Specification of Rasch-based Measures in Structural Equation Modelling (SEM) Thomas Salzberger

(1)

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

[email protected]

.

They will be integrated in future updates of this document.

Not covered in this document

Estimation of measures using the Rasch

model (e.g., RUMM2030)

Specification of Rasch measure as a

single-item indicator of a latent variable

in SEM (e.g. in Mplus)

Purification of model (assessment of fit of the

data to the Rasch model) using, for example,

RUMM 2030

Estimation of measures using IRT model

or probit-CFA (e.g., Mplus)

(2)

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.

Figure 1: The Rasch measure as a single indicator of a latent variable

ξ

Four elements have to be considered.

•

First, the Rasch measures [

;

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

[

;

ε

] is required.

•

Third, an estimate of the

regression coefficient

[

;

λ

] from the latent variable to the

Rasch measure is needed.

•

Fourth, an estimate of the

variance of the latent variable

[

;

_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}

... SEM model

Rasch measure

(as a single

indicator),

β

1

Latent variable

ξ

Error

ε

λ

(3)

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).

(4)

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

∙

(5)

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 [email protected]; !regression coefficient

[email protected]; !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 [email protected]; !regression coefficient [email protected]; !error variance f1@1; !variance of latent variable OUTPUT: sampstat standardized;

INPUT READING TERMINATED NORMALLY RASCH measure into SEM

(6)

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 F1 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

(7)

Value 0.000 Degrees of Freedom 0 P-Value 0.0000 CFI/TLI CFI 1.000 TLI 1.000 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

Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 1.255 0.000 999.000 999.000 Intercepts RASCH 0.113 0.062 1.837 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.335 0.000 999.000 999.000

(8)

STDYX Standardization

Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 0.908 0.000 999.000 999.000 Intercepts RASCH 0.082 0.045 1.837 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.176 0.000 999.000 999.000 STDY Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 0.908 0.000 999.000 999.000 Intercepts RASCH 0.082 0.045 1.837 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.176 0.000 999.000 999.000 STD Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 1.255 0.000 999.000 999.000 Intercepts RASCH 0.113 0.062 1.837 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.335 0.000 999.000 999.000 R-SQUARE Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value RASCH 0.824 0.000 999.000 999.000

(9)

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 ξ

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:

(10)

Error variance

ε

= 0.3354

Variance of the latent variable

ξ

= 1.5745

Regression coefficient of Rasch measure on latent variable

λ

= 1

ANALYSIS: TYPE = GENERAL;

MODEL: f1 by rasch@1; !regression coefficient

[email protected]; !variance of latent variable

ANALYSIS: TYPE = GENERAL;

MODEL: f1 by rasch@1; !regression coefficient [email protected]; !error variance [email protected]; !variance of latent variable OUTPUT: sampstat standardized;

INPUT READING TERMINATED NORMALLY RASCH measure into SEM

SUMMARY OF ANALYSIS

Number of groups 1

Number of continuous latent variables 1 Observed dependent variables

(11)

Continuous RASCH

Continuous latent variables F1

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 Value 0.000 Degrees of Freedom 0 P-Value 0.0000 CFI/TLI CFI 1.000 TLI 1.000

(12)

Loglikelihood

H0 Value -872.633 H1 Value -872.632 Information Criteria

90 Percent C.I. 0.000 0.000 Probability RMSEA <= .05 0.991 SRMR (Standardized Root Mean Square Residual) Value 0.001

MODEL RESULTS

STANDARDIZED MODEL RESULTS STDYX Standardization

(13)

STDY Standardization

Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 0.908 0.000 999.000 999.000 Intercepts RASCH 0.082 0.045 1.837 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.176 0.000 999.000 999.000 STD Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 1.255 0.000 999.000 999.000 Intercepts RASCH 0.113 0.062 1.837 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.335 0.000 999.000 999.000 R-SQUARE Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value 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.

(14)

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

.

Error variance

ε

= 0.3354

Variance of the latent variable

ξ

= 1.9099

Regression coefficient of Rasch measure on latent variable

λ

= 0.9079

MODEL: f1 by [email protected]; !regression coefficient

[email protected]; !variance of latent variable

(15)

ANALYSIS: TYPE = GENERAL;

MODEL: f1 by [email protected]; !regression coefficient [email protected]; !error variance [email protected]; !variance of latent variable OUTPUT: sampstat standardized;

INPUT READING TERMINATED NORMALLY

RASCH measure into SEM SUMMARY OF ANALYSIS

Number of groups 1

Observed dependent variables Continuous RASCH Continuous latent variables F1 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

(16)

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 Value 0.000 Degrees of Freedom 0 P-Value 0.0000 CFI/TLI CFI 1.000 TLI 1.000 Loglikelihood H0 Value -872.633 H1 Value -872.632 Information Criteria

(17)

MODEL RESULTS

STANDARDIZED MODEL RESULTS

Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 0.908 0.000 999.000 999.000 Intercepts RASCH 0.082 0.045 1.838 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.176 0.000 999.000 999.000 STDY Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 0.908 0.000 999.000 999.000 Intercepts RASCH 0.082 0.045 1.838 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.176 0.000 999.000 999.000 STD Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value

(18)

F1 BY RASCH 1.254 0.000 999.000 999.000 Intercepts RASCH 0.113 0.062 1.838 0.066 Variances F1 1.000 0.000 999.000 999.000 Residual Variances RASCH 0.335 0.000 999.000 999.000 R-SQUARE Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value 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.

Case 1

(Pallant, 2007)

Case 2

Case 3

(Mathieu et al., 1992)

General case

Variance

of latent

variable

ξ

1

Actual true

variance

∙

Actual total

variance

a

ξ= ∙ )

Regression

coefficient

λ

· √

1

√

λ= * · )

a

Error

variance

ε

Actual error

variance

· (1 − )

Actual error

variance

· (1 − )

Actual error

variance

· (1 − )

Actual error

variance

· (1 − )

Actual error

variance

· (1 − )

λ

∙ ξ

PSI ·

(19)

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

λ

∙ ξ

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]

USEVARIABLES ARE rasch v1 v2 v3;

MODEL: f1 by [email protected];

f2 by v1 v2 v3;

f2 on f1;

[email protected];

f1@1;

VARIABLE: NAMES ARE rasch dep v1 v2 v3; USEVARIABLES ARE rasch v1 v2 v3; ANALYSIS: TYPE = GENERAL;

MODEL: f1 by [email protected]; f2 by v1 v2 v3; f2 on f1;

[email protected]; f1@1;

OUTPUT: sampstat standardized; INPUT READING TERMINATED NORMALLY RASCH measure into SEM

(20)

SUMMARY OF ANALYSIS

Number of groups 1

Observed dependent variables Continuous RASCH V1 V2 V3 Continuous latent variables F1 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

(21)

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 Value 2008.088 Degrees of Freedom 6 P-Value 0.0000 CFI/TLI CFI 0.999 TLI 0.998 Loglikelihood H0 Value -3815.214 H1 Value -3812.877 Information Criteria

MODEL RESULTS

Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 1.255 0.000 999.000 999.000 F2 BY 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 F2 ON F1 1.449 0.144 10.075 0.000 Intercepts RASCH 0.113 0.062 1.837 0.066

(22)

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 Variances F1 1.000 0.000 999.000 999.000 Residual Variances 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

Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 0.908 0.000 999.000 999.000 F2 BY 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 F2 ON F1 0.463 0.040 11.694 0.000 Intercepts 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 Variances F1 1.000 0.000 999.000 999.000 Residual Variances 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 Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY

RASCH 0.908 0.000 999.000 999.000 F2 BY

(23)

V3 0.960 0.005 192.377 0.000 F2 ON F1 0.463 0.040 11.694 0.000 Intercepts 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 Variances F1 1.000 0.000 999.000 999.000 Residual Variances 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 Two-Tailed Estimate S.E. Est./S.E. P-Value F1 BY RASCH 1.255 0.000 999.000 999.000 F2 BY 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 F2 ON F1 0.463 0.040 11.694 0.000 Intercepts 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 Variances F1 1.000 0.000 999.000 999.000 Residual Variances 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

(24)

R-SQUARE

Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value 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 Two-Tailed Variable Estimate S.E. Est./S.E. P-Value F2 0.214 0.037 5.847 0.000

The latent correlation of the Rasch-based latent variable F1 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).

(25)

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.

(26)

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 F11.2 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 F11.2 type 97-107 F11.2 citiz 108-118 F11.2 abroad 119-129 F11.2 V18 130-130 A1 . EXECUTE.

(27)

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_case1.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