Working with Under-identified
Structural Equation Models
David A. Kenny
University of Connecticut
Website: davidakenny.net/kenny.htm
Paper download: davidakenny.net/doc/kandm.doc
Powerpoint download: davidakenny.net/doc/under.ppt
Introductory Comment
• Talk is about Structural Equation
Models (SEM).
• Nonetheless, the points apply to
many other types of modeling as
issues about identification apply to a
broad range of models.
Identification in SEM
• Specify a model.
• See if it is identified.
–If identified, estimate it.
–If under-identified, respecify the
model until it is identified.
• Models that are under-identified are
not estimated and are thought to be
useless models.
Quote
• If a model is not identified, it must be
made identified by increasing the
number of manifest variables or by
reducing the number of parameters to
be estimated (Blunch, 2008, p. 78).
What To Do
with Under-identified Models
• Make them identified:
– Add variables
– Make parameter constraints
• Estimate the range of possible values.
• Sensitivity analysis: Fix the “under-
identifying parameter” to a range of
reasonable values, and examine the
solutions.
Some Under-identified Models Contain
Useful Information
• A: Some model parameters can be
estimated even if the model as a whole is
under-identified.
– These might be theoretically or practically
important parameters.
• B: Fit can be evaluated sometimes even if
the model as a whole is under-identified.
– Can be a way for ruling out models.
A: Under-identified Models with
Identified Parameters
• A model is under-identified if not all the
parameters of the model are indentified.
However, some of the parameters of the model
might be identified.
• Those parameters may be of interest.
• Three Examples
– Outcome with a single indicator
– Stability of personality
– Growth curve model with just two waves
How to Estimate
Under-identified Models?
• Can set one or more of the under-identified
parameter estimate to "allowable" values.
• A fix: Turning an under-identified model into an
identified model by pretending something is true
which is not true.
• Some programs do estimate parameter
estimates, even if the model is not identified.
– With Amos: “Try to fit under-identified models.”
– Use MIIV.
Outcome with a Single Indicator:
Fishbein & Ajzen
Despite the model being under-
identified, paths a, b, and c (the key
parts of the model) are identified.
Usual Fix
W = V + E7
What to Do?
• Use the fix.
– Model is identified!
– But the model is wrong!
• Use the under-identified model.
– Obvious drawback: The model is under-
identified.
– But it does give information about key
parameters.
– It does not pretend to know something to that
it does not know.
Stability of Depression in Boys
10 knowns
11 unknowns
model under-identified
Standardized a is identified = .561
Fixes
• Add a third indicator.
• Fix one of the free loadings to one (it does
not matter which one).
• Fix both free loadings to one.
– Model now over-identified and fit may be
poor.
– The under-identified model might be better.
Growth-curve Model with Just
Two Waves
20 knowns
22 unknowns
model under-identified Red paths are identified!
Fix
W = U + E1
X = V + E2
Information Lost by
Not Having Three or More Waves
• Slope and intercept variances are not
identified. Thus, measures of variance
explained are not available.
• Linearity must be assumed and is not
tested.
Identified Parameters in
Under-identified Models
• Best to estimate the under-identified
model as it makes clear what is known
and what is unknown.
• One can find a “fix,” but the fix gives the
illusion that the model is identified, when in
fact it is not. The “fix” might make an
unreasonable assumption.
B: Under-identified Models for
Which Fit Can Be Evaluated
• For all models that meet or exceed the
minimum condition of identifiability but are
under-identified, the fit of the model can be
evaluated because all of these models
place some sort of constraint on the data.
– Two examples
• Longitudinal Models with No Cross-causal
Effects
Models that Meet or Exceed the
Minimum Condition of Identifiability
• Minimum condition of identifiability or the t
rule: The number of knowns (variances,
covariances, and means) must be greater
than or equal to the number of unknowns
(e.g., paths).
• Some models that meet this condition are
not identified.
Non-recursive Model
X3
X1 1
U
V
1
X2 X4
10 knowns
10 unknowns 2df: r23 − r12r13 = 0 and r24 − r12r14 = 0
model under-identified
Some Paths Can Be Estimated and
Fit Can Be Evaluated
U1
1
X2
c
e
X3 X4
a b
d
1 1
X1
U3 U4
1
U2
10 knowns Paths a and b not identified.
10 unknowns
Paths c, d, and e are identified.
model under-identified
Model has 2df,
Fix
Longitudinal Model of
Spuriousness
• Common Model for Two-wave Data Is to
Estimate Cross-causal Effects
– Whismam: Depression Causes Marital
Dissatisfaction vs. Marital Dissatisfaction
Causes Depression
– Alternative Model: Depression and Marital
Dissatisfaction Do Not Cause Each Other
• Zero paths model makes strong and implausible
assumption about spuriousness (Dwyer).
• Better might be an explicit model (under-identified,
but testable) of spuriousness
Model of Spuriousness
• Four or more measures at two or more times
• Assumptions
– Spuriousness
• The manifest variables are caused by latent
variables which explain all the covariation
in the variables.
• No lagged causal effects.
– Stationarity
• After a linear transformation, factor
structure and variances invariant over time.
Dumenci and Windle Example
• Four measures of depression (CESD) for 16 and
17 years olds, 372 males and 433 females.
• Chosen because the measures should not have
causal effects between them.
• Model Fit (p values)
Stationarity (df = 2) Spuriousness (df = 6)
Males .514 .079
Females .273 .990
• As expected, data are consistent with
spuriousness, i.e., no causal effects.
Conclusions
• Not all under-identified models are
hopeless.
• Sometimes key parameters can be
estimated in an under-identified model.
• Sometimes model fit can be estimated for
under-identified model which can be useful
in testing the model.
Suggestions
• SEM programs need to be able to
estimate under-identified models.
• Try to avoid “fixes”; estimate a more
realistic model even if part of it is under-
identified.
• Sometimes cheaper (e.g., fewer measures
or time points) designs can yield key
information with an under-identified model.
– e.g., two-wave growth models
Final Suggestion
• Kenny 1979, p. 40: "Making new
specifications to just to be able to identify
the parameters of a causal model is
perhaps the worst sin of causal modelers."
The End
• Download powerpoint:
davidakenny.net/doc/under.ppt
• Download Kenny & Milan Identification
Chapter for the forthcoming Handbook of
Structural Equation Modeling (Richard
Hoyle, David Kaplan, George Marcoulides,
and Steve West, Eds.), New York:
Guilford Press:
davidakenny.net/doc/kandm.doc