65129Computational Issues Related To CFA• Scale of observed variables – important to keep them on a similar scale• Convergence – often related to starting values or the type of model being estimated• Program stops because maximum number of iterations has been reached• If no negative residual variances, either increase the number of iterations or use the preliminary parameter estimates as starting values• If there are large negative residual variances, try better starting values• Program stops before the maximum number of iterations has been reached• Check if variables are on a similar scale• Try new starting values• Starting values – the most important parameters to give starting values to are residual variances130Mplus MODEL Command For CFAMODEL command is used to describe the model to be estimatedBY statement is used to define the latent variables or factorsBY is short for “measured by”Example 1 – standard parameterizationMODEL: f1 BY y1 y2 y3;f2 BY y4 y5 y6;Defaults• Factor loading of first variable after BY is fixed to one• Factor loadings of other variables are estimated• Residual variances are estimated• Residual covariances are fixed to zero• Variances of factors are estimated• Covariance between the exogenous factors is estimated66131Example 2 – Alternative parameterizationMODEL: f1 BY y1* y2 y3;f2 BY y4* y5 y6;f1@1 f2@1; ! or f1-f2@1;Mplus MODEL Command For CFA(Continued)132EFA In A CFA Framework*67133EFA In A CFA FrameworkJöreskog, K.G. (1969)• Purpose• To obtain standard errors to determine if factor loadings are statistically significant• To obtain modification indices to determine if residual covariances are needed to represent minor factors• Use the same number of restrictions as an exploratory factor analysis model – m2• Fix factor variances to one for m restrictions• Fix factor loadings to zero for the remaining restrictions• Find an anchor item for each factor – select an item that has a large loading for the factor and small loadings for other factors• Fix the loading of the anchor item to zero for all of the other factors• Allow all other factor loadings to be free• Will get the same model fit as EFA*134Promax Rotated Loadings – 3 Factor Solution0.0760.3510.081FIGUREW0.0350.0610.577VISUAL0.043-0.1140.0100.8160.5720.8910.7770.728-0.0100.033-0.0390.6780.5730.6110.0370.065-0.0150.080-0.029-0.0320.115-0.1140.127OBJECT0.116NUMBERR-0.023WORDR0.015WORDM0.149WORDC-0.060SENTENCE0.009PARAGRAP0.152GENERAL0.765LOZENGES0.434PAPER0.602CUBESVerbalMemorySpatialSelecting Anchor Items*68135ESTIMATOR = ML;ANALYSIS: FILE IS holzall.dat;FORMAT IS f3,2f2,f3,2f2/3x,13(1x,f3)/3x,11(1x,f3);DATA:NAMES ARE id female grade agey agem school visual cubes paper lozenges general paragrap sentence wordcwordm addition code counting straight wordr numberrfigurer object numberf figurew deduct numeric problemr series arithmet;USEV ARE visual cubes paper lozenges general paragrap sentence wordc wordm wordr numberr object figurew;USEOBS IS school EQ 0;VARIABLE:EFA in a CFA framework using 13 variables from Holzinger and Swineford (1939)TITLE:Input For Holzinger-Swineford EFA In A CFA Framework Using 13 Variables*136Input For Holzinger-Swineford EFA In A CFA Framework Using 13 Variables (Continued)spatial BY visual-figurew*0 ! start all items at 0lozenges*1 ! start anchor item at 1cubes*1 ! start other large items at 1sentence@0 wordr@0; ! remove 2 indeterminaciesverbal BY visual-figurew*0 ! start all items at 0sentence*1 ! start anchor item at 1wordm*1 ! start other large items at 1lozenges@0 wordr@0; ! remove 2 indeterminaciesspatial-verbal@1; ! remove 3 indeterminaciesSTANDARDIZED MODINDICES(3.84) SAMPSTAT FSDETERMINACY;OUTPUT:memory BY visual-figurew*0 ! start all items at 0wordr*1 ! start anchor item at 1object*1 ! start other large items at 1lozenges@0 sentence@0; ! remove 2 indeterminaciesMODEL:*69137Tests Of Model Fit0.949Probability RMSEA <= .05SRMR (Standardized Root Mean Square Residual)0.028Value0.00090 Percent C.I.0.000Estimate39.028Value42Degrees of Freedom0.6022P-Value1.009TLIChi-Square Test of Model FitCFIRMSEA (Root Mean Square Error Of Approximation)CFI/TLI1.000Output Excerpts Holzinger-Swineford EFA In A CFA Framework Using 13 Variables0.050Factor Determinacies0.869SPATIAL0.841MEMORY0.948VERBAL*138Model Results0.793-0.6610.9380.0000.7801.8890.0000.6922.1498.0713.7174.6204.8480.4410.6631.0190.0000.7100.5260.0000.3071.0600.7650.3270.5590.8110.0980.3500.350FIGUREW-0.096-0.439-0.439OBJECT0.1270.9560.956NUMBERR0.0000.0000.000WORDR0.0700.5540.554WORDM0.1860.9940.994WORDC0.0000.0000.000SENTENCE0.0630.2120.212PARAGRAP0.1962.2782.278GENERAL0.7456.1736.173LOZENGES0.4321.2161.216PAPER0.5832.5842.584CUBES0.5713.9333.933VISUALSPATIAL BYEst./S.E.S.E. StdYXStdEstimatesOutput Excerpts Holzinger-Swineford EFA In A CFA Framework Using 13 Variables (Continued)**Note that theory predicts that GENERAL loads on VERBAL only.*701392.9234.7044.3926.1800.5900.8080.0001.030-0.0910.0001.123-0.7120.7180.4330.6460.9771.0580.7200.5400.0000.3091.1030.0000.3330.5580.8080.3531.2641.264FIGUREW0.6683.0403.040OBJECT0.5714.2914.291NUMBERR0.6066.5416.541WORDR0.0540.4250.425WORDM0.0820.4360.436WORDC0.0000.0000.000SENTENCE0.0940.3180.318PARAGRAP-0.009-0.100-0.100GENERAL0.0000.0000.000LOZENGES0.1330.3740.374PAPER-0.090-0.398-0.398CUBES0.0840.5800.580VISUALMEMORY BYOutput Excerpts Holzinger-Swineford EFA In A CFA Framework Using 13 Variables (Continued)Est./S.E.S.E. StdYXStdEstimates*1400.5700.212-0.8420.0008.7515.65612.2638.2647.6820.0000.294-0.2360.3260.4330.6531.0330.0000.7070.5170.3220.3031.0580.0000.3270.5460.8110.0690.2470.247FIGUREW0.0300.1390.139OBJECT-0.116-0.870-0.870NUMBERR0.0000.0000.000WORDR0.7826.1916.191WORDM0.5482.9272.927WORDC0.8533.9543.954SENTENCE0.7442.5012.501PARAGRAP0.7008.1308.130GENERAL0.0000.000.000LOZENGES0.0340.0960.096PAPER-0.029-0.129-0.129CUBES0.0380.2650.265VISUALVERBAL BYOutput Excerpts Holzinger-Swineford EFA In A CFA Framework Using 13 Variables (Continued)Est./S.E.S.E. StdYXStdEstimates*71141Output Excerpts Holzinger-Swineford EFA In A CFA Framework Using 13 Variables (Continued)Variances1.0001.0000.0000.0001.000SPATIAL1.0001.0000.0000.0001.000VERBAL1.0001.0000.0000.0001.000MEMORY3.1812.1733.9370.1440.1710.1190.4590.4590.459VERBAL0.3710.3710.371SPATIALMEMORY