SEM BASIC MODELS
1
Regression Model
V3 = *V1 + *V2+ D
V1
*
D
*
*
V3
V2
2
Regression with error-in-variables
Ex. 3.1.2 of Fuller (1987)
Data from a sample of Iowa farm operators
Y = ln (farm size)
X1 = ln ( # years experience )
X2 = ln (# years education )
y    1 x 1   2 x 2  u
Y  y  e1
X1  x 1  e 2
X 2  x 2  e3
(to protect confidentiality,
random error was added to each variable)
3
Regression with error-in-variables
*
E1
V1
F1
*
F3
*
E2
V2
*
F2
D3
*
*
V3
F3 = *F1 + *F2 + D3
V1 = F1 + E1
V2 = F2 + E2
V3 = F3 + E3
E3
E1 = .0997;
E2 = .2013;
E3 = .1808;
Coeficientes de fiabilidad son .80, .83 y .89 respectivamente
4
N=176
Regression equation
 y  0 0.439 (.112) 0.313(.107)
 x   0 0

0
 1 

 x 2  0 0

0
y e
x   x 
 1  1
 x 2   x 2 
0
0
0.699

  0
0.805  0.449
0
 0.449
0.858
5
Regression with error-in-variables
/TITLE
MODELO DE REGRESION CON ERROR EN LAS VARIABLES
/SPECIFICATIONS
CAS=176; VAR=3; ME=ML;
/LABELS
V1=TAMANO; V2=EXPER; V3=EDUCAC;
F1=TAM; F2=EXP; F3=EDU;
/EQUATIONS
V1 = F1+ E1;
V2 = F2+ E2;
V3 = F3+ E3;
F1=*F2+*F3+D1;
/VARIANCES
F2 TO F3 = *;
D1=*;
E1 = 0.0997;
E2 = 0.2013;
E3 = 0.1808;
/COVARIANCES
F2,F3 = *;
/MATRIX
.9148
.2129 1.006
.0714 -.449 1.039
/PRINT
DIG=4;
/END
6
Path analysis model
V3 = *V1 + *V2 + D3
V4 = *V1 + *V2 + D4
D3
*
V1
*
*
V3
*
V2
D4
*
V4
7
Simultaneous equations
Education development, Sewell, Haller &
Ohlendorf (1970) sample of n = 3500
Y13212 
X
Y1
Y2
Y3
  11 X 1 
  21Y1   21 X 1   22 X 2
 31Y1  31Y2 
u1
 u2
u3
where:
Y1 = academic performance (AP),
Y2 = significant influences of others(SO),
Y3 = educational aspirations (EA),
X1 = mental ability (MA),
X2 = socioeconomic status (SES).
8
Y1
u3
X1
Y3
e2
X2
y2
Y2
u2
Model:
 Y1  1
 Y  0
 2 
 Y3   0
  
 X 1  0
X 2  0
0 0 0 0
1 0 0 0
0 1 0 0

0 0 1 0
0 0 0 1
 Y1   0 
 y  e 
 2   2
 Y3    0 
   
 X1   0 
X 2   0 
  diagonal0, *, 0, 0, 0
df =
chi2=7.14. Without introducing measurement error on Y2, chi2 is 186.39 with3 df, so …
9
Path analysis model
//TITLE
modified Sewell et al (1970) model
/SPECIFICATIONS
CAS=3500; VAR=5; ME=ML; MA=COR; ANAL=COR;
/LABELS
V1=HABMENT; V2=ESTATSOC; V3=EXACAD;
V4=INFOTROS; V5=ASPEDUC;
/EQUATIONS
V3 =*V1 +D1;
F1 =*V1+*V2+*V3 +D2;
V5 = *V3+*F1 +D3;
V4=F1+E1;
/VARIANCES
E1=*;
V1 TO V2 = *;
D1 TO D3 = *;
/COVARIANCES
V1 TO V2 = *;
/MATRIX
1.000
.288 1.000
.589 .194 1.000
.438 .359 .473 1.000
.418 .380 .459 .611 1.000
/PRINT
DIG=3;
/END
10
Mimic model
Joreskog & Goldberger, JASA (1979)
y =social participation
X1 = Income
X2 = Occupation
X3 = Education
Y1= Church attendance
Y2 = Membership
Y3 = Frieds Seen
X1
X2
X3
1
2
l1
l2
l3
y
3
e
Y1
u1
Y2
u2
Y3
u3
11
Mimic model
F1 = *V1 + *V2 + *V3 + D
V4 = *F1 + E4
V5 = *F1 + E5
V6 = *F1 + E6
V1
*
*
V2
*
*
*
*
*
F1
*
*
V3
D1
V4
E2
V5
E5
V6
E6 12
ML estimates:
y  .269 X 1  .114 X 2  .386 X 3  u
(.066)
(.065)
(.070)
Y1  .402 y 
(.046)
Y2  .634 y 
e2
(.060)
Y3  .346 y 
e3
e1
(.046)
6 overidentifying restrictions. The corresponding chi2 is
12.36 with “P-VALUE” 0.052.
13
Mimic model
/TITLE
Modelo MIMIC
/SPECIFICATIONS
VARIABLES=6; CASES=530;
METHODS=ML;
MATRIX=CORRELATION;
/LABELS
V1 = Income;
V2 = Occupa;
V3 = Educat;
V4 = Church;
V5 = Afiliat ; V6 = Friends;
/EQUATIONS
V4 = 1F1 + E4;
V5 = *F1 + E5;
V6 = *F1 + E6;
F1 = *V1 + *V2 + *V3 + D1;
/VARIANCES
V1 TO V3 = *;
E4 TO E6 = *;
D1 = *;
/COVARIANCES
V2 , V1 = *;
V3 , V1 = *;
V3 , V2 = *;
/MATRIX
1.000
0.304 1.000
0.305 0.344 1.000
0.100 0.156 0.158 1.000
0.284 0.192 0.324 0.360 1.000
0.176 0.136 0.226 0.210 0.265
/LMTEST
/WTEST
/PRINT
1.000
14
Panel data
xt = t + x(t-1) + l + mt
Xt = xt + vt
t = 1,2, ..., T
 = 1,2,..., N
Anderson (1986)
xt budget of household  at time t
l individual (unobserved) characteristic of
household 
15
Panel data
E2
E1
E3
E4
V1
V2
V3
V4
1
1
1
1
F1
*
F2
*
*
F3
D2
1
F4
In a stationary process,
Var(F1)=[Var(D) + Var F0 ]/(1-)
1
ET
V5
VT
….
1
D3
1
E5
1
*
1
*
F5
D4
D5
1
1
….
*
FT
DT
F0
16
17
Factor analysis
(Spearman, 1904)
Variables
CLASSIC
FRENCH
ENGLISH
MATH
DISCRIM
MUSIC
=
=
=
=
=
=
V1
V2
V3
V4
V5
V6
Correlation matrix
1
.83 1
.78 .67 1
.70 .64 .64 1
.66 .65 .54 .45 1
.63 .57 .51 .51 .40 1
cases = 23;
18
Single-Factor Model
*
V1
*
V2
*
*
*
V3
*
V4
*
*
*
V5
*
*
V6
*
F1
19
EQS code for a factor model
/Title
confirmatory factor analysis: 1 factor ! (Spearman, 1904 )
eqs/exer3.eqs
/Specifications
var = 6; cases = 23;
/Label
v1 = classic; v2 = french; v3 =english; v4 = math; V5 = discrim;
V6=music;
/equations
RESIDUAL COVARIANCE MATRIX (S-SIGMA) :
V1 = *f1 + e1;
V2 = *f1+ e2;
V3 = *f1 + e3;
CLASSIC
FRENCH
ENGLISH
MATH
V4 = *f1 + e4;
V 1
V 2
V 3
V 4
V5 = *f1 + e5;
CLASSIC V 1
0.000
FRENCH
V 2
-0.001
0.000
V6 = *f1 + e6;
ENGLISH V 3
0.005
-0.029
0.000
/variances
MATH
V
4
-0.006
0.003
0.046
0.000
f1 = 1; e1 to e6 = *;
DISCRIM V 5
-0.001
0.054
-0.015
-0.056
/matrix
MUSIC
V 6
0.003
0.005
-0.017
0.030
1
.83 1
.78 .67 1
MUSIC
V 6
.70 .64 .64 1
MUSIC
V 6
0.000
.66 .65 .54 .45 1
.63 .57 .51 .51 .40 1
CHI-SQUARE =
1.663 BASED ON
9 DEGREES OF FREEDOM
/LMTEST
PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS
0.99575
/end
THE NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION IS
.
DISCRIM
V 5
0.000
-0.049
1.648
20
Single-Factor Model
Loadings’ estimates , s.e. and z-test
statistics
Unique factors
E1
CLASSIC =V1
FRENCH
=V2
ENGLISH =V3
MATH
=V4
DISCRIM =V5
MUSIC
=V6
=
=
=
=
=
=
.960*F1
.160
6.019
+1.000 E1
.866*F1
.171
5.049
+1.000 E2
.807*F1
.178
4.529
+1.000 E3
.736*F1
.186
3.964
+1.000 E4
.688*F1
.190
3.621
+1.000 E5
.653*F1
.193
3.382
+1.000 E6
-CLASSIC
E2
-FRENCH
E3
-ENGLISH
E4
-
E5
-DISCRIM
E6
-MUSIC
MATH
.078*I
.064 I
1.224 I
I
.251*I
.093 I
2.695 I
I
.349*I
.118 I
2.958 I
I
.459*I
.148 I
3.100 I
I
.527*I
.167 I
3.155 I
I
.574*I
.180 I
3.184 I
I
21
Single-Factor Model
STANDARDIZED SOLUTION:
CLASSIC
FRENCH
ENGLISH
MATH
DISCRIM
MUSIC
=V1
=V2
=V3
=V4
=V5
=V6
=
=
=
=
=
=
.960*F1
.866*F1
.807*F1
.736*F1
.688*F1
.653*F1
+
+
+
+
+
+
.279
.501
.591
.677
.726
.758
E1
E2
E3
E4
E5
E6
22
Factor analysis
Vi = l Fi + Ei
Var Fi = 1
F1
Var Ei = f
F2
V1
V2
V3
V4
E1
E2
E3
E4
23
Lisrel example Analysis of Reader Reliability in Essay Scoring
Analysis of Reader Reliability in Essay Scoring
Congeneric model estimated by ML
DA NI=4 NO=126
LA
ORIGPRT1 WRITCOPY CARBCOPY ORIGPRT2
CM
25.0704
12.4363 28.2021
11.7257 9.2281 22.7390
20.7510 11.9732 12.0692 21.8707
MO NX=4 NK=1 LX=FR PH=ST
!EQ TD(1) - TD(4)
!EQ LX(1) - LX(4)LK
Esayabil
PD
OU
Votaw's Data
24