Specification of Rasch-based Measures in Structural Equation Modelling (SEM) Thomas Salzberger www.matildabayclub.net

Specification of Rasch-based Measures in Structural Equation Modelling (SEM) Thomas Salzberger www.matildabayclub.net This document deals with the specification of a latent variable - in the framework of structural equation modelling (SEM) - that is measured utilizing a Rasch model. The Rasch measure acts as a single item indicator in SEM. The approach is illustrated using RUMM 2030 (Andrich et al., 2010) and Mplus 5.1 (Muthen and Muthen, 1998-2008). Alternatively, the measurement model may be specified in Mplus as an IRT model or a CFA treating the variables as categorical using a probit link. This possibility will be dealt with in another document (in preparation). However, this implies that distributional assumptions in terms of the person measures are to be met. Comments and corrections are welcome and should be sent to Thomas.Salzberger@wu.ac.at. They will be integrated in future updates of this document. Purification of model (assessment of fit of the data to the Rasch model) using, for example, RUMM 2030 Estimation of measures using the Rasch model (e.g., RUMM2030) Estimation of measures using IRT model or probit-cfa (e.g., Mplus) Not covered in this document Specification of Rasch measure as a single-item indicator of a latent variable in SEM (e.g. in Mplus) Comparison Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 1

Specification of Rasch-based Measures in Structural Equation Modelling (SEM) 1 Researchers are often interested in complex relationships between multiple latent variables. Such networks of latent variables are typically analysed using structural equation modelling (SEM; e.g. Mplus, AMOS, Lisrel or EQS) 2. When the measurement models are to be analysed using the Rasch model, the question arises of how to integrate the Rasch measures into the SEM-framework. This can be done quite easily treating the Rasch measure of the latent variable as a single indicator of the latent variable (see Figure 1). The same procedure can be used for exogenous (independent) as well as endogenous latent variables. Error ε 2 1 Rasch measure (as a single indicator), β 1 3 λ 4 Latent variable ξ... SEM model Figure 1: The Rasch measure as a single indicator of a latent variable ξ Four elements have to be considered. First, the Rasch measures [1; β] need to be added to the data file to be used in structural equation modelling. Typically, the data are read into SPSS first. Therefore, the measures have to be exported from RUMM and then read into SPSS (see the appendix pp. 24ff explaining how this is done). Second, an estimate of the error variance [2; ε ] is required. Third, an estimate of the regression coefficient [3; λ] from the latent variable to the Rasch measure is needed. Fourth, an estimate of the variance of the latent variable [4; ξ ] is required. The Rasch measure is a linear combination of a true component and an error component ε. The true component is the product of regression coefficient λ and the latent variable ξ: [1] λ ξ ε If we take the variances, we get: [2] λ ξ ε 1 Includes Export Rasch measures from RUMM in the appendix (pp. 26ff) 2 The term SEM model is used to denote a model represented by structural equations, even though it actually translates to structural equation modelling model. Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 2

The error variance, the regression coefficient and the variance of the latent variable have to be chosen in a way that the right hand side of equation [2] corresponds to the left hand side, that is, the variance of the Rasch measure. Obviously, there is some arbitrariness involved as only the product of the squared regression coefficient and the variance of the latent variable have to equal the true component. The reason is that the metric of the latent variable is arbitrary. In an ordinary SEM model, it is reasonable to define the metric of the latent variable by fixing its variance to 1 (Case 1; Pallant, 2007). However, there are instances, where this should not be done. In a multi-group model the same model is applied to potentially different populations. Setting the variance to 1 in all groups implies the assumption that the variance in these groups is actually equal (Case 2). Finally, a third possibility (Case 3) will be discussed. Case 1: fixing the variance of the latent variable to 1 If your software package sets the variance of the latent variable to 1 by default, then you do not have to define it. Otherwise, the variance has to be fixed to 1. Next, we determine the variance of the error. Since the variance of the Rasch measure is the total variance, which in turn is sum of the error variance and the true variance (see equation [2]), the error variance is the difference between the total variance and the true variance. [3] ε λ ξ This can be rewritten as follows: [4] ε = λ ξ The squared regression coefficient times the variance of the latent variable divided by the variance of the Rasch measures (total variance) is the proportion of true variance in the total variance (highlighted in yellow in equation [4]. This expression corresponds to the Person Separation Index (PSI; Andrich, 1982), the Rasch equivalent of reliability/cronbach s alpha. Consequently, [5] ε = [6] ε = (1 ) Of course, this is immediately evident, as (1-PSI) is the share of the variance that is due to error. If we multiply that by the total variance, we get the actual error variance. Both the variance of the Rasch measures and the PSI are provided by the Rasch analysis. In RUMM (Andrich, Sheridan and Luo, 2010) the estimates of the standard deviation and the PSI are reported on the SUMMARY STATISTICS sheet in the TEST-OF-FIT DETAILS section (see Figure 2). Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 3

Figure 2: SUMMARY STATISTICS in RUMM (person measure standard deviation and PSI) In this example, the standard deviation amounts to 1.382. The square of this value is the variance (1.9099). The PSI is 0.82437. [7] ε = 1 = 1.382 (1 0.82437) = 0.3354 Next, we determine the regression coefficient λ. Since we fix the variance of the latent variable ξ to 1, equation [2] simplifies to: [8] λ ε This can be rewritten as follows: [9] 1 λ [10] λ 1 [11] λ 1 1 [12] λ 1 1 [13] λ Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 4

The regression coefficient is therefore the standard deviation of the Rasch measures times the square root of the PSI. In the example, this implies: [14] λ= 1.382 0.82437=1.2548 In summary, we derived the following parameters: Error variance ε = 0.3354 Variance of the latent variable (fixed) ξ = 1 Regression coefficient of Rasch measure on latent variable λ = 1.2548 Next, we specify this measurement model in Mplus. [file rasch_into_sem_case1.inp] TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@1.2548;!regression coefficient rasch@0.3354;!error variance f1@1;!variance of latent variable OUTPUT: sampstat standardized; There are five variables in the data file. The first is rasch, which is the Rasch person measure of the first latent variable (f1). The second is dep, which is a second latent variable used in the data generation. This variable is not used here and therefore it is not listed on the USEVARIABLES ARE subcommand. The same applies to the variables v1, v2 and v3, which are the indicators of the second latent variable. The output looks as follows: Mplus VERSION 5.1 MUTHEN & MUTHEN 07/22/2011 1:02 PM INPUT INSTRUCTIONS TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@1.2548;!regression coefficient rasch@0.3354;!error variance f1@1;!variance of latent variable OUTPUT: sampstat standardized; INPUT READING TERMINATED NORMALLY RASCH measure into SEM Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 5

SUMMARY OF ANALYSIS Number of groups 1 Number of observations 501 Number of dependent variables 1 Number of independent variables 0 Number of continuous latent variables 1 Observed dependent variables Continuous RASCH Continuous latent variables 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) C:\raschsem.dat Input data format FREE SAMPLE STATISTICS SAMPLE STATISTICS Means RASCH 1 0.113 Covariances RASCH RASCH 1.907 Correlations RASCH RASCH 1.000 THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 0.000 Degrees of Freedom 1 P-Value 0.9826 Chi-Square Test of Model Fit for the Baseline Model Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 6

Value 0.000 Degrees of Freedom 0 P-Value 0.0000 CFI/TLI Loglikelihood CFI 1.000 TLI 1.000 H0 Value -872.633 H1 Value -872.632 Information Criteria Number of Free Parameters 1 Akaike (AIC) 1747.265 Bayesian (BIC) 1751.482 Sample-Size Adjusted BIC 1748.308 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <=.05 0.991 SRMR (Standardized Root Mean Square Residual) Value 0.001 MODEL RESULTS RASCH 1.255 0.000 999.000 999.000 RASCH 0.113 0.062 1.837 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 STANDARDIZED MODEL RESULTS Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 7

STDYX Standardization RASCH 0.908 0.000 999.000 999.000 RASCH 0.082 0.045 1.837 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 STDY Standardization RASCH 0.908 0.000 999.000 999.000 RASCH 0.082 0.045 1.837 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 STD Standardization RASCH 1.255 0.000 999.000 999.000 RASCH 0.113 0.062 1.837 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 R-SQUARE Observed Variable RASCH 0.824 0.000 999.000 999.000 Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 8

The sections highlighted in green refer to the fit of the model, the parameter estimates we provided and the explained variance (r²). The fit is perfect in this case. It means that the estimates we put into Mplus correctly recover the actual variance of the Rasch measures. The values have been properly specified and the r² corresponds to the PSI. Case 2: fixing the variance of the latent variable to the actual true variance As mentioned above, in some cases, the variance of the latent variable should not be set to 1. Therefore, we know fix the variance of the latent variable at its actual value from the Rasch analysis. The error variance is determined as in case 1. Next, we determine the regression coefficient. We recall equation 2: [2] = λ ξ + ε [15] λ = ε Since the difference between the total variance and the error variance is the true variance, and since we want to fix the variance of the latent variance to the true variance, it follows that: [16] λ = 1 Therefore, in case 2 equation 2 simplifies to: [17] = ξ + ε [18] ξ = ε Since ε = (1 ): [19] ξ = (1 ) [20] ξ = (1 [1 ]) [21] ξ = Again, equation 21 is obvious as the share of the true variance in the total variance (PSI) multiplied by the total variance has to be the true variance. In the example, the variance of the latent variable is: [22] ξ = = 1.9099 0.82437 = 1.5745 ξ Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 9

In summary, we derived the following parameters: Error variance ε = 0.3354 Variance of the latent variable ξ = 1.5745 Regression coefficient of Rasch measure on latent variable λ = 1 Next, we specify this measurement model in Mplus. [file rasch_into_sem_case2.inp] TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@1;!regression coefficient rasch@0.3354;!error variance f1@1.5745;!variance of latent variable OUTPUT: sampstat standardized; The output looks as follows: Mplus VERSION 5.1 MUTHEN & MUTHEN 07/22/2011 1:30 PM INPUT INSTRUCTIONS TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@1;!regression coefficient rasch@0.3354;!error variance f1@1.5745;!variance of latent variable OUTPUT: sampstat standardized; INPUT READING TERMINATED NORMALLY RASCH measure into SEM SUMMARY OF ANALYSIS Number of groups 1 Number of observations 501 Number of dependent variables 1 Number of independent variables 0 Number of continuous latent variables 1 Observed dependent variables Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 10

Continuous RASCH Continuous latent variables 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) C:\raschsem.dat Input data format FREE SAMPLE STATISTICS SAMPLE STATISTICS Means RASCH 1 0.113 Covariances RASCH RASCH 1.907 Correlations RASCH RASCH 1.000 THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 0.000 Degrees of Freedom 1 P-Value 0.9828 Chi-Square Test of Model Fit for the Baseline Model CFI/TLI Value 0.000 Degrees of Freedom 0 P-Value 0.0000 CFI 1.000 TLI 1.000 Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 11

Loglikelihood H0 Value -872.633 H1 Value -872.632 Information Criteria Number of Free Parameters 1 Akaike (AIC) 1747.265 Bayesian (BIC) 1751.482 Sample-Size Adjusted BIC 1748.308 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <=.05 0.991 SRMR (Standardized Root Mean Square Residual) Value 0.001 MODEL RESULTS RASCH 1.000 0.000 999.000 999.000 RASCH 0.113 0.062 1.837 0.066 1.574 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 STANDARDIZED MODEL RESULTS STDYX Standardization RASCH 0.908 0.000 999.000 999.000 RASCH 0.082 0.045 1.837 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 12

STDY Standardization RASCH 0.908 0.000 999.000 999.000 RASCH 0.082 0.045 1.837 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 STD Standardization RASCH 1.255 0.000 999.000 999.000 RASCH 0.113 0.062 1.837 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 R-SQUARE Observed Variable RASCH 0.824 0.000 999.000 999.000 As can be seen from the Mplus output, the solution in case 2 is equivalent to case 1. However, the cases differ in terms of the unstandardized parameters. Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 13

Case 3: fixing the variance of the latent variable to the total variance Mathieu, Tannenbaum and Salas (1992) suggest that the path from a latent variable to its corresponding observed variable (lambda] is equal to the square root of the reliability of the observed score (p.837). The error variance is defined in the same way as in cases 1 and 2 ( In addition, the associated amount of random error variance (theta) is equal to one minus the reliability of the observed score times the variance of the observed score (p. 837)). Fixing the regression coefficient λ to the square root of the PSI implies that the variance of the latent variable equals total variance. We recall equation 2: [2] = λ ξ + ε If λ =, then λ =. [23] = ξ + ε [24] ξ = ε Since the PSI is the true variance (which is the total variance minus the error variance ) divided by the total variance, if follows that: [25] ξ = ε ε = Consequently, fixing the regression coefficient λ to means that we have to fix the variance of the latent variable ξ to the total variance. In summary, we derived the following parameters: Error variance ε = 0.3354 Variance of the latent variable ξ = 1.9099 Regression coefficient of Rasch measure on latent variable λ = 0.9079 Next, we specify this measurement model in Mplus. [file rasch_into_sem_case3.inp] TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@0.9079;!regression coefficient rasch@0.3354;!error variance f1@1.909;!variance of latent variable OUTPUT: sampstat standardized; Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 14

The output looks as follows: Mplus VERSION 5.1 MUTHEN & MUTHEN 07/22/2011 1:50 PM INPUT INSTRUCTIONS TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@0.9079;!regression coefficient rasch@0.3354;!error variance f1@1.909;!variance of latent variable OUTPUT: sampstat standardized; INPUT READING TERMINATED NORMALLY RASCH measure into SEM SUMMARY OF ANALYSIS Number of groups 1 Number of observations 501 Number of dependent variables 1 Number of independent variables 0 Number of continuous latent variables 1 Observed dependent variables Continuous RASCH Continuous latent variables 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) C:\raschsem.dat Input data format FREE SAMPLE STATISTICS SAMPLE STATISTICS Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 15

Means RASCH 1 0.113 Covariances RASCH RASCH 1.907 Correlations RASCH RASCH 1.000 THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 0.000 Degrees of Freedom 1 P-Value 0.9890 Chi-Square Test of Model Fit for the Baseline Model CFI/TLI Loglikelihood Value 0.000 Degrees of Freedom 0 P-Value 0.0000 CFI 1.000 TLI 1.000 H0 Value -872.633 H1 Value -872.632 Information Criteria Number of Free Parameters 1 Akaike (AIC) 1747.265 Bayesian (BIC) 1751.482 Sample-Size Adjusted BIC 1748.308 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 0.000 Probability RMSEA <=.05 0.994 SRMR (Standardized Root Mean Square Residual) Value 0.001 Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 16

MODEL RESULTS RASCH 0.908 0.000 999.000 999.000 RASCH 0.113 0.062 1.838 0.066 1.909 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 STANDARDIZED MODEL RESULTS STDYX Standardization RASCH 0.908 0.000 999.000 999.000 RASCH 0.082 0.045 1.838 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 STDY Standardization RASCH 0.908 0.000 999.000 999.000 RASCH 0.082 0.045 1.838 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 STD Standardization Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 17

RASCH 1.254 0.000 999.000 999.000 RASCH 0.113 0.062 1.838 0.066 1.000 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 R-SQUARE Observed Variable RASCH 0.824 0.000 999.000 999.000 Summary The three cases cover reasonable ways to fix the variance of the latent variable (see Table 1). However, since the metric is arbitrary, any value could be chosen. Then the regression coefficient has to be specified in a way that λ ξ = PSI is satisfied. Likewise, any value can be chosen for the regression coefficient as long as the variance of the latent variable is specified accordingly. Case 1 corresponds to a standard practice in SEM, where the variance of the latent variable is often fixed to 1. Case 2 might be preferred, as the variance of the latent variable actually reflects the true value. However, true has of course no absolute meaning. It just means that it refers to the Rasch metric, which is determined by the scaling constant of 1 in the Rasch model, regressed to the mean. In case 3, the metric of the latent variable corresponds to the Rasch metric without taking error variance into account. Variance of latent variable ξ Regression coefficient λ Error variance ε λ ξ Case 1 (Pallant, 2007) 1 Case 2 Case 3 (Mathieu et al., 1992) Actual true variance Actual total variance 1 λ= Actual error variance (1 ) PSI Actual error variance (1 ) PSI Table 1: Possible parameter specifications Actual error variance (1 ) PSI a General case Actual error variance (1 ) PSI ξ = a Actual error variance (1 ) PSI Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 18

Table 2 shows the actual values used in the example. Case 1 Case 2 Case 3 Variance latent variable ξ 1 1.5745 1.9099 Regression coefficient λ 1.2548 1 0.9079 Error variance ε 0.3354 0.3354 0.3354 λ ξ 1.575 1.575 1.575 Table 2: Parameter specifications used in the example Using Rasch measure in a two-factor structural model We may now use the latent variable based on the Rasch measure as a predictor of the dep variable. Dep is measured by three indicators v1, v2 and v3. In the following, case 1 specification is used. However, the other cases provide exactly the same standardized solution. [file rasch_into_sem_2f_case1.inp] TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch v1 v2 v3; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@1.2548; f2 by v1 v2 v3; f2 on f1; rasch@0.3354; f1@1; OUTPUT: sampstat standardized; The output looks as follows: Mplus VERSION 5.1 MUTHEN & MUTHEN 07/22/2011 9:10 PM INPUT INSTRUCTIONS TITLE: RASCH measure into SEM DATA: FILE IS C:\raschsem.dat; VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch v1 v2 v3; ANALYSIS: TYPE = GENERAL; MODEL: f1 by rasch@1.2548; f2 by v1 v2 v3; f2 on f1; rasch@0.3354; f1@1; OUTPUT: sampstat standardized; INPUT READING TERMINATED NORMALLY RASCH measure into SEM Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 19

SUMMARY OF ANALYSIS Number of groups 1 Number of observations 501 Number of dependent variables 4 Number of independent variables 0 Number of continuous latent variables 2 Observed dependent variables Continuous RASCH V1 V2 V3 Continuous latent variables F2 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) C:\raschsem.dat Input data format FREE SAMPLE STATISTICS SAMPLE STATISTICS Means RASCH V1 V2 V3 1 0.113 0.219 0.155 0.236 Covariances RASCH V1 V2 V3 RASCH 1.907 V1 1.731 10.890 V2 1.833 10.000 11.163 V3 1.959 10.143 10.321 11.377 Correlations RASCH V1 V2 V3 RASCH 1.000 V1 0.380 1.000 V2 0.397 0.907 1.000 V3 0.421 0.911 0.916 1.000 THE MODEL ESTIMATION TERMINATED NORMALLY Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 20

TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 4.675 Degrees of Freedom 3 P-Value 0.1972 Chi-Square Test of Model Fit for the Baseline Model CFI/TLI Loglikelihood Value 2008.088 Degrees of Freedom 6 P-Value 0.0000 CFI 0.999 TLI 0.998 H0 Value -3815.214 H1 Value -3812.877 Information Criteria Number of Free Parameters 11 Akaike (AIC) 7652.429 Bayesian (BIC) 7698.811 Sample-Size Adjusted BIC 7663.897 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.033 90 Percent C.I. 0.000 0.089 Probability RMSEA <=.05 0.612 SRMR (Standardized Root Mean Square Residual) Value 0.007 MODEL RESULTS F2 F2 RASCH 1.255 0.000 999.000 999.000 V1 1.000 0.000 999.000 999.000 V2 1.018 0.022 47.230 0.000 V3 1.034 0.021 48.449 0.000 ON 1.449 0.144 10.075 0.000 RASCH 0.113 0.062 1.837 0.066 Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 21

V1 0.219 0.147 1.483 0.138 V2 0.155 0.149 1.041 0.298 V3 0.236 0.151 1.564 0.118 1.000 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 V1 1.080 0.100 10.816 0.000 V2 0.994 0.099 10.079 0.000 V3 0.886 0.097 9.167 0.000 F2 7.712 0.573 13.466 0.000 STANDARDIZED MODEL RESULTS STDYX Standardization F2 F2 RASCH 0.908 0.000 999.000 999.000 V1 0.949 0.006 168.293 0.000 V2 0.954 0.005 179.386 0.000 V3 0.960 0.005 192.377 0.000 ON 0.463 0.040 11.694 0.000 RASCH 0.082 0.045 1.837 0.066 V1 0.066 0.045 1.482 0.138 V2 0.047 0.045 1.041 0.298 V3 0.070 0.045 1.562 0.118 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 V1 0.099 0.011 9.265 0.000 V2 0.089 0.010 8.763 0.000 V3 0.078 0.010 8.120 0.000 F2 0.786 0.037 21.475 0.000 STDY Standardization F2 RASCH 0.908 0.000 999.000 999.000 V1 0.949 0.006 168.293 0.000 V2 0.954 0.005 179.386 0.000 Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 22

V3 0.960 0.005 192.377 0.000 F2 ON 0.463 0.040 11.694 0.000 RASCH 0.082 0.045 1.837 0.066 V1 0.066 0.045 1.482 0.138 V2 0.047 0.045 1.041 0.298 V3 0.070 0.045 1.562 0.118 1.000 0.000 999.000 999.000 Residual RASCH 0.176 0.000 999.000 999.000 V1 0.099 0.011 9.265 0.000 V2 0.089 0.010 8.763 0.000 V3 0.078 0.010 8.120 0.000 F2 0.786 0.037 21.475 0.000 STD Standardization F2 F2 RASCH 1.255 0.000 999.000 999.000 V1 3.132 0.109 28.856 0.000 V2 3.189 0.109 29.156 0.000 V3 3.239 0.110 29.489 0.000 ON 0.463 0.040 11.694 0.000 RASCH 0.113 0.062 1.837 0.066 V1 0.219 0.147 1.483 0.138 V2 0.155 0.149 1.041 0.298 V3 0.236 0.151 1.564 0.118 1.000 0.000 999.000 999.000 Residual RASCH 0.335 0.000 999.000 999.000 V1 1.080 0.100 10.816 0.000 V2 0.994 0.099 10.079 0.000 V3 0.886 0.097 9.167 0.000 F2 0.786 0.037 21.475 0.000 Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 23

R-SQUARE Observed Variable RASCH 0.824 0.000 999.000 999.000 V1 0.901 0.011 84.146 0.000 V2 0.911 0.010 89.693 0.000 V3 0.922 0.010 96.189 0.000 Latent Variable F2 0.214 0.037 5.847 0.000 The latent correlation of the Rasch-based latent variable and the dependent latent variable F2 is 0.463 (highlighted in yellow). Without correction for error variance in the Rasch measure, the correlation is just 0.412 (from SPSS). Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 24

Appendix: Export Rasch measures from RUMM Person measures can be taken from the INDIVIDUAL PERSON FIT sheet (in the section TEST-OF-FIT DETAILS). Highlight the column LOCATION and click on the COPY button at the bottom right of the sheet. Figure 3: INDIVIDUAL PERSON-FIT in RUMM It is recommended to highlight and copy more columns than just the location column. Specifically the ID column (presuming the person id has been named so) should be chosen as this allows for a proper matching of the person measures when they are merged with the original data set in SPSS. Then copy the estimates into a new SPSS data file. If it does not work (because the first row contains the names of the variables), paste the estimates first into EXCEL and then into SPSS. Using the ID as a matching criterion the file can then be merged with the existing data file in SPSS (DATA/MERGE FILES/ADD VARIABLES ). You may also paste the location estimates from one SPSS file into the other. However this requires that both files are properly ordered by ID. Moreover, if there are persons whose entire response patterns consist of missing values, then these records will be missing in the RUMM sheet of person locations. If you also paste the ID (under a different name), the ID from RUMM can be correlated with the original ID in SPSS. If this correlation is not perfect, then the merging has failed. Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 25

A more elegant way to export person measures from RUMM is using the SAVE button at the bottom of the sheet (see Figure 3). The file has the extension.prn but is just a plain ASCI document that can be renamed to a.txt or a.dat file. It looks like this: RUMM2030 Project: MAIN Analysis: RUNALL Title: RUNALL Date: 20 Jul 2011 09:07:54 Display: INDIVIDUAL PERSON-FIT - Location Order ----------------------------------------------------------------------------------------------------------------------------- ID Total Max Miss Extreme Locn SE Residual DegFree DataPts id gender edu place age ----------------------------------------------------------------------------------------------------------------------------- 1 496 592 93 0.774 0.10 4.471 # 91.30 93 97 2 4 1 3 2 221 404 46 0.368 0.04 2.871 # 45.10 46 184 1 2 1 1 3 248 592 93 0.244 0.04 6.838 # 91.30 93 249 2 3 1 3 4 257 592 93-0.055 0.06 16.702 # 91.30 93 17 2 4 1 1 This file can be read into SPSS. However, before that scroll down to the very end of the file and delete the summary there. The header is no problem as SPSS can be instructed to read the first record in line 9 (the first line is empty in the RUMM file, so it is really the 9 th row). The file can be read in using SPSS syntax. In the following, the file exported from RUMM was renamed to somi_measures_to_spss.txt. All the specifications up to datapts 61-68 F8.2 can be used in any case, as these columns are identical for every project. The remaining variables (id and person factors) are user-defined. However, you do not need to import them into SPSS anyway. [file read_rumm_into_spss-syntax.sps] GET DATA /TYPE = TXT /FILE = 'C:\Users\Thomas\Desktop\somi_measures_to_spss.txt' /FIXCASE = 1 /ARRANGEMENT = FIXED /FIRSTCASE = 9 /IMPORTCASE = ALL /VARIABLES = /1 record 0-4 F5.2 Total 5-9 F5.2 Max 10-14 F5.2 Miss 15-19 F5.2 extreme 20-30 1.2 location 31-36 F6.2 se 37-42 F6.2 residual 43-51 F9.2 reswarn 52-54 A3 df 55-60 F6.2 datapts 61-68 F8.2 id 69-77 F9.2 gender 78-85 F8.2 study 86-96 1.2 type 97-107 1.2 citiz 108-118 1.2 abroad 119-129 1.2 V18 130-130 A1. EXECUTE. Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 26

Alternatively, the file can be read into SPSS by going through the SPSS menu FILE/OPEN/DATA/file type.dat or.txt and then either defining the variables or making use of a predefined format file (like the read_rumm_into_spss.tpf file which corresponds to the syntax above). References Andrich, D. (1982). An Index of Person Separation in Latent Trait Theory, the Traditional KR-20 Index, and the Guttman Scale Response Pattern. Education Research and Perspectives, 9 (1), 95-104. Andrich, D., Sheridan, B.S., Luo, G. (2010). Rumm 2030: Rasch Unidimensional Measurement Models [computer software], http://www.rummlab.com.au/, RUMM Laboratory Perth, Western Australia. Mathieu, J. E., Tannenbaum, S. I., & Salas, E. (1992). Influences of individual and situational characteristics on measures of training effectiveness. Academy of Management Journal, 35, 828-847. Muthén, B. and L. Muthén (1998, 2001, 2004, 2006, 2008), Mplus version 5.1 (http://www.statmodel.com/) [computer software], Muthén & Muthén, Los Angeles, CA. Pallant, J. (2007). Step by step guide to using AMOS. Prepared for the Psychometric Laboratory, Academic Department of Rehabilitation Medicine, Faculty of Medicine and Health, University of Leeds, http://www.leeds.ac.uk/medicine/rehabmed/psychometric/. Files associated with this document (in raschsem.zip) Mplus input files (.inp) and output files (.out) for the three cases: rasch_into_sem_case1.inp rasch_into_sem_case2.inp rasch_into_sem_case3.inp rasch_into_sem_case1.out rasch_into_sem_case2.out rasch_into_sem_case3.out Mplus input file (.inp) and output file (.out) for the example involving a dependent latent variable rasch_into_sem_2f_case1.inp rasch_into_sem_2f_case1.out rasch_into_sem_2f_case2.inp rasch_into_sem_2f_case2.out SPSS-files to read in data from RUMM read_rumm_into_spss-syntax.sps read_rumm_into_spss.tpf Data file used in the example raschsem.dat Excel file providing the required specifications of the Rasch-based latent variable in SEM rasch_measures_into_sem.xlsx Specification of Rasch-based Measures in Structural Equation Modelling, Thomas Salzberger, August 2011, www.matildabayclub.net 27