Latent variable models involve restrictions on the data that can be formulated in terms of "misspecifications": restrictions with a model-based meaning. Examples include zero cross-loadings and local dependencies, as well as “measurement invariance” or “differential item functioning”. If incorrect, misspecifications can potentially disturb the main purpose of the latent variable analysis—seriously so in some cases.
Recently, I proposed to evaluate whether a particular analysis at hand is such a case or not.
To do this, I define a measure based on the likelihood of the restricted model that approximates the change in the parameters of interest if the misspecification were freed, the EPC-interest. The main idea is to examine the EPC-interest and free those misspecifications that are “important” while ignoring those that are not. I have implemented the EPC-interest in the lavaan software for structural equation modeling and the Latent Gold software for latent class analysis.
This approach can resolve several problems and inconsistencies in the current practice of model fit evaluation used in latent variable analysis, something I illustrate using analyses from the “measurement invariance” literature and from item response theory.
Detecting local dependence in latent class modelsDaniel Oberski
Latent class (mixture) models are often used in wide range of fields. These models assume that the observed variables are independent given the latent classes: local independence. What if this assumption does not hold?
ESRA2015 course: Latent Class Analysis for Survey ResearchDaniel Oberski
Slides for a 3-hour short course I gave at the European Survey Research Association's 2015 meeting in Reykjavík, Iceland.
This course gives a short introduction to Latent Class Analysis (LCA) for survey methodologists. R code and some Latent GOLD input is also provided.
The R code and data for the examples can be found at http://daob.nl/wp-content/uploads/2015/07/ESRA-LCA-analyses-data.zip
Colloquium talk on modal sense classification using a convolutional neural ne...Ana Marasović
Modal sense classification (MSC) is a special case of sense disambiguation relevant for distinguishing facts from hypotheses and speculations, or apprehended, planned and desired states of affairs. Prior approaches showed that even with carefully designed semantic feature sets, the models have difficulties beating the majority sense baseline in cases of difficult sense distinctions and when applying the models to heterogeneous text genres. Another drawback of former approaches is that feature implementation heavily depends on a external language-specific resources such as dependency or constituency parse trees and lexical databases such as WordNet or CELEX. To alleviate manual crafting of the features and to obtain a model which is easily portable to novel languages, we propose to cast MSC as a sentence classification task with a fixed sense inventory in a convolutional neural network (CNN) architecture. Our performance study shows that CNN is an appropriate model for MSC and its special properties motivate us to investigate it as a formal framework for general word sense disambiguation tasks.
Detecting local dependence in latent class modelsDaniel Oberski
Latent class (mixture) models are often used in wide range of fields. These models assume that the observed variables are independent given the latent classes: local independence. What if this assumption does not hold?
ESRA2015 course: Latent Class Analysis for Survey ResearchDaniel Oberski
Slides for a 3-hour short course I gave at the European Survey Research Association's 2015 meeting in Reykjavík, Iceland.
This course gives a short introduction to Latent Class Analysis (LCA) for survey methodologists. R code and some Latent GOLD input is also provided.
The R code and data for the examples can be found at http://daob.nl/wp-content/uploads/2015/07/ESRA-LCA-analyses-data.zip
Colloquium talk on modal sense classification using a convolutional neural ne...Ana Marasović
Modal sense classification (MSC) is a special case of sense disambiguation relevant for distinguishing facts from hypotheses and speculations, or apprehended, planned and desired states of affairs. Prior approaches showed that even with carefully designed semantic feature sets, the models have difficulties beating the majority sense baseline in cases of difficult sense distinctions and when applying the models to heterogeneous text genres. Another drawback of former approaches is that feature implementation heavily depends on a external language-specific resources such as dependency or constituency parse trees and lexical databases such as WordNet or CELEX. To alleviate manual crafting of the features and to obtain a model which is easily portable to novel languages, we propose to cast MSC as a sentence classification task with a fixed sense inventory in a convolutional neural network (CNN) architecture. Our performance study shows that CNN is an appropriate model for MSC and its special properties motivate us to investigate it as a formal framework for general word sense disambiguation tasks.
Computer Generated Items, Within-Template Variation, and the Impact on the Pa...Quinn Lathrop
Computer Generated Items, Within-Template Variation, and the Impact on the Parameters of Response Models.
Master's thesis talk related to Lathrop, Q.N., Cheng, Y. Item Cloning Variation and the Impact on the Parameters of Response Models. Psychometrika 82, 245–263 (2017). https://doi.org/10.1007/s11336-016-9513-1
Rsqrd AI - ML Interpretability: Beyond Feature ImportanceAlessya Visnjic
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Two critical applications are reviewed: one-shot correction of errors in artificial intellectual systems and emergence of static and associative memories in ensembles of single neurons. Error correctors should be simple; not damage the existing skills of the system; allow fast non-iterative learning and correction of new mistakes without destroying the previous fixes. All these demands can be satisfied by new tools based on the concentration of measure phenomena and stochastic separation theory.
In several words, the stochastic separation theorems state that for an essentially high-dimensional distributions a random point can be separated from a random set by Fisher's linear discriminant with high probability. The number of points in this set can grow exponentially with dimension. Different versions of stochastic separation theorems use different definitions of `random sets' and `essentially high-dimensional distributions' but the essence of these definitions is simple: sets with very small (vanishing) volume should not have high probability even for large dimension.
The talk is based on the work: A.N. Gorban, V.A. Makarov, I.Y. Tyukin, The unreasonable effectiveness of small neural ensembles in high-dimensional brain. Physics of Life Reviews, 2019, https://doi.org/10.1016/j.plrev.2018.09.005
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The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
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A measure to evaluate latent variable model fit by sensitivity analysis
1. A measure to evaluate latent variable
model fit by sensitivity analysis
Daniel Oberski
Department of methodology and statistics
Dept of Statistics, Leiden University
Latent variable model fit by sensitivity analysis Daniel Oberski
2. Latent variable models
What do they assume and what are they good for?
Latent variable model fit by sensitivity analysis Daniel Oberski
5. Example
Goal: estimate false positives and false negatives in four
diagnostic tests for C. Trachomatis infection:
y1 Ligase chain reaction (LCR) test (Yes/No);
y2 Polymerase chain reaction (PCR) test (Yes/No);
y3 DNA probe test (DNAP) (Yes/No);
y4 Culture (CULT) (Yes/No).
Tool: 2-latent class model (diseased or non-diseased).
(Original data from Dendukuri et al. 2009)
Latent variable model fit by sensitivity analysis Daniel Oberski
6. Assume:
ξ
y1 y2 yJ...
But really:
ξ
y1 y2 yJ...
What difference does it make for the goal: false positives and
false negatives? (simulation by Van Smeden et al., submitted)
Latent variable model fit by sensitivity analysis Daniel Oberski
9. Example
Goal: Estimate gender differences in ”valuing Stimulation”:
(1) Very much like me; (2) Like me; (3) Somewhat like me; (4) A
little like me; (5) Not like me; (6) Not like me at all.
impdiff (S)he looks for adventures and likes to take risks.
(S)he wants to have an exciting life.
impadv (S)he likes surprises and is always looking for new
things to do. He thinks it is important to do lots of
different things in life.
Tool: Structural Equation Model for European Social Survey data
(n = 18519 men and 16740 women).
(Original study by Schwarz et al. 2005)
Latent variable model fit by sensitivity analysis Daniel Oberski
10. Assume:
ξ
y1 y2 yJ...
x
But really (?):
ξ
y1 y2 yJ...
x
What difference does it make for the goal: true gender
differences in values? (re-analysis of data by Oberski 2014)
q
q
q
q
q
q
q
q
Men value more
Women value more
−0.2
0.0
0.2
ACPO ST SD HE COTR SE UN BE
"Human value" factor
Latentmeandifferenceestimate±2s.e.
Model
q Scalar invariance
Free intercept 'Adventure'
Latent variable model fit by sensitivity analysis Daniel Oberski
11. PROBLEM
The original authors found that the conditional independence
model fit the data ”approximately” (p. 1013)...
”Chi-square deteriorated significantly, ∆χ2
(19) = 3313,
p < .001, but CFI did not change. Change in chi-square is
highly sensitive with large sample sizes and complex models.
The other indices suggested that scalar invariance might be
accepted (CFI = .88, RMSEA = .04, CI = .039.040, PCLOSE
= 1.0).”
... but unfortunately this ”acceptable” misspecification could
reverse their conclusions!
Latent variable model fit by sensitivity analysis Daniel Oberski
12. Numbers that indicate how well the model fits the data
• Likelihood Ratio vs. saturated
• Information-based criteria: AIC, BIC, CAIC, ...
• Bivariate residuals (Maydeu & Joe 2005; Oberski, Van Kollenburg &
Vermunt 2013)
• Score/Lagrange multiplier tests, “modification index”,
“expected parameter change” (EPC) (Saris, Satorra & Sörbom
1989; Oberski & Vermunt 2013; Oberski & Vermunt accepted)
“Fit indices”:
• RMSEA:
√
(χ2/df)−1)
N−1
• CFI:
[
(χ2
null − dfnull) − (χ2 − df)
]
/(χ2
null − dfnull)
• Lots of others: TLI, NFI, NNFI, RFI, IFI, RNI, RMR,
SRMR1-3, GFI, AGFI, MFI, ECVI, ...
Latent variable model fit by sensitivity analysis Daniel Oberski
13. What is the problem?
• We do latent variable modeling with a goal in mind.
• But the latent variable model might be misspecified.
• The appropriate question: ”will that affect my goal?”
• The actual question: ”do the data fit the model in the
population” (LR) or ”are the model and the data far apart
relative to model complexity” (RMSEA etc.)
What is the solution?
Evaluate directly what effect possible misspecifications
have on the goal of the analysis.
Latent variable model fit by sensitivity analysis Daniel Oberski
14. How to evaluate directly what effect possible misspecifications
have on the goal of the analysis.
Latent variable model fit by sensitivity analysis Daniel Oberski
15. Two ideas to evaluate the effect of misspecifications
1 Try out all possible models with misspecifications, calculate
the estimates of interest under these models and evaluate
whether these are substantively different.
Advantage: Does the job.
Disadvantage: There may be too many alternative models.
Also: are applied researchers really going to do this?
2 Use EPC-interest: expected change in free parameters
Advantage: Does the job without the need to estimate any
alternative models.
Disadvantage: Is an approximation (though a reasonable
one).
Latent variable model fit by sensitivity analysis Daniel Oberski
16. EPC-interest applied to Stimulation example
• After fitting the full scalar invariance model,
• Effect size estimate of sex difference in Stimulation is +0.214
(s.e. 0.0139).
• But EPC-interest of equal ”Adventure” item intercept is
-0.243.
• So EPC-interest suggests conclusion can be reversed by
freeing a misspecified scalar invariance restriction
• Actual change when freeing this intercept is very close to
EPC-interest: -0.235.
Latent variable model fit by sensitivity analysis Daniel Oberski
18. • Let’s say there is a restricted model whose purpose it is to
estimate its parameters, θ, or some linear function of them
such as a subselection, Pθ.
• We could parameterize these restrictions as ψ = 0.
For example: ψ could be direct effect of gender on
”Adventure”, or loglinear dependence between DNA tests.
• The maximum likelihood estimates are then
ˆθ = arg max L(θ, ψ = 0)
Question: How much would ˆθ change if we freed ψ?
Latent variable model fit by sensitivity analysis Daniel Oberski
19. How much would ˆθ change if we freed ψ?
The trick is to consider estimate of θ we would get under ψ ̸= 0;
that is, ˜θ = arg max L(θ, ψ).
As it turns out, we don’t actually need ˜θ, since
˜θ − ˆθ = ˆH
−1
θθ
ˆHθψD−1
[
∂L(θ, ψ)
∂ψ θ=ˆθ
]
+ O(δ′
δ),
where H is a Hessian, D = ˆHψψ − ˆH
′
θψ
ˆH
−1
θθ
ˆHθψ and δ is the
”overall wrongness” of the model (ψ′
, θ′
− ˆθ
′
)′.
Latent variable model fit by sensitivity analysis Daniel Oberski
20. How much would ˆθ change if we freed ψ?
Dropping the approximation term (assuming the model
parameters are not ”too far” from the truth) we get the
approximation
EPC-interest = −P ˆH
−1
θθ
ˆHθψ EPC-self ≈ −P ˆH
−1
θθ
ˆHθψ
(
ψ − ˆψ
)
For those of you familiar with Structural Equation Modeling (or
attending my 2013 MBC2 talk), ”EPC-self” is the usual ”expected
parameter change” in the fixed parameter vector, i.e. the size of
the misspecification.
Latent variable model fit by sensitivity analysis Daniel Oberski
21. Monte Carlo simulation: EPC-interest is a good
approximation to the actual change in parameters of
interest when freeing equality restriction
Average over 200 replications
∆ν1 ng EPC-self ∆ˆα ∆ˆα bias EPC-interest EPC-interest bias
0.1 50 0.064 0.240 -0.040 -0.034 0.005
0.3 50 0.213 0.313 -0.113 -0.113 -0.001
0.8 50 0.657 0.505 -0.305 -0.401 -0.096
0.1 100 0.058 0.231 -0.031 -0.031 0.000
0.3 100 0.203 0.323 -0.123 -0.109 0.014
0.8 100 0.619 0.492 -0.292 -0.370 -0.077
0.1 500 0.063 0.233 -0.033 -0.033 0.000
0.3 500 0.208 0.307 -0.107 -0.112 -0.005
0.8 500 0.598 0.501 -0.301 -0.349 -0.048
Latent variable model fit by sensitivity analysis Daniel Oberski
23. Ranking data in 48 WVS countries
Option # M/P Value wording
Set A
1. M A high level of economic growth
2. M Making sure this country has strong defense forces
3. P Seeing that people have more say about how things are done at
their jobs and in their communities
4. P Trying to make our cities and countryside more beautiful
Set B
1. M Maintaining order in the nation
2. P Giving people more say in important government decisions
3. M Fighting rising prices
4. P Protecting freedom of speech
Set C
1. M A stable economy
2. P Progress toward a less impersonal and more humane society
3. P Progress toward a society in which ideas count more than money
4. M The fight against crime
Latent variable model fit by sensitivity analysis Daniel Oberski
24. Figure: Graphical representation of the multilevel latent class regression
model for (post)materialism measured by three partial ranking tasks.
Observed variables are shown in rectangles while unobserved (“latent”)
variables are shown in ellipses.
Latent variable model fit by sensitivity analysis Daniel Oberski
25. Latent class ranking model with 4 choices
Each ranking set, for example, set A:
P(A1ic = a1, A2ic = a2|Xic = x) =
ωa1x
∑
k ωkx
ωa2x
∑
k̸=a1
ωkx
,
where ωkx is the “utility” of object k for respondents in class x.
Multilevel structure to account for the countries using group class
variable G:
P(Xic = x|Z1ic = z1ic, Z2ic = z2, Gc = g) =
=
exp(αx + γ1xz1 + γ2xz2 + βgx)
∑
t exp(αt + γ1tz1 + γ2tz2 + +βtg)
,
Latent variable model fit by sensitivity analysis Daniel Oberski
26. Multilevel latent class model w/ covariates for rankings
L(θ) = P(A1, A2, B1, B2, C1, C2|Z1, Z2) =
C∏
c=1
∑
G
P(Gc)
nc∏
i=1
∑
X
P(Xic|Z1ic, Z2ic, Gc)×
P(A1ic, A2ic|Xic)P(B1ic, B2ic|Xic)P(C1ic, C2ic|Xic),
Goal: estimate γ (especially its sign).
Possible problem: Violations of scalar and metric
measurement invariance (DIF), parameterized respectively as
τ∗ and λ∗.
Solution: See if these matter for the sign of γ.
Latent variable model fit by sensitivity analysis Daniel Oberski
27. Table: Full invariance multilevel latent class model: parameter estimates
of interest with standard errors (columns 3 and 4), as well as expected
change in these parameters measured by the EPC-interest when
freeing each of six sets of possible misspecifications (columns 5–10).
EPC-interest for...
τ∗
jkg λ∗
jkxg
Estimates Ranking task Ranking t
Est. s.e. 1 2 3 1 2
Class 1 GDP -0.035 (0.007) -0.013 0.021 -0.002 0.073 0.252
Class 2 GDP -0.198 (0.012) -0.018 -0.035 0.015 -0.163 -0.058
Class 1 Women 0.013 (0.001) -0.006 0.002 0.000 -0.003 0.029
Class 2 Women -0.037 (0.001) 0.007 -0.003 0.002 -0.006 -0.013
Latent variable model fit by sensitivity analysis Daniel Oberski
28. Table: Partially invariant multilevel latent class model: parameter
estimates of interest with standard errors (columns 3 and 4), as well as
expected change in these parameters measured by the EPC-interest
when freeing each of four sets of remaining possible misspecifications
(columns 5–7 and 10).
EPC-interest for non-invariance of...
τ∗
kg λ∗
kxg
Ranking task Ranking task
Est. s.e. 1 2 3 1 2 3
Class 1 GDP -0.127 (0.008) -0.015 -0.003 0.002 0.097
Class 2 GDP 0.057 (0.011) -0.043 -0.013 0.002 0.161
Class 1 Women 0.008 (0.001) -0.002 0.000 0.002 0.001
Class 2 Women 0.020 (0.001) -0.007 -0.001 0.002 0.007
Latent variable model fit by sensitivity analysis Daniel Oberski
29. Mixed
Postmaterialist
Materialist
Mixed
Postmaterialist
Materialist
% Women in parliament GDP per capita
0.2
0.4
0.6
Minimum Maximum Minimum Maximum
Covariate level
ProbabilityofClass
Figure: Estimated probability of choosing each class as a function of the
covariates of interest under the final model.
Latent variable model fit by sensitivity analysis Daniel Oberski
30. ARM
AUS
AZE
BLR
CHL
CHNCOL
CYP
DEU
DZA
ECU
EGY ESPEST
GHA
IRQ
JOR
JPN
KAZ
KGZ
KOR
LBN
MAR
MEX
MYSNGA
NLD
NZL
PAK
PER
PHL
POLQAT ROU
RUS
RWA
SGPSVN
SWE
TTO
TUN
TUR
UKR
URY
USA
UZB
YEM
ZWE
ARM
AUS
AZE
BLR
CHL
CHN
COL
CYP
DEU
DZA
ECU
EGY
ESP
ESTGHA
IRQJOR
JPN
KAZ
KGZ
KOR
LBN MAR
MEX
MYSNGA NLD
NZL
PAK
PER
PHLPOL
QAT
ROU
RUS
RWA
SGP
SVN
SWE
TTO
TUN
TUR
UKR
URY
USA
UZB
YEM ZWE
ARM
AUS
AZE
BLR
CHL
CHN
COL
CYP
DEU
DZA
ECU
EGY
ESP
EST
GHA
IRQ
JOR
JPN KAZKGZ
KOR
LBN
MAR
MEX
MYSNGA
NLDNZL
PAK
PER
PHL
POL
QAT
ROU
RUS RWASGP
SVN
SWE
TTO
TUN
TUR
UKR
URY
USA
UZB
YEM
ZWE
Class 1
("Materialist")
Class 2
("Postmaterialist")
Class 3
("Mixed")
0.0
0.2
0.4
0.6
0.8
0 20 40 0 20 40 0 20 40
% Women in Parliament
Classposterior
ARM
AUS
AZE
BLR
CHL
CHNCOL
CYP
DEU
DZA
ECU
EGY ESPEST
GHA
IRQ
JOR
JPN
KAZ
KGZ
KOR
LBN
MAR
MEX
MYSNGA
NLD
NZL
PAK
PER
PHL
POL QATROU
RUS
RWA
SGPSVN
SWE
TTO
TUN
TUR
UKR
URY
USA
UZB
YEM
ZWE
ARM
AUS
AZE
BLR
CHL
CHN
COL
CYP
DEU
DZA
ECU
EGY
ESP
ESTGHA
IRQJOR
JPN
KAZ
KGZ
KOR
LBNMAR
MEX
MYSNGA NLD
NZL
PAK
PER
PHL POL
QAT
ROU
RUS
RWA
SGP
SVN
SWE
TTO
TUN
TUR
UKR
URY
USA
UZB
YEMZWE
ARM
AUS
AZE
BLR
CHL
CHN
COL
CYP
DEU
DZA
ECU
EGY
ESP
EST
GHA
IRQ
JOR
JPNKAZKGZ
KOR
LBN
MAR
MEX
MYSNGA
NLDNZL
PAK
PER
PHL
POL
QAT
ROU
RUSRWA SGP
SVN
SWE
TTO
TUN
TUR
UKR
URY
USA
UZB
YEM
ZWE
Class 1
("Materialist")
Class 2
("Postmaterialist")
Class 3
("Mixed")
0.0
0.2
0.4
0.6
0.8
7 8 9 10 11 7 8 9 10 11 7 8 9 10 11
Ln(GDP per capita)
Classposterior
Latent variable model fit by sensitivity analysis Daniel Oberski
31. What has been gained by using EPC-interest:
I am fairly confident here that there truly is ”approximate
measurement invariance”, in the sense that any violations of
measurement invariance do not bias the primary conclusions.
I think attaining this goal is the main purpose of model fit
evaluation.
Latent variable model fit by sensitivity analysis Daniel Oberski
33. Conclusion
• Latent variable modeling is often performed for a purpose;
• Model fit evaluation should then be done for the reason that
violations of assumptions can disturb this purpose.
• Introduced the EPC-interest to look into this;
• Evaluates the change in the parameter(s) of interest that
would result if a restriction is freed that parameterizes a
potential violation of assumptions.
Latent variable model fit by sensitivity analysis Daniel Oberski
34. Implemented in SEM software lavaan for R:
Oberski (2014). Evaluating Sensitivity of Parameters of Interest to Measurement
Invariance in Latent Variable Models. Political Analysis, 22 (1).
Implemented in LCA software Latent Gold:
Oberski, Vermunt & Moors (submitted). Evaluating measurement invariance in
categorical data latent variable models with the EPC-interest. Under
review.
Oberski & Vermunt (2014). A model-based approach to goodness-of-fit
evaluation in item response theory. Measurement, 11, 117–122.
Nagelkerke, Oberski, & Vermunt (accepted). ”Goodness-of-fit of Multilevel
Latent Class Models for Categorical Data”. Sociological Methodology.
Oberski & Vermunt (conditionally accepted). ”The Expected Parameter Change
(EPC) for Local Dependence Assessment in Binary Data Latent Class
Models”. Psychometrika.
Latent variable model fit by sensitivity analysis Daniel Oberski
35. Thank you for your attention!
Daniel Oberski
doberski@uvt.nl
See http://daob.nl/publications for full texts & code
Latent variable model fit by sensitivity analysis Daniel Oberski
36. SEM regression coefficient example
European Sociological Review 2008, 24(5), 583–599
Latent variable model fit by sensitivity analysis Daniel Oberski
38. SEM regression coefficient example
EPC-interest statistics of at least 0.1 in absolute value with
respect to the latent variable regression coefficients.
Metric invariance (loading) restriction
“Conditions → Work skills” in...
Slovenia France Hungary Ireland
EPC-interest w.r.t.:
Conditions →
Self-transcendence -0.073 -0.092 -0.067 0.073
Conservation 0.144 0.139 0.123 -0.113
SEPC-self 0.610 0.692 0.759 -0.514
Latent variable model fit by sensitivity analysis Daniel Oberski
39. SEM regression coefficient example
What has been gained by using EPC-interest
• Full metric invariance model: ”close fit”;
• EPC-interest still detects threats to cross-country
comparisons of regression coefficients;
• MI and EPC-self do not detect these particular
misspecifications;
• MI and EPC-self detect other misspecifications;
• Looking at EPC-interest reveals that these do not affect the
cross-country comparisons of regression coefficients.
Latent variable model fit by sensitivity analysis Daniel Oberski