# Estimation of the causal model with complex concepts

Now that we have estimates of the quality of all the variables in the model, including the composite score for democracy level, the estimation of the causal model with correction for measurement errors is exactly the same as for the model with variables based on single questions. Thus, we are ready to estimate the parameters of the causal model with the composite score correcting for measurement errors.

Below, we will illustrate how to run the causal model for the composite score specified before in Figure 7.1 using both LISREL1 and Stata2. As both programs produce very similar results, please select which program you want to continue the analysis with:

The variances in the disturbances of the explained variables are denoted as ps(1,1) and ps(2,2). For the details of the procedure, we refer to the LISREL manual [Jör96] and introductions to the program LISREL [Sar84]. First, the LISREL input for this analysis without corrections is presented in Syntax 7.1. Next, we present the same input corrected for measurement errors (see Syntax 7.2).

**Syntax 7.1:**The LISREL syntax for the estimation of the parameters of the causal model including a composite score without correction for measurement errors

**Syntax 7.2:**The LISREL syntax for the estimation of the parameters of the causal model including a composite score with correction for measurement errors

The most important point is that the coefficients that have to be estimated are presented in the lines starting with ‘free’. Comparing these two inputs, we see that only the matrix with the data to be analysed has been changed. Focusing on the input for the model with correction for measurement errors, the effects will be estimated on the basis of the covariance matrix in Table 7.5 (i.e. the matrix with the correlations corrected for cmv and with the qualities on the diagonal). Because we ask in the data line that the matrix to be analysed (ma) should be the correlation matrix (km), Table 7.5 is transformed by the program into the correlation matrix corrected for measurement errors, Table 7.6.

The nice feature of this approach, correcting the correlations for measurement errors before estimating the effects, is that the input for the analysis is exactly the same with and without correction for measurement errors, except for the matrix of correlations that is used in the analysis. This point is illustrated in the input for the analyses with and without correction for measurement errors presented in Syntaxes 7.1 and 7.2.

It is important in the estimation of causal models to test whether the model fits to the data, i.e. that the model is not misspecified. Without going into detail, see [Sar09], we can say that the model fits very well to the data corrected for measurement errors. So there is no reason to change the model.

However, analysing the matrix without correction for measurement errors, the program indicates that the fit of the model is not good. This suggests that the effect, ga(1,2), of the control variable Income on satisfaction with democracy has to be introduced in the model. If we do so, this model also fits well to the data and we obtain the results presented in Table 7.7.

Comparing Syntaxes 7.1 and 7.2, we can observe that all effects have been indicated using the Stata notation [Aco13]. Comparing the two inputs, we see that only the matrix with the data to be analysed has been changed. Focusing on the input for the model with correction for measurement errors, the effects will be estimated on the basis of the covariance matrix in Table 7.5 (i.e. the matrix with the correlations corrected for cmv and with the qualities on the diagonal).

**Syntax 7.1:**The Stata syntax for the estimation of the parameters of the causal model including a complex concept without correction for measurement errors

**Syntax 7.2:**The Stata syntax for the estimation of the parameters of the causal model including a complex concept with correction for measurement errors

The nice feature of this approach, correcting the correlations for measurement errors before estimating the effects, is that the input for the analysis is exactly the same with and without correction for measurement errors, except for the matrix of correlations that is used in the analysis. This point is illustrated in the input for the analyses with and without correction for measurement errors presented in Syntaxes 7.1 and 7.2.

However, analysing the matrix without correction for measurement errors, the program indicates that the fit of the model is not good. This suggests that the effect of the control variable income (Inc) on satisfaction with democracy has to be introduced in the model. If we do so, this model also fits well to the data and we get the results presented in Table 7.8.

If we compare the results with and without correction for measurement errors, we see first of all that the model is different. After correction for measurement errors, the effects of the control variable Inc on satisfaction with the democracy is not significantly different from zero, while, without correction for measurement errors, this effect is necessary to achieve a good fit of the model. In the latter case, we say that this variable has a direct effect on satisfaction with democracy, while in the former analysis we have to conclude that there is no direct effect, only an indirect effect.

Furthermore, comparing Figures 7.3 and 7.4, we see that, after correction for measurement errors, nearly all other effects are much bigger than without correction for errors. All significant effects are indicated in the figures by an asterisk (*). In this example, we see that we now have only one variable out of the three indicators of democracy level. The effect of this variable increases much more than before because the effect is not reduced by the correlations between these three indicators, i.e. the variable is now alone. Furthermore, without corrections, the explained variance in the variable Satdem is 21.4%, while, after corrections, it increases to 44.4%.3 This example again illustrates how different the results can be if measurement errors are corrected.

## Exercise 7.1

Compute the corrected correlation matrix for the variables introduced in exercise 3.1 using the composite score of the variables Economy and Culture as represented in the figure below. The composite score ‘Country threats’ is created as a simple sum:

**Country threats (CS) = Economic threat + Cultural threat**

The model to be estimated is:

Below, the correlation matrix is provided without corrections, together with the quality predictions obtained in exercise 3.1 using SQP and the descriptive statistics used in exercise 6.1 (adding the composite score descriptives).

- Correlation matrix (n=1801):
- Quality predictions obtained in exercise 3.1 using SQP:
- Descriptive statistics of the variables:

Use all this information to correct the correlation matrix for measurement errors.

In order to correct the correlation of a composite score, we need to first calculate the predicted quality. This cannot be done by SQP. The alternative is to use the following formula:

_{cs})/ var(CS))

_{cs}) = Σw

_{k}

^{2}var(e

_{k}) + 2Σw

_{k}w

_{k'}cov(e

_{k}e

_{k'}) over k where k≠k

^{'}

_{i}) = (1-q

_{i2})var(y

_{i})

_{i}e

_{j}) = cmv

_{ij}• s

_{i}s

_{j}= (r

_{i}m

_{i}m

_{j}r

_{j})(s

_{i}s

_{j})

Using the quality predictions obtained in exercise 3.1, we can derive the quality of the composite score (Threats). To do so, we need to first compute the variance in the error of the composite score using the quality estimates of the variables Economy and Culture and the information from the table above, where we have presented their variances. Computing var(ei) and cov(eiej), we get:

_{B38}) = (1 – q

_{B38}

^{2})var(B38) = (1-0.702) * 3.640 = 1.085

_{B39}) = (1 – q

_{B39}

^{2})var(B39) = (1-0.641) * 3.771 = 1.354

_{B38}e

_{B39}) = (cmv

_{B38,B39})(s

_{B38}s

_{B39}) = r

_{B38}m

_{B38}m

_{B39}r

_{B39}* s

_{B38}s

_{B39}=

Now we have all the components required to compute the variance in the errors of the composite score for the variable country threats, which is:

_{threats}) = Σw

_{k}

^{2}var(e

_{k}) + 2Σw

_{k}w

_{k'}cov(e

_{k}e

_{k'})

Finally, the quality of the composite score of country threats can be computed as follows:

_{threats}= 1 – (var(e

_{threats})/ var

_{threats})

**0.709**

In this case, the common method variance between the composite score (Threats) and the variable (Allow) also has to be taken into account. In this chapter , we have presented the formula for computing the cmv for this pair of variables:

_{B40,threats}= r

_{B40}m

_{B40}[(1/σ

_{threats}) m

_{B38}r

_{B38}+ (1/σ

_{threats})r

_{B39}m

_{B39}] =

**0.061**

As before, the common method variance has to be subtracted from the correlation between these two variables affected by the same method. So:

**0.563**

To conclude, we just have to change on the diagonal the variances in the variables Allow and Better for the qualities obtained in SQP. The variance in the variable Threats has to be substituted by the quality, and the correlation between Better and Threats has to be corrected as indicated above. This will result in the correlation matrix corrected for measurement errors.

## Exercise 7.2

Taking into account the results obtained in exercise 7.1, run the estimation of the same causal model with the complex concept. The correlation matrices with and without corrections obtained before are reproduced below. Use this information to compute, either in LISREL or Stata, the results for the analysis of the causal explanation of the opinion about immigration by people from outside Europe to the Netherlands, with and without correction for measurement errors.

**LISREL syntax for the estimation of the causal model without corrections for measurement errors:**Causal model without correction for measurement errorsdata ni=3 no=1801 ma=kmkm1.000-0.351 1.000-0.424 0.624 1.000labelsimpcntr imwbcnt threatsmodel ny=2 nx=1 be=fu,fi ga=fu,fi ps=sy,fifree be(1,2)free ga(2,1) ga(1,1)free ps(1,1) ps(2,2)pdout nd=3**LISREL syntax for the estimation of the causal model with corrections for measurement errors:**Causal model with correction for measurement errorsdata ni=3 no=1801 ma=kmkm0.763-0.351 0.639-0.424 0.563 0.709labelsimpcntr imwbcnt threatsmodel ny=2 nx=1 be=fu,fi ga=fu,fi ps=sy,fifree be(1,2)free ga(2,1) ga(1,1)free ps(1,1) ps(2,2)pdout nd=3

**Stata syntax for the estimation of the causal model without corrections for measurement errors:***Causal model without correction for measurement errorsclear allssd init impcntr imwbcnt threatssd set observations 1801*Correlation matrix#delimit ;ssd set correlation1.000\-0.351 1.000\-0.424 0.624 1.000;#delimit crsave ssdmatrix.dat, replace*Causal modelclearuse ssdmatrix.datssd listsem (impcntr <- imwbcnt) ///(impcntr imwbcnt <- threat), ///standardizedestat eqgof**Stata syntax for the estimation of the causal model with corrections for measurement errors:***Causal model with correction for measurement errorsclear allssd init impcntr imwbcnt threatssd set observations 1801*Covariance matrix#delimit ;ssd set covariance0.763\-0.351 0.639\-0.424 0.563 0.709;#delimit crsave ssdmatrix.dat, replace*Causal modelclearuse ssdmatrix.datssd listsem (impcntr <- imwbcnt) ///(impcntr imwbcnt <- threat), ///standardizedestat eqgof

The figure at the top presents the results for the model with the composite score Threats before corrections, while the lower figure presents the results of the model after corrections. In this case, we observe that the differences in the effects are considerable, i.e. one of the effects is no longer significant, while the other two are much larger. From the exercises, we can conclude that the differences are larger in this small model. It is obvious that the explained variance (R^{2}) in this case again increases after correcting for measurement errors.

This exercise has shown once again how different the results can be if it is corrected for measurement errors. Furthermore, it has also shown that the correction for measurement errors can be done not only in models with simple concepts, but also in models with complex concepts.

#### Footnotes

- [1] The following illustration and results are based on the LISREL 8.7 software version: Jöreskog, K.G. & Sörbom, D. (2004). LISREL 8.7 for Windows [Computer software]. Skokie, IL: Scientific Software International, Inc.
- [2] The following illustration and results are based on the Stata 12 software version: StataCorp. 2011.
**Stata Statistical Software: Release 12**. College Station, TX: StataCorp LP. - [3] The explained variance can be obtained in LISREL from the section Squared Multiple Correlations for Structural Equations (R2). Similarly, in Stata the command estat eqgof will show you the value of R2.

#### References

- [Aco13] Acock, A. C. (2013). Discovering Structural Equation Modeling Using Stata, Revised Edition.
*Stata press*. - [Jör96] Jöreskog, K. G. and Sörbom, D. (1996). LISREL 8 User’s Reference Guide.
*Scientific Software International*. - [Sar09] Saris, W. E., Satorra, A. and Van der Veld, W. M. (2009). Testing Structural Equation Models or Detection of Misspecifications?
*Structural Equation Modeling: A Multidisciplinary Journal, 16 (4), 561-582*. - [Sar84] Saris, W. E. and Stronkhorst, L. H. (1984). Causal modelling in nonexperimental research: an introduction to the LISREL approach.
*Sociometric Research Foundation*.