) (DF = 0). * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * RESIDUAL SUBJ Error PILL(1) Error PILL(2)

9 *WS Contrasts with BS Data Format, Numerators correct, but denominator not partitioned. * INCORRECT because no partitioning of Error term (although Ok with some researchers). * Note effect is more significant because of increased df. MANOVA mood BY subj(1 3) pill(1 3) /PRINT = SIGN(SINGL) /CONTRA(pill) = HELMERT /DESIGN subj pill. RESIDUAL SUBJ ST Parameter ND Parameter PILL ST Parameter ND Parameter (Model) (Total) R-Squared =.863 Adjusted R-Squared =.725 Estimates for mood --- Individual univariate.9500 confidence intervals SUBJ PILL *Given data in Between-S format, some relevant regression analyses are possible. * Generate indicator variables corresponding to Pill, Subject, and PxS contrasts. RECODE pill (1 = -2) (2 3 = 1) INTO p1. RECODE pill (1 = 0) (2 = -1) (3 = 1) INTO p2. RECODE subj (1 2 = -1) (3 = 2) INTO s1. RECODE subj (1 = -1) (2 = 1) (3 = 0) INTO s2. COMPUTE p1xs1 = p1*s1. COMPUTE p1xs2 = p1*s2. COMPUTE p2xs1 = p2*s1. COMPUTE p2xs2 = P2*s2. FORMAT p1 TO p2xs2 (F2.0). LIST pill subj mood p1 TO p2xs2. pill subj mood p1 p2 s1 s2 p1xs1 p1xs2 p2xs1 p2xs CORR mood TO p2xs2. mood p1 p2 s1 s2 p1xs1 p1xs2 p2xs1 p2xs2 p1 Pearson.728 p2 Pearson s1 Pearson s2 Pearson p1xs1 Pearson p1xs2 Pearson p2xs1 Pearson p2xs2 Pearson

10 *CHANGE statistics = significance of Pill effect; SS change = = SS pill. REGRE /STAT = DEFA CHANGE/DEP = mood /ENTER s1 s2 /ENTER p1 p2. Model R R Adjusted Std. Error of Change Statistics Square R Square the Estimate R Square Change F Change df1 df2 Sig. F Change 1.560(a) (b) Model Sum of df Mean Square F Sig. 1 Regression (a) Residual Regression (b) Residual *SS Residual for Model 2 above = 42.0 = PxS interaction. *SS Change below = 42.0 = PxS interaction. REGRE /DEP = mood /ENTER p1 p2 s1 s2 /ENTER p1xs1 p1xs2 p2xs1 p2xs2. Model Sum of df Mean Square F Sig. 1 Regression (a) Residual Regression (b) Residual *SS Change = = SS P1 contrast. REGRE /DEP = mood /ENTER p2 s1 s2 p1xs1 p1xs2 p2xs1 p2xs2 /ENTER p1. Model Sum of df Mean Square F Sig. 1 Regression (a) Residual Regression (b) Residual *SS Change = 21.0 = P1xS interaction. REGRE /DEP = mood /ENTER p1 p2 s1 s2 p2xs1 p2xs2 /ENTER p1xs1 p1xs2. Model Sum of df Mean Square F Sig. 1 Regression (a) Residual Regression (b) Residual

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