The analysis of complex nonlinear structural equation models: A comparison of LMS and QML

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The analysis of complex nonlinear structural
equation modelsA comparison of LMS and QML

• Polina Dimitruk, Helfried Moosbrugger,
• Augustin Kelava Karin Schermelleh-Engel
• J. W. Goethe University, Frankfurt am Main,
  Germany

Supported by a grant of the German Research
Foundation (DFG), Nr.Mo 474/6-1
2nd Conference of the European Association of
Methodology (EAM) July 2 5, 2006, Budapest
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Outline

• Nonlinear Regression Models
• LMS method
• QML method
• Simulations Studies
• Design of Monte-Carlo Study
• Results of Monte-Carlo Study
• Conclusions

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Nonlinear Regression Models

• There are 3 common effects in nonlinear
  regression models
• Interaction effect The slope of the regression
  of the criterion variable on a predictor variable
  varies with the realizations of a second
  predictor variable.
• Y ß0ß1Xß2Z ß3XZe
• 2. Quadratic effectsThe slope of the regression
  of the criterion variable on a predictor variable
  varies with the realizations of the predictor
  variable itself.
• Y ß0ß1Xß2X2e
• 3. Mixed nonlinear effects The slope of the
  regression of the criterion variable on a
  predictor variable varies not only with the
  realizations of the predictor variable itself,
  but also with the realizations of a second
  predictor variable.
• Y ß0ß1Xß2Zß3XZß4X2e

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Nonlinear Models

• Nonlinear regression models can contain several
  nonlinear effects, i.e., one or more interaction
  effects and one or more qudratic effects. In a
  simple case a full nonlinear model includes a
  criterion variable h, two predictor variables x1
  and x2, a product term x1x2 and two quadratic
  terms x12 and x22
• h g1x1 g2x2 w12x1x2 w11x12 w22x22
  z
• Linear model results when all nonlinear effects
  (w12, w11 and w22) are zero, interaction model
  results when all quadratic effects (w11 and w22)
  are zero, quadratic model results when the
  interaction effect (w12) is zero, and finally
  full model results when all nonlinear effects
  (w12, w11 and w22) are unequal to zero.

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LMS method(Latent Moderated Structural
Equations, Klein, 2000 Klein Moosbrugger,
2000)

• LMS was developed especially for the estimate and
  check of latent interaction effects in structural
  equation models.
• LMS is a maximum likelihood estimation technique
  for latent nonlinear models with multiple latent
  product terms. The LMS method assumes
  normal-distributed predictor variables and error
  variables, and this implies the normal
  distribution of the X variables.
• implemented in Mplus (Muthén Muthén, 2004).

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QML method(Quasi-Maximum Likelihood, Klein
Muthén, 2006)

• The QML approach has been developed for an
  efficient and computationally feasible estimation
  of multiple nonlinear effects in structural
  equation models with quadratic forms.
• The QML method has been specifically developed
  to provide robust, efficient, and computationally
  inexpensive analysis of structural equation
  models with multiple nonlinear effects.
• QML only assumes that the conditional
  distribution of the latent criterion given the
  X-variables can be approximated as a normal
  distribution.

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Research questions

• Although simulation studies for small latent
  interaction models have shown that LMS and QML
  estimators are consistent, asymptotically
  unbiased, asymptotically efficient, and
  asymptotically normally distributed, it is not
  known up to now, whether these methods perform
  equally well when larger models are analyzed
  including multiple nonlinear terms.
• Up to now it has not been investigated whether
  QML is as efficient as the ML estimator of LMS
  when the distributional assumptions are met and a
  complex model with several nonlinear effects is
  analyzed.
• Furthermore, it was not systematically
  investigated how LMS and QML behave with more
  complex models with several nonlinear effects
  with smaller sample sizes (N200 and N400).

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Design of the Monte-Carlo Study

• An extensive simulation study was carried out to
  the analysis of complex nonlinear structural
  equation models with one interaction term and two
  quadratic terms.
• The structural equation models were analysed
  using the LMS and QML methods.
• In the simulation study a structural equation
  model with two latent predictor variables x1 and
  x2 and a single criterion variable h is used.

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Design of Monte-Carlo Study

• Data were generated from a population with known
  parameter values.
• 500 data sets per condition combination were
  drawn.
• We chose sample size of N200 and N400.
• Latent predictor variables were correlated (f12
  .50)

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Design of Monte-Carlo Study The following
parameters have remained constant in our study

• - linear effects g1 g2 .316 (10 unique
  variance of the criterion variable h )
• - predictor correlation f12 .50
• - reliability of indicators rel(X) rel(Y)
  .80
• - 3 indicators per each latent variable

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Design of Monte-Carlo Study Data Generation
In our models the size of the latent interaction
effects and the size of both quadratic effects
were varied across three levels The unique
variance of the criterion variable explained by
the interaction term and/or both quadratic terms
are 0 , 2 , or 5 Interaction effect w12
Quadratic effects w11 and w22
Each of these 9 conditions was examined by LMS
and QML using 500 data sets of sample sizes N200
and N400. It is a simulation design with
9x2x236 conditions
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Results of Monte-Carlo Study

• In the following, well present 12 conditions, 4
  conditions for each model
• Interaction model
• Quadratic model
• Full model with one interaction effect and two
  quadratic effects

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Results of Monte-Carlo StudyInteraction model
with interaction effect w12.127 DR2 2 The
selection of ?12  .127 gives a model where the
percentage of variance of ? explained by the
interaction term is 2.
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Results of Monte-Carlo StudyQuadratic model with
two quadratic effects w11w22.071DR2 2The
selection of ?11   ?22 .071 gives a model
where the percentage of variance of ? explained
by both quadratic term is 2.
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Results of Monte-Carlo StudyFull model with
interaction effect w12.200DR2 5 andtwo
quadratic effects w11w22.112DR2 5
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Results of Monte-Carlo Study

• All linear parameters are unbiased.
• The nonlinear parameter estimates of QML for N
  200 show also no substantial bias.
• But the Monte-Carlo standard deviations (SD) are
  somewhat larger for QML than they are for LMS.
  And also the standard errors (SE) of the
  parameter estimates for QML are a little larger
  than for LMS. For N 400 both methods show
  practically no difference.
• The study reveals a small tendency to an
  overestimation of the Monte-Carlo standard
  deviations (SD) for QML for N200, for which the
  SE/SD ratios lie between .979 and 1.483. Opposed
  to this, for LMS SE/SD ratios lie between .914
  and 1.009 for sample size 200. For sample size
  400 SE/SD ratios lie for LMS between .936 and
  1.022 and for QML between 0.960 and 1.031.

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Conclusions

• This matches the expectation about the possibly
  somewhat lower efficiency of QML under the normal
  condition, because QML is only an approximate ML
  estimator
• Both methods LMS and QML have been developed for
  an efficiency of multiple nonlinear effects in
  structural equation models with quadratic forms
  and the sample size from N 200 and N 400.
• The results of the simulation study show that
  parameters of the nonlinear effects of LMS and
  QML for the sample sizes of N 200 and N 400
  were estimated unbiased.
• A small advantage for LMS for small sample sizes
  exists compared to QML that the standard errors
  are estimated more accurately.

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Conclusions

• The QML and LMS estimation of standard errors
  showed no substantial bias which supports precise
  significance testing of multiple nonlinear
  effects. They are very efficient, computationally
  feasible, and practically adequate approaches,
  which is of particular relevance when rather
  complex structural equation models with 3
  nonlinear effects are to be analyzed.
• Further, simulation studies are needed in order
  to analyze more complex models with one ore more
  interaction and several quadratic terms.
  Furthermore, the nonnormality of predictor
  variables should also be systematically varied in
  order to investigate whether methodological
  problems may even be aggravated using nonnormally
  variables compared to normally distributed
  predictor variables.

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• Thank you for your attention

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References

• Aiken, L. S., West, S. G. (1991). Multiple
  Regression Testing and interpreting
  interactions. Newbury Park, CA Sage.
• Klein, A. Moosbrugger, H. (2000). Maximum
  likelihood estimation of latent interaction
  effects with the LMS method. Psychometrika, 65,
  457-474.
• Klein, A. G. Muthén, B. O. (2006). Quasi
  Maximum Likelihood Estimation of Structural
  Equation Models with Multiple Interaction and
  Quadratic Effects. Journal of Multivariate and
  Behavioral Research (in press).
• Muthén, L. K. Muthén, B. O. (2004). Mplus
  user's guide version 3. Los Angeles, CA Muthén.
  Muthén.