1. Causal Relationships with measurement error in the data
A brief introduction by
Willem E.Saris

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Basic concepts
Direct effect y x y
Indirect effect z x y
Spurious relation z x x z
Joint effect w y
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An example of a model
How can these effects be estimated ?
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Decomposition rule
The correlation between two variables is equal to the sum of
- the direct effect,
- indirect effects,
- spurious relationships and
- joint effects between these variables.
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5. Expression for the different components
The indirect effect, spurious relations and joint effects are equal to the products of the coefficients along the path going from one variable to the other while one can not pass the same variable twice and can not go against the direction of the arrows.
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Derivations
These derivations can also be used to estimate the parameters of this model. How ?
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A second example
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8. A Structural Equations Model
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Derivations
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The Proof
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11. The correlations between the variables
The effects are equal to the correlations with x1
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12. What if x1 is not observed ? Can we still estimate the effects ?
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13. What happens if we have 4 observed variables ?
With extra info
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Identification
Of these three equations we need only one to
determine the value of b41 when we have solved
b11 and the other coefficients from the first three
correlation coefficients
This model is called overidentified or the
degrees of freedom or df= 2
df= # correlations - # parameters to be estimated
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A test is possible
If we know that b11 = .7 and that
r(y1y4) = b11b41 =.35 it follows that b41= .5
Now we know all coefficients and two correlations are not used yet and can be used to test the model
r(y2y4) = b21b r(y3y4) = b31b41
r(y2y4) - r(y2y4) = r(y2y4) - b21b41=.3- .6x.5 = .0
r(y3y4) - r(y3y4) = r(y3y4) - b31b41= x.5 = .1
These differences are called residuals.
If these residuals are big the model must be wrong.
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Identification again
With 3 observed variables df=0 and no test is possible
With 2 observed variables df=-1 and no test is possible but even the effects can not be estimated
If df<0 the model is not identified
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Estimation
The decomposition rules only hold for the population correlations and not for the sample correlations
But , normally, we know only the sample correlations
It is easily shown that the solution is different depending of the equations used
So an efficient estimation procedure is needed.
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Estimation
There are several general principles.
We will discuss:
- the Unweighted Least Squares (ULS) procedure
- the Weighted Least Squares (WLS) procedure.
Both procedures are based on the residuals between the sample correlations and the expected values of the correlations.
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Estimation
The expected correlations are a function of the parameters fij(p)
where p represents the set of parameters of the model
and fij the specific function which gives the link between the population correlations and the parameters for the variables i and j.
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ULS estimators
The ULS procedure suggests to look for the parameter values that minimize the unweighted sum of squared residuals:
FULS = S(rij –fij(p))2
where the summation is over all unique elements of the correlation matrix.
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21. Estimation in this specific case
The program looks for the values of all the parameters that
minimize the function Fuls
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WLS estimators
The WLS procedure suggests to look for the parameter values that minimize the weighted sum of squared residuals:
FWLS = Swij(rij –fij(p))2 where the summation is also over all unique elements of the correlation matrix.
These weights can be chosen in different ways.
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ADF estimator
Using weights derived from the Variance Covariances of the covariances the Asymptotic Distribution Free estimator is specified.
For any distribution of the observed variables this estimator is consistent and provides standard errors and a test statistic
The problem is that it requires very large samples
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ML estimator
The most commonly used procedure, the Maximum Likelihood (ML) estimator, can be specified as a special case of the WLS estimator.
The ML estimator provides standard errors for the parameters and a test statistic for the fit of the model for much smaller samples
but this estimator is developed under the assumption that the observed variables have a multivariate normal distribution.
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25. Standard Procedure for testing S E Models
Testing is essential for S E Models
The test statistic t used is the value of the fitting function at its minimum
If the model is correct, t is c2 (df) distributed
Normally the model is rejected if t > Ca
where Ca is the value of the c2 for which
pr(c2df > Ca) =a
We come back to this issue later
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LISREL input
estimation and testing a factor model
data ni=4 no=400 ma=km km
1.0
model ny=4 ne=1 ly=fu,fi te=di,fi ps=di,fi
free ly 1 1 ly 2 1 ly 3 1 ly 4 1
free te 1 1 te 2 2 te 3 3 te 4 4
value 1 ps 1 1
out ULS
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27. LISREL estimates of the effects of the latent factor
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28. LISREL estimates of the error variances
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Goodness of fit test
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30. LISREL input for different correlation matrix
estimation and testing a factor model
data ni=4 no=400 ma=km km
1.0
model ny=4 ne=1 ly=fu,fi te=di,fi ps=di,fi
free ly 1 1 ly 2 1 ly 3 1 ly 4 1
free te 1 1 te 2 2 te 3 3 te 4 4
value 1 ps 1 1
out ULS
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31. Estimates of the effects of the latent variable
estimation and testing a factor model
Number of Iterations = 9
LISREL Estimates (Unweighted Least Squares)
LAMBDA-Y
ETA 1
VAR
(0.05)
14.18
VAR
(0.04)
15.43
VAR
15.75
VAR
14.28
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32. Goodness of fit test of the model on the new correlation matrix
Goodness of Fit Statistics
W_A_R_N_I_N_G: Chi-square, standard errors, t-values and standardized residuals are calculated under the assumption of multi-variate normality.
Degrees of Freedom = 2
Normal Theory Weighted Least Squares Chi-Square = (P = 0.00)
Estimated Non-centrality Parameter (NCP) = 17.62
90 Percent Confidence Interval for NCP = (6.96 ; 35.72)
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General Approach
A model is specified with observed and latent variables
Correlations (covariances) between the observed variables can be expressed in the parameters of the model (decomposition rules)
If the model is identified the parameters can be estimated
A test of the model can be performed if df>0
Eventual misspecifications can be detected
Corrections in the models can be introduced
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Important Result
The distinction between observed and latent variables makes the estimation of error variances possible
The errors in social science survey data can be quite large.
These errors will bias the estimates if not taken into account
So the SEM approach has important advantages
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