What if there are also measurement errors in the outcome variable Y? How can you write such an extended model? The following model would take measurement errors in both X and Y into account:









with the following assumption:









Again, the first equation, expressing the relationship between two latent truescore variables, defines the structural or causal model. The next two equations express the observed variables in terms of a true score plus error; these two equations define the measurement model. This is essentially the same form as the socalled LISREL model (Keesling, 1972; Wiley, 1973; Jöreskog, 1973), which has been popularized by the LISREL program (Jöreskog and Sörbom, 1988). Typically, there are several X and Y variables in a LISREL model. For the moment, however, the focus is on the current regression form in which there is only a single predictor and a single outcome variable. The LISREL model is considered in the section Fitting LISREL Models by the LISMOD Modeling Language.
With the intercept term left out for modeling, you can use the following statements for fitting the regression model with measurement errors in both X and Y:
proc calis data=corn; lineqs Fy = beta * Fx + DFy, Y = 1. * Fy + Ey, X = 1. * Fx + Ex; run;
Again, you do not need to specify the zerocorrelation assumptions in the LINEQS model because they are set by default given the latent factors and errors in the LINEQS modeling language. When you run this model, PROC CALIS issues the following warning:
WARNING: Estimation problem not identified: More parameters to estimate ( 5 ) than the total number of mean and covariance elements ( 3 ).
The five parameters in the model include beta
and the variances for the exogenous variables: Fx
, DFy
, Ey
, and Ex
. These variance parameters are treated as free parameters by default in PROC CALIS. You have five parameters to estimate,
but the information for estimating these five parameters comes from the three unique elements in the sample covariance matrix
for X and Y. Hence, your model is in the socalled underidentification situation. Model identification is discussed in more detail in
the section Model Identification.
To make the current model identified, you can put constraints on some parameters. This reduces the number of independent parameters
to estimate in the model. In the errorsinvariables model for the corn data, the variance of Ex
(measurement error for X
) is given as the constant value 57, which was obtained from a previous study. This could still be applied in the current
model with measurement errors in both X and Y. In addition, if you are willing to accept the assumption that the structural equation model is (almost) deterministic, then
the variance of Dfy
could be set to 0. With these two parameter constraints, the current model is justidentified. That is, you can now estimate
three free parameters from three distinct covariance elements in the data. The following statements show the LINEQS model
specification for this justidentified model:
proc calis data=corn; lineqs Fy = beta * Fx + Dfy, Y = 1. * Fy + Ey, X = 1. * Fx + Ex; variance Ex = 57., Dfy = 0.; run;
Figure 17.6 shows the estimation results.
Figure 17.6: Regression Model With Measurement Errors in X and Y for Corn Data
Linear Equations  

Fy  =  0.4232  *  Fx  +  1.0000  Dfy  
Std Err  0.1658  beta  
t Value  2.5520  
y  =  1.0000  Fy  +  1.0000  Ey  
x  =  1.0000  Fx  +  1.0000  Ex 
Estimates for Variances of Exogenous Variables  

Variable Type 
Variable  Parameter  Estimate  Standard Error 
t Value 
Error  Ex  57.00000  
Disturbance  Dfy  0  
Latent  Fx  _Add1  247.85450  136.33508  1.81798 
Error  Ey  _Add2  43.29105  23.92488  1.80946 
In Figure 17.6, the estimate of beta
is 0.4232, which is basically the same as the estimate for beta
in the errorsinvariables model shown in Figure 17.4. The estimated variances for Fx
and Ey
match for the two models too. In fact, it is not difficult to show mathematically that the current constrained model with
measurements errors in both Y and X is equivalent to the errorsinvariables model for the corn data. The numerical results merely confirm this fact.
It is important to emphasize that the equivalence shown here is not a general statement about the current model with measurement
errors in X and Y and the errorsinvariables model. Essentially, the equivalence of the two models as applied to the corn data is due to those
constraints imposed on the measurement error variances for DFy
and Ex
. The more important implication from these two analyses is that for the model with measurement errors in both X and Y, you need to set more parameter constraints to make the model identified. Some constraints might be substantively meaningful,
while others might need strong or risky assumptions.
For example, setting the variance of Ex
to 57 is substantively meaningful because it is based on a prior study. However, setting the variance of Dfy
to 0 implies the acceptance of the deterministic structural model, which could be a rather risky assumption in most practical
situations. It turns out that using these two constraints together for the model identification of the regression with measurement
errors in both X and Y does not give you more substantively important information than what the errorsinvariables model has already given you
(compare Figure 17.6 with Figure 17.4). Therefore, the set of identification constraints you use might be important in at least two aspects. First, it might lead
to an identified model if you set them properly. Second, given that the model is identified, the meaningfulness of your model
depends on how reasonable your identification constraints are.
The two identification constraints set on the regression model with measurement errors in both X and Y make the model identified. But they do not lead to model estimates that are more informative than that of the errorsinvariables
regression. Some other sets of identification constraints, if available, might have been more informative. For example, if
there were a prior study about the measurement error variance of corn yields (Y), a fixed constant for the variance of Ey
could have been set, instead of the unrealistic zero variance constraint of Dfy
. This way the estimation results of the regression model with measurement errors in both X and Y would offer you something different from the errorsinvariables regression.
Setting identification constraints could be based on convention or other arguments. See the section Illustration of Model Identification: Spleen Data for an example where model identification is attained by setting constant error variances for X and Y in the model. For the corn data, you have seen that fixing the error variance of the predictor variable led to model identification of the errorsinvariables model. In this case, prior knowledge about the measurement error variance is necessary. This necessity is partly due to the fact that each latent true score variable has only one observed variable as its indicator measure. When you have more measurement indicators for the same latent factor, fixing the measurement error variances to constants for model identification would not be necessary. This is the modeling scenario assumed by the LISREL model (see the section Fitting LISREL Models by the LISMOD Modeling Language), of which the confirmatory factor model is a special case. The confirmatory factor model is described and illustrated in the section The FACTOR and RAM Modeling Languages.