lecture 16 structural equation modeling. sem purpose model phenomena from observed or theoretical...
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LECTURE 16
• STRUCTURAL EQUATION MODELING
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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
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Relationships to other quantitative methods
STRUCTURAL EQUATION MODELS (SEM)
LATENT MANIFEST
Factor analysis Structural path models Confirmatory Exploratory Canonical analysis Discriminant True Score Theory Analysis GLM Validity Reliability Multiple ATI ANOVA (concurrent/ (generalizability) regression predictive) ANCOVA 2 group t-test IRT bivariate partial correlation correlation logistic models Causal (Grizzle et al)
Loglinear Models Associational (Holland,et al)
HLM
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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.
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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 2000. Thus,
TAAS
Grade3 Reading
TAAS
Grade4 Reading
.647
error.768
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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;
TAAS
Grade3 Reading
TAAS
Grade4 Reading
.674
errorH: =.647
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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:
1 .674
.674 1 =
1 r
r 1
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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,
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Decomposition of Covariance/Correlation
• Under SEM, the model is represented asF = log + tr(S-1 ) - logS - (p – q) = log | | + tr{ -1 }
- log | | - (2-1)
= log (1-.6472) + tr{ } -log (1-.6742)-1 = -.23553 + 1.94 - .23552-1 = .94 = 1.94 X2 = .94 , df=1, p > .33
1 .647
.647 1
1 .674
.674 1
1 .647
.647 1
1 .674
.674 1
1 .674
.674 1
1/(1-.6472) - .647/(1-.6472)
-.647/(1-.6472) 1/(1-.6472)
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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)
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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
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MEASUREMENT MODELS
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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
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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)
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x x
e
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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 / 2
x
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Var()
Var(x)
Var(e)
reliability
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Reliability: parallel forms
• x1 = + e1 , x2 = + e2
(x1 ,x2 ) = reliability
• = xx’
• = correlation between parallel forms
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x1 x
e
x2
e
x
xx’ = x * x
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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)
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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
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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
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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
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x1
x
e
x2
e
x
xx’ = xi * xj
x3
e
x
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RELIABILITY FROM SEM• TRUE SCORE VARIANCE OF THE
COMPOSITE IS OBTAINABLE FROM THE LOADINGS:
K = 2
i i=1
K = # items or subtests
• = K2x
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RELIABILITY FROM SEM
• RELIABILITY OF THE COMPOSITE IS OBTAINABLE FROM THE LOADINGS:
= K/(K-1)[1 - 1/ ]
• example 2x = .8 , K=11
= 11/(10)[1 - 1/8.8 ] = .975
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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.
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x1
x1
e1
x2
e2
x2
(x1 , x2 )= x1 * x2
x3
e3
x3
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COEFFICIENT ALPHA
xx’ = 1 - 2E /2
X
• = 1 - [2i (1 - ii )]/2
X ,
• since errors are uncorrelated = K/(K-1)[(1 - s2
i )/ s2X ]
• where X = xi (composite score)
s2i = variance of subtest xi
sX = variance of composite
• Does not assume knowledge of subtest ii
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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;
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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?
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1
x1
x11
e1
x2
e2
x22
x3
e3
x31
MINIMAL CORRELATED FACTOR STRUCTURE
2
x4e4
x42
12
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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
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Path analysis for latent variables
X1
Y3
X2
X3
X4
1
2
1
2
4
3
1
2
Y4
Y2Y1
Y6
Y5
3
3 4
6
5
21
y = By + x +
12
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Measurement Model first
X1
Y3
X2
X3
X4
1
2
1
2
4
3
1
2
Y4
Y2
Y1
Y6
Y5
3
3 4
6
5
21
12
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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
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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)
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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