General Structural Equations

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General Structural Equations

• Week 1 Class 5

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Today

• More on estimation
• More on block tests
• Out of range solutions what they mean how to
  deal with them
• Higher order latent variable models
• When a latent variable has other latent variables
  as indicators
• Using the PRELIS program to generate covariance
  matrices for LISREL
• A quick look at SAS-CALIS
• Item parcels (a controversy?)
• An extended discussion of an example set
• If time permits re-expressing latent variable
  structural equation models in matrix terms

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First computer assignment

• Due Tuesday
• Short answer responses (submit some of the
  programs if AMOS, diagrams or diskette)
• REQUIRED for credit, letter with grade
  participants
• Choice of AMOS or SIMPLIS. If using SIMPLIS, a
  system (.dsf) file has already been created.

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PRELIS demonstrationReview of SIMPLIS programs
(ReligSexMor problem)
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Estimation (notes)

• Parameter estimates are obtained through
  iterative methods
• Start with start values
• Could be user guesses (early versions of
  LISREL)
• Could use some other single-step estimation
  method (eg 2SLS)
• Use start values to calculate reproduced
  covariance matrix

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Estimation (notes)

• Start with start values
• Could be user guesses (early versions of
  LISREL)
• Could use some other single-step estimation
  method (eg 2SLS)
• Use start values to calculate reproduced
  covariance matrix
• Calculate first order derivatives for each free
  parameter
• Will tell us for any given parameter whether next
  iteration value should be higher or lower e.g.,
  positive derivative means value is too high
• Optionally, calculate second order derivatives
• Computationally intensive (usually)
• Trade off between extra effort at re-calculation
  (sometimes, matrix is merely updated with an
  approximation and only fully re-calculated every
  X iterations) and precision
• Sometimes, programs unable to calculate matrix of
  2nd order derivatives with given start values and
  use Steepest Descent methods (esp. initially)

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Estimation (notes)

• Convergence declared when estimates of fit
  function become sufficiently similar from one
  iteration to next and/or parameter estimates
  dont differ by more than x (convergence
  criterion)
• Occasionally, a model will not converge
• Check the model to make sure there are no
  implausible paths, model is identified
• Check to make sure there arent any ve error
  variances how to deal with these will be
  discussed later
• If otherwise OK except for non-convergence, ask
  for more iterations (most software will allow
  this)

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Problems

• The negative error variance.
• In theory, it is impossible for a variance to be
  negative
• But, SEM models can be estimated where the fit
  function minimum occurs when one of the
  parameters is improperly negative.

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Problems

• The negative error variance.
• Main reasons
• Improperly specified model (e.g., missing a
  parameter that would improve the fit
  considerably)
• Frequently occurs in models with LVs with 2
  indicators
• Sampling distribution (the real parameter is
  positive in the population, but in our sample, it
  ends up being ve)
• More likely to happen in smaller samples

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Problems

• The negative error variance.
• How the software responds to it
• Allow the parameter to go negative, perhaps
  providing a warning (matrix not
  positive-definite)
• Allow a finite number of further iterations, and
  if the parameter doesnt become positive, stop
  the iteration process and generate a warning
  message (LISREL default)
• Impose an inequality constraint

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Problems

• The negative error variance.
• How the investigator should respond to it
• Check the model carefully. Are there any
  inappropriate paths? Major (likely) paths that
  are missing? In 2-indicator LV models, can a 3rd
  indicator be found?
• Check the significance of the ve parameter (if
  necessary, re-run the model over-riding the
  defaults so that the model is allowed to reach
  convergence)
• Constrain the variance parameter to zero or some
  small value.

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Higher order models

• 2nd order models where the indicators for a
  latent variable are themselves latent variables
• many of the same principles apply treat 1st
  level latent variables as indicators of 2nd level
  latent variables
• will need reference indicator, for example
• first-level latent variables must be
  sufficiently correlated (or model will not work)

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Higher order models start with more modest
ambitions and test a correlated LV
If correlations among LVs are low, it may not be
reasonable or even possible to estimate a
higher-order model
Only two indicators? Same issues as with
lower-level LVs.
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Is this model adequate?
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Or is something like this required?
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Block tests
SAME ISSUES WHETHER L1-L3 are LVs, single
indicator exogenous vars or dummy variables
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Block Tests

• Matrix
• L1 L2 L3 exogenous
• L4 b1 b2 b3 (Model 1)
• L5 b4 b5 b6
• Test of whether L1 has effect on endog.
  variables
• Model 2, as above but b10 b40
• Model 3, b1 through b6 0
• Model 4, b1?0, b4 ?0, b2 b3 b5 b6 all 0t
• Test of equation with L4 dependent
• Model 5, as above but b10, b20, b30
• Compare models using chi-square difference (df
  difference in of degrees of freedom between
  models)

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Tests involving dummy (exogenous) variable
contrasts

• Example
• Variable religion
• Categories Protestant D11
• Fundamentalist Prot D21
• Catholic D31
• Muslim D41
• Atheist/agnostic reference
• (D1D2D3D40)

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Tests involving dummy (exogenous) variable
contrasts

• Example
• Variable religion
• Categories Protestant D11
• Fundamentalist Prot D21
• Catholic D31
• Muslim D41
• Atheist/agnostic reference
• (D1D2D3D40)
• Important dummy variables are not (usually)
  orthogonal. Make sure to allow for covariances
  among them exception orthogonally coded or
  effects coded in balanced designs
• Coefficients
• Protestant b1 Tests Protestant vs. Atheist
• Fund. Prot b2 Tests Fund. Prot. Vs. Atheist
• Catholic b3 Tests Cath. Vs. atheist
• Muslim b4 Tests Muslim vs. atheist
• Other tests
• Protestant vs. Catholic? Run a new model with
  b1b3, compare chi-sq.
• Protestants AND fund. Prot. TOGETHER vs.
  Catholic?
• Model 1 b1b2

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Item parcels
Versus
Assume 0 error variance or estimate from
reliability coeff.
X9

Add scores of X1X2X3X4X5X6X7X8 to get an
item parcel
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Item parcels

• Less extreme

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Reasons for using item parcels

• With single indicator models, will get fits that
  are very close to perfect (bad reason!)
• Individual indicators may be non-normally
  distributed (e.g., 4-category, 5-category
  attitude scale items tend to be kurtotic)
  summing indicators will often help
• Individual indicators may be extreme (e.g,
  dichotomies, tricotomies)
• The model may be monstrous (dozens of
  indicators per construct hundreds of variables
  in model) with a lot of somewhat redundant
  information (alternative would be to randomly
  select indicators, but why throw away data?)

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Reasons for NOT using parcels

• One of the big advantages of LV SEM models is
  discarded (at least in extreme cases where items
  reduced to a single item)
• Hypothesized pattern of inter-relationships among
  indicators may be incorrect (suggestion test by
  running model on items to be parceled, if
  possible)
• Even if items internally consistent, assumes
  internal consistency will imply consistency with
  respect to other variables in the model

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LV Structural Equation Models in Matrix terms

• Thus far, our work has involved scalar
  equations.
• one equation at a time
• Specify a model (e.g, with software) by writing
  these equations out, one line per equation

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Matrix form

• We can represent the previous 2 equations in
  matrix form

Matrix Form (single, double subscript)
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There are other matrices in this model
Variance-covariance matrix of error terms (es)
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(other matrices, continued)
Variance covariance matrix of exogenous
(manifest) variables
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Two scalar equations re-written
scalar
Matrix
Contents of matrices
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More generic form (combines all exogenous
variables into single matrix)
More generic
Where E1 ? X1, E2 ? X2 and E3 ? X3
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More generic form
All exogenous variables part of a single
variance-covariance matrix
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Reproduced covariances (the formula in matrix
terms)
T above elements of which are called ?theta
is not the same as ? in S(?). Latter refers to
all parameters in a model. Theta above refers to
elements in the variance-covariance matrix of
errors/exogenous variables.
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A simple model
B
Continued..
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Reproduced covariances (observed variable model
without latent variables)
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(proof of inverse quick aside)
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Measurement (factor) model
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Alternative notation systems for coefficients
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