SAS Code for Data Manipulation: SPSS Code for Data Manipulation: STATA Code for Data Manipulation: Psyc 945 Example 1 page 1

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1 Psyc 945 Example page Example : Unconditional Models for Change in Number Match 3 Response Time (complete data, syntax, and output available for SAS, SPSS, and STATA electronically) These data come from a short-term longitudinal study of 6 observations over weeks for 0 adults age years. The goal is to see how performance on this processing speed task ( number match 3 ), as measured by response time in milliseconds, declines (improves) over the 6 practice sessions. We will examine both polynomial and piecewise models to describe individual differences in within-person change across sessions. SAS Code for Data Manipulation: * SAS code to import data; DATA work.example34; SET example.example34; * Center time for polynomial models; csess = Session - ; LABEL csess = "csess: Session Centered at "; * Create piecewise slopes; IF Session = THEN DO; Slope = 0; Slope6 = 0; END; ELSE IF Session = THEN DO; Slope = ; Slope6 = 0; END; ELSE IF Session = 3 THEN DO; Slope = ; Slope6 = ; END; ELSE IF Session = 4 THEN DO; Slope = ; Slope6 = ; END; ELSE IF Session = 5 THEN DO; Slope = ; Slope6 = 3; END; ELSE IF Session = 6 THEN DO; Slope = ; Slope6 = 4; END; LABEL Slope = "Slope: Early Practice Slope (Session -)" Slope6 = "Slope6: Later Practice Slope (Session -6)"; RUN; SPSS Code for Data Manipulation: * SPSS code to import data. GET FILE = "example/example34.sav". DATASET NAME example34 WINDOW=FRONT. * Center time for polynomial models. COMPUTE csess = session -. VARIABLE LABELS csess "csess: Session Centered at ". * Create piecewise slopes. RECODE session (=0) ( THRU HI=) INTO Slope. RECODE session (LO THRU =0) (3=) (4=) (5=3) (6=4) INTO Slope6. VARIABLE LABELS Slope "Slope: Early Practice Slope (Session -)" Slope6 "Slope6: Later Practice Slope (Session -6)". STATA Code for Data Manipulation: * Center time for polynomial models (and make quadratic version) gen csess = session label variable csess "csess: Session Centered at " gen csess = csess * csess label variable csess "csess: Quadratic Session Centered at " * Create piecewise slopes gen slope = session gen slope6 = session recode slope (=0) if session== recode slope (=) if session== recode slope (3=) if session==3 recode slope (4=) if session==4 recode slope (5=) if session==5 recode slope (6=) if session==6 recode slope6 (=0) if session== recode slope6 (=0) if session== recode slope6 (3=) if session==3 recode slope6 (4=) if session==4 recode slope6 (5=3) if session==5 recode slope6 (6=4) if session==6 label variable slope "slope: Early Practice Slope (Session -)" label variable slope6 "slope6: Later Practice Slope (Session -6)"

2 Psyc 945 Example page Model a. Most Conservative Baseline Empty Means, Random Intercept Level : yti 0i eti Level : Intercept: U 0i 00 0i TITLE "SAS Model a: Empty Means, Random Intercept Only"; PROC MIXED DATA=work.example34 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT / G V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; RUN; TITLE; TITLE "SPSS Model a: Empty Means, Random Intercept". MIXED nm3rt BY ID session /PRINT = SOLUTION TESTCOV G R /FIXED = /RANDOM = INTERCEPT SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID). * STATA Model a: Empty Means, Random Intercept xtmixed nm3rt, id:, /// variance reml covariance(unstructured) residuals(independent,t(session)), estat ic, n(0) estat recovariance, level(id) METHOD = ML or REML (default) CLASS = categorical predictors, nesting MODEL dv = fixed effects / print solution RANDOM = person variances in G REPEATED = residuals in R matrix MIXED dv BY categorical predictors WITH continuous predictors or ML /PRINT = regression solution /FIXED = predictors for means model /RANDOM = person variances in G DV = nm3rt, random part after Level ID is PersonID, random intercept by default Print variances instead of SD, use reml covariance(unstructured) refers to G matrix residuals(independent) refers to R matrix by session estat ic Print IC given N = 0 persons Estimated R Matrix for ID Estimated G Matrix Participant Row Effect ID Col Intercept This is the level- G matrix, just a random intercept variance so far. Estimated V Matrix for ID Estimated V Correlation Matrix for ID This level- R matrix (with equal variance over time, no covariance of any kind, known as VC or independence) will be used repeatedly as we add fixed The V matrix is the total variance-covariance matrix after combining the level- G and level- R matrices.

3 Psyc 945 Example page 3 Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 Session ID <.000 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq <.000 NegLogLike Parms AIC AICC HQIC BIC CAIC Effect Estimate Error DF t Value Pr > t Intercept <.000 Model b. Most Liberal Baseline Saturated Means, Unstructured Variances (Model Answer Key) TITLE "SAS Model b: Saturated Means, Unstructured Variances"; PROC MIXED DATA=work.example34 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = session / SOLUTION DDFM=Satterthwaite; REPEATED session / R RCORR TYPE=UN SUBJECT=ID; LSMEANS session /; RUN; TITLE; TITLE "SPSS Model b: Saturated Means, Unstructured Variances". MIXED nm3rt BY ID session /PRINT = SOLUTION TESTCOV R /FIXED = session /REPEATED = session SUBJECT(ID) COVTYPE(UN) /EMMEANS = TABLES(session). * STATA Model b: Saturated Means, Unstructured Variances xtmixed nm3rt ib(last).session, id:, noconstant /// variance reml residuals(unstructured, t(session)), estat ic, n(0), contrast session, // omnibus test of mean differences margins i.session, // observed means per session marginsplot name(observed_means, replace) // plot observed means Calculate the ICC for the Number Match 3 outcome: ICC This null model LRT tells us that the random intercept variance is significantly greater than 0, and thus so is the ICC. REML only counts the # parameters in the model for the variance (not fixed effects). This is the fixed intercept (just the grand mean so far). Placing session on the CLASS/BY statements and in the FIXED/MODEL statements treats it as a categorical predictor. So this is an ANOVA means model. No RANDOM statements mean no random effects. i. indicates categorical predictor of session (ref=last to match others) noconstant = no random intercept (just R matrix) Estimated R Matrix for ID Estimated R Correlation Matrix for ID This Unstructured R matrix estimates all variances and covariances separately. THIS IS THE DATA we are trying to duplicate with our model for the variance

4 Psyc 945 Example page 4 NegLogLike Parms AIC AICC HQIC BIC CAIC Effect Session # Estimate Error DF t Value Pr > t Intercept <.000 Session <.000 Session <.000 Session Session Session Session Mean diffs relative to session 6 (which is the intercept given that it is the highest value) Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Session <.000 This is the omnibus test of mean differences across 6 sessions. Least Squares Means Effect Session # Estimate Error DF t Value Pr > t Session <.000 Session <.000 Session <.000 Session <.000 Session <.000 Session <.000 These are the means per session that the fixed effects will be trying to reproduce. So here is what are we trying to model means and variances, where model b is the data: Predicted Means By Model Predicted Variance By Model Saturated Means Model b Random Intercept Model a Unstructured Model b Random Intercept Model a, ,000, ,000,800 50,000 Mean RT,700 RT Variance 00,000,600 50,000, Session 00, Session

5 Psyc 945 Example page 5 Model a. Fixed Linear Time, Random Intercept Level : y Session e Level : Intercept: 0i 00 U0i Linear Session: ti 0i i ti ti i 0 TITLE "SAS Model a: Fixed Linear Time, Random Intercept"; PROC MIXED DATA=work.example35 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = csess / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT / G V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; RUN; TITLE; TITLE "SPSS Model a: Fixed Linear Time, Random Intercept". MIXED nm3rt BY ID session WITH csess /PRINT = SOLUTION TESTCOV G R /FIXED = csess /RANDOM = INTERCEPT SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID). * STATA Model a: Fixed Linear Time, Random Intercept xtmixed nm3rt c.csess, id:, /// variance reml covariance(un) residuals(independent,t(session)), estat ic, n(0), estimates store FixLin Estimated V Matrix for ID Estimated V Correlation Matrix for ID Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 Session ID <.000 NegLogLike Parms AIC AICC HQIC BIC CAIC The predictor of csess will be treated as continuous given that it is not on the CLASS statement (SAS) and it is on WITH (SPSS). DV = nm3rt, c. means continuous fixed slope for csess Level ID is id, random intercept by default estimates save results as FixLin for next LRT Relative to the empty means, random intercept model a, the fixed linear effect of session explained ~% of the residual variance (which made the random intercept variance increase due to its smaller residual variance correction factor). Effect Estimate Error DF t Value Pr > t Intercept <.000 Csess <.000 The fixed linear effect of csess is significant according to the Wald test (p-value for fixed effect).

6 Psyc 945 Example page 6 Model b. Random Linear Time Level : y Session e Level : Intercept: 0i 00 U0i Linear Session: U ti 0i i ti ti i 0 i TITLE "SAS Model b: Random Linear Time"; PROC MIXED DATA=work.example35 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = csess / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT csess / G V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; RUN; TITLE; TITLE "SPSS Model b: Random Linear Time". Now there are random effects: intercept and linear MIXED nm3rt BY ID session WITH csess slope, given by csess on the RANDOM statements. /PRINT = SOLUTION TESTCOV G R /FIXED = csess /RANDOM = INTERCEPT csess SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID). * STATA Model b: Random Linear Time xtmixed nm3rt c.csess, id: csess, /// variance reml covariance(un) residuals(independent,t(session)), estat ic, n(0), estat recovariance, level(id), estimates store RandLin, lrtest RandLin FixLin DV = nm3rt, c. means continuous fixed slope for csess Level ID is id, random intercept and csess now estimates save results as RandLin for LRT Estimated R Matrix for ID Estimated G Matrix Participant Row Effect ID Col Col Intercept Csess Estimated G Correlation Matrix ID: Participant Row Effect ID Col Col Intercept csess GCORR shows the correlation among random effects. Estimated V Matrix for ID The V matrix is the total variance-covariance matrix after combining the level- G and level- R matrices. Now the variances and covariances are predicted to change based on time.

7 Psyc 945 Example page 7 How the V matrix variances and covariances get calculated in a random linear time model: Vi matrix: VarianceytimeU Session 0 U Session U 0 e V i matrix: Covariance y A,yB U A B U AB U 0 0 Estimated V Correlation Matrix for ID The VCORR matrix is the correlation version. The ICC is now predicted to change over time, too (and conditional on linear time). Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 UN(,) ID UN(,) ID <.000 Session ID <.000 NegLogLike Parms AIC AICC HQIC BIC CAIC Effect Estimate Error DF t Value Pr > t Intercept <.000 Csess <.000 Model 3a. Fixed Quadratic, Random Linear Time Level : y Session Session e Level : Intercept: 0i 00 U0i Linear Session: i 0 Ui Quadratic Session: ti 0i i ti i ti ti i 0 TITLE "SAS Model 3a: Fixed Quadratic, Random Linear Time"; PROC MIXED DATA=work.example35 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = csess csess*csess / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT csess / G V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; RUN; TITLE; TITLE "SPSS Model 3a: Fixed Quadratic, Random Linear Time". MIXED nm3rt BY ID session WITH csess /PRINT = SOLUTION TESTCOV G R /FIXED = csess csess*csess /RANDOM = INTERCEPT csess SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID). * STATA Model 3a: Fixed Quadratic, Random Linear Time xtmixed nm3rt c.csess c.csess#c.csess, id: csess, /// variance reml covariance(un) residuals(independent,t(session)), estat ic, n(0), Is the random linear time model (b) better than the fixed linear time, random intercept model (a)? Yep, ΔLL= 43, which is bigger than the critical value of 5.99ish on df =~ish Interactions can be defined on the fly in SAS and SPSS using *. Interactions can be defined on the fly in STATA using # for fixed effects, but not for random effects.

8 Psyc 945 Example page 8 estat recovariance, level(id), estimates store FixQuad Estimated R Matrix for ID Estimated G Matrix ID: Participant Row Effect ID Col Col Intercept csess GCORR shows intercept slope correlation r =.53 Estimated V Matrix for ID Estimated V Correlation Matrix for ID Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 UN(,) ID UN(,) ID <.000 session ID <.000 Relative to the random linear time model b, the fixed quadratic effect of session explained another ~6% of the residual variance (which made the random intercept variance increase due to its residual variance correction factor). NegLogLike Parms AIC AICC HQIC BIC CAIC Effect Estimate Error DF t Value Pr > t Intercept <.000 Csess <.000 Csess*Csess <.000 The fixed quadratic effect of csess is significant according to the Wald test (p-value for fixed effect).

9 Psyc 945 Example page 9 Model 3b. Random Quadratic Time (and an example of ESTIMATE/TEST/LINCOM statements) Level : y Session Session e Level : Intercept: 0i 00 U0i Linear Session: i 0 Ui Quadratic Session: U ti 0i i ti i ti ti i 0 i TITLE "SAS Model 3b: Random Quadratic Time"; PROC MIXED DATA=work.example35 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = csess csess*csess / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT csess csess*csess / G V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; ESTIMATE "Intercept at Session " intercept csess 0 csess*csess 0; ESTIMATE "Intercept at Session " intercept csess csess*csess ; ESTIMATE "Intercept at Session 3" intercept csess csess*csess 4; ESTIMATE "Intercept at Session 4" intercept csess 3 csess*csess 9; ESTIMATE "Intercept at Session 5" intercept csess 4 csess*csess 6; ESTIMATE "Intercept at Session 6" intercept csess 5 csess*csess 5; * Predicting linear rate of change at each session (linear changes by *quad); ESTIMATE "Linear Slope at Session " csess csess*csess 0; ESTIMATE "Linear Slope at Session " csess csess*csess ; ESTIMATE "Linear Slope at Session 3" csess csess*csess 4; ESTIMATE "Linear Slope at Session 4" csess csess*csess 6; ESTIMATE "Linear Slope at Session 5" csess csess*csess 8; ESTIMATE "Linear Slope at Session 6" csess csess*csess 0; RUN; TITLE; TITLE "SPSS Model 3b: Random Quadratic Time". MIXED nm3rt BY ID session WITH csess /PRINT = SOLUTION TESTCOV G R /FIXED = csess csess*csess /RANDOM = INTERCEPT csess csess*csess SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID) /TEST = "Intercept at Session " intercept csess 0 csess*csess 0 /TEST = "Intercept at Session " intercept csess csess*csess /TEST = "Intercept at Session 3" intercept csess csess*csess 4 /TEST = "Intercept at Session 4" intercept csess 3 csess*csess 9 /TEST = "Intercept at Session 5" intercept csess 4 csess*csess 6 /TEST = "Intercept at Session 6" intercept csess 5 csess*csess 5 /TEST = "Linear Slope at Session " csess csess*csess 0 /TEST = "Linear Slope at Session " csess csess*csess /TEST = "Linear Slope at Session 3" csess csess*csess 4 /TEST = "Linear Slope at Session 4" csess csess*csess 6 /TEST = "Linear Slope at Session 5" csess csess*csess 8 /TEST = "Linear Slope at Session 6" csess csess*csess 0. * STATA Model 3b: Random Quadratic Time The random statement will xtmixed nm3rt c.csess c.csess#c.csess, id: csess csess, /// not accept interaction terms, variance reml covariance(un) residuals(independent,t(session)), so we are using the csess estat ic, n(0), created manually before. estat recovariance, level(id), estimates store RandQuad, lrtest RandQuad FixQuad, margins, at(c.csess=(0()5)) vsquish // intercepts per session marginsplot, name(predicted_means, replace) // plot intercepts margins, at(c.csess=(0()5)) dydx(c.csess) vsquish // linear slope per session marginsplot, name(change_in_linear_slope, replace) // plot quadratic effect

10 Psyc 945 Example page 0 Estimated R Matrix for ID Estimated G Matrix Participant Row Effect ID Col Col Col3 Intercept Csess Csess*Csess Estimated G Correlation Matrix ID: Participant Row Effect ID Col Col Col3 Intercept csess csess*csess Estimated V Matrix for ID The V matrix is the total variance-covariance matrix after combining the level- G and level- R matrices. The variances and covariances are predicted to change based on time, but differently. How the V matrix variances and covariances get calculated in a random quadratic time model: Predicted Variance at Time T: Var(y T ) = σ + τ + *T*τ + T *τ + *T *τ + *T 3 *τ + T 4 *τ Predicted Covariance between Time A and B: Cov(y A, y B ) = τ + (A+B)*τ + (AB)*τ + (A +B )*τ + (AB )+(A B)*τ + (A B )*τ Estimated V Correlation Matrix for ID Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 UN(,) ID UN(,) ID <.000 UN(3,) ID UN(3,) ID <.000 UN(3,3) ID Session ID <.000

11 Psyc 945 Example page NegLogLike Parms AIC AICC HQIC BIC CAIC Effect Estimate Error DF t Value Pr > t Intercept <.000 Csess <.000 Csess*Csess <.000 Is the random quadratic model (3b) better than the fixed quadratic, random linear model (3a)? Yep, ΔLL= 39, which is bigger than the critical value of 7.8ish on df=~3ish Computing random effects confidence intervals for each random effect: Random Effect 95% CI = fixed effect ±.96* Random Variance 0 0 U 0 U Intercept 95% CI = γ ±.96* τ,945.9 ±.96* 76,09 = 96 to, U Linear Time Slope 95% CI = γ ±.96* τ 0.9 ±.96* 5,840 = 436 to 94 Quadratic Time Slope 95% CI = γ ±.96* τ 3.9 ±.96* 634 = 36 to 63 Estimates Label Estimate Error DF t Value Pr > t Intercept at Session <.000 Intercept at Session <.000 Intercept at Session <.000 Intercept at Session <.000 Intercept at Session <.000 Intercept at Session <.000 These are the quadraticmodel-predicted means (intercepts) per session. Linear Trend at Session <.000 Linear Trend at Session <.000 Linear Trend at Session <.000 Linear Trend at Session <.000 Linear Trend at Session Linear Trend at Session These are the instantaneous linear slopes at each session. Note how the SEs narrow towards the middle sessions. How well do the predicted means, variances, and covariances from the random quadratic model (3b) match the original means, variances, and covariances from the saturated means model (b)?,000 Predicted Means By Model Saturated Means Model b Random Intercept Model a Random Linear Model b Random Quadratic Model 3b 350,000 Predicted Variance By Model Unstructured Model b Random Intercept Model a Random Linear Model b Random Quadratic Model 3b, ,000 Meant RT,800,700 RT Variance 50,000 00,000,600 50,000, Session 00, Session

12 Psyc 945 Example page The quadratic model appears to be a good contender, but let s examine how well a two-slope piecewise model might fit these same data Model 4a: Fixed Slope, Fixed Slope6, Random Intercept Model Level : y Slope Slope6 e Level : Intercept: 0i 00 U0i Slope: i 0 Slope6: ti 0i i ti i ti ti i 0 TITLE "Model 4a: Fixed Slope, Fixed Slope6, Random Intercept Model"; PROC MIXED DATA=work.example34 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = Slope Slope6 / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT / V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; RUN; TITLE; TITLE "SPSS Model 4a: Fixed Slope, Fixed Slope6, Random Intercept Model". MIXED nm3rt BY ID session WITH Slope Slope6 /PRINT = SOLUTION TESTCOV G R /FIXED = Slope Slope6 /RANDOM = INTERCEPT SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID). * STATA Model 4a: Fixed Slope, Fixed Slope6, Random Intercept Model xtmixed nm3rt c.slope c.slope6, id:, /// variance reml covariance(un) residuals(independent,t(session)), estat ic, n(0), estimates store FixPiece Estimated R Matrix for ID This R matrix VC structure (equal variance over time, no covariance of any kind) will be used repeatedly as we add fixed and random piecewise slopes to the model. Estimated G Matrix Row Effect Person ID Col Intercept Estimated V Matrix for ID This random intercept model predicts a compound symmetry pattern for the V matrix (equal variance, equal covariance over time). Estimated V Correlation Matrix for ID

13 Psyc 945 Example page Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr > Z UN(,) ID <.000 Session ID <.000 NegLogLike Parms AIC AICC HQIC BIC CAIC Effect Estimate Error DF t Value Pr > t Intercept <.000 Slope <.000 RATE OF CHANGE FROM SESSION - Slope <.000 RATE OF CHANGE FROM SESSION -6 Session Number-Match 3 Slope + Slope Deviation Slope Slope Observed Means Piecewise Linear Two Slopes Slope 0 Slope RT Two Direct Slopes 96 64(Slope) 33(Slope6) Session Model 4b: Random Slope, Fixed Slope6 Model Level : y Slope Slope6 e Level : Intercept: 0i 00 U0i Slope: i 0 Ui Slope6: ti 0i i ti i ti ti i 0 TITLE "Model 4b: Random Slope, Fixed Slope6 Model"; PROC MIXED DATA=work.example34 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = Slope Slope6 / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT Slope / G GCORR V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; RUN; TITLE; TITLE "SPSS Model 4b: Random Slope, Fixed Slope6 Model".

14 Psyc 945 Example page 4 MIXED nm3rt BY ID session WITH Slope Slope6 /PRINT = SOLUTION TESTCOV G R /FIXED = Slope Slope6 /RANDOM = INTERCEPT Slope SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID). * STATA Model 4b: Random Slope, Fixed Slope6 Model xtmixed nm3rt c.slope c.slope6, id: slope, /// variance reml covariance(un) residuals(independent,t(session)), estat ic, n(0), estat recovariance, level(id), estimates store RandP, lrtest RandP FixPiece Estimated R Matrix for ID Estimated G Matrix Row Effect Person ID Col Col Intercept Slope Estimated G Correlation Matrix Row Effect Person ID Col Col Intercept Slope Estimated V Matrix for ID The pattern of variance and covariances in V is now compound symmetry from session onward because slope6 is fixed. Estimated V Correlation Matrix for ID Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 UN(,) ID UN(,) ID <.000 Session ID <.000

15 Psyc 945 Example page 5 NegLogLike Parms AIC AICC HQIC BIC CAIC Is random slope significant? Yep, ΔLL= 63, which is bigger than the critical value of 5.99ish on df =~ish Effect Estimate Error DF t Value Pr > t Intercept <.000 Slope <.000 RATE OF CHANGE FROM SESSION - Slope <.000 RATE OF CHANGE FROM SESSION -6 Model 4c: Random Slope, Random Slope6 Model Level : y Slope Slope6 e Level : Intercept: 0i 00 U0i Slope: i 0 Ui Slope6: U ti 0i i ti i ti ti i 0 i TITLE "Model 4c: Random Slope, Random Slope6 Model"; PROC MIXED DATA=work.example34 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = Slope Slope6 / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT Slope Slope6 / G GCORR V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; ESTIMATE "Session Predicted Mean" Intercept Slope 0 Slope6 0; ESTIMATE "Session Predicted Mean" Intercept Slope Slope6 0; ESTIMATE "Session 3 Predicted Mean" Intercept Slope Slope6 ; ESTIMATE "Session 4 Predicted Mean" Intercept Slope Slope6 ; ESTIMATE "Session 5 Predicted Mean" Intercept Slope Slope6 3; ESTIMATE "Session 6 Predicted Mean" Intercept Slope Slope6 4; ESTIMATE "Difference between slopes" Slope - Slope6 ; RUN; TITLE; TITLE "SPSS Model 4c: Random Slope, Random Slope6 Model". MIXED nm3rt BY ID session WITH Slope Slope6 /PRINT = SOLUTION TESTCOV G R /FIXED = Slope Slope6 /RANDOM = INTERCEPT Slope Slope6 SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID) /TEST = "Session Predicted Mean" Intercept Slope 0 Slope6 0 /TEST = "Session Predicted Mean" Intercept Slope Slope6 0 /TEST = "Session 3 Predicted Mean" Intercept Slope Slope6 /TEST = "Session 4 Predicted Mean" Intercept Slope Slope6 /TEST = "Session 5 Predicted Mean" Intercept Slope Slope6 3 /TEST = "Session 6 Predicted Mean" Intercept Slope Slope6 4 /TEST = "Difference between slopes" Slope - Slope6. * STATA Model 4c: Random Slope, Random Slope6 Model xtmixed nm3rt c.slope c.slope6, id: slope slope6, /// variance reml covariance(un) residuals(independent,t(session)), estat ic, n(0), estat recovariance, level(id), estimates store RandP, lrtest RandP RandP lincom -*c.slope + *c.slope6 // difference between slopes

16 Psyc 945 Example page 6 Estimated R Matrix for ID Estimated G Matrix Row Effect Person ID Col Col Col3 Intercept Slope Slope Estimated G Correlation Matrix Row Effect Person ID Col Col Col3 Intercept Slope Slope Estimated V Matrix for ID Estimated V Correlation Matrix for ID Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 UN(,) ID UN(,) ID <.000 UN(3,) ID UN(3,) ID UN(3,3) ID <.000 Session ID <.000 NegLogLike Parms AIC AICC HQIC BIC CAIC Is random slope6 significant? Yep, ΔLL= 44, which is bigger than the critical value of 7.8ish on df =~3ish Effect Estimate Error DF t Value Pr > t Intercept <.000 Slope <.000 RATE OF CHANGE FROM SESSION - Slope <.000 RATE OF CHANGE FROM SESSION -6

17 Psyc 945 Example page 7 Estimates Label Estimate Error DF t Value Pr > t Session Predicted Mean <.000 Session Predicted Mean <.000 Session 3 Predicted Mean <.000 Session 4 Predicted Mean <.000 Session 5 Predicted Mean <.000 Session 6 Predicted Mean <.000 Difference between slopes Random Effect 95% CI = fixed effect ±.96* Random Variance 0 Intercept 95% CI = γ ±.96* τ,96.9 ±.96* 84,3 = 97 to 3, U Slope 95% CI = γ ±.96* τ 63.6 ±.96* 63,954 = 659 to 3 0 U Slope6 95% CI = γ ±.96* τ 3.9 ±.96*,67 = 33 to 67 0 U So far we ve examined one way to fit piecewise slopes models direct slopes that represent the change during each time period. Let s now examine an alternative specification: slope + deviation slope, which can be useful in examining individual differences in differential change between time periods. Model 5c: Random Slope, Random Deviation Slope Model Level : y Slope6 Slope6 e Level : Intercept: 0i 00 U0i Slope: i 0 Ui Slope6: U ti 0i i ti i ti ti i 0 i TITLE "Model 5c: Random Slope6, Random Slope6 Model"; PROC MIXED DATA=work.example34 NOCLPRINT NOITPRINT COVTEST NAMELEN=00 IC METHOD=REML; MODEL nm3rt = csess Slope6 / SOLUTION DDFM=Satterthwaite; RANDOM INTERCEPT csess Slope6 / G GCORR V VCORR TYPE=UN SUBJECT=ID; REPEATED session / R TYPE=VC SUBJECT=ID; ESTIMATE "Slope between sessions -6" csess Slope6 ; RUN; TITLE; TITLE "SPSS Model 5c: Random Slope6, Random Slope6 Model". MIXED nm3rt BY ID session WITH csess Slope6 /PRINT = SOLUTION TESTCOV G R /FIXED = csess Slope6 /RANDOM = INTERCEPT csess Slope6 SUBJECT(ID) COVTYPE(UN) /REPEATED = session SUBJECT(ID) COVTYPE(ID) /TEST = "Slope between sessions -6" csess Slope6. * STATA Model 5c: Random Slope6, Random Slope6 Model xtmixed nm3rt c.csess c.slope6, id: csess slope6, /// variance reml covariance(un) residuals(independent,t(session)), estat ic, n(0) estat recovariance, level(id) lincom *c.csess + *c.slope6 // slope between sessions to 6

18 Psyc 945 Example page 8 Estimated R Matrix for ID Estimated G Matrix Row Effect Person ID Col Col Col3 Intercept csess Slope Estimated G Correlation Matrix Row Effect Person ID Col Col Col3 Intercept csess Slope Estimated V Matrix for ID Estimated V Correlation Matrix for ID Covariance Parameter Estimates Z Cov Parm Subject Estimate Error Value Pr Z UN(,) ID <.000 UN(,) ID UN(,) ID <.000 UN(3,) ID UN(3,) ID <.000 UN(3,3) ID <.000 Session ID <.000 NegLogLike Parms AIC AICC HQIC BIC CAIC Effect Estimate Error DF t Value Pr > t Intercept <.000 csess <.000 RATE OF CHANGE FROM SESSION - Slope DIFFERENCE IN RATE OF CHANGE FROM SESSION -6 Estimates Label Estimate Error DF t Value Pr > t Slope from session <.000

19 Random Effect 95% CI = fixed effect ±.96* Random Variance 0 Intercept 95% CI = γ ±.96* τ,96.9 ±.96* 84,3 = 97 to 3, U Slope6 95% CI = γ ±.96* τ 63.6 ±.96* 63,954 = 659 to 3 0 U Slope6 95% CI = γ ±.96* τ 30.8 ±.96* 69,96 = 338 to U Psyc 945 Example page 9 So how did we do? Let s compare model predictions in terms of means (top) and variances (bottom)?,000 Predicted Means by Session by Model (from REML),900 Response Time (milliseconds),800,700 Saturated Means, Unstructured Variance (Model 0) Random Quadratic Time (Model 3b),600,500 35, , Session Piecewise Slope and Slope6 (Model 4c) Predicted Total Variance by Session by Model (from REML) See Hoffman chapter 6 for an example results section and complete description of these models, as well as a negative exponential model for these data. Response Time (milliseconds) 75,000 50,000 5,000 Saturated Means, Unstructured Variance (Model 0) Random Quadratic Time (Model 3b) Random Slope -, Fixed Slope -6 (Model 4b) 00,000 Random Slope -, Random Slope -6 (Model 4c) 75, Session