Confirmatory factor analysis (CFA) is a measurement model that estimates continuous latent variables based on observed indicator variables (also called manifest variables). The observed indicator variables may be either categorical or continuous. One way to think about confirmatory factor analysis is that each case has a "true score" on the (continuous) latent variable, and that each of the observed values is a result of that "true score" plus measurement error. The model attempts to estimate that "true score" based on the relationships among the observed values.
Mplus version 5.2 was used for these examples.
1.0 A Measurement Model for a Single Latent Variable
The examples on this page use data on the attributes of a group of students (see note at the bottom of the page for information on the source). The dataset (https://stats.idre.ucla.edu/wp-content/uploads/2016/02/wordland_data.dat) contains 12 observed variables, which can be used to estimate four latent variables. The 12 observed variables have all been standardized to have a mean of zero and a standard deviation of one. The four latent variables are students’ family “risk factors” (family), cognitive ability based on standardized tests (cognitive/cog), achievement, that is grades, in school (achieve), and classroom adjustment based on ratings by each student’s teacher (adjust). As a first step, we will estimate a model for a single latent variable. The diagram below shows the measurement model for the adjustment latent variable (adjust). The observed variables, represented as empty boxes are motivation (motiv), extraversion (extra), harmony (harm), and stability (stabi).
The input file shown below estimates the model described above. In the model: command, the keyword by indicates that the latent variable named before the by is measured by the manifest variables listed after it.
Title: Measurement model for one latent variable Data: File is worland_data.dat ; Variable: Names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; usevariables are motiv extra harm stabi; Model: adjust by motiv extra harm stabi;
The output based on this input file is shown below.
INPUT READING TERMINATED NORMALLY SUMMARY OF ANALYSIS Number of groups 1 Number of observations 500 Number of dependent variables 4 Number of independent variables 0 Number of continuous latent variables 1 Observed dependent variables Continuous MOTIV EXTRA HARM STABI Continuous latent variables ADJUST Estimator ML Information matrix OBSERVED Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Maximum number of iterations for H1 2000 Convergence criterion for H1 0.100D-03 Input data file(s) worland_data.dat Input data format FREE SUMMARY OF DATA Number of missing data patterns 1 COVARIANCE COVERAGE OF DATA Minimum covariance coverage value 0.100 PROPORTION OF DATA PRESENT Covariance Coverage MOTIV EXTRA HARM STABI ________ ________ ________ ________ MOTIV 1.000 EXTRA 1.000 1.000 HARM 1.000 1.000 1.000 STABI 1.000 1.000 1.000 1.000 THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 218.606 Degrees of Freedom 2 P-Value 0.0000 Chi-Square Test of Model Fit for the Baseline Model Value 927.867 Degrees of Freedom 6 P-Value 0.0000 CFI/TLI CFI 0.765 TLI 0.295 Loglikelihood H0 Value -2481.245 H1 Value -2371.942 Information Criteria Number of Free Parameters 12 Akaike (AIC) 4986.489 Bayesian (BIC) 5037.065 Sample-Size Adjusted BIC 4998.976 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.465 90 Percent C.I. 0.414 0.519 Probability RMSEA <= .05 0.000 SRMR (Standardized Root Mean Square Residual) Value 0.113 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value ADJUST BY MOTIV 1.000 0.000 999.000 999.000 EXTRA 0.211 0.053 4.002 0.000 HARM 0.954 0.056 17.086 0.000 STABI 0.722 0.050 14.582 0.000 Intercepts MOTIV 0.000 0.045 0.000 1.000 EXTRA 0.000 0.045 0.000 1.000 HARM 0.000 0.045 0.000 1.000 STABI 0.000 0.045 0.000 1.000 Variances ADJUST 0.811 0.074 11.016 0.000 Residual Variances MOTIV 0.187 0.041 4.505 0.000 EXTRA 0.962 0.061 15.693 0.000 HARM 0.259 0.040 6.499 0.000 STABI 0.575 0.041 14.055 0.000 QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.385E-01 (ratio of smallest to largest eigenvalue)
In the MODEL RESULTS section of the above output, the first block of estimates labeled ADJUST ON contains the path coefficients for the relationship between the individual items and the latent variable. All of the path coefficients (shown in the Estimates column) are positive, indicating a positive relationship between the latent variable adjustment, and our four observed measures of adjustment. In the far right column, we can also see that each of the path coefficients is significantly different from zero. The subsequent blocks show the intercepts for the observed variables (labeled Intercepts), the variance of the latent variable adjust (labeled Variances), and the estimates of the error variance for each of the observed variables (labeled Residual Variances).
2.0 A Measurement Model with Multiple (Correlated) Latent Variables
In this example, the model estimates all four latent variables at the same time, and allows the latent variables to covary, without imposing additional structure. A model with all of the latent variables allowed to covary is often run as a precursor to a model with a more specific set of relationships among the latent variables. The desired model is shown in the diagram below. Note that the curved double-headed arrows denote covariances.
The input file for this model is similar to the last. This model contains instructions for four latent variables, each measured by a series of observed variables (e.g. family by ppsych ses;).
Title: Measurement model with correlations Data: File is worland_data.dat ; Variable: Names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; Model: adjust by motiv extra harm stabi; family by ppsych ses; cog by verbal vissp mem; achieve by read arith spell;
The output based on this input file is shown below.
INPUT READING TERMINATED NORMALLY SUMMARY OF ANALYSIS Number of groups 1 Number of observations 500 Number of dependent variables 12 Number of independent variables 0 Number of continuous latent variables 4 Observed dependent variables Continuous PPSYCH SES VERBAL VISSP MEM READ ARITH SPELL MOTIV EXTRA HARM STABI Continuous latent variables FAMILY COG ACHIEVE ADJUST Estimator ML Information matrix OBSERVED Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Input data file(s) worland_data.dat Input data format FREE THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 600.106 Degrees of Freedom 48 P-Value 0.0000 Chi-Square Test of Model Fit for the Baseline Model Value 4124.707 Degrees of Freedom 66 P-Value 0.0000 CFI/TLI CFI 0.864 TLI 0.813 Loglikelihood H0 Value -6745.325 H1 Value -6445.272 Information Criteria Number of Free Parameters 42 Akaike (AIC) 13574.649 Bayesian (BIC) 13751.663 Sample-Size Adjusted BIC 13618.352 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.152 90 Percent C.I. 0.141 0.163 Probability RMSEA <= .05 0.000 SRMR (Standardized Root Mean Square Residual) Value 0.063 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value FAMILY BY PPSYCH 1.000 0.000 999.000 999.000 SES -1.107 0.115 -9.657 0.000 COG BY VERBAL 1.000 0.000 999.000 999.000 VISSP 0.833 0.045 18.393 0.000 MEM 0.972 0.044 22.326 0.000 ACHIEVE BY READ 1.000 0.000 999.000 999.000 ARITH 0.842 0.034 24.840 0.000 SPELL 0.954 0.027 35.622 0.000 ADJUST BY MOTIV 1.000 0.000 999.000 999.000 EXTRA 0.233 0.048 4.813 0.000 HARM 0.857 0.042 20.295 0.000 STABI 0.662 0.045 14.615 0.000 FAMILY WITH COG -0.411 0.046 -8.852 0.000 ACHIEVE -0.363 0.044 -8.151 0.000 ADJUST -0.245 0.040 -6.099 0.000 COG WITH ACHIEVE 0.740 0.056 13.305 0.000 ADJUST 0.508 0.048 10.510 0.000 ACHIEVE WITH ADJUST 0.567 0.051 11.102 0.000 Intercepts PPSYCH 0.000 0.045 0.000 1.000 SES 0.000 0.045 0.000 1.000 VERBAL 0.000 0.045 0.000 1.000 VISSP 0.000 0.045 0.000 1.000 MEM 0.000 0.045 0.000 1.000 READ 0.000 0.045 0.000 1.000 ARITH 0.000 0.045 0.000 1.000 SPELL 0.000 0.045 0.000 1.000 MOTIV 0.000 0.045 0.000 1.000 EXTRA 0.000 0.045 0.000 1.000 HARM 0.000 0.045 0.000 1.000 STABI 0.000 0.045 0.000 1.000 Variances FAMILY 0.379 0.061 6.201 0.000 COG 0.739 0.063 11.678 0.000 ACHIEVE 0.897 0.064 14.002 0.000 ADJUST 0.901 0.070 12.842 0.000 Residual Variances PPSYCH 0.619 0.053 11.681 0.000 SES 0.534 0.055 9.652 0.000 VERBAL 0.259 0.024 10.967 0.000 VISSP 0.485 0.035 13.679 0.000 MEM 0.300 0.026 11.643 0.000 READ 0.101 0.014 7.142 0.000 ARITH 0.362 0.027 13.612 0.000 SPELL 0.181 0.016 11.387 0.000 MOTIV 0.097 0.032 3.049 0.002 EXTRA 0.949 0.060 15.702 0.000 HARM 0.336 0.033 10.318 0.000 STABI 0.604 0.042 14.269 0.000 QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.320E-02 (ratio of smallest to largest eigenvalue)
Looking at the MODEL RESULTS section of the output, the first four blocks of estimates give the path coefficients for the relationship between the latent variables and the observed variables (e.g. FAMILY BY). After the path coefficients for the four latent variables, the covariances between the latent variables (indicated using the keyword WITH) are shown. Looking at the first block of covariances (labeled FAMILY WITH) we see that the latent variable family (i.e. family risk factors) has a negative relationship with cog (cognitive ability), achieve (academic achievement), and adjust (classroom adjustment). Note that our input file does not explicitly include these covariances, Mplus includes them by default.
3.0 Saving Factor Scores
In addition to the output file produced by Mplus, it is possible to save factor scores for each case in a text file that can later be used by Mplus or read into another statistical package. To do this the savedata: command is added to the input file. The file option gives the name of the file in which the factor scores should be saved (i.e. scores.txt). Whenever the file option is used, all of the variables used in the analysis are saved in an external file. The save = fscores; option specifies that the factor scores should be saved, in addition to the variables used in estimation. Additional variables that were not used in the analysis, but which you wish to include in the saved file, for example, an id variable, can be included by adding the auxiliary option (e.g. auxiliary = id;) to the variable: command.
Title: Saving Factor Scores Data: File is worland_data.dat ; Variable: Names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; Model: adjust by motiv extra harm stabi; family by ppsych ses; cog by verbal vissp mem; achieve by read arith spell; Savedata: file is scores.txt; save = fscores;
The output file for this model contains all of the information contained in the output for the previous model, plus additional output associated with the savedata: command. This additional output appears towards the end of the output file, and is shown below.
SAVEDATA INFORMATION Order and format of variables PPSYCH F10.3 SES F10.3 VERBAL F10.3 VISSP F10.3 MEM F10.3 READ F10.3 ARITH F10.3 SPELL F10.3 MOTIV F10.3 EXTRA F10.3 HARM F10.3 STABI F10.3 ADJUST F10.3 FAMILY F10.3 COG F10.3 ACHIEVE F10.3 Save file scores.txt Save file format 16F10.3 Save file record length 5000
The additional output associated with the savedata: command lists the variables in the order in which they appear in the saved dataset. Note that the 12 observed variables used in estimation are listed first, followed by four variables containing the factor scores associated with each of the four latent variables. Below the list of variables the name of the file, and information on the format of the file are shown.
The file class.txt is a text file that can be read by a large number of programs. The first few lines of this file are shown below. This file contains 16 variables, each in its own column. Based on the information in the output file, we know that the first 12 columns contain each student's value on the 12 observed variables, and the final four columns are each student's factor score for each of the four latent variables.
-1.780 0.477 -0.790 -0.363 0.311 -0.349 -0.999 -0.657 -0.791 -0.496 -0.508 -0.314 -0.693 -0.318 -0.239 -0.509 0.701 -0.605 -0.955 -0.769 -0.398 -0.452 0.820 0.878 0.175 -0.240 -0.416 0.352 0.055 0.452 -0.421 -0.013 2.373 -1.697 -0.130 -0.391 0.146 -0.482 0.753 -0.569 1.447 0.293 -0.454 0.407 0.926 0.809 -0.333 -0.262 0.149 0.140 1.752 2.141 -0.189 -0.314 0.573 -0.292 -0.117 -0.174 -0.567 0.260 -0.134 -0.315 0.536 0.011 -0.599 -1.838 0.675 -0.144 -0.246 -0.201 -0.062 -0.102 -0.422 0.366 -1.007 -0.603 -0.498 0.253 -0.069 -0.123
The data for these examples is based on a correlation matrix published in Worland et al. (1984). Although the correlation matrix would have been sufficient to specify these models, 500 cases were randomly drawn from the distribution described by the published correlation matrix. The models below do not necessarily match those specified in Worland et al. (1984), they are intended as examples only.
Worland, Julien, David G. Weeks, Cynthia L. Janes, and Barbara D. Strock (1984) Intelligence, classroom behavior, and academic achievement in children at high and low risk for psychopathology: A structural equation analysis. Journal of Abnormal Child Psychology Vol. 12, No. 3, pp. 437-454.