ANOVA Longitudinal Models for the Practice Effects Data: via GLM
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1 Psyc 943 Lecture 25 page 1 ANOVA Longitudinal Models for the Practice Effects Data: via GLM Model 1. Saturated Means Model for Session, E-only Variances Model (BP) Variances Model: NO correlation, EQUAL variance across time SPSS GLM: TITLE "SPSS Saturated Means, E-Only Variances Model via GLM". SUBTITLE "BP ANOVA". UNIANOVA nm3rt BY session /EMMEANS = TABLES(session) /DESIGN = session. SAS GLM: TITLE1 "SAS: Saturated Means, E-Only Variances Model via GLM"; TITLE2 "BP ANOVA"; PROC GLM DATA=stacked; CLASS session; MODEL nm3rt = session / SOLUTION; LSMEANS session / STDERR; RUN; QUIT; Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE nm3rt Mean Parameter Estimate Error t Value Pr > t Intercept B <.0001 Session B <.0001 Session B Session B Session B Session B Session B... 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. Session nm3rt LSMEAN Error Pr > t < < < < < <.0001
2 Psyc 943 Lecture 25 page 2 Model 2. Saturated Means Model for Session, U0 + e (CS) Variances Model (WP) Model 3. Saturated Means Model for Session, Saturated (Multivariate) Variances Model (WP) Variances Model: EQUAL correlation, EQUAL variance across time CS Univ. RM ANOVA Variances Model: Completely UNEQUAL correlation and variance across time UN Multiv. RM ANOVA SPSS GLM: DATASET ACTIVATE chapmultiv WINDOW=FRONT. TITLE "SPSS Saturated Means, CS and Unstructured Variances Models via GLM". SUBTITLE "RM ANOVA". GLM nm3rt1 nm3rt2 nm3rt3 nm3rt4 nm3rt5 nm3rt6 /WSFACTOR = session 6 SAS GLM: TITLE "SAS Saturated Means, CS and Unstructured Variances Models via GLM"; TITLE2 "RM ANOVA"; PROC GLM DATA=multiv; MODEL nm3rt1-nm3rt6 = / SOLUTION NOUNI; REPEATED session / PRINTE; RUN; QUIT; Sphericity Tests Mauchly's Variables DF Criterion Chi-Square Pr > ChiSq Transformed Variates <.0001 Orthogonal Components <.0001 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no session Effect H = Type III SSCP Matrix for session E = Error SSCP Matrix S=1 M=1.5 N=47 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda <.0001 Pillai's Trace <.0001 Hotelling-Lawley Trace <.0001 Roy's Greatest Root <.0001 Repeated Measures Analysis of Variance Univariate Tests of Hypotheses for Within Subject Effects Adj Pr > F Source DF Type III SS Mean Square F Value Pr > F G - G H - F session <.0001 <.0001 <.0001 Error(session) Greenhouse-Geisser Epsilon Huynh-Feldt Epsilon
3 SAS Syntax: Psyc 943 Lecture 25 page 3 ANOVA Longitudinal Models for the Practice Effects Data: via MIXED Syntax Model 1. Saturated Means Model for Session, E-only Variances Model (BP no covariance over time) TITLE1 "SAS: Saturated Means, E-Only Variances Model via MIXED"; REPEATED session / R TYPE=VC SUBJECT=ID; Model 2. Saturated Means Model for Session, U0 + eti (CS) Variances Model (WP constant covariance) TITLE1 "SAS: Saturated Means, CS Variances Model"; TITLE2 "Univariate RM ANOVA"; REPEATED session / R RCORR TYPE=CS SUBJECT=ID; Model 3. Saturated Means Model for Session, Saturated (Multivariate) Variances Model (WP any covariance) TITLE1 "SAS: Saturated Means, Unstructured Variances Model"; TITLE2 "Multivariate RM ANOVA"; REPEATED session / R RCORR TYPE=UN SUBJECT=ID; SPSS Syntax: Model 1. Saturated Means Model for Session, E-only Variances Model (BP no covariance over time) TITLE "SPSS Saturated Means, E-Only Variances Model via MIXED". MIXED nm3rt BY ID session /METHOD = REML /FIXED = session /PRINT = SOLUTION TESTCOV R /REPEATED = session COVTYPE(ID) SUBJECT(ID) Model 2. Saturated Means Model for Session, U0 + eti (CS) Variances Model (WP constant covariance) TITLE "SPSS Saturated Means, CS Variances Model". SUBTITLE "Univariate RM ANOVA". MIXED nm3rt BY ID session /METHOD = REML /FIXED = session /PRINT = SOLUTION TESTCOV R /REPEATED = session COVTYPE(CS) SUBJECT(ID) Model 3. Saturated Means Model for Session, Saturated (Multivariate) Variances Model (WP any covariance) TITLE "SPSS Saturated Means, Unstructured Variances Model". SUBTITLE "Multivariate RM ANOVA". MIXED nm3rt BY ID session /METHOD = REML /FIXED = session /PRINT = SOLUTION TESTCOV R /REPEATED = session COVTYPE(UN) SUBJECT(ID)
4 Psyc 943 Lecture 25 page 4 Model 1. Saturated Means Model for Session, E-only Variances Model (BP no covariance over time) Variances Model: NO correlation, EQUAL variance across time TITLE1 "SAS: Saturated Means, E-Only Variances Model via MIXED"; REPEATED session / R TYPE=VC SUBJECT=ID; Dimensions Covariance Parameters 1 still just e in model for variances Columns in X 7 should be 6 but it counts the unidentified one Columns in Z 0 still no U s yet Estimated R Matrix for ID The R matrix is the unstandardized matrix of the error variances and covariances for each session. So far no covariance is allowed across time, with equal variance across time. Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z Session ID <.0001 E variance after accounting for means Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Solution for Fixed Effects Intercept <.0001 Session <.0001 Session Session Session Session Session Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Session The test of the fixed effects (p <.05) tells us that there is a significant effect of time overall. This test matches the results from BP ANOVA. Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001
5 Psyc 943 Lecture 25 page 5 Model 2. Saturated Means Model for Session, U0 + eti (CS) Variances Model (WP constant covariance) Variances Model: EQUAL correlation, EQUAL variance across time TITLE1 "SAS: Saturated Means, CS Variances Model"; TITLE2 "Univariate RM ANOVA"; REPEATED session / R RCORR TYPE=CS SUBJECT=ID; Dimensions Covariance Parameters 2 Columns in X 7 Columns in Z 0 Estimated R Matrix for ID 101 COMBINED U 0i + e ti VARIANCE AFTER DIFFERENT MEANS Estimated R Correlation =.8559 Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z CS ID <.0001 Var(U 0i ) after accounting for means Residual <.0001 Var(e ti ) after accounting for means Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Does saturated means, CS variances model fit better than the saturated means, no correlation variances model? Model 1 vs. Model vs diff for df=1 > 2.7, so yes Solution for Fixed Effects Intercept <.0001 Session <.0001 Session <.0001 Session Session Session Session Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001 The test of the fixed effects (p <.05) tells us that there is a significant effect of time overall. This test matches the results from Univ. RM ANOVA.
6 Psyc 943 Lecture 25 page 6 Model 3. Saturated Means Model for Session, Saturated (Multivariate) Variances Model (WP any covariance) Variances Model: Completely UNEQUAL correlation and variance across time TITLE1 "SAS: Saturated Means, Unstructured Variances Model"; TITLE2 "Multivariate RM ANOVA"; REPEATED session / R RCORR TYPE=UN SUBJECT=ID; Dimensions Covariance Parameters 21 Columns in X 7 Columns in Z 0 Estimated R Matrix for ID 101 TOTAL COVARIANCE MATRIX AFTER DIFFERENT MEANS Estimated R Correlation Matrix for ID 101 TOTAL CORRELATION MATRIX AFTER DIFFERENT MEANS Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Solution for Fixed Effects Does saturated means, unstructured variances model fit better than the saturated means, CS variances model? Model 2 vs. Model vs diff for df=19 > 30.1, so yes Intercept <.0001 Session <.0001 Session <.0001 Session Session Session Session Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001 Session <.0001 The test of the fixed effects (p <.05) tells us that there is a significant effect of time overall. This test matches the results from Multiv. RM ANOVA.
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