working with under-identified structural equation models david a. kenny university of connecticut...
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Working with Under-identified Structural Equation Models
David A. KennyUniversity of Connecticut
Website: davidakenny.net/kenny.htmPaper download: davidakenny.net/doc/kandm.doc
Powerpoint download: davidakenny.net/doc/under.ppt
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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.
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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.
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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).
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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.
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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.
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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
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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.
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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.
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Usual Fix
W = V + E7
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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.
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Stability of Depression in Boys
10 knowns11 unknownsmodel under-identified
Standardized a is identified = .561
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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.
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Growth-curve Model with Just Two Waves
20 knowns22 unknownsmodel under-identified Red paths are identified!
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Fix
W = U + E1
X = V + E2
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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.
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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.
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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
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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.
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Non-recursive Model
X1
X2
X3
X4
U
V
1
1
2df23 − 1213 = 0 and 24 − 1214 = 0
10 knowns10 unknownsmodel under-identified
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Some Paths Can Be Estimated and Fit Can Be Evaluated
X2
X3
U3
X1
c
d
a
U1
1
1
X4
U4
1
b
U2
1
e
Paths a and b not identified.
Paths c, d, and e are identified.
Model has 2df,
10 knowns10 unknownsmodel under-identified
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Fix
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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
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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.
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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 .079Females .273 .990
• As expected, data are consistent with spuriousness, i.e., no causal effects.
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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.
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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
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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."
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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