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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. Major approaches:. - PowerPoint PPT Presentation
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1
General Structural Equation (LISREL) Models Week 4 #1
Non-normal data: summary of approaches Missing data approaches: summary, review and computer examplesLongitudinal data analysis: lagged dependent variables in LISREL models
2
Major approaches:
1. Transform data to normality before using in SEM software
• Can be done with any stats packages• Common transformations: log, sqrt, square
2. ADF (also called WLS [in LISREL] AGLS [EQS]) estimation
• Requires construction of asymptotic covariance matrix
• Requires large Ns
3
Major approaches to non-normal data
1. Transform data to normality before using in SEM software2. ADF (also called WLS [in LISREL] AGLS [EQS]) estimation
3. Scaled test statistics (Bentler-Satorra)• also referred to as “robust test statistics”
4. Bootstrapping5. New approaches (Muthen) 6. Polychoric correlations (PM matrix)
• Require asympt. Cov. Matrix• Not suitable for small Ns
4
Scaled test statistics
Generate an asymptotic covariance matrix in PRELIS as well as the usual covariance matrix
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Scaled Test Statistics
Added statistics provided when asymptotic covariance matrix specified in LISREL program
Part 2A: ML estimation but scaled chi-square statisticDA NI=14 NO=1456 CM FI=e:\classes\icpsr2004\Week3Examples\nonnormaldata\relmor1.covAC 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)
6
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)
7
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.
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Categorical Variable Model
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Categorical Variable Model
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Categorical Variable Model
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Categorical Variable Model
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Categorical Variable Model
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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)
14
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)
15
Categorical Variable Model
What if underlying bivariate normality does not hold approximately?
- reduce # of categories
- eliminate offending variables
- assess if conditional on covariates
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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
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Insert if time permits: brief overview of LISREL CVM approach Subdirectory Week4Examples\
OrdinalData
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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)
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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.
20
Bootstrapping in AMOS
Under analysis options, Bootstrapping tab
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
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.
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Bootstrapping in AMOS
|--------------------
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 |
449.772 |*
|--------------------
22
Bootstrapping in AMOS
Parameter 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
23
Missing Data
The major approaches we discussed last class:EM algorithm to “replace” case values
and estimate Σ, zNearest neighbor imputationFIML
24
The “mechanics” of working with missing data in PRELIS/LISREL
Nearest Neighbor:
In PRELIS syntax:IM (V356 SEX ) (V147 V176 V355) VR=.5 XN or XL
25
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).
26
PRELIS imputation
Can save results of imputation in raw data file
27
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”).
(would need to read in raw data file back into PRELIS)
28
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 variablesNumber of Missing Values per Variable After Imputation
V9 V147 V151 V175 V176 V304 V305 V307 -------- -------- -------- -------- -------- -------- -------- -------- 16 13 54 38 9 21 35 56 V308 V309 V310 V355 V356 SEX OCC1 OCC2 -------- -------- -------- -------- -------- -------- -------- -------- 32 37 36 29 62 13 0 0 OCC3 OCC4 OCC5 -------- -------- -------- 0 0 0Distribution of Missing Values Total Sample Size = 1839 Number of Missing Values 0 1 2 3 4 5 6 7 8 9 Number of Cases 1584 162 50 17 10 5 7 2 1 1
29
EM algorithm: PRELIS
30
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 -2 Ln(L) = 98714.48572
31
Multiple Group ApproachAllison Soc. Methods&Res. 1987Bollen, p. 374 (uses old LISREL matrix notation)
32
Multiple Group Approach
Note: 13 elements of matrix have “pseudo” values
- 13 df
33
Multiple group approach
Disadvantage:
- Works only with a relatively small number of missing patterns
34
Other missing data option:FIML estimationLISREL 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 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
LISREL IMPLEMENTATION
35
FIML GAMMA
V355 V356 SEX -------- -------- -------- ETA 1 -0.0137 0.0604 0.4172 (0.0024) (0.0202) (0.0828) -5.7192 2.9812 5.0358 ETA 2 -0.0066 0.1583 -0.3198 (0.0025) (0.0215) (0.0871) -2.6128 7.3654 -3.6705 GAMMA -- regular ML, listwise
AGE EDUC SEX -------- -------- -------- ETA 1 -0.0130 0.0732 0.4257 (0.0025) (0.0205) (0.0904) -5.2198 3.5626 4.7098 ETA 2 -0.0076 0.1562 -0.3112 (0.0028) (0.0227) (0.0970) -2.7180 6.8715 -3.2087
36
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
37
FIML
38
FIML
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FIML
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(INSERT PRELIS/LISREL DEMO HERE)
EM covariance matrix Nearest neighbour imputation FIML
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EM algorithm: in SAS
PROC MI
Example: religiosity/morality problem.
/Week4Examples/MissingData/SAS
SASMIProc1.sas
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SAS MI procedure
libname in1 'e:\classes\icpsr2005\Week4Examples\MissingData2\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]
43
SAS MI procedure
Data Set WORK.ONE Method MCMC Multiple Imputation Chain Single Chain Initial Estimates for MCMC EM Posterior Mode Start Starting Value Prior Jeffreys Number of Imputations 5 Number of Burn-in Iterations 200 Number of Iterations 100 Seed for random number generator 1254
Missing Data Patterns
Group V9 V151 V175 V176 V147 V304 V305 V307 V308 V309 V310 V355 V356 SEX Freq
1 X X X X X X X X X X X X X X 1456 2 X X X X X X X X X X X X . X 173 3 X X X X X X X X X X X . X X 10
44
SAS MI procedure
Missing Data Patterns
Group V9 V151 V175 V176 V147 V304 V305 V307 V308 V309 V310 V355 V356 SEX Freq
4 X X X X X X X X X X X . . X 10 5 X X X X X X X X X X . X X X 5 6 X X X X X X X X X . X X X X 9 7 X X X X X X X X X . X X . X 1 8 X X X X X X X X X . . X X X 2 9 X X X X X X X X . X X X X X 3 10 X X X X X X X X . X . X X X 1 11 X X X X X X X . X X X X X X 13 12 X X X X X X X . X X X X . X 2 13 X X X X X X X . X X X . . X 1 14 X X X X X X X . X X . X X X 3 15 X X X X X X X . X X . . X X 1 16 X X X X X X X . X . X X X X 1 17 X X X X X X X . X . X X . X 1
45
SAS MI procedure
Initial Parameter Estimates for EM
_TYPE_ _NAME_ V9 V151 V175 V176 V147
MEAN 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
V356 SEX
7.775246 0.489396
46
SAS MI procedure
Initial Parameter Estimates for EM
_TYPE_ _NAME_ V9 V151 V175 V176 V147
COV V9 0.808388 0 0 0 0 COV V151 0 0.168983 0 0 0 COV V175 0 0 0.483982 0 0 COV V176 0 0 0 6.783348 0 COV V147 0 0 0 0 6.575298
47
SAS MI procedure
EM (MLE) Parameter Estimates
_TYPE_ _NAME_ V9 V151 V175 V176 V147
MEAN 1.721840 1.180968 1.420315 8.046136 3.959583 COV V9 0.807215 0.184412 0.307067 -1.599731 1.301326 COV V151 0.184412 0.170271 0.137480 -0.626684 0.454568 COV V175 0.307067 0.137480 0.485803 -1.073616 0.753307 COV V176 -1.599731 -0.626684 -1.073616 6.805023 -3.428576 COV V147 1.301326 0.454568 0.753307 -3.428576 6.567477 COV V304 0.390792 0.165856 0.263160 -1.368173 1.069671 COV V305 0.455902 0.114129 0.249936 -1.353161 0.993579
48
SAS PROC mi
Multiple Imputation Variance Information
Relative Fraction -----------------Variance----------------- Increase Missing Variable Between Within Total DF in Variance Information
V9 0.000000239 0.000439 0.000439 1834.4 0.000653 0.000653 V151 0.000002904 0.000092789 0.000096275 1120.1 0.037561 0.036832 V175 0.000002180 0.000264 0.000266 1741.7 0.009913 0.009863 V176 0.000002364 0.003710 0.003713 1834.1 0.000765 0.000764 V147 0.000025982 0.003571 0.003602 1760.1 0.008731 0.008692 V304 0.000002260 0.001621 0.001623 1830.6 0.001674 0.001672 V305 0.000034129 0.001946 0.001987 1509.7 0.021050 0.020824 V307 0.000027451 0.003995 0.004028 1767.2 0.008245 0.008211
49
Sas PROC mi
SAS log:115 proc mi; em outem=in1.cov; varNOTE: 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
50
CALIS (SAS)
proc calis data=in1.cov cov nobs=1836 mod; nobs= not needed if working with raw datalineqs 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;stde1-e10 = errvar:, - special convention for more than 1 at a time (generates warning msg.)v355=vv355, v356 = vv356, sex = vsex,d1 = vd1, d2= vd2;covd1 d2 = covD1D2;run;
51
SAS - CALIS The CALIS Procedure Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
V9 = 1.0000 F1 + 1.0000 e1 V175 = 0.6232*F1 + 1.0000 e2 Std Err 0.0223 b1 t Value 27.9048 V176 = -3.0284*F1 + 1.0000 e3 Std Err 0.0835 b2 t Value -36.2766 V147 = 2.2839*F1 + 1.0000 e4 Std Err 0.0822 b3 t Value 27.7987 V304 = 1.0000 F2 + 1.0000 e5 V305 = 1.0732*F2 + 1.0000 e6 Std Err 0.0671 b4 t Value 15.9949 V307 = 2.1959*F2 + 1.0000 e7 Std Err 0.1118 b5 t Value 19.6468 V308 = 1.6376*F2 + 1.0000 e8 Std Err 0.0863 b6 t Value 18.9819 V309 = 2.3768*F2 + 1.0000 e9 Std Err 0.1184 b7 t Value 20.0708 V310 = 1.9628*F2 + 1.0000 e10 Std Err 0.1037 b8 t Value 18.9346
52
SAS - CALIS
Variances of Exogenous Variables
Standard Variable Parameter Estimate Error t Value
V355 vv355 314.89289 9.85535 31.95 V356 vv356 4.80150 0.14968 32.08 SEX vsex 0.24989 0.00821 30.44 e1 errvar1 0.27695 0.01365 20.29 e2 errvar2 0.27984 0.01049 26.67 e3 errvar3 1.94179 0.11086 17.52 e4 errvar4 3.80162 0.14232 26.71 e5 errvar5 2.20316 0.07811 28.21 e6 errvar6 2.68880 0.09493 28.32 e7 errvar7 3.58210 0.14913 24.02 e8 errvar8 2.60696 0.10216 25.52 e9 errvar9 3.40464 0.15093 22.56 e10 errvar10 3.81099 0.14885 25.60 d1 vd1 0.50262 0.02572 19.54 d2 vd2 0.70781 0.06601 10.72
53
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 0.0000 0.0000 0.1671 9.0161 . 1.0000 1.0000 0.6827 0.0027 . 0.0000 -0.0000 -0.1071 0.6931 [vv355]
V356 0.0000 1029.0776 0.0000 3.6310 0.4885 1.0000 . 1.0000 0.0567 0.4846 0.0000 . 0.0000 0.0610 -0.0197 [vv356]
SEX 0.0000 0.0000 926.8575 1.3898 0.1477 1.0000 1.0000 . 0.2384 0.7007 -0.0000 0.0000 . 0.0089 0.0026 [vsex]
e1 0.1671 3.6310 1.3898 411.7985 29.2403 0.6827 0.0567 0.2384 . 0.0000 -0.1071 0.0610 0.0089 . -0.0528 [errvar1]
e2 9.0161 0.4885 0.1477 29.2403 711.2491 0.0027 0.4846 0.7007 0.0000 . 0.6931 -0.0197 0.0026 -0.0528 . [errvar2]
54
A general “what to do when” outline (see handout)
55
Longitudinal data
I. Modeling of latent variable mean differences over time
II. More complicated tests (linear growth, quadratic growth, etc.)
56
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
X5 = a5 + b3 L2 + e5
X6 = a6 + b4 L2 + 36
Constraints:
b1=b3 b2=b4 LX=IN
a1=a4 a2=a5 a3=a6 TX=IN
Ka1 = 0 ka2 = (to be estimated)
57
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
X5 = a5 + b3 L2 + e5
X6 = a6 + b4 L2 + 36
Constraints:
b1=b3 b2=b4 LX=IN
a1=a4 a2=a5 a3=a6 TX=IN
Ka1 = 0 ka2 = (to be estimated)
Correlated errors
58
Applications to longitudinal data
Model for assessing latent variable mean change
Ksi-1
x11
1
x2
1
x3
1
Ksi-2
x4 x5 x61
1 1 1
Ksi-3
x7 x8 x91
1 1 1
Usual parameter constraints:
TX(1)=TX(4)=TX(7)
LISREL: EQ TX 1 TX 4 TX 7
AMOS: same parameter name
0,
Ksi-1
a1
x1
0,
1
1a2
x2
0,
1a3
x3
0,
1
0,
Ksi-2
a1
x4
0,
a2
x5
0,
a3
x6
0,
1
1 1
0,
Ksi-3
a1
x7
0,
a2
x8
0,
a3
x9
0,
1
1 1 1
59
Applications to longitudinal data
Model for assessing latent variable mean change
Ksi-1
x11
1
x2
1
x3
1
Ksi-2
x4 x5 x61
1 1 1
Ksi-3
x7 x8 x91
1 1 1
Usual parameter constraints:
TX(1)=TX(4)=TX(7)
LISREL: EQ TX 1 TX 4 TX 7
AMOS: same parameter name
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
IN AMOS:
0,
Ksi-1
a1
x1
0,
1
1a2
x2
0,
1a3
x3
0,
1
kappa1,
Ksi-2
a1
x4
0,
a2
x5
0,
a3
x6
0,
1
1 1
kappa2,
Ksi-3
a1
x7
0,
a2
x8
0,
a3
x9
0,
1
1 1 1
60
Applications to longitudinal data
Model for assessing latent variable mean change
Ksi-1
x11
1
x2
1
x3
1
Ksi-2
x4 x5 x61
1 1 1
Ksi-3
x7 x8 x91
1 1 1
Usual parameter constraints:
TX(1)=TX(4)=TX(7)
LISREL: EQ TX 1 TX 4 TX 7
AMOS: same parameter name
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
61
As a causal model:
• Beta 1 “stability coefficient”
Eta-1
1
1 1 1
Eta-2
1
1 1 1
Beta-1 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)
62
Causal models:
Ksi-1
Ksi-2 Eta-1
gamma1,1
gamma1,2
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
63
Causal models:
Ksi-1
Ksi-2 Eta-2
ga2,1
Eta-1
ga1,2
1
1
Cross-lagged panel coefficients
[Reduced form of model on next slide]
64
Causal models:
Reciprocal effects, using lagged values to achieve model identification
Ksi-1
Ksi-2 Eta-2
Eta-1
1
1
65
Causal models:
TV Use
PoliticalTrust
Pol TrustTime 2
gamma 1,1 gamma2,1
Beta 2,1
A variant
Issue: what does ga(1,1) mean given concern over causal direction?
66
Lagged and contemporaneous effects
1
1
This model is underidentified
67
Lagged and contemporaneous effects
Three wave model with constraints:
a
e f
b
d
c
1
1
a
b
e f
1
1
d
c
68
Lagged effects model
ksi-2 eta-1 eta-2
ksi-1
Ksi-1 could be an “event”
1/0 dummy variable
69
Lagged effects model
1
1
1
1
70
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)