An Introduction to Structural Equation Modeling SEM

1
An Introduction to Structural Equation
Modeling(SEM)
Albert Dionne Laval University Nov. 2000
2
An Introduction to Structural Equation
Modeling(SEM)

• INTRODUCTION
• PATH DIAGRAMS
• GLOBAL MODEL
• SEM in SEVEN STEPS
• EQS commands
• GOODNESS-OF-FIT INDICES
• MODIFICATION INDICES
• EXAMPLE

3
INTRODUCTION

• SEM Multivariate technique to study causal
  relationships among variables
• Observed variables only path analysis
• Observed and Latent (unobserved) variables SEM
• Applications biology, economics, education,
• marketing, psychology, sociology, etc.
• Software
• LISREL (LInear Structural RELations, Jöreskog
  Sörbom,1972)
• EQS (EQuationS, Bentler, 1985)
• AMOS in SPSS
• PROC CALIS in SAS
• PLS (partial least squares) , etc.

4
PATH DIAGRAMS

• 1- Simple Regression

Quantity sold
Earnings
Error
5
PATH DIAGRAMS

• 2- Multiple Regression

of Hours
of Clients
Earnings
of Calls
Error
6
PATH DIAGRAMS

• 3- Path Analysis System of simultaneous
  equations

Error
Fathers SES
Sons SES at time 1
Sons SES at time 2
Sons PSYCH at time 1
Error
Error
7
PATH DIAGRAMS

• 4- Factor Analysis

of Hours
Error
Ambition
of Clients
Error
of Calls
Error
8
PATH DIAGRAMS
Error

• 5- SEM

Hours
Error
Sales
Ambition
Performance
Error
Clients
New Clients
Error
Calls
Error
9
GLOBAL MODEL

• It consists in a series of linear equations
• 1- Equation with 2 observed variables (path
  analysis)
• Y b0 b1X e
• Example Y earnings X quantity sold
• 2- Equation with 1 observed var. and 1 latent
  var. (SEM)
• Y l ? e
• Example Y earnings ? job
  satisfaction
• 3- Equation with 2 latent variables (SEM)
• ? ? ? ?
• Example ? job satisfaction ?
  motivation

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GLOBAL MODEL

• It consists in a series of linear equations in
  terms of observed variables , of latent
  variables and of structural parameters linking
  them.
• Structural parameters are unknown constants which
  indicate the strength of the causal relationships
  between
• 1- latent variables, as in ? ? ? ?
• 2- latent variables and observed variables,
• as in Y l ? e

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GLOBAL MODEL (LISREL notation)

• It is composed of sub-models
• 1- A structural model between latent variables ?
  and ?
• ? B ? ?? ?
• 2- Two measurement models between observed
  variables and latent variables
• X Lx? ? Y Ly? ?

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GLOBAL MODEL (LISREL notation)
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SEM in SEVEN STEPS

• (1) Develop a model based on theory
• (2) Build the path diagram representing the
  causal relationships
• (3) Write the set of linear equations (EQS) or
  specify the nature of the 8 matrices (LISREL) and
  check the identification status of the model
• (4) Estimate the model with an appropriate
  iterative estimation method like ML, GLS, ULS,
  etc.
• (5) Evaluate the goodness-of-fit indices
• (6) Take the modification indices into
  consideration
• (7) Interpret the results

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EQS (a CFA example)
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EQS (a CFA example file MULE99MOD)
1 /TITLE
2 CFA
ANALYSIS WITH 2 FACTORS (adapted from MUELLER p.
99) 3 /SPECIFICATIONS
4
CASES 3094 VARIABLES 6 MATRIX
CORRELATION 5 /LABELS

6 V1 AcAbility V2 SelfConf V3
DegreAsp V4 Selctvty 7 V5
Degree V6 OcPrestg
8 F1 AcMotiv F2
ColgPres
9 /EQUATIONS

10 V1 F1E1
11 V2
F1E2
12 V3 F1E3

13 V4 F2E4

14 V5 F2E5
15 V6
F2E6
16 /VARIANCES

17 F1 TO F2

18 E1 TO E6
19 /
COVARIANCES
20 F2,F1
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EQS (a CFA example)
21 /MATRIX
22 1
23 .487 1
24 .236
.206 1
25 .382 .216 .214 1
26 .242 .179 .253 .254 1
27 .163 .090 .125 .155 .481
1 28 /STANDARD DEVIATIONS
29 .744 .782 1.014 1.99 .962 1.591
30 /LMTEST
31 Set PEE
32 /WTEST
33 /PRINT
34
Digit 2
35 /END
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GOODNESS-OF-FIT INDICES

• (1) H0 Model is ok H1 Model is not ok
• (2) Many statistics compare S, the
  variance-covariance matrix of the observed
  variables, with the reproduced var-cov matrix
• (3) In particular
• 1. T1 Khi-square test. Sensitive to sample
  size.
• 2. T2 (Khi-square) / degrees of freedom
• Model OK if T2 lt 2 (or 3, or 4 or 5 ?).
  Sensitive to sample size.
• 3. Others GFI, AGFI, NFI, NNFI, CFI, PGFI, etc.

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EQS (a CFA example)
GOODNESS OF FIT SUMMARY   INDEPENDENCE MODEL
CHI-SQUARE 2832.727 ON 15 DEGREES OF
FREEDOM   INDEPENDENCE AIC 2802.72721
INDEPENDENCE CAIC 2697.16891 MODEL
AIC 427.14190 MODEL CAIC
370.84414   CHI-SQUARE 443.142 BASED ON
8 DEGREES OF FREEDOM PROBABILITY VALUE FOR
THE CHI-SQUARE STATISTIC IS LESS THAN 0.001 THE
NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION
IS 433.136.   BENTLER-BONETT NORMED
FIT INDEX 0.844 BENTLER-BONETT NONNORMED
FIT INDEX 0.710 COMPARATIVE FIT INDEX
(CFI) 0.846      
ITERATIVE SUMMARY  
PARAMETER ITERATION ABS CHANGE
FUNCTION 1 0.357097
0.57598 2 0.162575
0.25114 3
0.071338 0.14738 4
0.041108 0.14395
5 0.009730
0.14338 6 0.008100
0.14329 7 0.002992
0.14328 8
0.001486 0.14327 9
0.000665 0.14327

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EQS (a CFA example)
MEASUREMENT EQUATIONS WITH STANDARD ERRORS
AND TEST STATISTICS     ACABILITV1
1.00 F1 1.00 E1



SELFCONFV2
.85F1 1.00 E2
.05

18.02
DEGREASPV3
.61F1 1.00 E3
.04

14.10
SELCTVTYV4 1.00 F2
1.00 E4



DEGREE V5 1.02F2
1.00 E5
.07

14.51

OCPRESTGV6 1.28F2 1.00 E6

.08

15.38

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MODIFICATION INDICES

• Are there some parameters that should be dropped
  from the estimated model ?
• WALD TEST (FOR DROPPING PARAMETERS)
• MULTIVARIATE WALD TEST BY SIMULTANEOUS PROCESS
  CUMULATIVE MULTIVARIATE STATISTICS
  UNIVARIATE INCREMENT
  --------------------------------------------------
  --------- -----------------------------
  -------
  STEP PARAMETER CHI-SQUARE D.F. PROB.
  CHI-SQUARE PROB.
  
  ----
  ----------------- ---------- ----
  --------- ----------------
  -----------
  
  
• NONE OF THE FREE PARAMETERS IS DROPPED IN THIS
  PROCESS.

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MODIFICATION INDICES

• Are there some parameters that should be added to
  the estimated model ?
• LAGRANGE MULTIPLIER TEST (FOR ADDING PARAMETERS)
   
• ORDERED UNIVARIATE TEST STATISTICS
  
  
  NO
  PARAMETER CHI-SQUARE PROBABILITY
  PARAMETER CHANGE 1 E6,E5
  306.228 0.000
  1.470
  2 E4,E1
  203.719 0.000
  0.315 3 E2,E1 83.180
  0.000 0.321 4
  E5,E3 58.523
  0.000 0.112 5 E5,E4
  48.642 0.000
  -0.349 .............................etc...
  .....................
• MULTIVARIATE LAGRANGE MULTIPLIER TEST
  CUMULATIVE MULTIVARIATE STATISTICS
  UNIVARIATE INCREMENT STEP
  PARAMETER CHI-SQUARE D.F. PROB.
  CHI-SQUARE PROB. 1 E6,E5
  306.228 1 0.000 306.228
  0.000 2 E2,E1 389.408
  2 0.000 83.180
  0.000 3 E4,E1 413.299 3
  0.000 23.891 0.000
  .............................etc.................
  .......

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EXAMPLE PEDHAZUR p. 727

Error
MA1
Mental Ability
Error
MA2
Error
MA3
Self Concept
Academic Achievement
SC1
SC3
SC2
AA3
AA1
AA2
Error
Error
Error
Error
Error
Error
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EXAMPLE EQS NOTATION

E7
V7
F3 Mental Ability
E8
V8
E9
V9
F2 Self Concept
F1 Academic Achievement
V4
V6
V5
V3
V1
V2
E5
E6
E4
E3
E2
E1
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EQS (a SEM example)
GOODNESS OF FIT SUMMARY     INDEPENDENCE MODEL
CHI-SQUARE 805.064 ON 36 DEGREES OF
FREEDOM   INDEPENDENCE AIC 733.06418
INDEPENDENCE CAIC 578.32476 MODEL
AIC -18.18121 MODEL CAIC
-121.34082   CHI-SQUARE 29.819 BASED ON
24 DEGREES OF FREEDOM PROBABILITY VALUE FOR
THE CHI-SQUARE STATISTIC IS 0.19083 THE
NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION
IS 28.662.   BENTLER-BONETT NORMED
FIT INDEX 0.963 BENTLER-BONETT NONNORMED
FIT INDEX 0.989 COMPARATIVE FIT INDEX
(CFI) 0.992     ITERATIVE
SUMMARY   PARAMETER
ITERATION ABS CHANGE
FUNCTION 1 42.292736
29.16202 2
528.183411 19.39098 3
43272.484400
19.81291 4 37024.593800
18.68308 5
5342.509280 17.77084 6
4374.743650
14.70848 17
0.001240
0.14984 18 0.000574
0.14984

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EQS (a SEM example)
MAXIMUM LIKELIHOOD SOLUTION (NORMAL DISTRIBUTION
THEORY)     STANDARDIZED SOLUTION

R-SQUARED     AA1 V1 .65 F1 .76
E1 .429
AA2 V2 .77F1 .64 E2
.595 AA3 V3
.74F1 .67 E3
.547 SC1 V4 .73 F2 .69
E4 .526
SC2 V5 .69F2 .73 E5
.472 SC3 V6
.68F2 .73 E6
.467 MA1 V7 .86 F3 .51
E7 .737
MA2 V8 .82F3 .57 E8
.678 MA3 V9
.89F3 .46 E9
.784 ACADEMICF1 .65F3 .76
D1 .426
SELFCONCF2 .31F1 .44F3 .73
D2 .465