General Structural Equation (LISREL) Models

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General Structural Equation (LISREL) Models
ICPSR – D. Baer Week 1 #2

Today’s class Translating Diagrams to Equations and vice-versa
Implications of measurement error (more) Testable vs. non-testable hypotheses Covariance algebra for structural equation models (the scalar version) Introduction to the AMOS computer program

A short AMOS exercise will be provided at the end of the class (this is not a formal assignment – not to be handed in) A short covariance algebra exercise (again, not a formal assignment)

Path Models Familiar to some, though prior to the widespread diffusion of SEM models in the social sciences, path models were seldom seen after the 1970s Early path models based on OLS estimation; other estimation methods possible More sophisticated path models involved simultaneous equations

Path Models A Regular path model Variables in rectangles (V1, V2, V3)
Error terms in circles (alternative convention: leave error terms unenclosed) This model can be estimated with 2 OLS equations: V2 = b1 V1 + e2 V3 = b2 V1 + b3 V2 + e3 No intercepts in equations: variables are mean-centered.

Path Models With 3 variables, V1, V2 and V3, other models are possible
Model on bottom is non-recursive, makes different assumptions about the causal relationships among variables.

Path Models With 3 variables, V1, V2 and V3, other models are possible

Path Models With 3 variables, V1, V2 and V3, other models are possible
In previous slide, models not testable. Here, models are “nested” and thus can be tested against each other: Model 1 H0: b3=0 Model 2 b3≠ 0 Test between 2 models involves test of 1 parameter (it is a df=1 test)

Implications of Measurement error: Three quick examples
A two-variable example LV1, LV2 imperfectly measured Assume reliability of .80 How does b1 compare with the coefficient we would have obtained if we had regressed X2 on X1? If true b1 = .80 Observed X2 = b1*X1 + e b1* = .512 If true b1 = .5 b1* = .32

Implications of Measurement error: A three-variable example
Different degrees of measurement error for LV1, LV2, LV3 LV3 .70 LV2 .90 LV1 .60

Implications of Measurement error: A three-variable example
Parameter True Observed Cov(LV1,LV2) -.40 -.22 b1 .86 .31 B2 .54 .27

Implications of Measurement error: A second three-variable example
Parameter True Observed Cov(LV1,LV2) -.60 -.32 b1 .50 .16 B2 .00 -.13 Same reliabilities/measurement error variances as previous examples: LV1 .6 LV LV3 .70

Implications of Measurement error:
Previous examples all assumed random measurement error Error can be systematic too. We will discuss hypothetical and actual examples of systematic measurement error later

Translating Diagrams to Equations and Equations to Diagrams Two quick exercises
Write out equations List all model parameters (enlarged version on next slide)

Write out equations for this model:

Write out equations for this model:
Measurement equations: X1 = 1.0 LV1 + e1 (or: X1 = LV1 + e1) X2 = b1 LV1 + e2 X3 = b2 LV2 + e3 X4 = 1.0 LV2 + e4 X5 = 1.0 LV3 + e5 X6 = b3 LV3 + e6 Structural equations among latent variables: (“construct equations”) LV3 = b4 LV1 + b5 LV2 + d1 Notes: It is common to have one indicator with a fixed measurement equation parameter value of 1.0. The error term d1 could have been called e7 instead. Usually, we distinguish between measurement equation errors and construct eq. errors, but not necessary to do so.

Model parameters: b1, b2, b3, b4, b5 Error term variances:
Var (e1) var (e2) var (e3) var (e4) Var (e5) var (e6) var (d1) Exogenous latent variable variances: Var (LV1) var (LV2) Exogenous latent variable covariance: Cov (LV1, LV2) Error term covariances: (NONE IN THIS MODEL) Total number of model parameters: 15 (error term is a type of exogenous variable)

Another example:

Equations: X4 = 1.0 LV1 + e4 X5 = b1 LV1 + e5 X6 = b2 LV1 + e6 X7 = b3 LV1 + e7 X8 = b4 LV2 + e8 X9 = b5 LV2 + e9 X10 = 1.0 LV2 + e10 LV1 = b6 X1 + b7 X2 + D1 LV2 = b8 X2 + b9 X3 + D2 Parameters in addition to 9 coefficients: Variances of : e4,e5,e6,e7,e8,e9,e10, d1, d2 Variances of: X1,X2,X3 Covariances: X1,X2 ; X1,X3 ; X2,X3 ; D1,D2 ; e5, e6

Taking equations  diagrams
X1 = 1.0 LV1 + e1 X2 = b2 LV1 + e2 X3 = b3 LV1 + b4 LV2 + e3 X4 = 1.0 LV2 + e4 X5 = b5 LV2 + e5 X6 = b6 LV2 + e6 X7 = 1.0 LV3 + e7 LV1 = b7 LV2 + b8 LV3 + d1 LV2 = b9 LV1 + d2

Taking equations  diagrams
Equations alone don’t specify whether there are any exogenous variable covariances (probably needed: cov(D1,D2)

Different notation systems
AMOS: Indic1 = b1 LVar1 + 1 e1 EQS V1 = * F1 + E1 LISREL X1 = λ1 ξ1 + δ1

Covariance algebra for structural equation models
WHY? For any model, we use as input the observed covariance matrix, S But we also use the model parameters to calculate a “reproduced covariance matrix”, Σ. This matrix is important for a number of reasons (for one thing, the best estimates of model parameters are those that in some way minimize Σ- S).

Covariance algebra for structural equation models
For any model, we can take the model parameters, and calculate the elements of Σ as a function of these model parameters using the rules of covariance algebra.

Covariance algebra for structural equation models: Four simple rules
Null rule: covariance between a variable and a constant is 0 COV(X,a) = X = variable a = constant Variance definition: covariance of a variable with itself is its variance VAR(X) = COV(X,X)

Covariance algebra for structural equation models: Four simple rules
Constant Rule: COV(X,aY) = a · COV(X,Y) We can factor constants out of covariances The covariance between 2 variables is multiplied by a constant when either of the two variables is multiplied by that constant

Covariance algebra for structural equation models: Four simple rules
Sum Rule: COV(X,{Y+Z}) = COV(X,Y) + COV(X,Z) The covariance of a variable with the sum of two variables is the sum of the covariances between that variable and each of these two variables.

Covariance algebra for structural equation models: Four simple rules
COV(X,a) = 0 [null rule] VAR(X) = COV(X,X) [variance def.] COV(X,aY) = a * COV(x,y) [constant rule] COV(X,{Y+Z}) = COV(X,Y) + COV(X,Z) [sum rule]

Applications: X2 = b1 X1 + e2 X3 = b2 X2 + b3 X1 + e3
We can use covariance algebra to “solve” for the reproduced covariance for any of the 6 variances-covariances among manifest variables.

Application: X2 = b1 X1 + e2 X3 = b2 X2 + b3 X1 + e3
Example: cov(x2,X1) X1 is already exogenous, can be left. X2 is endogenous, so we must replace: COV(X2,X1) = COV (b1 X1 + e2, X1) = COV (B1 X1,X1) + COV(e2,X1) [sum rule] = b1 * COV (X1,X1) + COV(e2,X1) =b1 * VAR (X1) [variance definition + model specification]

Application: X2 = b1 X1 + e2 X3 = b2 X2 + b3 X1 + e3
We can solve for each of the 6 variances and covariances VAR(X3) = COV(X3,X3) = COV (b2 X2 + b3 X1 + e3, b2 X2 + b3 X1 + e3) Note that X2 is not exogenous, so we still have work to do! = COV (b2 {b1 X1 + e2} + b3 X1 + e 3, b2 {b1 X1 + e2} + b3 X1 + e 3) = COV (b2*b1 X1 + b2*e2 + b3 X1 + e3, b2*b1 X1 + b2*e2 + b3 X1 + e3) = b12b22 COV(X1,X1) + b22b1 COV(e2,X1) + b3*b2*b1 (COV(X1,X1) + b22*b1 COV(e2,X1) + b22 COV (e2,e2) + b2*b3 COV(e2,X1) + b2 COV(e2,e3) + b3*b2*b1 COV(X1,X1) + b3*b2 COV(X1,e2) + b32 COV(X1,X1) + b3(COV(X1,e3) + b2*b1 COV(e3,X1) + b2 COV(e3,e2) + b3 COV(e3,X1) + COV(e3,e3)

Application: X2 = b1 X1 + e2 X3 = b2 X2 + b3 X1 + e3
We can solve for each of the 6 variances and covariances VAR(X3) = b12b22 COV(X1,X1) + b22b1 COV(e2,X1) + b3*b2*b1 (COV(X1,X1) + b22*b1 COV(e2,X1) + b22 COV (e2,e2) + b2*b3 COV(e2,X1) + b2 COV(e2,e3) + b3*b2*b1 COV(X1,X1) + b3*b2 COV(X1,e2) + b32 COV(X1,X1) + b3(COV(X1,e3) + b2*b1 COV(e3,X1) + b2 COV(e3,e2) + b3 COV(e3,X1) + COV(e3,e3) Red = reduces to zero Result: VAR(X3) = b12b22VAR(X1) + b3b2b1 VAR(x1) + b22var(e2) + b3b2b1 VAR(X1) + b32var(X1) + VAR(e3) = (b12b22 + 2b3b2b1 + b32) VAR(X1) + b22var(e2) + var(e3)

Application Equations: X1 = 1.0 LV1 + e1 X2 = b2 LV1 + e2
COV (X1,X3) = COV(LV1 + e1, b3 LV1 + e3) = b3 COV(LV1,LV1) + b3 COV(e1,LV1) + COV(LV1,e3) + COV(e1,e3) = b3 VAR(LV1) Note: cov(e1,e3) may be non-zero in some models; look for curved arrow connecting error terms

Application Equations: X1 = 1.0 LV1 + e1 X2 = b2 LV1 + e2
VAR(X3) = COV(X3,X3) = COV(b3 LV1 + e3, b3 LV1 + e3) = b32 COV(L1,L1) + b3 COV(LV1,e3) + b3 COV(LV1,e3) + COV(e3,e3) = b32 VAR(L1) + VAR(e3)

Homework (to be discussed in class tomorrow) *not a formal assignment
Express the following covariances as a function of model parameters: COV(X1,X2) COV(X3,X4) VAR(x6)

Computer programming: The AMOS program
We will start with an SPSS file, which AMOS will read AMOS calculates covariance matrices “internally” The SPSS file was set up with program ReligSexMoral1.sps located in a folder called “Setups”. Missing cases have been deleted listwise (see program to see how this is done)

Amos example 1990 World Values Study Models:
Religiosity and Sexual Morality Example The Variables: Religiosity Indicators: V9 Importance of Religion 1=Very important; 2=quite imp.;3=not very; 4=not at all V147 Church attendance: 1=more than 1/week; 2=1/week;3=1/month;4=Christmas+Easter; 5=other holy days; 6=once/year; 7=less; 8=never V175 Views: 1=there is a personal God; 2=some sort of spirit/life force; 3=don’t know; 4= don’t think there is any sort of God V176 Importance of God in your life 1=Not at all through 10=very Sexual Morality Can each be always justified (=10), never justified (=1) or something in between. V304 Married men/women having an affair. V305 Sex under legal age of consent V307 Homosexuality V308 Prostitution V309 Abortion V310 Divorce

AMOS example File : USA1M Location: \baer\Week1Examples\ReligSexMoral
Do NOT use file USA1; there are missing cases in this file USA1M has had cases deleted listwise (no variable has missing cases) We will discuss how to deal with files with missing cases later. Location: \baer\Week1Examples\ReligSexMoral

LAST Slide AMOS DEMONSTRATION