Causal Relationships with measurement error in the data

1
Causal Relationships with measurement error in
the data

• A brief introduction
• by
• Willem E.Saris

2
Basic concepts
Direct effect
y
x
y
Indirect effect
z
x
y
Spurious relation
z
x
x
z
Joint effect
w
y
3
An example of a model

• How can these effects be estimated ?

4
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.

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.

6
Derivations

• These derivations can also be used to estimate
  the parameters of this model. How ?

7
A second example
8
A Structural Equations Model
9
Derivations
10
The Proof
11
The correlations between the variables

• The effects are equal to the correlations with x1

12
What if x1 is not observed ? Can we still
estimate the effects ?
13
What happens if we have 4 observed variables ?

• With extra info

14
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

15
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) b21b41 r(y3y4) b31b41
• r(y2y4) - r(y2y4) r(y2y4) - b21b41.3- .6x.5
  .0
• r(y3y4) - r(y3y4) r(y3y4) - b31b41.5 - .8x.5
  .1
• These differences are called residuals.
• If these residuals are big the model must be
  wrong.

16
Identification again

• With 3 observed variables df0 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 dflt0 the model is not identified

17
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.

18
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.

19
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.

20
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.

21
Estimation in this specific case
The program looks for the values of all the
parameters that minimize the function Fuls
22
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.

23
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

24
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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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 gt Ca
• where Ca is the value of the c2 for which
• pr(c2df gt Ca) a
• We come back to this issue later

26
LISREL input

• estimation and testing a factor model
• data ni4 no400 makm
• km
• 1.0
• .42 1.0
• .56 .48 1.0
• .35 .30 .40 1.0
• model ny4 ne1 lyfu,fi tedi,fi psdi,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

27
LISREL estimates of the effects of the latent
factor
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LISREL estimates of the error variances
29
Goodness of fit test
30
LISREL input for different correlation matrix
estimation and testing a factor model data ni4
no400 makm km 1.0 .42 1.0 .56 .48 1.0 .35 .50
.50 1.0 model ny4 ne1 lyfu,fi tedi,fi
psdi,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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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
1 0.64 (0.05)
14.18 VAR 2 0.67 (0.04)
15.43 VAR 3 0.79
(0.05) 15.75 VAR 4
0.64 (0.05) 14.28
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 19.62 (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 dfgt0
• 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