General Structural Equations (LISREL)

1
General Structural Equations(LISREL)

• Week 3 4
• Mean Models Reviewed
• Non-parallel slopes
• Non-normal data

2
Models for Means and Intercepts (continued)

• Multiple Group Models
• For zero order latent variable mean
  differences
• free individual measurement equation intercepts
  but constrain them to equality across groups
• Fix the latent variable means to 0 in group 1
• Free the latent variable means in groups 2-gtk
• If the latent variables of interest are
  endogenous and if there are exogenous latent
  variables in the model, constrain construct
  equation path coefficients to zero.

3
Models for Means and Intercepts (continued)

• For zero order latent variable mean
  differences
• free individual measurement equation intercepts
  but constrain them to equality across groups
• Fix the latent variable means to 0 in group 1
• Free the latent variable means in groups 2-gtk
• If the latent variables of interest are
  endogenous and if there are exogenous latent
  variables in the model, constrain construct
  equation path coefficients to zero.
• Individual LV mean parameters represent contrast
  with (differerence from) reference group (group
  with LV mean set to zero LR tests requested for
  joint hypotheses (e.g, constrain means to zero in
  all groups vs. model with groups 2-gtk freed)
• Check modification indices on measurement
  equation intercepts to verify proportional
  indicator differences assumption holds (or at
  least holds approximately)

4
AMOS Programming

• Check off means and intercepts
• Means and intercepts will now appear on diagram.
  Where variances used to appear, there will now be
  two parameters (mean variance) where the
  variable is dependent, one parameter (intercept)
  will appear.
• Impose appropriate parameter constraints
• insert brief demonstration here!

5
Review yesterdays slides from slide 52

• Uses World Values Study 1990 data for an example
• Well use an updated version (new data, some
  difference in countries) today
• Refer to handout (slides not reproduced)

6
Means1a.LS8 - tau-x elements allowed to vary
between countries. Must fix kappa (mean of
ksis) to 0 since otherwise not identified.
Chi-square233.65 df42 United States TAU-X
A006 F028 F066
F063 F118 F119
-------- -------- -------- --------
-------- -------- 1.6191
3.6383 2.2287 8.5530 4.7504
2.9739 (0.0263) (0.0688) (0.0563)
(0.0733) (0.0941) (0.0749)
61.4969 52.8937 39.5980 116.6334
50.4717 39.6838 TAU-X
F120 F121 --------
-------- 4.3443 5.9000
(0.0883) (0.0757) 49.2263
77.9553 CANADA TAU-X A006
F028 F066 F063 F118
F119 -------- --------
-------- -------- -------- --------
2.1202 4.7402 3.2042 7.4657
5.4974 3.3091 (0.0236) (0.0612)
(0.0551) (0.0706) (0.0812) (0.0646)
89.9232 77.4453 58.1887 105.7780
67.7035 51.2445 TAU-X
F120 F121 --------
-------- 4.4986 6.0079
(0.0713) (0.0636)
7

• Means1b.ls8 Measurement model like means1a, but
  now we are expressing group 1 versus group 2
  differences in means by 2 parameters (1 for each
  latent variable) as opposed to calculating them
  for each indicator using, e.g., TX 1 1 TX1
  2.
• Chi-square276.27 df48
• KAPPA in Group 2 (Canada) Kappa in Group 1
  is fixed to zero
• KSI 1 KSI 2
• -------- --------
• 1.0712 0.3236
• (0.0731) (0.0948)
• 14.6538 3.4138
• Above provides significance tests for
• Canada-U.S. differences in religiosity
  (z14.6538, plt.001)
• Canada-U.S. differences in sex/morality attitudes
  (z3.4138, plt.001)
• For a joint significance test to see if both the
  means for Religiosity and Sex/morality are
  different (null hypothesis, differences both
  0), see program Means1c.ls8. Chi-square
  512.9661 df50 for this model subtract
  chi-squares (512-276) for test (df2).

8

• Diagnostics for this model See Modification
  Indices for TX vectors
• USA
• Modification Indices for TAU-X
• A006 F028 F066
  F063 F118 F119
• -------- -------- --------
  -------- -------- --------
• 0.6495 0.2995 2.8724
  8.2808 27.0494 2.0749
• Modification Indices for TAU-X
• F120 F121
• -------- --------
• 12.1313 5.2727
• CANADA
• Modification Indices for TAU-X
• A006 F028 F066
  F063 F118 F119
• -------- -------- --------
  -------- -------- --------
• 0.6495 0.2995 2.8725
  8.2808 27.0495 2.0749
• Modification Indices for TAU-X
• F120 F121
• -------- --------
• 12.1312 5.2728

9
Means2a Model with exogenous single-indicator
variables. Single indicator ksi-variables
gender, age, education. Specification GAIN in
group 2 implies a parallel slopes model. Thus,
the AL parameters in group 2 can be interpreted
as group 1 vs. group 2 differences,
controlling for differences in sex, education and
age. TAU-X GENDER
AGE EDUC --------
-------- -------- 0.4217
42.3840 4.5365 (0.0146)
(0.4750) (0.0413) 28.9360
89.2300 109.8409 ALPHA
ETA 1 ETA 2 --------
-------- 1.2272 0.5898
(0.0714) (0.0954) 17.1899
6.1819 KAPPA
GENDER AGE EDUC
-------- -------- --------
-0.0196 3.3360 -0.4151
(0.0187) (0.6297) (0.0504)
-1.0482 5.2977 -8.2333
10

• Diagnostics Test of equal slopes (GAIN)
  assumption
• Modification Indices for GAMMA
• GENDER AGE EDUC
• -------- -------- --------
• ETA 1 7.7083 6.9705 0.2122
• ETA 2 3.1923 0.1765 9.3836
• A global test will require the estimation of a
  separate model (Means2b) with GAPS (parallel
  slopes assumption relaxed).
• Chi-square df CFI
• Chi-square comparisons Means2a 699.807 90 .963
  5
• Means2b 669.594 84 .9649

11

• Means2b
• ALPHA CANADA (FIXED TO 0 IN US)
• ETA 1 ETA 2
• -------- --------
• 1.2545 0.6371
• (0.0725) (0.0968)
• 17.3057 6.5809
• GAMMA - USA
• GENDER AGE EDUC
• -------- -------- --------
• ETA 1 0.6845 -0.0170 0.0817
• (0.1003) (0.0031) (0.0352)
• 6.8230 -5.5398 2.3209
• ETA 2 0.0624 -0.0144 0.3074
• (0.1462) (0.0045) (0.0520)
• GAMMA-Canada
• GENDER AGE EDUC
• -------- -------- --------

12

• Expressing effects when parallel slope assumption
  is relaxed
• is pattern diverging, converging, crossover?
• Equations
• Eta1 alpha1 gamma1 Ksi 1 gamma2 Ksi2
  gamma3 Ksi 3 zeta1
• Hold constant at the 0 values of all Ksi
  variables except one. Not quite the overall mean
  (Ksi0 in group 1, but in group 2 its 0
  kappa), but close enough.
• In group 1, alpha1 0, equation is
• Eta1 gamma1 1Ksi1 alpha10 gamma2
  Ksi20 gamma3 Ksi30 zeta1
• where E(zeta1)0
• In group 2, alpha1 alpha12
• Eta1 alpha12 gamma12 Ksi1 other terms
  0
• Now, the question is, at what values do we
  evaluate the equation?
• 1. Ksi10 This is the Ksi1 mean in group 1.
  (we could, alternatively
• use something like kappa12/2, which is half
  way between
• the group 1 and the group 2 mean of kappa1
  or even a
• weighted version)
• 2. Ksi1 0 k standard deviations, where k can
  be any reasonable number
• 1? 1.5? 2.0?
• 3. Ksi1 0 k standard deviations.

13

• How do we find the standard deviation of Ksi?
• Look at the PHI matrix to obtain variances, and
  take the square root of these!
• PHI USA
• GENDER AGE EDUC
• -------- -------- --------
• GENDER 0.2441
• (0.0102)
• 23.9687
• AGE -0.4381 259.2400
• (0.2350) (10.8158)
• -1.8642 23.9687
• EDUC 0.0251 1.7457 1.9599
• (0.0204) (0.6670) (0.0818)

14

• For education, if we had a pooled estimate
  (Canada US) we could use it, otherwise, we can
  be approximate 1.9599, 1.4733 1.72
  sqrt(1.72) 1.3. So we will want to evaluate
  at EDUC0, EDUC1.3 (or perhaps 2.6?),
  EDUC-1.3 (or perhaps -2.6?).
• At Educ0, Canada-US difference is 1.2545 (see
  alpha parameter, above) USA0 Canada1.2545
• At Educ-2.6, USA 0 (-2.6 .0817) usa gamma
  for educ .0817
• -.2124
• Canada 1.2545 (-2.6 .1525) Canadian gamma
  for educ .1525
• 858
• At Educ 2.6, USA 0 (2.6 .0817) .2124
• Canada 1.2545 (2.6 .1525) 1.651

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16
For age, approximate variance is sqrt (270)
16.43. We could thus use 0 16.43 or 0 ? 32.86
(or 0 ? (1.5 16.43) or if we knew that the mean
was approximately 42 (see tau-x parameter), we
could simply do something like 20 years (more
intuitive)
17

• Models for Four Groups
• Canada
• U.S.A.
• Germany
• U.K.
• Means3a GAPS Chi-square 1892.25 df180
• Means3b GAIN Chi-square 1986.94 df198

18
Formulas USA 0.0738B8 Canada
1.087(B80.1457) UK 2.4339(B8-0.116
7) Germany 1.8139(B80.0957) B8
refers to the first education row. Formula
becomes B9, B10 For rows below
19
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20
Dealing with data that are not normally
distributed within the traditional LISREL
framework

• Questions
• how bad is it if our data are not normally
  distributed?
• what can we do about it?
• are there easy fixes?

21
Non-Normal Data

• How about just ignoring the problem?
• Early 1980s Robustness studies.
• Major findings
• In almost all cases, using LV models better than
  OLS even if data non-normal
• (assumes multiple indicators available)
• some discussion of conditions under which
  parameters might not be accurate (e.g., low
  measurement coefficient models)

22
Non-Normal Data

• Early articles
• A. Boomsa, On the Robustness of LISREL
• Johnson and Creech, American Sociological Review,
  48(3), 1983, 398-403
• Henry, ASR, 47 299-307
• (related Bollen and Barb, ASR, 46 232-39)
• See a good summary of early and later simulation
  studies West, Finch and Curran in Hoyle.

23
Non-Normal Data

• See a good summary of early and later simulation
  studies West, Finch and Curran in Hoyle.
• Formal properties

Consistent? Asymp. Efficient? Acov(?) X2
Multinormal (no kurtosis) v v v v
Elliptical v v X X
Arbitrary v X X X
24
Non-Normal Data

• Many of the studies have involved CFA models
• E.g., Curran, West, Finch, Psych. Methods, 1(1),
  1996.
• General findings (non-normal data)
• ML, GLS produce X2 values too high
• Overestimated by 50 in simulations
• GLS, ML produce X2 value slightly larger when
  sample sizes small, even when data are normally
  distributed
• Underestimation of NFI, TLI, CFI
• Also underestimated in small samples esp. NFI
• Moderate underestimation of std. errors (phi 25,
  lambda 50)

25
Non-Normality

• Detection
• ur E(x ur)r kurtosis ? 4th moment
• Mean of 3 standardized u4 / u22
• Standardized 3rd moment u3/ (u2)3/2
• Tests of statistical significance usually
  available (Bollen, p. 421) b1, b2 (skew,kurt)
• N(0,1) test statistic for Kurtosis (H0 B2 3
  0)
• Different tests (one approx. requires Ngt1000)
• Joint test ?2 Approx. distr. as X2, df2
• Mardias multivariate test skewedness, kurtosis,
  joint.

26
Non-Normality

• An alternative estimator
• Fwls (also called Fagls)
• s s(?) w-1 s s(?)
• Browne, British Journal of Mathematical and
  Statistical Psychology, 41 (1988) 193ff.
• also 37 (1984), 62-83
• Optimal weight matrix?
• asymptotic covariance matrix of sij
• Acov(sij,sgh) N-1 (sijgh - sij sgh)
• Sijgh 1/N S (zi)(zj)(zg)(zh)
• where zi is the mean-deviated value
• If multinomial sijgh sij sgh sjg sjh sjh
  sjg
• (reduces to GLS)
• W-1 is ½ (k)(k1) ½ (k)(K1)

27
Non-Normality

• An alternative estimator
• Fwls (also called Fagls)
• s s(?) w-1 s s(?)
• W-1 is ½ (k)(k1) ½ (k)(K1)
• Computationally intense
• 20 variables 22,155 distinct elements
• To be non-singular,
• N must be gt p ½ (p)(p1)
• 20 variables minimum 230
• 30 variables minimum 495
• Older versions of LISREL used to impose higher
  restrictions (refused to run until thresholds
  well above the minima shown above were reached)

28
Non-Normality

• An alternative estimator
• Fwls (also called Fagls)
• s s(?) w-1 s s(?)
• W-1 is ½ (k)(k1) ½ (k)(K1)
• The AGLS estimator is commonly available in SEM
  software
• LISREL 8
• AMOS
• SAS-CALIS
• EQS
• Be careful! Not really suitable for small N
  problems
• Good idea to have sample sizes in the thousands,
  not hundreds.

29
Non-Normality

• An alternative estimator
• Fwls (also called Fagls)
• s s(?) w-1 s s(?)
• W-1 is ½ (k)(k1) ½ (k)(K1)
• The AGLS estimator is commonly available in SEM
  software
• LISREL 8 MEWL in OU statement must also
  provide asymptotic covariance matrix generated by
  PRELIS
• AC FI statement follows CM FI statement
• AMOS check box on analysis options
• Again, the problem is that this estimator can be
  unstable given the size of the matrix (acov) that
  needs to be inverted (especially in moderate
  sample sizes)

30
Non-Normality

• Sample program in LISREL with adf estimator
• LISREL model for religiosity and moral
  conservatism
• Part 2 ADF estimation
• DA NI14 NO1456
• CM FIh\icpsr2003\Week4Examples\nonnormaldata\re
  lmor1.cov
• ACC FIh\icpsr2003\Week4Examples\nonnormaldata\r
  elmor1.acc
• SE
• 1 2 3 4 5 6 7 8 9 10 11 12 13 14/
• MO NY11 NX3 NE2 Nk3 fixedx lyfu,fi gafu,fr
  c
• pssy,fr tesy
• va 1.0 ly 1 1 ly 8 2
• fr ly 2 1 ly 3 1 ly 4 1 ly 5 1
• fr ly 6 2 ly 7 2 ly 9 2 ly 10 2 ly 11 2
• fr te 2 1 te 11 10 te 7 6
• ou meml se tv sc nd3 mi

31
Non-Normality

• Generating asymptotic covariance matrix in PRELIS

32
Non-Normality

• Generating asymptotic covariance matrix in PRELIS
• Resultant matrix will be much larger than
  covariance matrix

33
Non-Normality ADF estimation
LISREL model for religiosity and moral
conservatism Part 2 ADF estimation DA NI14
NO1456 CM FIh\icpsr99\nonnorm\relmor1.cov ACC
FIh\icpsr99\nonnorm\relmor1.acc SE 1 2 3 4 5 6
7 8 9 10 11 12 13 14/ MO NY11 NX3 NE2 Nk3
fixedx lyfu,fi gafu,fr c pssy,fr tesy va
1.0 ly 1 1 ly 8 2 fr ly 2 1 ly 3 1 ly 4 1 ly 5
1 fr ly 6 2 ly 7 2 ly 9 2 ly 10 2 ly 11 2 fr te 2
1 te 11 10 te 7 6 ou mewl se tv sc nd3 mi
34
Non-Normality ML, scaled statistics
LISREL model for religiosity and moral
conservatism Part 2 ADF estimation DA NI14
NO1456 CM FIh\icpsr2003\Week4Examples\nonnorma
ldata\relmor1.cov ACC FIh\icpsr2003\Week4Example
s\nonnormaldata\relmor1.acc SE 1 2 3 4 5 6 7 8 9
10 11 12 13 14/ MO NY11 NX3 NE2 Nk3 fixedx
lyfu,fi gafu,fr c pssy,fr tesy va 1.0 ly 1
1 ly 8 2 fr ly 2 1 ly 3 1 ly 4 1 ly 5 1 fr ly 6
2 ly 7 2 ly 9 2 ly 10 2 ly 11 2 fr te 2 1 te 11
10 te 7 6 ou meml se tv sc nd3 mi
35
Non-Normality

• Low tech solutions
• For variables that are continuous,
• TRANSFORMATION
• See classic regression texts such as Fox
• Common transformations
• X ? log(X) (usually natural log)
• X ? sqrt (X)
• X ? X2
• X ? 1/ X (even harder to interpret since this
  will result in sign reversal)
• Transforming to remove skewedness often/usually
  removes kurtosis, but this is not guaranteed
• Normalization as an extreme option (e.g., map
  rank-ordered data onto N(0,1) distribution).

36
Non-Normality

• Generally, if kurtosis between 1 and -1, not
  considered too problematic
• (See Bollen, 1989)
• From this.

37
Transformations

• AMOS Transformations must be performed on SPSS
  dataset. Save new dataset, and work from
• this. (e.g, COMPUTE X1 LOG(X1).)

LISREL Transformations can be performed in
PRELIS. PRELIS already provides distribution
information on variables as a check
PRELIS compute dialogue box under
transformations
Remember to SAVE the Prelis dataset after each
transformation. Use of stat package (SPSS,
Stata, SAS) may be preferable
38
Transformations

• All the usual caveats apply
• If a variable only has 4-5 values, transformation
  will not normalize a variable (at the very least,
  will still have tucked-in tails) though it
  could help bring it closer to within the 1 ? -1
  range (Kurtosis)
• If a categorized variable has one value with a
  majority of cases, then no transformation will
  work
• If the variable has negative values, make sure to
  add a constant (offset) before logging

39
Other solutions

• Robust test statistics
• (Bentler) Implementation EQS, LISREL
• 2. Muthen has recently developed a WLSM
  (mean-adjusted) and WLSMV (mean and variance
  adjusted) estimator
• Implementation MPLUS only
• 3. Bootstrapping
• Implementation AMOS (easy to use)
• LISREL (awkward)
• 4. CATEGORICAL VARIABLE MODELS (CVM).

40
Bootstrapping

• Computationally intensive
• Sampling with replacement from resampling space
  R draw bootstrap sample Sn,j where j of
  samples, nbootstrap n
• Typically, bootstrap N sample N
• Repeat resampling B times, get set of values
• Issue what if, across 200 resamples, 2 of them
  have ill-defined matrices?
• Usually, these are discarded

41
Bootstrapping

• Computationally intensive
• Sampling with replacement from resampling space
  R draw bootstrap sample Sn,j where j of
  samples, nbootstrap n
• Typically, bootstrap N sample N
• Repeat resampling B times, get set of values
• Issue what if, across 200 resamples, 2 of them
  have ill-defined matrices?
• Usually, these are discarded
• Tests 5 confidence intervals (want large of
  samples confidence intervals do not need to be
  symmetric (can look to value at 95th percentile
  and at 5th among bootstrapped samples).
• More common to compute standard errors

42
Bootstrapping

• Overall model X2 correction (available in AMOS)..
  Bollen and Stine.
• Yang and Bentler (chapter in Marcoulides
  Schumacker)
• faith in bootstrap based on its appropriateness
  in other apps
• Simulation study, 1995, if explor. factor
  analysis rotated solutions close, but not so
  with unrotated solutions
• It seems that in the present stage of
  development, the use of the bootstrap estimator
  in covariance structure analysis is still
  limited. It is not clear whether one can trust
  the bias estimates.

43
Bootstrapping

• Ichikawa and Konishi, 1995
• When data multinormal, bootstrap ses not as good
  as ML
• Bootstrap doesnt seem to work when Nlt150
  consistent overestimation (at N300, not a
  problem though).

44
The Categorical Variable Model

• Conceptual background
• We observe y interested in latent y
• with C discrete values
• Yi Ci 1 if vi,ci-1 lt yi where v is a
  threshhold
• Yi Ci 2 if vi,ci-2 lt yi vi,ci-1
• Yi Ci 3 if vi,ci-3 lt yi vi,ci-2
• ..
• If v1,1 if vi,1 lt yi vi,2
• 0 if yi vi,1 vs are threshhold parameters
• to be estimated.

45
The Categorical Variable Model

• Observed and Latent Correlations
• X-variable scale y-variable scale Observed
  correl. Latent corr.
• Continuous continuous pearson pearson
• Contiuous categorical pearson polyserial
• Continuous dichtoomous point-biserial biserial
• Categorical categorical pearson polychoric
• Dichotomous dichotomous phi tetrachoric
• If it is reasonable to assume that continuous and
  normally distributed y variables underlie the
  categorical y variables a variety of latetn
  correlations can be specified.

46
The Categorical Variable Model

• If it is reasonable to assume that continuous and
  normally distributed y variables underlie the
  categorical y variables a variety of latetn
  correlations can be specified.
• First step estimate thresholds using ML
• Second step latent correlations estimated
• Third step obtain a consistent estimator of
  the asymptotic covariance matrix of the latent
  correlations (for use in a weighted least squares
  estimator in the SEM model).
• Extreme case ability to recover y model when
  variables split into 25/75 dichotomies
  promising (though X2 underestimated)