Transcript STRUCTURAL EQUATION WORKSHOP COURSE TEXAS A&M …

ESTIMATION AND MODEL
FIT
LECTURE 2
EPSY 651
ORDINARY LEAST SQUARES
Minimize sum of squared errors:
SSe = (y – yhat)2
Best Linear Unbiased Estimates (BLUE) are
B = (X’X)-1X’y
In SEM this is translated as
F = (s-)’ (s-)
Also termed Unweighted Least Squares
ORDINARY LEAST SQUARES
ASSUMPTIONS
Normality of endogenous variables
Uncorrelated errors
Homogeneity of errors for conditional
values of predictors
Not robust with respect to violations,
particularly skewness for estimating mean
differences, kurtosis for estimating
covariances among variables
WEIGHTED LEAST SQUARES
Minimize
f = ½ tr [ I – S-1 () ]
Where I = Identity matrix and
S-1 = inverse of sample covariance
matrix ( S S-1 = I )
tr is the “trace” of a matrix (sum of the
diagonal elements of the matrix)
Simpler, but may not be as robust overall
MAXIMUM LIKELIHOOD
minimize
FML = ln| ()| + tr)S-1())– ln|S| - (p+q)
Where () is the hypothesized model covariance matrix
S is the sample covariance matrix
p is the number of exogenous var’s
q is the number of endogenous var’s
MAXIMUM LIKELIHOOD
ASSUMPTIONS
Multivariate normality on the p+q variables
Mardia’s statistic is used to test this
some evidence this is too strict
local independence: estimates are asymptotically independent
i / (ij ) = 0
Correct model specification
All data present
MAXIMUM LIKELIHOOD
VIOLATIONS OF ASSUMPTIONS
Nonnormality: little effect on estimates or
standard errors for moderate violations (Lei
& Lomax, 2005 SEM) but n> 100
Better than GLS, WLS under nonnormality
(Olsson, Foss, Troye & Howell, 2000 SEM)
Errors correlated: can affect estimates and
standard errors (Yuan, 2004; Box, 1954)
RESTRICTED MAXIMUM LIKELIHOOD
This method uses OLS or related estimates
for fixed effects (all conditions of the
population present in the study) and ML
for the residuals of the fixed effects
Used in multilevel modeling (HLM analysis),
discussed later in the course
ASYMPTOTICALLY DISTRIBUTION FREE
(ADF)
Minimizes
F = (s-)’ W-1 (s-)
Where s’ = (s11, s21, s31, …spp)
’ = (11, 21, 31, … pp)
W = some weight function
Browne (1984) showed all fit functions are special
cases of this function
Theoretically it should work well, but does not
appear as robust or efficient even for large
sample size as ML and its variants
ML VARIANTS
ML with Satorra-Bentler ADF approximations for
standard errors and chi square statistic;
approximates chi square as a sum of chi
squares weighted by functions of a WLS type
estimate
Used with nonnormal data to estimate fit statistics
based on the chi square, works better than
standard estimates with nonnormal data (Chou, Bentler, & Satorra, 1991; Hu,
Bentler, & Kano, 1992; Curran, West, & Finch, 1996)
ML VARIANTS
ML with Satorra-Bentler : in MPLUS termed
MLM
LISREL: Diagonally weighted least
squares
EQS: option available
AMOS 7: does not include
SAS Proc CALIS: not available
ML VARIANTS
ML with Yuan-Bentler chi square T-square
test statistic; used for nonnormal data with
so-called sandwich estimator; useful for
small sample data
MODEL EVALUATION
Statistical analysis
Fit indices
FIT AND LIKELIHOOD
FUNCTIONS
Fit functions (equivalent to log(L), log of
maximum likelihood function)
FML = log|()| + tr{S-1()|} -log|S| - (p+q)
FGLS = ½ tr{[ I - ()S-1 ]2}
FGLS = ½ tr{ [S - () ]2 }
FIT AND LIKELIHOOD
FUNCTIONS
-2log(L0/L1) = (N-1) FML ~2 (df = df1 – df0)
log L0 = -{(N-1)/2}{ log |(0)| + tr{-1(0)S}
= -{(N-1)/2}{ log |S)| + p+q}
under perfect fit and df0 = 0
log L1 = -{(N-1)/2}{ log |(1)| + tr{-1(1)S}
GOODNESS OF FIT INDEX
GFI = F1/F0
note: GFI = 1.0 if fit is perfect; it is the ratio
of the chi square under the fitted model to
the chi square under no fit
AGFI = 1 – [q(q+1)/2df1] [ 1 – GFI ]
RMR = [ Sum(Sij -( ))2 /(p+q)(p+q+1)/2)]½
= square root of mean of residuals of variances and
covariances between observed and fitted covariance
matrices
SRMR = SQRT ( average {residual/sresid})
INCREMENTAL FIT INDICES
NFI = (b2 - m2 )/ b2 where b=baseline,
m = model
Tucker-Lewis = (
 2 /dfb -  2 /dfm) /( 2 /(dfb-1))
b
m
b
= NNFI (nonnormed fit index)
CFI = 1-{max(m2 -dfm,0)} / {max(( b2 -dfb),
(b2 - dfb),0}
PARSIMONY-ADJUSTED FIT
INDICES COMPARED TO S
RMSEA = ½ ((m2 /(n-1)dfm)-(dfm/((N-1)dfm)))
Parsimony ratio
PR = df(model)/df(maximum possible df)
Parsimonious GFI
PGFI = GFI * dfm/dfb
PNFI = NFI*PR
PCFI = CFI*PR
INFORMATION-BASED INDICES
Akaike Information Criterion
AIC = (m2 /N) + (2k/(N-1)),
k= (.5v(v+1))-dfm,
v = # parameters fitted
Bayesian Information Criterion
BIC = m2 /N) + ((2k)/(N-v-2)
Both are used to compare non-nested models – lower values indicate
better fit
Other related statistics:
The Consistent AIC (CAIC) adjusts for
both sample size and model complexity and is interpreted in the same
way as the AIC
Browne-Cudeck Criterion (BCC) is a cross-validation statistic
BCC = F(ML) + 2q/n
EVALUATING A MODEL
• Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit
indexes in covariance structure analysis: Conventional
criteria versus new alternatives. Structural Equation
Modeling, 6, 1-55.
– use CFI> .95, TLI>.95, RMSEA<.06
• Fan, X. and Sivo, S.A. (2005) Sensitivity of fit indices to
misspecified structural or measurement model
components: rationale of two-index strategy revisited.
Structural Equation Modeling, 12, 3, 343-367.
• Yuan, K.H. (2005) Fit indices versus test statistics.
Multivariate Behavioral Research, 40, 1, 115-148.
• Their conclusions: depends on the model and on
misspecifying the model. NO ONE CRITERION WILL
WORK
EVALUATING A MODEL
• Hayduk: chi square statistic is the only test
of significance; all other statistics are
derivatives and not necessarily statistically
better
• Chi square tests’ power increases with
sample size; SEM is a large sample
method; therefore, you will almost always
reject a model with large N
• Conclusion: chi square testing is useless
EVALUATING A MODEL
• Chi square DIFFERENCE testing between
competing, nested models is the most
useful statistical test and approach to
model evaluation
• Compare a model with simpler ones, not
with a null model- you will almost certainly
always reject the null model, but so what?
MODIFICATION INDICES
• WALD STATISTICS: test if a path should be
removed (parameter = 0)
• LAGRANGE MULTIPLIER: test if a path should
be added (parameter changed to nonzero value)
• Chi Square statistics equivalent to holding model
constant except for the parameter being
considered (ie chi square difference between
baseline and model in slides above)
• Can only be computed with full data (no missing
data elements)
MODIFICATION INDICES
• Use as evidence to change a model
• Recommended statistical approach:
– Liberal alpha level for statistical test if parameter was
theoretically specified
– Conservative alpha level for statistical test if
parameter was not theoretically specified
• Only one change can be made at a time, each
change affects all other parameters and
modification indices
• MI chi square will NOT equal model chi square
improvement, should be close, however
Evaluating a data analysis with
AMOS
Predict Teacher Salary from predictors:
Enrollment
Local revenues available
Percent majority students
Student-Teacher ratio
AMOS REGRESSION ANALYSIS- OUTPUT
Your model contains the following variables (Group number 1)
Observed, endogenous variables
LOGPAY
Observed, exogenous variables
LOGENROLL
LOGREVLOC
MAJPCT01
TCHSTD01
Unobserved, exogenous variables
Epay
Variable counts (Group number 1)
Number of variables in your model:6Number of observed
variables:5Number of unobserved variables:1Number of
exogenous variables:5Number of endogenous variables:1
AMOS REGRESSION ANALYSIS- OUTPUT
Parameter summary (Group number 1)
WeightsCovariancesVariancesMeansInterceptsTotalFixed200002L
abeled000000Unlabeled3654119Total5654121Models
Default model (Default model)
Notes for Model (Default model)
Computation of degrees of freedom (Default model)
Number of distinct sample moments:20
Number of distinct parameters to be estimated:19
Degrees of freedom (20 - 19):1
Result (Default model)
Minimum was achieved
Chi-square = .047
Degrees of freedom = 1
Probability level = .829
AMOS REGRESSION ANALYSIS- OUTPUT
Regression Weights: (Group number 1 - Default model)
Estimate S.E. C.R.
P
Label
LOGPAY<---LOGENROLL
.145
.010
14.589 ***
LOGPAY<---LOGREVLOC
.060
.007
8.698 ***
LOGPAY<---MAJPCT0
1.000
LOGPAY<---TCHSTD01
-.017 .004
-4.696 ***
Standardized Regression Weights: (Group number 1 - Default
model)
Estimate
LOGPAY<---LOGENROLL.695
LOGPAY<---LOGREVLOC.317
LOGPAY<---MAJPCT01.000
LOGPAY<---TCHSTD01-.129
AMOS REGRESSION ANALYSIS- OUTPUT
Means: (Group number 1 - Default model)
Estimate
S.E. C.R.
P
Label
LOGENROLL
7.198
.047 153.236 ***
LOGREVLOC
15.040
.0522 90.870 ***
MAJPCT01
61.018
.907 67.263 ***
TCHSTD01
12.598
.074 171.216 ***
Intercepts: (Group number 1 - Default model)
Estimate
S.E. C.R.
P
Label
LOGPAY
9.613
.0651 48.875 ***
AMOS REGRESSION ANALYSIS- OUTPUT
Covariances: (Group number 1 - Default model)
Estimate
S.E.
Label
LOGENROLL<-->LOGREVLOC
1.890
.0961
MAJPCT01<-->TCHSTD01
-11.956 2.007
LOGREVLOC<-->TCHSTD01
2.121
.133
LOGREVLOC<-->MAJPCT01
-9.888 1.421
LOGENROLL<-->MAJPCT01
-12.684 1.327
LOGENROLL<-->TCHSTD01
2.427
.1311
Correlations: (Group number 1 - Default model)
Estimate
LOGENROLL<-->LOGREVLOC
.897
MAJPCT01<-->TCHSTD01
-.207
LOGREVLOC<-->TCHSTD01
.643
LOGREVLOC<-->MAJPCT01
-.243
LOGENROLL<-->MAJPCT01
-.343
LOGENROLL<-->TCHSTD01
.810
C.R.
P
9.665
-5.957
15.927
-6.957
-9.561
8.531
***
***
***
***
***
***
AMOS REGRESSION ANALYSIS- OUTPUT
Variances: (Group number 1 - Default model)
EstimateS.E.C.R.P
Label
LOGENROLL
1.913 .092
20.821
LOGREVLOC
2.317 .111
20.817
MAJPCT01
713.50 34.269 20.821
TCHSTD01
4.694 .225
20.821
Epay
.017
.001
20.819
***
***
***
***
***
Squared Multiple Correlations: (Group number 1 - Default
model)
Estimate
LOGPAY
.799
AMOS REGRESSION ANALYSIS- OUTPUT
Model Fit Summary
CMIN
Model
NPAR CMIN DF
PCMIN/DF
Default model 19.
0471.829.047
Saturated model20.0000
Independence model53990.05815.000266.004
Baseline Comparisons
Model
NFI RFI
IFI
TLI CFI
Delta1 rho1 Delta2 rho2
Default
1.000 1.000 1.000 1.004 1.000
Saturated model
1.000
1.000 1.000
Independence .000 .000 .000 .000 .000
Parsimony-Adjusted Measures
Model
PRATIO PNFI PCFI
Default
.067
.067
.067
Saturated
.000
.000
.000
Independence 1.000
.000
.000
AMOS REGRESSION ANALYSIS- OUTPUT
RMSEA
Model
RMSEA
Default
.000
Independence .553
LO 90 HI 90
.000
.054
.538
.567
PCLOSE
.941
.000
AIC
Model
Default
Saturated
Independence
BCC
38.311
40.279
4000.128
BIC
AIC
38.047
40.000
4000.058
CAIC
MPLUS REGRESSION analysis
INPUT CODE:
TITLE: TEXAS A&M SUMMER INSTITUTE REGRESSION
EXAMPLE PREDICTING TEACHER SALARY FROM 3
PREDICTORS
DATA: FILE IS SUMMER INSTITUE EXAMPLE
REGRESSION.dat;
VARIABLE:
NAMES ARE TCHSAL01 TCHSTD01 WHTPCT01
TAAS801 LPROPTX LLOCTAX LENROL01 LSUPPY01;
USEVARIABLES ARE TCHSAL01 WHTPCT01 LPROPTX
LENROL01;
MODEL:
TCHSAL01 ON WHTPCT01 LPROPTX
LENROL01;
OUTPUT: STANDARDIZED;
•
MPLUS REGRESSION analysis
OUTPUT
MODEL RESULTS
•
Estimates
S.E.
Est./S.E.
•
•
•
•
TCHSAL01 ON
WHTPCT01
LPROPTX
LENROL01
0.502
•
•
Residual Variances
TCHSAL01
********* ******* 21.201
•
R-SQUARE
-17.820 2.481
1510.289 88.070
899.791 46.613
•
•
Observed
Variable R-Square
•
TCHSAL01
0.465
Std
-7.183
17.149
19.304
*******
StdYX
-17.820 -0.189
*******
0.425
899.791
0.535
Evaluating a data analysis with
MPLUS
Reanalysis of the Salary data
•
including student-teacher ratio
•
restricting the stdt-tchr path to salary to
zero to test its necessity
• Why include a variable with zero path?
– 1. evaluate its spurious contribution
– 2. suppressor effect potential
– 3. evaluate complete model with all variables
PATH MODEL- ML ESTIMATES IN STANDARDIZED
FORM
R2 = 0.465
LENRO
LL
.80
-.097
LPROP
TAX
-.346
.169
-.267
.502
.425
-.204
TCHR
SALARY
% MAJ
ORITY
 1 - .465 = .731
e
-.189
0
STUDENT
TEACHER
RATIO
2e = .7312 = .535
MPLUS Model Fit
Chi-Square Test of Model Fit
Value
2.249
Degrees of Freedom
1
P-Value
0.1337
Chi-Square Test of Model Fit for the Baseline Model
Value
564.867
Degrees of Freedom
4
P-Value
0.0000
CFI/TLI
CFI
0.998
TLI
0.991
Loglikelihood
H0 Value
-16207.536
H1 Value
-16206.412
Model Fit continued
Information Criteria
Number of Free Parameters
4
Akaike (AIC)
32423.072
Bayesian (BIC)
32442.278
Sample-Size Adjusted BIC
32429.574
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate
0.037
90 Percent C.I.
0.000 0.105
Probability RMSEA <= .05
0.501
SRMR (Standardized Root Mean Square Residual)
Value
0.005
Model Revision
• Theoretical issue: should you revise the
model?
– What about sample specificity (changes hold
for the sample but not for the population)
– What happened to confirmation of the theory?
– How much revision?
– Where to revise- error covariances vs. path
changes
Model Revision
Some considerations:
1. Split sample, model and revise on 1 sample, confirm
on the other
2. Start with confirmation; if it fits, stop; if it fits poorly,
revise
3. Begin with error covariances that are nonzero: they
imply reliable covariance not modeled in your study
4. Consider paths (or factor loadings, to be discussed
tomorrow) from theoretical perspectives first: do they
make sense to change?
5. Retain general theoretical direction of the model rather
than change directions inconsistent with previous
research
LISREL
TUTORIAL:
http://www.utexas.edu/its/rc/tutorials/stat/lisrel/lisrel8.html
Syntax approach
INPUT SYNTAX
Stability of Alienation
! See LISREL8 manual, p. 207
! Chapter 6.4: Two-wave models
DA NI=6 NO=932
LA
ANOMIA67 POWERL67 ANOMIA71 POWERL71 EDUC SEI
CM FI=ex64.cov
MO NY=4 NX=2 NE=2 NK=1 BE=SD TE=SY,FI
LE
ALIEN67 ALIEN71
LK
SES
FR LY(2,1) LY(4,2) LX(2,1) TE(3,1) TE(4,2)
VA 1 LY(1,1) LY(3,2) LX(1,1)
OU
GOODNESS OF FIT STATISTICS
CHI-SQUARE WITH 4 DEGREES OF FREEDOM = 4.73 (P = 0.32)
ESTIMATED NON-CENTRALITY PARAMETER (NCP) = 0.73
90 PERCENT CONFIDENCE INTERVAL FOR NCP = (0.0 ; 10.53)
MINIMUM FIT FUNCTION VALUE = 0.0051
POPULATION DISCREPANCY FUNCTION VALUE (F0) = 0.00079
90 PERCENT CONFIDENCE INTERVAL FOR F0 = (0.0 ; 0.011)
ROOT MEAN SQUARE ERROR OF APPROXIMATION (RMSEA) =
0.014
90 PERCENT CONFIDENCE INTERVAL FOR RMSEA = (0.0 ; 0.053)
P-VALUE FOR TEST OF CLOSE FIT (RMSEA < 0.05) = 0.93
EXPECTED CROSS-VALIDATION INDEX (ECVI) = 0.042
90 PERCENT CONFIDENCE INTERVAL FOR ECVI = (0.041 ; 0.052)
ECVI FOR SATURATED MODEL = 0.045
ECVI FOR INDEPENDENCE MODEL = 2.30
CHI-SQUARE FOR INDEPENDENCE MODEL WITH 15 DEGREES OF
FREEDOM = 2131.40
(Further output not shown)
• If you plan to have LISREL read full matrices you create
with other software products, you must use the FU
keyword.
• CM FI=ex64.cov FU
•
•
•
The MO line defines the model. NY and NX define the number of
measurement or observed variables present. Notice that you can have two
sets of measured variables, one set on each end of a structural model. A
structural model is the portion of your model composed exclusively of latent
variables (as opposed to measurement variables). NX identifies the
"starting" or "upstream" side of your model; NY refers to the "finishing" or
"downstream" side of your model.
NE and NK define the number of latent variables associated with the
observed variables of NY and NX, respectively. In the example program,
there are four observed variables (NY) influenced by two latent NE
variables. There are two observed variables (NX) influenced by one NK
latent variable.
The LE and LK commands allow you to label the latent variables numbered
in the NE and NK commands, respectively. In this example, the two NE
latent variables are called ALIEN67 and ALIEN71. The single NK variable is
called SES.
Stability of Alienation
! See LISREL8 manual, p. 207
! Chapter 6.4: Two-wave models
DA NI=6 NO=932
LA LABELS
ANOMIA67 POWERL67 ANOMIA71 POWERL71 EDUC SEI
CM FI=ex64.cov COVARIANCE MATRIX INPUT
MO NY=4 NX=2 NE=2 NK=1 BE=SD TE=SY,FI MODEL
WITH 4 ENDOGENOUS 2 EXOGENOUS MANIFEST VARIABLES, 2
ENDOGENOUS LATENT VARIABLES (NE), ONE EXOGENOUS
LATENT VARIABLES (NK), FULL SUBDIAGONAL B-MATRIX,
ENDOGENOUS ERROR MATRIX (TE) IS SYMMETRIC AND
LE LATENT ENDOGENOUS LABELS
ALIEN67 ALIEN71
LK LATENT EXOGENOUS LABELS
SES
FR LY(2,1) LY(4,2) LX(2,1) TE(3,1) TE(4,2) FREE PARAMETERS
VA 1 LY(1,1) LY(3,2) LX(1,1) VALUES FOR PARAMETERS
OU OUTPUT
LISREL MATRIX MODES
• Matrix
LY
LX
BE
GA
PH
PS
TE
TD
Default Form Default Mode
FU
FI
FU
FI
ZE
FI
FU
FR
SY
FR
DI
FR
DI
FR
DI
FR
Possible Forms
ID, IZ, ZI, DI, FU
ID, IZ, ZI, DI, FU
ZE, SD, FU
ID, IZ, ZI, DI, FU
ID, DI, SY, ST
ZE, DI, SY
ZE, DI, SY
ZE, DI, SY