General Structural Equation (LISREL) Models

1
General Structural Equation(LISREL) Models

• Week 3 3
• MODELS FOR MEANS AND INTERCEPTS

2

• Refer to slides from previous class (Week 3 2)
  if not covered in full on Tuesday.

3
Models with Means and Intercepts

• Review of material from last class
• (detail of coverage to depend on progress from
  Tuesdays class)
• Consider a measurement model

Equations V1 1.0 L1 E1 V2 b1L1 E2 V3
b2L1 E3 V4 b3L1 E4
4
Models with Means and Intercepts

• The covariance matrix upon which this model is
  based

5
Models with Means and Intercepts

• Simple replacements in this matrix
• 1. For any element, covariance replaced by
  moment

2. And an augmented moment matrix is created by
letting the first (or the last) element of the
data matrix (the X in XX) be a vector of 1s
6
Models for Means and Intercepts

• Augmented moment matrix

Each of the above divided by N-1
7
Means and intercepts in SEM Models
Working from this matrix instead of working from
S, we can add intercepts back into equations
(reproduce M instead of S).
8
Models for Means and Intercepts

• MEASUREMENT EQUATIONS NOW BECOME
• V1 a1 1.0L1 E1
• V2 a2 b1 L1 E2
• V3 a3 b2 L1 E3
• V4 a4 b3 L1 E4
• And there is a final equation for the mean of the
  latent variable
• L1 a5

9
Means and intercepts in SEM Models
Conventional Model X1 1.0 LV1 e1 X2 b2 LV1
e2 X3 b3 LV1 e3
Extended to include intercepts X1 a1 1.0 LV1
e1 X2 a2 b2 LV1 e2 X3 a3 b3 LV1
e3 LV1 a4
EQS calls this V999. Other programs do not
explicitly model 1 as if it were a variable
10
Means and intercepts in SEM Models
Three new pieces of information Means of X1, X2,
X3 Equations X1 a1 1.0 L1 e1 X2 a2
b2 L1 e2 X3 a3 b3 L1 e3 Other
parameters Var(e1) Var(e2) Var(e3)
Var(L1) Mean(L1) One of the following
parameters needs to be fixed a1,a2,a3,
mean(L1)
11
Models for Means and Intercepts

• From the augmented moment matrix, 4 new pieces of
  information
• 5 new (possible) parameters
• a1 through a5
• ? cannot identify equation intercepts
  (under-identified)
• ? but we can identify differences between
  intercepts.

12
Means and intercepts in SEM Models
Equations X1 a1 1.0 L1 e1 X2 a2 b2
L1 e2 X3 a3 b3 L1 e3 Conventions a1
0 Then Mean(L1) Mean(X1) and a2 is
difference between means X1,X2 (not usually of
interest) a3 is difference between means X1,
X3 (not usually of interest)
13
Means and intercepts in SEM Models
Equations X1 a1 1.0 L1 e1 X2 a2 b2
L1 e2 X3 a3 b3 L1 e3
Conventions Mean(L1) 0 Then a1mean of
X1 a2 mean of X2 a3 mean of X3 Not
particularly useful means of LVs by
definition 0
14
Means and intercepts in SEM Models
In longitudinal case, more interesting
possibilities
Equations X1 a1 1.0 L1 e1 X2 a2 b1
L1 e2 X3 a3 b2 L1 e3 X4 a4 1.0 L2
e4 X5 a5 b3 L2 e5 X6 a6 b4 L2 e6
Constrain measurement models b1b3 b2b4 Constr
ain intercepts a1 a4 a2 a5 a3 a6 Fix
Mean(L1) to 0 Can now estimate parameter for Mean
(L2)
15
Means and intercepts in SEM Models
Equations X1 a1 1.0 L1 e1 X2 a2 b1
L1 e2 X3 a3 b2 L1 e3 Y1 a4 1.0 L2
e4 Y2 a5 b3 L2 e5 Y3 a6 b4 L2 e6
Constrain measurement models b1b3 b2b4 Constr
ain intercepts a1 a4 a2 a5 a3 a6 Fix
Mean(L1) to 0 Can now estimate parameter for Mean
(L2)
Example X1 X2 X3 X4 X5 X6 Means 2
3 2.5 3 4 3.5 Y1 a4 1.0
L2 e4 (E(L2)a7 Estimate a71.0 Y1 2
1.01 0 (expected value of L21.0) Y2 3
b31 0 (expected value of L2 1.0)
New parametera7
16
Means and intercepts in SEM Models
Equations X1 a1 1.0 L1 e1 X2 a2 b1
L1 e2 X3 a3 b2 L1 e3 Y1 a4 1.0 L2
e4 Y2 a5 b3 L2 e5 Y3 a6 b4 L2 e6
There can be a construct equation intercept
parameter in causal models
L2 a7 b1 L1 D2
If mean(L1) fixed to 0 E(L2) a7 b10 a7
As before, a7 represents the expected difference
between the mean of L1 and the mean of L2
17
Means and intercepts in SEM Models
L2 a7 b1 L1 D2
If mean(L1) fixed to 0 E(L2) a7 b10 a7
In practice, if L1 and L2 represent time 1 and
time 2 measures of the same thing, we would
expect correlated errors
18
Means and intercepts in SEM Models
Same principle can be applied to multiple group
models
Group 1
a11 a12
X1 a1 1.0 L1 e1 X2 a2 b2 L1 e2 X3
a3 b3 L1 e3
a21a22
a31a32
Mean(L1)0
Group 2
X1 a1 1.0 L1 e1 X2 a2 b2 L1 e2 X3
a3 b3 L1 e3
We usually constrain measurement
coefficients b21b22 b31b32
Mean(L1) a4
19
Models for Means and Intercepts

• Applications
• 1 A two-group model

Group 1 V1 a1 1.0L1 E1 V2 a2 b1 L1
E2 V3 a3 b2 L1 E3 V4 a4 b3 L1
E4 Group 2 V1 a1 1.0 L1 E1 V2 a2 b1 L1
E2 V3 a3 b2 L1 E3 V4 a4 b3 L1 E4
Group 1 Group 2
Mean(L1) a5
Mean(L1) a5
20
Models for Means and Intercepts

• Group 1 Group 2

Group 1 V1 a1 1.0L1 E1 V2 a2 b1 L1
E2 V3 a3 b2 L1 E3 V4 a4 b3 L1
E4 Group 2 V1 a1 1.0 L1 E1 V2 a2 b1 L1
E2 V3 a3 b2 L1 E3 V4 a4 b3 L1 E4
Mean(L1) a5
Mean(L1) a5
Constraints 1. Measurement model 2. intercepts
a11 a12 a21 a22 etc. 3. a51
0 THIS MEANS THAT a52 represents between-group
mean differences.
21
A practical example

• Differences in religiosity, World Values Study
  1990

In PRELIS, generate mean vectors as well as
covariances
22
Looking at item means

• Means
• U.S.
• Means
• v9 v147 v175 v176
  
• -------- -------- --------
  --------
• 1.700 3.854 1.401 8.126
• Canada
• Means high less religious except for V176
• v9 v147 v175
  v176
• -------- -------- -------- --------
• 2.193 4.811 1.750 7.005

23
Factor Means

• We cannot establish a factor mean for each group,
  but we CAN get a coefficient representing the
  difference between the factor means
• (factor mean in each group can be established
  trivially as equal to the mean of one of the
  indicators not particularly helpful though).

24
LISREL TERMINOLOGY

• Equations
• X1 tx1 ?11?1 d1
• X2 tx2 ?21?1 d2
• X3 tx3 ?31?1 d3
• X4 tx4 ?41?1 d4
• New vector Tau-X (TX)
• Normally, ?11 1.0 (reference indicator)
• Variances, covariances, means
• VAR(d1), VAR(d2), VAR(d3), VAR(d4), MEAN(?1)
• New vector Kappa (vector of means of ?s)

25
LISREL TERMINOLOGY

• Constraints
• Group 1 Group 2
• TX(1) TX(1)
• TX(2) TX(2)
• TX(3) TX(3)
• TX(4) TX(4)
• Kappa1 0 Kappa1 free

26
LISREL TERMINOLOGY

• Constraints
• Group 1 Group 2
• TX(1) TX(1)
• TX(2) TX(2)
• TX(3) TX(3)
• TX(4) TX(4)
• Kappa1 0 Kappa1 free
• Tau-X vector of manifest variable intercepts
• Kappa vector of latent (exogenous) variable
  means
• PROGRAMMING
• Group 1 TXFR KAFI
• Group 2 TXIN KAFR

27
LISREL TERMINOLOGY

• Equivalent for Y-variables
• Tau-Y intercepts for manifest variable eqs
• Alpha intercepts for construct equations
• Eta1 alpha1 gamma ksi zeta
• Important Note
• When gammas are constrained to equality across
  groups, alphas represent a between-group
  differences in means controlling for differences
  in Ksi.

28
Factor Mean differences

• Variances
• PHI USA Canada
• USA KSI 1 KSI 1
• --------
• 2.751 3.268
• (0.187) (0.162)
• TAU-X TAU-X is constrained to equality (both
  groups)
• v9 v147 v175
  v176
• -------- -------- --------
  --------
• 1.715 3.828 1.428
  8.197
• (0.023) (0.058) (0.016)
  (0.065)
• 74.484 65.924 86.556
  127.025
• KAPPA Kappa is zero in group 1
• KSI 1 Lambda-X V9 .458
• -------- V147 1.00
• 1.005 V175 .276
• (0.072) V176 -1.289

29
Models for Means and Intercepts

• Testing assumptions
• we have assumed that the pattern of differences
  between corresponding measurement equation
  intercepts can be expressed by a single
  coefficient

V1 a1 1.0 L1 e1 V2 a2 b2 L1 e2 V3
a3 b3 L1 e3 V4 a4 b4 L1 e4 L1a5
We constrain a1,a2,a3,a4 to equality across
groups and estimate a5 to represent between-group
differences
30
Models for Means and Intercepts

• What if the pattern is
• Group 1 Group 2
• v1 3.2 4.2
• v2 2.2 3.2
• v3 1.9 2.8
• v4 2.0 1.5
• a5 will be positive, but the fact that the
  group1-group2 difference on V4 is not consistent
  will lead to poorer fit

Could estimate model with a11a12,
a21a22, a31a32 BUT a41?a42
31
LISREL TERMINOLOGY

• LISREL
• Equations
• X1 tx1 ?11?1 d1
• X2 tx2 ?21?1 d2
• X3 tx3 ?31?1 d3
• X4 tx4 ?41?1 d4
• Normally, tx1 1.0 (reference indicator)
• Variances, covariances, means
• VAR(d1), VAR(d2), VAR(d3), VAR(d4), MEAN(?1)

32
Models for Means and Intercepts

• Modification Indices for TAU-X
• v9 v147 v175
  v176
• -------- -------- --------
  --------
• 4.941 0.613 13.006
  16.218
• We could estimate a model with TX 4 free (not
  essential would be more important if chi-square
  really large)
• Expected Change for TAU-X
• v9 v147 v175
  v176
• -------- -------- --------
  --------
• 0.045 -0.036 0.063
  0.236
• TAU-X (repeated from previous slide)
• v9 v147 v175
  v176
• -------- -------- --------
  --------
• 1.715 3.828 1.428
  8.197

33
LISREL PROGRAMMING CODE FOR PREVIOUS EXAMPLE

• 2 group model for relig 1USA
• DA NG2 NI23 NO1456
• CM FIg\MeansIntercepts\usa.cov
• ME FIg\MeansIntercepts\usa.mn
• LABELS
• v9 v147 v151 v175 v176 v304 v305 v307 v308 v309
  v310 v355 v356 sex
• occup1 occup2 occup3 occup4 occup5 occup6 occup7
  occup8 occup9
• SE
• 1 2 4 5 /
• MO NX4 NK1 LXFU,FI PHSY,FR TDSY C
• TXFR KAFI
• VA 1.0 LX 2 1
• FR LX 1 1 LX 3 1 LX 4 1
• OU MEML SE TV MI SC ND3
• Group 2 Canada
• DA NI23 NO1474
• CM FIg\MeansIntercepts\cdn.cov
• ME FIg\MeansIntercepts\cdn.mn
• LABELS

Do not include MACM
New
34
Doing it in AMOS
Add this
35
AMOS
For each exogenous variable, the Object
Properties box will now have Mean
and Variance
36
AMOS
For each endogenous variable, the Object
Properties box will now have an Intercept
For all indicators, type in a parameter name here.
For all indicators, click all groups to impose
equality constraint.
37
AMOS
Constraints Group 1 Group 2 b1 b1 b2
b2 b3 b3 a1 a1 a2 a2 a3 a3 a4
a4 a50 a5 free (parameter for
mean differences)
38
AMOS

• Group Canada
• Means
• Estimate S.E. C.R. P Label
• RELIG 1.005 0.072 13.931 0.000 a5
• Group United States
• Intercepts
• Estimate S.E. C.R. P Label
• V9 1.715 0.023 74.512 0.000 a1
• V147 3.828 0.058 65.947 0.000 a2
• V175 1.428 0.016 86.585 0.000 a3
• V176 8.197 0.065 127.068 0.000 a4

REFER TO Model2.amw for more extended example
39
Moving to Y, Eta and adding a 3rd country

• 2 group model for relig 1USA
• DA NG3 NI23 NO1456
• CM FIH\MeansIntercepts\usa.cov
• ME FIH\MeansIntercepts\usa.mn
• LABELS
• v9 v147 v151 v175 v176 v304 v305 v307 v308 v309
  v310 v355 v356 sex
• occup1 occup2 occup3 occup4 occup5 occup6 occup7
  occup8 occup9
• SE
• 1 2 4 5 /
• MO NY4 NE1 LYFU,FI PSSY,FR TESY C
• TYFR ALFI
• VA 1.0 LY 2 1
• FR LY 1 1 LY 3 1 LY 4 1
• OU MEML SE TV MI SC ND3
• Group 2 Canada
• DA NI23 NO1474
• CM FIH\MeansIntercepts\cdn.cov
• ME FIH\MeansIntercepts\cdn.mn
• LABELS

40
Mean comparisons USA0

• ALPHA Canada
• ETA 1
• --------
• 0.896
• (0.064)
• 14.077
• ALPHA Netherlands
• ETA 1
• --------
• 2.069
• (0.087)
• 23.889

Chi-square 280.733, df18 With
AL(1)AL(1)AL(1) 3 groups (i.e., AL0 in all
three groups) Chi-square 888. 794 df20
41
Mean comparisons USA0

In USA Modification Indices for TAU-Y
v9 v147 v175
v176 -------- --------
-------- -------- 0.855
6.130 13.121 2.882 In Canada
Modification Indices for TAU-Y
v9 v147 v175 v176
-------- -------- --------
-------- 20.003 18.756
3.873 69.008 In the Netherlands Modification
Indices for TAU-Y
v9 v147 v175 v176
-------- -------- -------- --------
19.044 60.570 4.629 62.667
42
Models for Means and Intercepts Interpreting
Mean differences with exogenous variables
GROUP 1
GROUP 2
Equations L3 a1 b1 L1 b2 L 2 D3 In
group 1, we will hold a1 fixed to 0. In group 2,
a1 will be free. IF b1 group 1 b1 group 2 AND
b2 group 1 b2 group 2 THEN a1 is the
between-group difference in L3, controlling for
the effects of L1 and L2
43
Lisrel model for mean comparisons with controls

• 2 group model for relig 1USA
• DA NG3 NI23 NO1456
• CM FIH\MeansIntercepts\usa.cov
• ME FIH\MeansIntercepts\usa.mn
• LABELS
• v9 v147 v151 v175 v176 v304 v305 v307 v308 v309
  v310 v355 v356 sex
• occup1 occup2 occup3 occup4 occup5 occup6 occup7
  occup8 occup9
• SE
• 1 2 4 5 12 13 14 /
• MO NY4 NX3 NK3 FIXEDX NE1 LYFU,FI PSSY,FR
  TESY C
• TYFR ALFI GAFU,FR KAFI TXFR
• VA 1.0 LY 2 1
• FR LY 1 1 LY 3 1 LY 4 1
• OU MEML SE TV MI SC ND3
• Group 2 Canada
• DA NI23 NO1474
• CM FIH\MeansIntercepts\cdn.cov
• ME FIH\MeansIntercepts\cdn.mn
• LABELS

44
Lisrel model for mean comparisons with controls
GROUP 1 SPECIFICATION MO NY4 NX3 NK3 FIXEDX
NE1 LYFU,FI PSSY,FR TESY C TYFR ALFI
GAFU,FR KAFI TXFR
TX parameters constrained to

• Group 3 Netherlands
• DA NI23 NO909
• CM FIH\MeansIntercepts\neth.cov
• ME FIH\MeansIntercepts\neth.mn
• LABELS
• v9 v147 v151 v175 v176 v304 v305 v307 v308 v309
  v310 v355 v356 sex
• occup1 occup2 occup3 occup4 occup5 occup6 occup7
  occup8 occup9
• SE
• 1 2 4 5 12 13 14 /
• MO LYIN PSPS TEPS ALFR TYIN FIXEDX GAIN
  KAFR TXIN
• OU MEML SE TV MI SC ND3

Exogenous variable mean 0 in group 1
Exogenous variable mean reflects difference from
group 1
45
Lisrel model for mean comparisons with controls
GROUP 1 SPECIFICATION MO NY4 NX3 NK3 FIXEDX
NE1 LYFU,FI PSSY,FR TESY C TYFR ALFI
GAFU,FR KAFI TXFR

• Group 3 Netherlands
• DA NI23 NO909
• CM FIH\MeansIntercepts\neth.cov
• ME FIH\MeansIntercepts\neth.mn
• LABELS
• v9 v147 v151 v175 v176 v304 v305 v307 v308 v309
  v310 v355 v356 sex
• occup1 occup2 occup3 occup4 occup5 occup6 occup7
  occup8 occup9
• SE
• 1 2 4 5 12 13 14 /
• MO LYIN PSPS TEPS ALFR TYIN FIXEDX GAIN
  KAFR TXIN
• OU MEML SE TV MI SC ND3

Alpha zero in group 1 Group m coefficient
represents differences from group 1
GA matrix fixed to invariance (ksi- variables
have same effect in each group)
46
Mean differences, with controls

• ALPHA CANADA
• ETA 1
• --------
• 0.854
• (0.062)
• 13.822
• KAPPA
• v355 v356 sex
• -3.465 -0.391 0.008
• (0.621) (0.085) (0.018)
• -5.579 -4.592 0.410
• ALPHA
• ETA 1
• --------
• 2.080
• (0.086)
• 24.324

47
Models for Means and Intercepts
GROUP 1
GROUP 2
L3 a11 b11L1 b21L2 D3
L3 a12 b12L1 b22L2 D3
Models/constraints 1 a110 (always) 2
b11 b12 and b21b22 (normally parallel
slopes) a120 vs. a12 ? 0 under 2
mean diffs controlling for L1,L2 a120
vs. a12? 0 under b1b20 mean diffs without
controls
48
Models for Means and Intercepts

• If slopes of all exogenous variables (L1 and L2
  in this example) are parallel, a1 is the mean
  difference controlling for exog. vars

b1
b1
a1
49
Models for Means and Intercepts

• What if slopes are not parallel?

L3
L1
A1 only represents between-group difference when
L10
Between-group difference contingent upon value of
L1
50
Models for Means and Intercepts

• 2 A longitudinal model

Fix measurement model intercepts to equality
(We would also normally fix measurement model b
coefficients to equality)
b5
Equations LVTime2 a6 b5LVTime1 D2
LVTime1 a5
Fix a50 a6 represents change in level over time
51
Models for Means and Intercepts

• An example

a1 to a11 between groups
Coeffs b13-b17 between groups
MODEL 3A
Measurement (b1 to b9) between groups
Latent var. intercepts 0 in group 1 free in
group 2
52
LISREL EXAMPLES

• Group 1 specification
• DA NG2 NI23 NO1456
• CM FIh\icpsr2003\Week3Examples\usacov1.cov
• ME FIh\icpsr2003\Week3Examples\usa.mn
• LABELS
• v9 v147 v151 v175 v176 v304 v305 v307 v308 v309
  v310 v355 v356 sex
• occup1 occup2 occup3 occup4 occup5 occup6 occup7
  occup8 occup9
• SE
• 1 2 4 5 6 7 8 9 10 11 /
• MO NY10 NE2 LYFU,FI PSSY,FR C
• TESY TYFR ALFI
• FR LY 2 1 LY 3 1 LY 4 1
• VA 1.0 LY 1 1 LY 5 2
• FR LY 6 2 LY 7 2 LY 8 2 LY 9 2 LY 10 2
• OU MEML SE TV MI SC ND3

53
LISREL EXAMPLES

• Group 2 specification

Group 2 Canada DA NI23 NO1474 CM
FIh\icpsr2003\Week3Examples\cdncov1.cov ME
FIH\ICPSR2003\Week3Examples\Cdn.mn LABELS v9
v147 v151 v175 v176 v304 v305 v307 v308 v309 v310
v355 v356 sex occup1 occup2 occup3 occup4 occup5
occup6 occup7 occup8 occup9 SE 1 2 4 5 6 7 8 9
10 11 / MO NY10 LYIN PSPS TEPS TYIN
ALFR OU MEML SE TV MI SC ND3
54
LISREL EXAMPLES

• Summary
• DA NG2 NI23 NO1456
• CM FIh\icpsr2003\Week3Examples\usacov1.cov
• ME FIh\icpsr2003\Week3Examples\usa.mn
• MO NY10 NE2 LYFU,FI PSSY,FR C
• TESY TYFR ALFI
• OU.
• Group 2 Canada
• DA NI23 NO1474
• CM FIh\icpsr2003\Week3Examples\cdncov1.cov
• ME FIH\ICPSR2003\Week3Examples\Cdn.mn
• MO NY10 LYIN PSPS TEPS TYIN ALFR

55
Group 2 printout

• TAU-Y
• v9 v147 v175
  v176 v304 v305
• -------- -------- --------
  -------- -------- --------
• 1.715 3.827 1.429
  8.195 1.908 2.246
• (0.023) (0.058) (0.016)
  (0.064) (0.040) (0.045)
• 74.370 65.859 86.728
  127.177 47.806 49.412
• TAU-Y
• v307 v308 v309
  v310
• -------- -------- --------
  --------
• 3.034 2.431 3.921
  4.792
• (0.065) (0.053) (0.065)
  (0.058)
• 46.933 45.617 60.368
  82.751
• ALPHA

PSI ETA 1 ETA 2
-------- -------- ETA 1
0.688 (0.034) 20.416
ETA 2 0.525 1.153
(0.036) (0.088) 14.790
13.134
56
Group 2 printout

• ALPHA
• ETA 1 ETA 2
• -------- --------
• 0.460 0.555
• (0.032) (0.045)
• 14.279 12.277

PSI ETA 1 ETA 2
-------- -------- ETA 1
0.688 (0.034) 20.416
ETA 2 0.525 1.153
(0.036) (0.088) 14.790
13.134
Modification Indices for TAU-Y
v9 v147 v175 v176
v304 v305 --------
-------- -------- -------- --------
-------- 4.900 0.666
13.544 16.089 11.448 16.359
Modification Indices for TAU-Y
v307 v308 v309 v310
-------- -------- --------
-------- 0.006 6.567
11.094 18.489 Expected Change for TAU-Y
v9 v147 v175
v176 v304 v305
-------- -------- -------- --------
-------- -------- 0.044
-0.037 0.064 0.235 0.161
0.234 Expected Change for TAU-Y
v307 v308 v309
v310 -------- --------
-------- -------- -0.005
0.141 -0.194 -0.217
V176 lambda is -ve
57

• We can perform block tests (both latent variables
  at a time
• MODEL 1 Group 1 TYFR ALFI
• Group 2 TYIN ALFR
• MODEL 2 Group 1 TYFR ALFI
• Group 2 TYIN ALFI

58
With exogenous variables
3 ksi variables (single indicator) ? 2
eta variables KSI variables There is
insufficient information to separately estimate
latent variable means differences using Tau
equality constraints as was done previously. We
CAN fix the mean of Ksis to the mean of the
manifest (single-indicator) variable, as
follows TXFI (i.e., fixed to TX0, both
groups) KAFR (Group 1) (Will register as mean
of corresponding X-variable) KAFR (Group
2) Important note When controlling for the
effects of the X-variables, we certainly want to
allow between-group differences in X (Ksi)
variables. Hence we usually impose no equality
constraint.
59
With exogenous variables
TXFI (i.e., fixed to TX0) KAFR (Group 1)
(Will register as mean of corresponding
X-variable) KAFR (Group 2) Important note
When controlling for the effects of the
X-variables, we certainly want to allow
between-group differences in X (Ksi) variables.
Hence we usually impose no equality
constraint. To test for significance of
differences of individual X/Ksi variables in a
2-group model, we can run another model that sets
KA in group 2 to KA in group 1 (Group 2
KAIN). We would not keep this equality
constraint in place when testing alpha parameters
for equivalence.
60
With exogenous variables
TXFI (i.e., fixed to TX0) KAFR (Group 1)
(Will register as mean of corresponding
X-variable) KAFR (Group 2) An alternative
specification TXFR KAFI (set to zero in
group 1) Group 2 TXIN KAFR (represents
between-group differences in the exogenous
single-indicator variables)
61
With exogenous variables
2 group mean model for relig sexual moral group
1USA ! adding exogenous variables DA NG2 NI23
NO1456 CM FIh\icpsr2003\Week3Examples\usacov1.
cov ME FIh\icpsr2003\Week3Examples\usa.mn LABELS
v9 v147 v151 v175 v176 v304 v305 v307 v308 v309
v310 v355 v356 sex occup1 occup2 occup3 occup4
occup5 occup6 occup7 occup8 occup9 SE 1 2 4 5 6 7
8 9 10 11 12 13 14 / MO NY10 NX3 NE2 NK3
LXID TDZE PHSY,FR LYFU,FI PSSY,FR C GAFU,FR
TYFR ALFI KAFR TXFI TESY GAFU,FR FR LY 2 1
LY 3 1 LY 4 1 VA 1.0 LY 1 1 LY 10 2 FR LY 7 2 LY
8 2 LY 9 2 LY 6 2 ly 5 2 FR TE 2 1 TE 10 9 TE 6
5 OU MEML SE TV MI SC ND3 Group 2 Canada DA
NI23 NO1474 CM FIh\icpsr2003\Week3Examples\cdn
cov1.cov ME FIH\ICPSR2003\Week3Examples\Cdn.mn L
ABELS v9 v147 v151 v175 v176 v304 v305 v307 v308
v309 v310 v355 v356 sex occup1 occup2 occup3
occup4 occup5 occup6 occup7 occup8 occup9 SE 1 2
4 5 6 7 8 9 10 11 12 13 14 / MO NY10 NX3 LYIN
LXIN PSPS PHPS TDIN TEPS C GAIN LXID TDZE
PHSY,FR TXFI KAFR TYIN ALFR OU MEML SE TV
MI SC ND3

62
With exogenous variables

MO NY10 NX3 NE2 NK3 LXID TDZE PHSY,FR
LYFU,FI PSSY,FR C GAFU,FR TYFR ALFI KAFR
TXFI TESY GAFU,FR Group 2 MO NY10 NX3
LYIN LXIN PSPS PHPS TDIN TEPS C GAIN LXID
TDZE PHSY,FR TXFI KAFR TYIN
ALFR Alternative specification in program
MMODEL3.ls8 yields same estimates for alpha (but
different estimates for kappa) Group 1 KAFI
TXFR Group 2 KAFR TXIN
63
With exogenous variables

Group 2 (Canada) results ALPHA
ETA 1 ETA 2
-------- -------- 0.426
0.839 (0.031) (0.064)
13.857 13.082 KAPPA
v355 v356 sex
-------- -------- --------
43.035 7.383 0.498 (0.421)
(0.063) (0.013) 102.251
117.894 38.210
Covariance Matrix of ETA and KSI
ETA 1 ETA 2 v355 v356
sex -------- --------
-------- -------- -------- ETA 1
0.615 ETA 2 0.701 2.411 v355
-2.678 -5.809 260.920 v356
0.340 1.028 -12.609 5.776 sex
0.062 0.009 0.014 0.012
0.250
64
With exogenous variables

Modification Indices for TAU-Y
v9 v147 v175 v176
v304 v305 --------
-------- -------- -------- --------
-------- 9.272 0.057
13.759 29.114 6.016 9.526
Modification Indices for TAU-Y
v307 v308 v309 v310
-------- -------- --------
-------- 2.967 2.309
1.943 5.638
65
With exogenous variables

So far, we have assumed GAIN What if this
assumption is unreasonable?
Modification index for Gamma Modification
Indices for GAMMA
v355 v356 sex
-------- -------- -------- ETA 1
11.943 1.530 4.826 ETA 2 0.097
0.036 0.950
Rerun model with FR GA 1 1 in group 2 BUT alpha
will no longer represent the between-group
difference in eta1, eta2, controlling for age,
educ, sex. alpha 1 will be the INTERCEPT (when
V3550)
66
With exogenous variables

Rerun model with FR GA 1 1 in group 2 BUT alpha
will no longer represent the between-group
difference in eta1, eta2, controlling for age,
educ, sex. alpha 1 will be the INTERCEPT (when
V3550)
Interpretation will depend on coding of KA and
TX If we specified TXFI and KAFR in groups 1
2, then V355 measured in YEARS so we would work
out the equation at Ksi20 Ksi40 Ksi40 If we
specified TXFR and KAFI, we have effectively
mean centred in group 1 and have centred the data
at the value of the between-group difference in
group 2. Would work out equation at Ksi-20
Ksi0 Ksi20 (still using same age metric)
67
With exogenous variables

With GA(1,1) free ALPHA
ETA 1 ETA 2 --------
-------- 0.657 0.840
(0.077) (0.064) 8.501
13.091 GAMMA group 1 v355
v356 sex --------
-------- -------- ETA 1 -0.006
0.039 0.246 (0.001) (0.007)
(0.030) -5.401 5.746
8.205 GAMMA group 2 GAMMA
v355 v356 sex
-------- -------- -------- ETA 1
-0.011 0.039 0.246 (0.001)
(0.007) (0.030) -8.666
5.746 8.205
US EQUATION Eta1 0 -.006Age CDN
EQUATION Eta1 .657 - .001Age
68
With exogenous variables

US EQUATION Eta1 0 -.006Age CDN
EQUATION Eta1 .657 - .001Age
In this model, AGE is expressed in the same
metric as the manifest variable (years). KAFR
TXFI (Different model, age would be mean
deviated group1 and centred on kappa, mean
difference from group 1in gr. 2 ) KAFI
TXFR Group 1 KAFR TXIN Group 2
69
With exogenous variables
In this model, AGE is expressed in the same
metric as the manifest variable (years). KAFR
TXFI (Different model, age would be mean
deviated group1 and centred on kappa, mean
difference from group 1in gr. 2 ) KAFI
TXFR Group 1 KAFR TXIN Group 2

US EQUATION Eta1 0 -.006Age CDN
EQUATION Eta1 .657 - .011Age
At Ksi1 20 (age average age 20) US 0 -
.00620 -.120 Cdn .657 - .01120 .657 -
.220 .437 (difference of .557 from US) At
Ksi1 0 (age average age US 0 Cdn .657
(.657 difference from US) At Ksi1 -20
(average age 20) US 0 - .006-20
.120 Cdn .657 - .011-20 .657 .220
.877 (difference of .757 from US)
70
With exogenous variables MMODEL4.lS8

At Ksi1 20 (age average age 20) US 0 -
.00620 -.120 Cdn .657 - .01120 .657 -
.220 .437 (difference of .557 from US) At
Ksi1 0 (age average age US 0 Cdn .657
(.657 difference from US) At Ksi1 -20
(average age 20) US 0 - .006-20
.120 Cdn .657 - .011-20 .657 .220
.877 (difference of .757 from US)
Cdn.
Less relig.
US
Age
71
LAST SLIDE