# Regression analysis without correction for measurement errors

Below, the regression analysis without correction for measurement errors is done using LISREL and Stata. The estimates obtained using both programs are very similar (except for the rounding off errors). Thus, as both programs provide very similar results, please select which program you want to continue with:

First, we ask LISREL1 to run the regression model without correction for measurement errors by using the data from the correlation matrix shown in Table 4.2 as input. The following LISREL syntax can be used to perform this analysis. For more detailed information about the LISREL procedure and notation, we refer to the LISREL 8 manual [Jör96] and LISREL introductory books [Sar84].

Syntax 5.1: The LISREL syntax for the estimation of the parameters of the regression model without correction for measurement errors

Regression analysis without correction for measurement error !Title

da ni=6 no=1468 ma=km !ni=number of variables no=number of observations ma=matrix

km !km=correlation matrix
1.00
.395 1.00
.268 .474 1.00
.310 .299 .271 1.00
.188 .070 .018 .094 1.00
.163 .227 .174 .064 .009 1.00

labels
satdem free critic equal lrplace inc !Labels of the variables
model ny=1 nx=5 !ny=number of dependent variable nx=number of independent variables
pd !To obtain a path diagram
out nd=3 !out=output nd=number of decimals

In Syntax 5.1, we can see that the correlation matrix and a model have to be specified. In this case, the model is a simple regression model. If this program is run, the result presented in Output 5.1 will be obtained.

Output 5.1: The LISREL output for the regression analysis without correction for measurement errors

Figure 5.2: The estimated standardized effects for the evaluation of democracy without correction for measurement errors

In Figure 5.2, we see that all standardized effects are relatively moderate, and that the variable freedom and fairness of elections (Free) has the largest effect. Second in size is the effect of the variable equality by law (Equal). The asterisks (*) indicate that, according to the analysis, all variables have a significant effect testing on the 5% level. It is also relevant to mention that, together, these variables explain only 22.7%2 of the variance in the dependent variable, which means that 77.3% of the variance is unexplained. This suggests that the model is rather incomplete or incorrect. It is also possible, however, that the explained variance is so small because all the variables contain measurement errors that have not been taken into account.

First, we ask Stata3 to run the regression model without correction for measurement errors by using as input the data from the correlation matrix shown in Table 4.2. The following Stata syntax can be used to perform this analysis. For more detailed information about Stata procedure and notation, we refer to Stata books [Aco13]. Appendix A is especially important if one wants to use the correlation or covariance matrix as the data to be analysed.

Syntax 5.1: The Stata syntax for the estimation of the parameters of the regression model without correction for measurement errors

*Regression analysis without correction for measurement errors

clear all
ssd init satdem free critic equal lrplace inc /*variables*/
ssd set observations 1468 /*observations*/

*Correlation matrix
#delimit ;
ssd set correlations
1.00\
.395 1.00\
.268 .474 1.00\
.310 .299 .271 1.00\
.188 .070 .018 .094 1.00\
.163 .227 .174 .064 .009 1.00;
#delimit cr
save ssdmatrix.dat, replace

*Regression model
clear
use ssdmatrix.dat /*use the correlation matrix input as data*/
ssd list
sem (satdem <- free critic equal lrplace inc), standardized /*model*/
estat eqgof /*Equation-level goodness of fit*/

In Syntax 5.1, we can see that the correlation matrix and the model have to be specified. In this case, the model is a simple regression model. If this program is run, the result presented in Output 5.1 will be obtained.

Output 5.1: The Stata output for the regression analysis without correction for measurement errors

Figure 5.2: The estimated standardized effects for the evaluation of democracy without correction for measurement errors

In Figure 5.2, we see that all standardized effects are relatively moderate, and that the variable freedom and fairness of elections (Free) has the largest effect. Second in size is the effect of the variable equality by law (Equal). The asterisks (*) indicate that, according to the analysis, all variables have a significant effect testing on the 5% level. It is also relevant to mention that, together, these variables explain only 22.7%4 of the variance in the dependent variable, which means that 77.3% of the variance is unexplained. This suggests that the model is rather incomplete or incorrect. It is also possible, however, that the explained variance is so small because all the variables contain measurement errors that have not been taken into account.

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#### 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 explained variance can be obtained in LISREL from the section Squared Multiple Correlations for Structural Equations (R2).
• [3] 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.
• [4] The explained variance can be obtained in Stata by the command estat eqgof 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.
• [Sar84] Saris, W. E. and Stronkhorst, L. H. (1984). Causal modelling in nonexperimental research: an introduction to the LISREL approach. Sociometric Research Foundation.