ANOVA Longitudinal Models for the Practice Effects Data: via GLM

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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 5 5606381.8 1121276.4 4.73 0.0003 Error 600 142087780.8 236813.0 Corrected Total 605 147694162.6 R-Square Coeff Var Root MSE nm3rt Mean 0.037959 27.48257 486.6343 1770.701 Parameter Estimate Error t Value Pr > t Intercept 1672.136012 B 48.42192532 34.53 <.0001 Session 1 289.757355 B 68.47894350 4.23 <.0001 Session 2 143.036372 B 68.47894350 2.09 0.0371 Session 3 77.898636 B 68.47894350 1.14 0.2558 Session 4 45.660447 B 68.47894350 0.67 0.5052 Session 5 35.039719 B 68.47894350 0.51 0.6091 Session 6 0.000000 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 1 1961.89337 48.42193 <.0001 2 1815.17238 48.42193 <.0001 3 1750.03465 48.42193 <.0001 4 1717.79646 48.42193 <.0001 5 1707.17573 48.42193 <.0001 6 1672.13601 48.42193 <.0001

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 14 0.1004957 225.39847 <.0001 Orthogonal Components 14 0.3154422 113.18581 <.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 0.54465555 16.05 5 96 <.0001 Pillai's Trace 0.45534445 16.05 5 96 <.0001 Hotelling-Lawley Trace 0.83602279 16.05 5 96 <.0001 Roy's Greatest Root 0.83602279 16.05 5 96 <.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 5 5606381.77 1121276.35 32.85 <.0001 <.0001 <.0001 Error(session) 500 17068104.97 34136.21 Greenhouse-Geisser Epsilon 0.6550 Huynh-Feldt Epsilon 0.6798

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)

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 101 1 236813 2 236813 3 236813 4 236813 5 236813 6 236813 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 236813 13672 17.32 <.0001 E variance after accounting for means Fit Statistics -2 Res Log Likelihood 9155.4 AIC (smaller is better) 9157.4 AICC (smaller is better) 9157.4 BIC (smaller is better) 9160.0 Solution for Fixed Effects Intercept 1672.14 48.4219 600 34.53 <.0001 Session 1 289.76 68.4789 600 4.23 <.0001 Session 2 143.04 68.4789 600 2.09 0.0371 Session 3 77.8986 68.4789 600 1.14 0.2558 Session 4 45.6604 68.4789 600 0.67 0.5052 Session 5 35.0397 68.4789 600 0.51 0.6091 Session 6 0.... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Session 5 600 4.73 0.0003 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 1 1961.89 48.4219 600 40.52 <.0001 Session 2 1815.17 48.4219 600 37.49 <.0001 Session 3 1750.03 48.4219 600 36.14 <.0001 Session 4 1717.80 48.4219 600 35.48 <.0001 Session 5 1707.18 48.4219 600 35.26 <.0001 Session 6 1672.14 48.4219 600 34.53 <.0001

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 1 236813 202677 202677 202677 202677 202677 2 202677 236813 202677 202677 202677 202677 3 202677 202677 236813 202677 202677 202677 4 202677 202677 202677 236813 202677 202677 5 202677 202677 202677 202677 236813 202677 6 202677 202677 202677 202677 202677 236813 Estimated R Correlation =.8559 Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z CS ID 202677 29470 6.88 <.0001 Var(U 0i ) after accounting for means Residual 34136 2158.96 15.81 <.0001 Var(e ti ) after accounting for means Fit Statistics -2 Res Log Likelihood 8353.4 AIC (smaller is better) 8357.4 AICC (smaller is better) 8357.4 BIC (smaller is better) 8362.6 Does saturated means, CS variances model fit better than the saturated means, no correlation variances model? Model 1 vs. Model 2 9155.4 vs. 8353.4 diff for df=1 > 2.7, so yes Solution for Fixed Effects Intercept 1672.14 48.4219 129 34.53 <.0001 Session 1 289.76 25.9993 500 11.14 <.0001 Session 2 143.04 25.9993 500 5.50 <.0001 Session 3 77.8986 25.9993 500 3.00 0.0029 Session 4 45.6604 25.9993 500 1.76 0.0797 Session 5 35.0397 25.9993 500 1.35 0.1784 Session 6 0.... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Session 5 500 32.85 <.0001 Session 1 1961.89 48.4219 129 40.52 <.0001 Session 2 1815.17 48.4219 129 37.49 <.0001 Session 3 1750.03 48.4219 129 36.14 <.0001 Session 4 1717.80 48.4219 129 35.48 <.0001 Session 5 1707.18 48.4219 129 35.26 <.0001 Session 6 1672.14 48.4219 129 34.53 <.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.

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 1 301985 235659 217994 202607 192154 195360 2 235659 259150 230217 213232 202092 193268 3 217994 230217 233368 205209 196919 188604 4 202607 213232 205209 217544 193676 185321 5 192154 202092 196919 193676 212098 187840 6 195360 193268 188604 185321 187840 196733 Estimated R Correlation Matrix for ID 101 TOTAL CORRELATION MATRIX AFTER DIFFERENT MEANS 1 1.0000 0.8424 0.8212 0.7905 0.7593 0.8015 2 0.8424 1.0000 0.9361 0.8981 0.8620 0.8559 3 0.8212 0.9361 1.0000 0.9108 0.8851 0.8802 4 0.7905 0.8981 0.9108 1.0000 0.9016 0.8958 5 0.7593 0.8620 0.8851 0.9016 1.0000 0.9196 6 0.8015 0.8559 0.8802 0.8958 0.9196 1.0000 Fit Statistics -2 Res Log Likelihood 8229.8 AIC (smaller is better) 8271.8 AICC (smaller is better) 8273.4 BIC (smaller is better) 8326.7 Solution for Fixed Effects Does saturated means, unstructured variances model fit better than the saturated means, CS variances model? Model 2 vs. Model 3 8353.4 vs. 8229.8 diff for df=19 > 30.1, so yes Intercept 1672.14 44.1345 100 37.89 <.0001 Session 1 289.76 32.7000 100 8.86 <.0001 Session 2 143.04 26.2031 100 5.46 <.0001 Session 3 77.8986 22.8842 100 3.40 0.0010 Session 4 45.6604 20.7853 100 2.20 0.0303 Session 5 35.0397 18.1168 100 1.93 0.0559 Session 6 0.... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Session 5 100 16.72 <.0001 Session 1 1961.89 54.6805 100 35.88 <.0001 Session 2 1815.17 50.6541 100 35.83 <.0001 Session 3 1750.03 48.0684 100 36.41 <.0001 Session 4 1717.80 46.4101 100 37.01 <.0001 Session 5 1707.18 45.8255 100 37.25 <.0001 Session 6 1672.14 44.1345 100 37.89 <.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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