1. LECTURE 16
STRUCTURAL EQUATION MODELING

2. SEM PURPOSE Model phenomena from observed or theoretical stances
Develop and test constructs not directly observed based on observed indicators
Test hypothesized relationships, potentially causal, ordered, or covarying

3. Relationships to other quantitative methods

4. Decomposition of Covariance/Correlation
Most hypotheses about relationship can be represented in a covariance matrix
SEM is designed to reproduce the observed covariance matrix as closely as possible
How well the observed matrix is fitted by the hypothesized matrix is Goodness of Fit
Modeling can be either entirely theoretical or a combination of theory and revision based on imperfect fit of some parts.

5. Decomposition of Covariance/Correlation
Example: Correlation between TAAS reading level at grades 3 and 4 in 1999 was .647 for 3316 schools that gave the test. Suppose this is taken as the theoretical value for year Thus,
.768
error
TAAS
Grade3 Reading
.647
TAAS
Grade4 Reading

6. Decomposition of Covariance/Correlation
Example: Correlation between TAAS reading level at grades 3 and 4 in 2000 was .674 for 3435 schools that gave the test. We then test the theory that the relationship is stable across years;
error
H: =.647
TAAS
Grade3 Reading
.674
TAAS
Grade4 Reading

7. Decomposition of Covariance/Correlation
In classical statistics this problem is solved through Fisher’s Z-transform
Zr=tanh-1 r = 1/2 ln[1+  r  /(1 -  r |)
And a normal statistic developed, z=Zr - ZH
In SEM this is a covariance problem of fitting the observed covariance matrix to the theoretical matrix:
.674 r
r 1 =

8. Decomposition of Covariance/Correlation
The test is based on large sample multivariate normality under either maximum likelihood or generalized least squares estimation. In this case there is no estimation required, since all parameters are known. For the Fisher Z-transform, the statistic is z=1.044, p >.29. For the SEM method,

9. Decomposition of Covariance/Correlation
Under SEM, the model is represented as
F = log  + tr(S-1 ) - logS - (p – q)
= log | | + tr{ }
- log | | - (2-1)
= log ( ) + tr{ } -log ( )-1
= = .94
= X2 = .94 , df=1, p > .33
.647
.674
.647
.674
.674
1/( ) /( )
-.647/( ) 1/( )

10. Developing Theories
Previous research- both model and estimates can be used to create a theoretical basis for comparison with new data
Logical structures- time, variable stability, construct definition can provide order
1999 reading in grade 3 can affect 2000 reading in grade 4, but not the reverse
Trait anxiety can affect state anxiety, but not the reverse
IQ can affect grade 3 reading, but grade 3 reading is unlikely to alter greatly IQ (although we can think of IQ measurements that are more susceptible to reading than others)

11. Developing Theories Experimental randomized design- can be part of SEM
What-if- compare competing theories within a data set. Are all equally well explained by the data covariances?
Danger- all just-identified models equally explain all the data (ie. If all degrees of freedom are used, any model reproduces the data equally well)
Parsimony- generally simpler models are preferred; as simple as needed but not simple minded

12. MEASUREMENT MODELS

13. BASIC EQUATION x =  + e x = observed score
 = true (latent) score: represents the score that would be obtained over many independent administrations of the same item or test
e = error: difference between y and 

14. ASSUMPTIONS  and e are independent (uncorrelated)
The equation can hold for an individual or a group at one occasion or across occasions:
xijk = ijk + eijk (individual)
x*** = *** + e*** (group)
combinations (individual across time)

15. x
x  e

16. RELIABILITY
Reliability is a proportion of variance measure (squared variable)
Defined as the proportion of observed score (x) variance due to true score ( ) variance:
2x = xx’
= 2 / 2x

17. Var()
Var(e)
Var(x)
reliability

18. Reliability: parallel forms
x1 =  + e1 , x2 =  + e2
(x1 ,x2 ) = reliability
= xx’
= correlation between parallel forms

19. x1 x2
x
x  e e
xx’ = x * x

20. ASSUMPTIONS  and e are independent (uncorrelated)
The equation can hold for an individual or a group at one occasion or across occasions:
xijk = ijk + eijk (individual)
x*** = *** + e*** (group)
combinations (individual across time)

21. Reliability: Spearman-Brown
Can show the reliability of the composite is
kk’ = [k xx’]/[1 + (k-1) xx’ ]
k = # times test is lengthened
example: test score has rel=.7
doubling length produces rel = 2(.7)/[1+.7] = .824

22. Reliability: parallel forms
For 3 or more items xi, same general form holds
reliability of any pair is the correlation between them
Reliability of the composite (sum of items) is based on the average inter-item correlation: stepped-up reliability, Spearman-Brown formula

23. COMPOSITES AND FACTOR STRUCTURE
3 MANIFEST VARIABLES REQUIRED FOR A UNIQUE IDENTIFICATION OF A SINGLE FACTOR
PARALLEL FORMS REQUIRES:
EQUAL FACTOR LOADINGS
EQUAL ERROR VARIANCES
INDEPENDENCE OF ERRORS

24. e e x1 x2
x
x e 
x x3
xx’ = xi * xj

25. RELIABILITY FROM SEM
TRUE SCORE VARIANCE OF THE COMPOSITE IS OBTAINABLE FROM THE LOADINGS: K  =  2i i=1 K = # items or subtests
= K2x

26. RELIABILITY FROM SEM
RELIABILITY OF THE COMPOSITE IS OBTAINABLE FROM THE LOADINGS:  = K/(K-1)[1 - 1/  ]
example 2x = .8 , K=  = 11/(10)[1 - 1/8.8 ] = .975

27. CONGENERIC MODEL
LESS RESTRICTIVE THAN PARALLEL FORMS OR TAU EQUIVALENCE:
LOADINGS MAY DIFFER
ERROR VARIANCES MAY DIFFER
MOST COMPLEX COMPOSITES ARE CONGENERIC:
WAIS, WISC-III, K-ABC, MMPI, etc.

28. e2 e1 x1 x2
x1
x2 e3
x3  x3
(x1 , x2 )= x1 * x2

29. COEFFICIENT ALPHA xx’ = 1 - 2E /2X = 1 - [2i (1 - ii )]/2X ,
since errors are uncorrelated
 = K/(K-1)[(1 - s2i )/ s2X ]
where X = xi (composite score)
s2i = variance of subtest xi
sX = variance of composite
Does not assume knowledge of subtest ii

30. SEM MODELING OF CONGENERIC FORMS
PROC CALIS COV CORR MOD;
LINEQS
X1 = L1 F1 + E1,
X2 = L2 F1 + E2, …
X10 = L10 F1 + E10;
STD E1-E10=THE:, F1= 1.0;

31. MULTIFACTOR STRUCTURE
Measurement Model: Does it hold for each factor?
PARALLEL VS. TAU-EQUIVALENT VS. CONGENERIC
How are factors related?
What does reliability mean in the context of multifactor structure?

32. MINIMAL CORRELATED FACTOR STRUCTUREx1 x2
x11
x22 e3 1 2
x31 x3
x42 x4
12 e4

33. STRUCTURAL MODELS
Path analysis for latent variables- but can include recursive models
Begins with measurement model
Theory-based model of relationships among all variables
Modification of model at path level: LaGrange and Wald modification indices

34. Path analysis for latent variables1 Y1 Y2 2 1 X1 1 1 Y5 5 X2
12 3 Y6 6 2 2 2 X3 Y3 Y4 X4 3 4 3 4
y = By + x + 

35. Measurement Model firstY1 1 Y2 2 1 X1 1 1 Y5 5 X2
12 3 Y6 6 2 2 2 X3 Y3 Y4 X4 3 4 3 4

36. Modify Measurement Model as needed
Modification indices:
Wald Index: release constrained parameter (usually 0 path)
* chi square statistic with df=#releases
LaGrange Multiplier Index: restrict to 0 a free parameter
* chi square statistic with df =# restrictions

37. Test Full Model Examine overall Fit Examine Modification Indices
Decide if there is evidence and theoretical justification for dropping or adding a path (Note- one path at a time- select most critical/theoretically important to start with)
Liberal rule for keeping, conservative rule for adding (VW recommendation)

38. Computer Programs AMOS 5.0 – both drawing and syntax, SPSS based
Mplus 3.0 text data input, syntax only
EQS 7.8 both drawing and syntax, similar to SAS
LISREL 8.7 both drawing and syntax, difficult to use
SAS Proc Calis syntax based, easiest to integrate with other data analysis procedures