# Regression analysis with correction for measurement error

Now we will run the analysis of the same regression model, but this time with correction for measurement errors. To do so, we just need to use the covariance matrix from Table 4.7 as the data. The rest of the specification remains the same in both LISREL and Stata.

In the same way as before, since both programs provide very similar results, please select which program you want to use to continue the analysis, LISREL1 or Stata2. The syntax of the regression model analysis with correction for measurement errors is available for the statistical computing program, R3, in Syntax A2.1 of Appendix 2.

In the previous chapter, after correction for common method variance in the correlation matrix and putting the qualities on the diagonal of the correlation matrix, we obtained the corrected correlation matrix presented in Table 4.7. LISREL can be asked to analyse this matrix in the same way as we have done above. The input for the regression analysis used in LISREL is presented in Syntax 5.2.

Syntax 5.2: The LISREL syntax for the estimation of the parameters of the regression model with correction for measurement errors

Regression analysis with correction for measurement errors
da ni=6 no=1468 ma=km
km
.710 !The correlation matrix corrected for measurement errors
.395 .643
.268 .333 .604
.310 .160 .114 .605
.112 .070 .018 .094 .682
.163 .277 .174 .064 .009 .624
labels
satdem free critic equal lrplace inc
model ny=1 nx=5
pd
out nd=3

In Syntax 5.2, we can see that the input in LISREL is the same as was used in Syntax 5.1, but this time, since our data is a correlation matrix corrected for measurement error, the diagonal does not contain 1s everywhere. Because ma=km is indicated in the program, the program transforms this matrix into a correlation matrix with 1s on the diagonal. This is exactly the transformation needed to correct for random errors because all correlations are divided by the quality coefficients of the two variables for which the correlation is corrected. Through this process, the program creates the matrix that was presented in Table 4.8. It will be clear that this is far more efficient and less risky than making these corrections for random errors by hand.

In the next step, the program performs the estimation of the regression coefficients on the basis of this correlation matrix corrected for measurement errors. The result is presented in Output 5.2.

Output 5.2: The LISREL output of the regression analysis with correction for measurement errors

Figure 5.3: The estimated standardized effects for the evaluation of democracy with correction for measurement errors

Comparing the results with and without correction for measurement error (see Figure 5.2), we see that there is a huge increase, by almost a factor two, of the effect of the evaluation variables Free, Critic and Equal, and a decrease in the effects of the control variables LRplace and Inc. Moreover, the Income control variable is no longer significant4.

Besides, the explained variance in the dependent variable has increased from 22.7% to 46.6%5 through correction for measurement errors. This also means that we now know that the unexplained variance of 53.4% cannot be due to measurement error, so it must be due to omitted explanatory variables or non-linear relationships between these variables.

In the previous chapter, after correction for common method variance in the correlation matrix and putting the qualities on the diagonal of the correlation matrix, we obtained the corrected covariance matrix presented in Table 4.7. Stata can be asked to analyse this matrix in the same way as we have done above. The regression analysis notation used in Stata is presented in Syntax 5.2.

Syntax 5.2: The Stata syntax for the estimation of the parameters of the regression model with correction for measurement errors

*Regression analysis with correction for measurement errors
clear
ssd init satdem free critic equal lrplace inc
ssd set observations 1468

*Covariance matrix
#delimit ;
ssd set covariance /*The correlation matrix corrected for measurement errors*/
.710\
.395 .643\
.268 .333 .604\
.310 .160 .114 .605\
.112 .070 .018 .094 .682\
.163 .277 .174 .064 .009 .624;
#delimit cr
save ssdmatrix.dat, replace

*Regression analysis
clear
use ssdmatrix.dat
ssd list
sem (satdem <- free critic equal lrplace inc), standardized
estat eqgof /*Equation-level goodness of fit*/

In Syntax 5.2, we can see that the input in Stata is the same as we ran in Syntax 5.1, but this time, since our data is a correlation matrix corrected for measurement error, we will also obtain the regression estimates corrected for measurement errors (see Output 5.2).

Output 5.2: The Stata output of the regression analysis with correction for measurement errors

Figure 5.3: The estimated standardized effects for the evaluation of democracy with correction for measurement errors

Comparing the results with and without correction for measurement error (see Figure 5.2), we see that there is a huge increase, by almost a factor two, of the effect of the evaluation variables Free, Critic and Equal, and a decrease in the effects of the control variables LRplace and Inc. Moreover, the Income control variable is no longer significant6.

Besides, the explained variance in the dependent variable has increased from 22.7% to 46.6%7 through correction for measurement errors. This also means that we now know that the unexplained variance of 53.4% cannot be due to measurement error, so it must be due to omitted explanatory variables or non-linear relationships between these variables.

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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 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] R: A Language and Environment for Statistical Computing, R Core Team, R Foundation for Statistical Computing. http://www.R-project.org.
• [4] Although most of the coefficients are significant, it has to be realized that the standard errors in this approach are underestimated for two reasons. The first reason is that the observed correlation is assumed to be a perfect estimate of the population correlation, and the second reason is that the quality estimates are assumed to be perfect measures of the quality of the questions (Obe11). In both cases, the uncertainty of the estimation is ignored. However, in most of the cases, this will not harm the conclusions because the effects are relative large or relative small, as it is the case in this example.
• [5] The explained variance can be obtained in LISREL from the section Squared Multiple Correlations for Structural Equations (R2).
• [6] Although most of the coefficients are significant, it has to be realized that the standard errors in this approach are underestimated for two reasons. The first reason is that the observed correlation is assumed to be a perfect estimate of the population correlation, and the second reason is that the quality estimates are assumed to be perfect measures of the quality of the questions (Obe11). In both cases, the uncertainty of the estimation is ignored. However, in most of the cases, this will not harm the conclusions because the effects are relative large or relative small, as it is the case in this example.
• [7] In Stata, the command estat eqgof will show you the value of R2 (The explained variance).