1. ESTIMATION AND MODEL FIT
LECTURE 2
EPSY 651

2. 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

3. 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

4. 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

5. MAXIMUM LIKELIHOOD
minimize
FML = ln| ()| + tr)S-1())– ln|S| - (p+q)
Where () is the hypothesized model cov-
ariance matrix
S is the sample covariance matrix
p is the number of exogenous var’s
q is the number of endogenous var’s

6. 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 asymp-
totically independent
i / (ij ) = 0
Correct model specification
All data present

7. VIOLATIONS OF ASSUMPTIONS
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)

8. 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

9. 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

10. 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 non-
normal data (Chou, Bentler, & Satorra, 1991; Hu, Bentler, & Kano, 1992; Curran, West, & Finch, 1996)

11. 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

12. 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

13. Statistical analysis Fit indices
MODEL EVALUATION
Statistical analysis
Fit indices

14. 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 }

15. 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}

16. 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})

17. INCREMENTAL FIT INDICES
NFI = (b2 - m2 )/ b2 where b=baseline,
m = model
Tucker-Lewis = (b2 /dfb - m2 /dfm) /(b2 /(dfb-1))
= NNFI (nonnormed fit index)
CFI = 1-{max(m2 -dfm,0)} / {max(( b2 -dfb), (b2 - dfb),0}

18. 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

19. 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

20. 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,
Yuan, K.H. (2005) Fit indices versus test statistics. Multivariate Behavioral Research, 40, 1,
Their conclusions: depends on the model and on misspecifying the model. NO ONE CRITERION WILL WORK

21. 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

22. 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?

23. 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)

24. 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

25. Evaluating a data analysis with AMOS
Predict Teacher Salary from predictors:
Enrollment
Local revenues available
Percent majority students
Student-Teacher ratio

26. 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

27. AMOS REGRESSION ANALYSIS- OUTPUT
Parameter summary (Group number 1)
WeightsCovariancesVariancesMeansInterceptsTotalFixed200002Labeled000000Unlabeled Total Models
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 ( ):1
Result (Default model)
Minimum was achieved
Chi-square = .047
Degrees of freedom = 1
Probability level = .829

28. AMOS REGRESSION ANALYSIS- OUTPUT
Regression Weights: (Group number 1 - Default model)
Estimate S.E. C.R. P
Label
LOGPAY<---LOGENROLL ***
LOGPAY<---LOGREVLOC ***
LOGPAY<---MAJPCT
LOGPAY<---TCHSTD ***
Standardized Regression Weights: (Group number 1 - Default model)
Estimate
LOGPAY<---LOGENROLL.695
LOGPAY<---LOGREVLOC.317
LOGPAY<---MAJPCT01.000
LOGPAY<---TCHSTD

29. AMOS REGRESSION ANALYSIS- OUTPUT
Means: (Group number 1 - Default model)
Estimate S.E. C.R. P
Label
LOGENROLL ***
LOGREVLOC ***
MAJPCT ***
TCHSTD ***
Intercepts: (Group number 1 - Default model)
Estimate S.E. C.R. P
LOGPAY ***

30. AMOS REGRESSION ANALYSIS- OUTPUT
Covariances: (Group number 1 - Default model)
Estimate S.E. C.R. P
Label
LOGENROLL<-->LOGREVLOC ***
MAJPCT01<-->TCHSTD ***
LOGREVLOC<-->TCHSTD ***
LOGREVLOC<-->MAJPCT ***
LOGENROLL<-->MAJPCT ***
LOGENROLL<-->TCHSTD ***
Correlations: (Group number 1 - Default model)
Estimate
LOGENROLL<-->LOGREVLOC .897
MAJPCT01<-->TCHSTD
LOGREVLOC<-->TCHSTD
LOGREVLOC<-->MAJPCT
LOGENROLL<-->MAJPCT
LOGENROLL<-->TCHSTD

31. AMOS REGRESSION ANALYSIS- OUTPUT
Variances: (Group number 1 - Default model)
EstimateS.E.C.R.P
Label
LOGENROLL ***
LOGREVLOC ***
MAJPCT ***
TCHSTD ***
Epay ***
Squared Multiple Correlations: (Group number 1 - Default model)
Estimate
LOGPAY .799

32. AMOS REGRESSION ANALYSIS- OUTPUT
Model Fit Summary
CMIN
Model NPAR CMIN DF PCMIN/DF
Default model
Saturated model
Independence model
Baseline Comparisons
Model NFI RFI IFI TLI CFI
Delta1 rho1 Delta2 rho2
Default
Saturated model
Independence
Parsimony-Adjusted Measures
Model PRATIO PNFI PCFI
Default
Saturated
Independence

33. AMOS REGRESSION ANALYSIS- OUTPUT
RMSEA
Model RMSEA LO 90 HI 90 PCLOSE
Default
Independence
AIC
Model AIC BCC BIC CAIC
Default
Saturated
Independence

34. 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;

35. MPLUS REGRESSION analysis OUTPUT
MODEL RESULTS
Estimates S.E. Est./S.E. Std StdYX
TCHSAL01 ON
WHTPCT
LPROPTX *******
LENROL
Residual Variances
TCHSAL ********* ******* *******
R-SQUARE
Observed
Variable R-Square
TCHSAL

36. 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

37. PATH MODEL- ML ESTIMATES IN STANDARDIZED FORM
LENROLL
.502
.80
-.097
 = .731
.425
LPROPTAX
TCHR
SALARY
-.346 e
2e = = .535
.169
% MAJ
ORITY
-.189
-.267
STUDENT TEACHER RATIO
-.204

38. MPLUS Model Fit Chi-Square Test of Model Fit Value 2.249
Degrees of Freedom
P-Value
Chi-Square Test of Model Fit for the Baseline Model
Value
Degrees of Freedom
P-Value
CFI/TLI
CFI
TLI
Loglikelihood
H0 Value
H1 Value

39. Model Fit continued Information Criteria Number of Free Parameters 4
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate
90 Percent C.I
Probability RMSEA <=
SRMR (Standardized Root Mean Square Residual)
Value

40. 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

41. Model Revision Some considerations:
Split sample, model and revise on 1 sample, confirm on the other
Start with confirmation; if it fits, stop; if it fits poorly, revise
Begin with error covariances that are nonzero: they imply reliable covariance not modeled in your study
Consider paths (or factor loadings, to be discussed tomorrow) from theoretical perspectives first: do they make sense to change?
Retain general theoretical direction of the model rather than change directions inconsistent with previous research

42. LISREL
TUTORIAL:
Syntax approach

43. 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 

44. GOODNESS OF FIT STATISTICS CHI-SQUARE WITH 4 DEGREES OF FREEDOM = 4
GOODNESS OF FIT STATISTICS     CHI-SQUARE WITH 4 DEGREES OF FREEDOM = 4.73 (P = 0.32)     ESTIMATED NON-CENTRALITY PARAMETER (NCP) =     90 PERCENT CONFIDENCE INTERVAL FOR NCP = (0.0 ; 10.53)        MINIMUM FIT FUNCTION VALUE =       POPULATION DISCREPANCY FUNCTION VALUE (F0) =     90 PERCENT CONFIDENCE INTERVAL FOR F0 = (0.0 ; 0.011)     ROOT MEAN SQUARE ERROR OF APPROXIMATION (RMSEA) =     90 PERCENT CONFIDENCE INTERVAL FOR RMSEA = (0.0 ; 0.053)      P-VALUE FOR TEST OF CLOSE FIT (RMSEA < 0.05) =        EXPECTED CROSS-VALIDATION INDEX (ECVI) =     90 PERCENT CONFIDENCE INTERVAL FOR ECVI = (0.041 ; 0.052)          ECVI FOR SATURATED MODEL = 0.045          ECVI FOR INDEPENDENCE MODEL =        CHI-SQUARE FOR INDEPENDENCE MODEL WITH 15 DEGREES OF FREEDOM =     (Further output not shown)

45. 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.

46. Stability of Alienation. See LISREL8 manual, p. 207. Chapter 6
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 

48. LISREL MATRIX MODES
Matrix      Default Form   Default Mode       Possible Forms        LY                   FU                   FI                   ID, IZ, ZI, DI, FU    LX                   FU                   FI                   ID, IZ, ZI, DI, FU    BE                   ZE                   FI                   ZE, SD, FU            GA                   FU                   FR                   ID, IZ, ZI, DI, FU    PH                   SY                   FR                   ID, DI, SY, ST        PS                   DI                   FR                   ZE, DI, SY            TE                   DI                   FR                   ZE, DI, SY            TD                   DI                   FR                   ZE, DI, SY