Independence (Null) Baseline Model: Item means and variances, but NO covariances
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1 CFA Example Using Forgiveness of Situations (N = 1103) using SAS MIXED (See Example 4 for corresponding Mplus syntax and output) SAS Code to Read in Mplus Data: * Import data from Mplus, becomes var1-var23 without names at top; PROC IMPORT OUT=work.Situation DATAFILE= "&example.\study2.dat" DBMS=TAB REPLACE; GETNAMES=NO; DATAROW=1; RUN; * Rename variables; DATA Situation; SET Situation; ARRAY old(23) var1-var23; ARRAY new(23) PersonID Self1 Self2r Self3 Self4r Self5 Self6r Other1r Other2 Other3r Other4 Other5r Other6 Sit1r Sit2 Sit3r Sit4 Sit5r Sit6 Selfsub Othsub Sitsub HFSsum; DO i=1 TO 23; new(i)=old(i); IF new(i)=99999 THEN new(i)=.; END; DROP i var1-var23; RUN; * Stack situation items; DATA SituationStacked; SET Situation; ARRAY aitem(6) Sit1r Sit2 Sit3r Sit4 Sit5r Sit6; DO i=1 TO 6; itemnum=i; response=aitem(i); OUTPUT; END; DROP i; RUN; Independence (Null) Baseline Model: Item means and variances, but NO covariances TITLE "Independence (Null) CFA Model in MIXED"; REPEATED itemnum / TYPE=TOEPH(1) SUBJECT=PersonID R; RUN; Cov Parm Subject Estimate Error Value Pr > Var(1) PersonID <.0001 Var(2) PersonID <.0001 Var(3) PersonID <.0001 Var(4) PersonID <.0001 Var(5) PersonID <.0001 Var(6) PersonID <.0001 TYPE=TOEPH(1) predicts a diagonal matrix (would be the same as TYPE=UN(1). unconditional variances per item repeated in the next piece of output as Var(item). Note that this independence model predicts no covariances between items Model fit is given as 2LL rather than LL (but otherwise is the same as given from Mplus). The fixed effects show the unconditional means per item. Example 4 CFA in MIXED 1 of 5
2 Saturated (Unstructured) Baseline Model: Item means, variances, and covariances in original data TITLE "Saturated (Unstructured) CFA Model in MIXED"; REPEATED itemnum / TYPE=UN(6) SUBJECT=PersonID R RCORR; RUN; TYPE=UN(6) predicts a fullyestimated matrix without any constraints whatsoever. unconditional variances and covariances for the items. RCORR is the unconditional correlation matrix. Estimated R Correlation Matrix for PersonID Cov Parm Subject Estimate Error Value Pr UN(1,1) PersonID <.0001 UN(2,1) PersonID <.0001 UN(2,2) PersonID <.0001 UN(3,1) PersonID <.0001 UN(3,2) PersonID <.0001 UN(3,3) PersonID <.0001 UN(4,1) PersonID <.0001 UN(4,2) PersonID <.0001 UN(4,3) PersonID <.0001 UN(4,4) PersonID <.0001 UN(5,1) PersonID <.0001 UN(5,2) PersonID <.0001 UN(5,3) PersonID <.0001 UN(5,4) PersonID <.0001 UN(5,5) PersonID <.0001 UN(6,1) PersonID <.0001 UN(6,2) PersonID <.0001 UN(6,3) PersonID <.0001 UN(6,4) PersonID <.0001 UN(6,5) PersonID <.0001 UN(6,6) PersonID <.0001 Note THIS IS THE DATA the only discrepancies you d see relative to descriptive statistics would be from missing data, as these are ML estimates (that assume MAR rather than MCAR as in listwise deletion) The fixed effects again show the unconditional means per item. Example 4 CFA in MIXED 2 of 5
3 Model 1. Fully -Scored Factor Model Identification (Factor Variance = 1, Factor Mean = 0, All Loadings and Intercepts Estimated) Hoffman Psyc 948 TITLE "Single-Factor CFA Model (Factor Variance=1, Factor Mean=0) in MIXED"; RUN; REPEATED itemnum / TYPE=FA(1) SUBJECT=PersonID R RCORR; TYPE=FA(1) creates the covariance matrix that would be predicted by a single-factor model Estimated R Correlation Matrix for PersonID predicted variances and covariances for the items. RCORR is the single-factor predicted correlation matrix. THIS IS NO LONGER THE DATA. So the objective is to see how close this predicted covariance matrix is from the one given by the saturated model (which was the data). Cov Parm Subject Estimate Error Value Pr FA(1) PersonID <.0001 FA(2) PersonID <.0001 FA(3) PersonID <.0001 FA(4) PersonID <.0001 FA(5) PersonID <.0001 FA(6) PersonID <.0001 FA(1,1) PersonID <.0001 FA(2,1) PersonID <.0001 FA(3,1) PersonID <.0001 FA(4,1) PersonID <.0001 FA(5,1) PersonID <.0001 FA(6,1) PersonID < The FA(item) terms are the item residual variances. The FA(item, factor) terms are the item factor loadings. So the total variance per item is given by: loading 2 (1) + error variance, as shown in the R matrix above. Item 1 = ^ = The covariance between items is given by their loadings multiplied together. Item 1 and 2 cov = * = The fixed effects now show the Example 4 CFA in MIXED 3 of 5
4 Tau-Equivalent Items Single-Factor Model with Marker Item Factor Model Identification (Factor Variance =?, Factor Mean = 0, All Loadings Equal at 1) TITLE "Tau-Equivalent Items Single-Factor CFA Model (Factor Variance=1, Factor Mean=0) in MIXED"; RANDOM INTERCEPT / TYPE=UN SUBJECT=PersonID G V VCORR; REPEATED itemnum / TYPE=TOEPH(1) SUBJECT=PersonID R; RUN; Estimated G Matrix Person Row Effect ID Col1 1 Intercept Estimated V Matrix for PersonID Estimated V Correlation Matrix for PersonID Cov Parm Subject Estimate Error Value Pr > UN(1,1) PersonID <.0001 Var(1) PersonID <.0001 Var(2) PersonID <.0001 Var(3) PersonID <.0001 Var(4) PersonID <.0001 Var(5) PersonID <.0001 Var(6) PersonID < A random intercept creates a constant source of covariance across all items. item residual variances. The G matrix shows the variance due to the factor for all items. V is the predicted covariance matrix from putting G and R back together, and VCORR is the predicted correlation matrix. itemnum <.0001 itemnum <.0001 itemnum <.0001 itemnum <.0001 itemnum <.0001 itemnum <.0001 The fixed effects still show the Example 4 CFA in MIXED 4 of 5
5 Parallel Items Single-Factor Model with Marker Item Factor Model Identification (Factor Variance =?, Factor Mean = 0, All Loadings = 1 and All Error Variances Equal) Hoffman Psyc 948 TITLE "Parallel Items Single-Factor CFA Model (Factor Variance=1, Factor Mean=0) in MIXED"; RANDOM INTERCEPT / TYPE=UN SUBJECT=PersonID G V VCORR; REPEATED itemnum / TYPE=VC SUBJECT=PersonID R; RUN; Estimated G Matrix Person Row Effect ID Col1 1 Intercept Estimated V Matrix for PersonID A random intercept creates a constant source of covariance across all items. A Type=VC R matrix means equal residual variance across items. item residual variances. The G matrix shows the variance due to the factor for all items. V is the predicted covariance matrix from putting G and R back together, and VCORR is the predicted correlation matrix. Estimated V Correlation Matrix for PersonID This type of predicted covariance matrix has a special name: compound symmetry. Cov Parm Subject Estimate Error Value Pr > UN(1,1) PersonID <.0001 itemnum PersonID < itemnum <.0001 itemnum <.0001 itemnum <.0001 itemnum <.0001 itemnum <.0001 itemnum <.0001 The fixed effects still show the Unfortunately, multiple factor models in MIXED appear to be EFA models instead of CFA models, so no examples of two-factor models are given. Example 4 CFA in MIXED 5 of 5
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