General Structural Equation (LISREL) Models Week 4 #1
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Transcript General Structural Equation (LISREL) Models Week 4 #1
General Structural Equation
(LISREL) Models
Week 4 #1
Non-normal data: summary of
approaches
Missing data approaches: summary,
review and computer examples
Longitudinal data analysis: lagged
dependent variables in LISREL models
1
Major approaches:
Transform data to normality before using in
SEM software
1.
•
•
Can be done with any stats packages
Common transformations: log, sqrt, square
ADF (also called WLS [in LISREL] AGLS
[EQS]) estimation
2.
•
•
Requires construction of asymptotic covariance
matrix
Requires large Ns
2
Major approaches to non-normal data
1.
Transform data to normality before using in SEM software
2.
ADF (also called WLS [in LISREL] AGLS [EQS]) estimation
3.
Scaled test statistics (Bentler-Satorra)
•
also referred to as “robust test statistics”
Bootstrapping
New approaches (Muthen)
Polychoric correlations (PM matrix)
4.
5.
6.
•
•
Require asympt. Cov. Matrix
Not suitable for small Ns
3
Scaled test statistics
Generate an
asymptotic
covariance
matrix in
PRELIS as well
as the usual
covariance
matrix
4
Scaled Test Statistics
Added statistics provided when asymptotic covariance matrix specified in
LISREL program
Part 2A: ML estimation but scaled chi-square statistic
DA NI=14 NO=1456
CM FI=e:\classes\icpsr2004\Week3Examples\nonnormaldata\relmor1.cov
AC FI=e:\classes\icpsr2004\Week3Examples\nonnormaldata\relmor1.acc
…PROGRAM MATRIX SPECIFICATION LINES
ou me=mll sc nd=3 mi
Degrees of Freedom = 67
Minimum Fit Function Chi-Square = 407.134 (P = 0.0)
Normal Theory Weighted Least Squares Chi-Square = 409.627 (P = 0.0)
Satorra-Bentler Scaled Chi-Square = 319.088 (P = 0.0)
Chi-Square Corrected for Non-Normality = 342.559 (P = 0.0)
5
Scaled Test Statistics
Added statistics provided when asymptotic covariance matrix
specified in LISREL program
Caution: LISREL manual suggests standard errors are
“robust” se’s but in version 8.54, identical to regular ML.
Use nested chi-square LR tests if needed
Degrees of Freedom = 67
Minimum Fit Function Chi-Square = 407.134 (P = 0.0)
Normal Theory Weighted Least Squares Chi-Square = 409.627 (P = 0.0)
Satorra-Bentler Scaled Chi-Square = 319.088 (P = 0.0)
Chi-Square Corrected for Non-Normality = 342.559 (P = 0.0)
6
Categorical Variable Model
Joreskog: with ordinal variables, “no units of
measurement.. Variances and covariances
have no meaning.. the only information we
have is counts of cases in each cell of a
multiway contingency table.
7
Categorical Variable Model
8
Categorical Variable Model
9
Categorical Variable Model
10
Categorical Variable Model
11
Categorical Variable Model
12
Categorical Variable Model
Bivariate normality: not testable 2x2
Issue: zero cells (skipped)
Too many zero cells: imprecise estimates
Only one non-zero cell in a row or column:
estimation breaks down
(in tetrachoric, PRELIS replaces 0 with 0.5;
will affect estiamtes)
13
Categorical Variable Model
Polychoric correlation very robust to violations
of underlying bivariate normality
- doctoral dissert. Ana Quiroga, 1992,
Upsala)
LR chi-square very sensitive
RMSEA measure:
- no serious effects unless RMSEA >1
(PRELIS will issue warning)
14
Categorical Variable Model
What if underlying bivariate normality does
not hold approximately?
- reduce # of categories
- eliminate offending variables
- assess if conditional on covariates
15
Bivariate data patterns not fitting the
model
Agr St Agree Neutr
Dis
Dis St
Agr St 20
10
10
5
0
Agree 10
20
20
10
5
Neut
40
20
20
20
10
Dis
0
5
20
40
10
Dis St 0
5
5
10
20
16
Insert if time permits:
brief overview of LISREL CVM
approach
17
Subdirectory
Week4Examples\OrdinalData
Bootstrapping
Hasn’t caught on as much as one might
have thought
Sample with replacement, repeat B times,
get set of values for parameters and
observe the distribution across “draws”
Typically, bootstrap N = sample N
(some literature suggestinng m<n might
be preferred, but n is standard)
18
Bootstrapping
Notes on technique:
Yung and Bentler in Marcoulides and
Schumaker, Advanced SEM (text supp.)
+ article in Br. J. Math & Stat Psych. 47:
63-84 1994
Important development: see Bollen and
Stine in Long, Testing Structural
Equation Models.
19
Bootstrapping in AMOS
Iterations
Method 0
Method 1
Method 2
1
0
0
0
2
0
0
0
3
0
0
0
4
0
0
0
5
0
0
0
6
0
7
0
7
0
99
0
8
0
206
0
9
0
114
0
10
0
46
0
11
0
18
0
12
0
7
0
13
0
2
0
14
0
0
0
15
0
1
0
16
0
0
0
17
0
0
0
18
0
0
0
19
0
0
0
Total
0
500
0
Under analysis options,
Bootstrapping tab
0 bootstrap samples were unused because of a singular covariance matrix.
0 bootstrap samples were unused because a solution was not found.
500 usable bootstrap samples were obtained.
20
|-------------------207.235
|*
224.559
|
241.883
|*
259.207
|***
276.531
|*******
293.855
|**********
311.179
|*******************
N = 500
328.503
|****************
Mean = 325.643
345.827
|*************
S. e. = 1.627
363.151
|********
380.476
|*****
397.800
|***
415.124
|*
432.448
|
Bootstrapping in AMOS
21
Bootstrapping in AMOS
Parameter
22
SE
SE-SE
Mean
Bias
SE-Bias
Relig
<---
V368
.029
.001
.060
-.001
.001
Relig
<---
V363
.032
.001
.042
.001
.001
Relig
<---
V356
.032
.001
.082
-.002
.001
Relig
<---
V355
.028
.001
-.094
-.003
.001
Relig
<---
V353
.030
.001
.126
.000
.001
Env2
<---
Relig
.044
.001
.131
-.003
.002
Env1
<---
Relig
.043
.001
-.084
.002
.002
Env1
<---
V368
.035
.001
-.010
-.002
.002
Env2
<---
V368
.036
.001
.070
.000
.002
Env2
<---
V363
.039
.001
.063
.000
.002
Env1
<---
V363
.038
.001
-.111
.000
.002
Env1
<---
V356
.038
.001
-.145
.002
.002
Env2
<---
V356
.040
.001
.227
-.002
.002
Env1
<---
V355
.034
.001
.005
.001
.002
Missing Data
The major approaches we discussed
last class:
EM algorithm to “replace” case values
and estimate Σ, z
Nearest neighbor imputation
FIML
23
The “mechanics” of working
with missing data in
PRELIS/LISREL
Nearest
Neighbor:
In PRELIS syntax:
24 ) (V147 V176 V355) VR=.5 XN or XL
IM (V356 SEX
The “mechanics” of working with missing data
in PRELIS/LISREL
The “matching variables” should have
relatively few missing cases (for a given case,
imputation will fail if any of the matching
variables is missing). Matching variables may
include variables in the “imputed variables” list
(though if any of these variables has a large
number of missing cases, this would not be a
good idea).
25
PRELIS imputation
Can save results
of imputation in
raw data file
26
Imputation
It is even possible to then
re-run PRELIS and do
other imputations.
(Although not advised,
a variable that has
been imputed can now
be used as a
“matching variable”. It
is also possible to
make another attempt
at imputation for the
same variable using
different “matching
variables”).
27
(would need to read in
raw data file back into
PRELIS)
Sample listing (IM)
SAMPLE listing:
Case
13 imputed with value
7 (Variance Ratio = 0.000), NM=
1
Case
14 not imputed because of Variance Ratio = 0.939 (NM=
2)
Case
21 not imputed because of missing values for matching variables
Number of Missing Values per Variable After Imputation
V9
V147
V151
V175
V176
-------- -------- -------- -------- -------16
13
54
38
9
V308
V309
V310
V355
V356
-------- -------- -------- -------- -------32
37
36
29
62
OCC3
OCC4
OCC5
-------- -------- -------0
0
0
Distribution of Missing Values
Total Sample Size =
1839
Number of Missing Values
0
1
2
3
Number of Cases 1584
162
50
17
28
V304
-------21
SEX
-------13
4
10
5
5
V305
-------35
OCC1
-------0
6
7
V307
-------56
OCC2
-------0
7
2
8
1
9
1
EM algorithm: PRELIS
29
EM algorithm: PRELIS
syntax:
!PRELIS SYNTAX: Can be edited
SY='G:\Missing\USA5.PSF'
SE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
EM CC = 0.00001 IT = 200
OU MA=CM SM=emcovar1.cov RA=usa6.psf
AC=emcovar1.acm XT XM
------------------------------EM Algoritm for missing Data:
------------------------------Number of different missing-value patterns=
80
Convergence of EM-algorithm in
4 iterations
30
-2 Ln(L) = 98714.48572
Multiple Group Approach
Allison Soc. Methods&Res. 1987
Bollen, p. 374 (uses old LISREL matrix notation)
31
Multiple Group Approach
Note: 13 elements of matrix have “pseudo”
values - 13 df
32
Multiple group approach
Disadvantage:
- Works only with a relatively small
number of missing patterns
33
Other missing data option:
FIML estimation
LISREL PROGRAM FOR SEXUAL MORALITY AND RELIGIOSITY EXAMPLE
DA NI=19 NO=1839 MA=CM
RA FI='G:\MISSING\USA1.PSF'
-------------------------------EM Algorithm for missing Data:
-------------------------------Number of different missing-value patterns=
80
Convergence of EM-algorithm in 5 iterations
-2 Ln(L) = 98714.48567
Percentage missing values= 1.81
LISREL
IMPLEMENTATION
Note:
The Covariances and/or Means to be analyzed are estimated
by the EM procedure and are only used to obtain starting
values for the FIML procedure
SE
V9 V151 V175 V176 V147 V304 V305 V307 V308 V309 V310 V355 V356 SEX/
MO NY=11 NE=2 LY=FU,FI PS=SY TE=SY BE=FU,FI NX=3 NK=3 LX=ID C
PH=SY,FR TD=ZE GA=FU,FR
VA 1.0 LY 5 1 LY 8 2
FR LY 1 1 LY 2 1 LY 3 1 LY 4 1
FR LY 11 2 LY 7 2 LY 6 2 LY 9 2 LY 10 2
FR BE 2 1
OU ME=ML MI SC ND=4
34
FIML
GAMMA
ETA 1
ETA 2
V355
--------0.0137
(0.0024)
-5.7192
V356
-------0.0604
(0.0202)
2.9812
SEX
-------0.4172
(0.0828)
5.0358
-0.0066
(0.0025)
-2.6128
0.1583
(0.0215)
7.3654
-0.3198
(0.0871)
-3.6705
GAMMA
-- regular ML, listwise
ETA 1
ETA 2
35
AGE
--------0.0130
(0.0025)
-5.2198
EDUC
-------0.0732
(0.0205)
3.5626
SEX
-------0.4257
(0.0904)
4.7098
-0.0076
(0.0028)
-2.7180
0.1562
(0.0227)
6.8715
-0.3112
(0.0970)
-3.2087
FIML (also referred to as
“direct ML”)
Available in AMOS and in LISREL
AMOS implementation fairly easy to
use (check off means and intercepts,
input data with missing cases and …
voila!)
LISREL implementation a bit more
difficult: must input raw data from
PRELIS into LISREL
36
FIML
37
FIML
38
FIML
39
(INSERT PRELIS/LISREL
DEMO HERE)
EM covariance matrix
Nearest neighbour imputation
FIML
40
EM algorithm: in SAS
PROC MI
Example: religiosity/morality problem.
/Week4Examples/MissingData/SAS
SASMIProc1.sas
41
SAS MI procedure
libname in1
'e:\classes\icpsr2005\Week4Examples\MissingData
2\SAS';
data one; set in1.wvssub3a;
proc mi; em outem=in1.cov; var
V9 V151 V175 V176 V147 V304 V305 V307 V308
V309 V310 v355 v356 SEX; run;
proc calis data=in1.cov cov mod;
[calis procedure specifications]
42
SAS MI procedure
Data Set
WORK.ONE
Method
Multiple Imputation Chain
Initial Estimates for MCMC
Start
Prior
Number of Imputations
Number of Burn-in Iterations
Number of Iterations
Seed for random number generator
MCMC
Single Chain
EM Posterior Mode
Starting Value
Jeffreys
5
200
100
1254
Missing Data Patterns
Group
1
2
3
V9
V151
V175
V176
V147
V304
V305
V307
V308
V309
V310
V355
V356
SEX
Freq
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
.
X
.
X
X
X
X
1456
173
10
43
SAS MI procedure
Missing Data Patterns
Group
4
5
6
7
8
9
10
11
12
13
14
15
16
17
V9
V151
V175
V176
V147
V304
V305
V307
V308
V309
V310
V355
V356
SEX
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
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X
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X
X
X
X
X
X
X
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X
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X
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X
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X
X
X
X
44
Freq
10
5
9
1
2
3
1
13
2
1
3
1
1
1
SAS MI procedure
Initial Parameter Estimates for EM
_TYPE_
_NAME_
MEAN
V9
V151
V175
V176
V147
1.720790
1.174790
1.414770
8.058470
3.958927
Initial Parameter Estimates for EM
V304
V305
V307
V308
V309
V310
V355
1.876238
2.151885
3.049916
2.395683
4.001110
4.896284
46.792265
Initial Parameter Estimates for EM
45
V356
SEX
7.775246
0.489396
SAS MI procedure
Initial Parameter Estimates for EM
_TYPE_ _NAME_
V147
COV
COV
COV
COV
COV
V9
V151
V175
V176
V147
46
V9
V151
V175
V176
0.808388
0
0
0
0
0
0.168983
0
0
0
0
0
0.483982
0
0
0
0
0
6.783348
0
0
0
0
0
6.575298
SAS MI procedure
EM (MLE) Parameter Estimates
_TYPE_
_NAME_
MEAN
COV
COV
COV
COV
COV
COV
COV
V9
V151
V175
V176
V147
V304
V305
47
V9
V151
V175
V176
V147
1.721840
0.807215
0.184412
0.307067
-1.599731
1.301326
0.390792
0.455902
1.180968
0.184412
0.170271
0.137480
-0.626684
0.454568
0.165856
0.114129
1.420315
0.307067
0.137480
0.485803
-1.073616
0.753307
0.263160
0.249936
8.046136
-1.599731
-0.626684
-1.073616
6.805023
-3.428576
-1.368173
-1.353161
3.959583
1.301326
0.454568
0.753307
-3.428576
6.567477
1.069671
0.993579
SAS PROC mi
Multiple Imputation Variance Information
Variable
V9
V151
V175
V176
V147
V304
V305
V307
-----------------Variance----------------Between
Within
Total
0.000000239
0.000002904
0.000002180
0.000002364
0.000025982
0.000002260
0.000034129
0.000027451
48
0.000439
0.000092789
0.000264
0.003710
0.003571
0.001621
0.001946
0.003995
0.000439
0.000096275
0.000266
0.003713
0.003602
0.001623
0.001987
0.004028
DF
Relative
Increase
in Variance
Fraction
Missing
Information
1834.4
1120.1
1741.7
1834.1
1760.1
1830.6
1509.7
1767.2
0.000653
0.037561
0.009913
0.000765
0.008731
0.001674
0.021050
0.008245
0.000653
0.036832
0.009863
0.000764
0.008692
0.001672
0.020824
0.008211
Sas PROC mi
SAS log:
115 proc mi; em outem=in1.cov; var
NOTE: This is an experimental version of the MI
procedure.
116 V9 V151 V175 V176 V147 V304 V305 V307 V308
V309 V310 v355 v356 SEX; run;
NOTE: The data set IN1.COV has 15 observations and
16 variables.
NOTE: PROCEDURE MI used:
real time
2.77 seconds
cpu time
2.65 seconds
49
CALIS (SAS)
proc calis data=in1.cov cov nobs=1836 mod; nobs= not needed if working with raw data
lineqs
v9 = 1.0 F1 + e1,
V175 = b1 F1 + e2,
V176 = b2 F1 + e3,
V147 = b3 F1 + e4,
V304 = 1.0 F2 + e5,
V305 = b4 F2 + e6,
V307 = b5 F2 + e7,
V308 = b6 F2 + e8,
V309 = b7 F2 + e9,
V310 = b8 F2 + e10,
F1 = b9 V355 + b10 V356 + b11 SEX + d1,
F2 = b12 V355 + b13 V356 + b14 SEX + d2;
std
e1-e10 = errvar:,
- special convention for more than 1 at a time (generates warning msg.)
v355=vv355, v356 = vv356, sex = vsex,
d1 = vd1, d2= vd2;
cov
d1 d2 = covD1D2;
run;
50
SAS - CALIS
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
51
V9
= 1.0000 F1
V175 = 0.6232*F1
Std Err 0.0223 b1
t Value 27.9048
V176 = -3.0284*F1
Std Err 0.0835 b2
t Value -36.2766
V147 = 2.2839*F1
Std Err 0.0822 b3
t Value 27.7987
V304 = 1.0000 F2
V305 = 1.0732*F2
Std Err 0.0671 b4
t Value 15.9949
V307 = 2.1959*F2
Std Err 0.1118 b5
t Value 19.6468
V308 = 1.6376*F2
Std Err 0.0863 b6
t Value 18.9819
V309 = 2.3768*F2
Std Err 0.1184 b7
t Value 20.0708
V310 = 1.9628*F2
Std Err 0.1037 b8
t Value 18.9346
+ 1.0000 e1
+ 1.0000 e2
+ 1.0000 e3
+ 1.0000 e4
+ 1.0000 e5
+ 1.0000 e6
+ 1.0000 e7
+ 1.0000 e8
+ 1.0000 e9
+ 1.0000 e10
SAS - CALIS
Variances of Exogenous Variables
Variable Parameter
V355
V356
SEX
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
d1
d2
52
vv355
vv356
vsex
errvar1
errvar2
errvar3
errvar4
errvar5
errvar6
errvar7
errvar8
errvar9
errvar10
vd1
vd2
Estimate
Standard
Error
t Value
314.89289
4.80150
0.24989
0.27695
0.27984
1.94179
3.80162
2.20316
2.68880
3.58210
2.60696
3.40464
3.81099
0.50262
0.70781
9.85535
0.14968
0.00821
0.01365
0.01049
0.11086
0.14232
0.07811
0.09493
0.14913
0.10216
0.15093
0.14885
0.02572
0.06601
31.95
32.08
30.44
20.29
26.67
17.52
26.71
28.21
28.32
24.02
25.52
22.56
25.60
19.54
10.72
SAS - CALIS
Lagrange Multiplier and Wald Test Indices _PHI_
[15:15]
Symmetric Matrix
Univariate Tests for Constant Constraints
Lagrange Multiplier or Wald Index / Probability / Approx Change of Value
V355
V356
SEX
e1
e2
V355
1020.8953
.
.
[vv355]
0.0000
1.0000
0.0000
0.0000
1.0000
-0.0000
0.1671
0.6827
-0.1071
9.0161
0.0027
0.6931
V356
0.0000
1.0000
0.0000
1029.0776
.
.
[vv356]
0.0000
1.0000
0.0000
3.6310
0.0567
0.0610
0.4885
0.4846
-0.0197
SEX
0.0000
1.0000
-0.0000
0.0000
1.0000
0.0000
926.8575
.
.
[vsex]
1.3898
0.2384
0.0089
0.1477
0.7007
0.0026
e1
0.1671
0.6827
-0.1071
3.6310
0.0567
0.0610
1.3898
0.2384
0.0089
411.7985
.
.
[errvar1]
29.2403
0.0000
-0.0528
e2
9.0161
0.0027
0.6931
0.4885
0.4846
-0.0197
0.1477
0.7007
0.0026
29.2403
0.0000
-0.0528
711.2491
.
.
[errvar2]
53
A general “what to do when”
outline (see handout)
54
Longitudinal data
I.
II.
Modeling of latent variable mean
differences over time
More complicated tests (linear growth,
quadratic growth, etc.)
55
Applications to longitudinal data
Basic model for assessing latent variable
mean change:
Can run this model on
X or Y side (LISREL)
Equations:
X1 = a1 + 1.0L1 + e1
X2 = a2 + b1 L1 + e2
X3 = a3 + b2 L1 + e3
X4 = a4 + 1.0 L2 + e4
Constraints:
b1=b3 b2=b4 LX=IN
a1=a4 a2=a5 a3=a6 TX=IN
Ka1 = 0 ka2 = (to be estimated)
X5 = a5 + b3 L2 + e5
X6 = a6 + b4 L2 + 36
56
Applications to longitudinal data
Basic model for assessing latent variable mean
change:
Constraints:
b1=b3 b2=b4 LX=IN
a1=a4 a2=a5 a3=a6 TX=IN
Ka1 = 0
ka2 = (to be estimated)
Can run this model on
X or Y side (LISREL)
Equations:
X1 = a1 + 1.0L1 + 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 + 36
Correlated errors
57
Applications to longitudinal data
Model for assessing latent variable mean change
Usual parameter constraints:
1
x1
1
1
1
x2
x3
x4
1
1
1
1
x5
x6
x7
1
x8
x9
TX(1)=TX(4)=TX(7)
LISREL: EQ TX 1 TX 4 TX 7
1
1
Ksi-1
1
Ksi-2
AMOS: same parameter name
Ksi-3
0,
0,
0,
1
a1
1
a2
1
a3
x1
x2
1
x3
0,
Ksi-1
0,
a1
x4
0,
0,
0,
0,
0,
1
a2
1
a3
1
a1
1
a2
1
a3
x5
1
x6
0,
Ksi-2
x7
x8
1
x9
0,
Ksi-3
58
Applications to longitudinal data
Model for assessing latent variable mean change
Usual parameter constraints:
1
1
1
1
1
1
1
1
1
x1
x2
x3
x4
x5
x6
x7
x8
x9
1
1
1
Ksi-1
LISREL: EQ TX 1 TX 4 TX 7
AMOS: same parameter name
Ksi-3
Ksi-2
TX(1)=TX(4)=TX(7)
KA(1) = 0
KA(2) = mean difference parameter #1
KA(3) = mean difference parameter #2
LISREL: KA=FI group 1 KA=FR groups 2,3
0,
0,
0,
1
a1
1
a2
1
a3
x1
x2
1
x3
0,
Ksi-1
0,
a1
x4
0,
0,
0,
0,
0,
1
a2
1
a3
1
a1
1
a2
1
a3
x5
1
x6
kappa1,
Ksi-2
x7
x8
1
x9
kappa2,
IN AMOS:
Ksi-3
59
Applications to longitudinal data
Model for assessing latent variable mean change
Usual parameter constraints:
1
1
1
1
1
1
1
1
1
x1
x2
x3
x4
x5
x6
x7
x8
x9
1
1
1
Ksi-1
Ksi-2
TX(1)=TX(4)=TX(7)
LISREL: EQ TX 1 TX 4 TX 7
AMOS: same parameter name
Ksi-3
KA(1) = 0
KA(2) = mean difference parameter #1
KA(3) = mean difference parameter #2
LISREL: KA=FI group 1 KA=FR groups 2,3
Some tests:
Test for change: H0: ka1=ka2=0
Linear change model:
ka2 = 2*ka1
Quadratic change model: ka2 = 4*ka1
60
As a causal model:
• Beta 1
“stability
coefficient”
1
1
1
1
1
1
1
1
Eta-1
Beta-1
Eta-2
1
• Stability coefficient is high if relative rankings
preserved, even if there has been massive change
with respect to means
• In model with AL1=0 and AL2=free, can have
high Beta2,1 with a) AL(1)=AL(2) or AL(1)
massively different from AL(2)
61
Causal models:
Ksi-1
gamma1,1
Ksi-2
gamma1,2
Eta-1
Ksi-2 as lagged (time 1) version of eta-1
(could re-specify as an eta variable)
Temporal order in Ksi-1 Eta-1 relationship
62
Causal models:
1
Eta-1
Ksi-1
ga2,1
1
ga1,2
Ksi-2
Eta-2
Cross-lagged panel coefficients
[Reduced form of model on next slide]
63
Causal models:
1
Ksi-1
Eta-1
1
Ksi-2
Eta-2
Reciprocal effects, using lagged values to achieve model
identification
64
Causal models:
A variant
Issue: what does ga(1,1) mean given
concern over causal direction?
TV Use
gamma 1,1
Political
Trust
gamma2,1
Beta 2,1Pol
Trust
Time 2
65
Lagged and contemporaneous effects
This model is underidentified
1
1
66
Lagged and contemporaneous effects
Three wave model with constraints:
1
1
a
a
d
e
d
f
c
e
f
c
b
b
1
1
67
Lagged effects model
Ksi-1 could be an “event”
1/0 dummy variable
ksi-1
ksi-2
eta-1
eta-2
68
Lagged effects model
1
1
1
1
69
Re-expressing parameters:
GROWTH CURVE MODELS
Intercept & linear (& sometimes quadratic)
terms
Exogenous variables
Alternative: HLM, subjects as level-2
observations within subjects as level-1
(mixed models: discussed elsewhere)
70