G89'2247 Lecture 8

1
G89.2247Lecture 8

• Comparing Measurement Models across Groups
• Reducing Bias with Hybrid Models
• Setting the Scale of Latent Variables
• Thinking about Hybrid Model Fit
• Recap of Measurement Model Issues

2
Strategies for Comparing Groups

• Are all paths exactly the same?
• Are all paths except the residual variances the
  same?
• Are residual variances and factor variances
  different?
• Are loadings the same, but factor correlations,
  variances and residual variances different?
• Are correlations the same, but variances and
  loadings different?

3
Payoff of Measurement Models in SEMBias Reduction

• Measures often are contaminated by noise
• Transient changes in subjects, random variation
  in rater selection, instrument failures, and so
  on.
• Noise in explanatory variables leads to biased
  structural effects
• Noise in covariates or control variables leads to
  underadjusted (biased) structural effects of
  variables that are measured perfectly
• Hybrid models (when properly specified) reduce or
  eliminate bias

4
Psychometric details of bias

• Suppose we have the simple structural model
  Y b0 b1X r.
• b0 and b1 can be estimated without bias if X is
  measured without error
• Suppose XXe is a contaminated version of X
• Suppose eN(0,s2), Then E(X)X
• If we estimate Y b0 b1X r
• b1 RXXb1 , where RXX is the reliability
  coefficient for X

5
Example of Bias Single Predictor

• I simulated a data set with N1000 with
• X2 24 - .5X1 r
• I then created added a version X1A of X (by
  adding error variance equal to the variance of
  X). Several noisy versions can be created.
• These have RXX of .50
• See SPSS listing
• Compare regressions of error free variables with
  noisy variables

6
Example of Bias Single Predictor

• Simulated (true) modelX2 24 - .5X1
• Estimate of true modelX2 25.154 .480X1
• Estimate of model when variables have reliability
  of .5X2A 20.863 .273X1A
• Note that coefficient is about .5 less than true
  value

7
Example of Bias Two Predictors

• Measurement error of an independent variable in
  multiple regression has two effects
• Its own coefficient is biased
• It incompletely adjusts other variables for its
  effect
• Adjustment is an important reason for multiple
  regression
• We may want to adjust for selection effects by
  including measures of social class, IQ, depressed
  affect and so on

8
Example of Bias continuedTwo Predictors

• Simulated (true) modelY -14.1 .4X1
  .6X2
• Estimate of true modelY -12.76 .375X1
  .546X2
• Estimate of model when variables have reliability
  of .5Y -1.587 .096X1 .194X2
• Note that the bias is not a simple function of RXX

9
Hybrid Model Taking Measurement Error into
Account
10
Setting the Scale of Latent Variables

• The latent variables can be scaled to have
  variance 1
• They can also be scaled to have units like the
  original indicators
• The indicators themselves do not have to be in
  the same units, so long as the units are linearly
  related
• E.G. inches, mm, cm

11
Setting the Scale, Continued

• In simple confirmatory factor analysis we tend to
  set the variance of latent variables to 1.0
• In hybrid models, we tend to set the scale of the
  latent variables to the units that have meaning
  for the structural model
• Choosing the "best" indicator to have loading set
  to 1.0
• Standardized versions of the analysis are always
  available as well

12
Fit of Hybrid Models

• Hybrid models may not fit for two reasons
• Measurement part
• Structural part
• Kline and others recommend a two step model fit
  exercise
• Test fit of CFA with no structural model
• Impose additional constraints due to structural
  model
• Don't claim validity for structural model that
  arises from a good fitting measurement model

13
How Fitted Variances and Covariances are
Represented in EQS, AMOS, LISREL

• Consider nine variable system with three latent
  variables

LISREL
EQS
X1
d1
d2
X2
x1
d3
X3
Y1
e1
e2
Y2
h1
E5
Y3
e3
z1
Y4
e4
e5
Y5
h2
e6
Y6
z2
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Structural Equation Forms
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Matrix Algebra Version of LISREL
X LXx d
Y LYh e
h Gx Bh z
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Variances of Exogenous Variables (in LISREL)

• The variances of the predicted (endogenous)
  variables are calculated from the structural
  models
• The variances of the exogenous variables,
  however, must be specified (defined)
• Var(d) qd
• Var(e) qe
• Var(z) Y
• Var(x) F

17
Variances of Endogenous Variables Can be
Expressed as Functions of Parameters

• Var(X) Var(LXx d) LXFLXT qd
• Var(Y) Var(LYh e) LYVar(h)LYT qe
• Var(h) Var(I-B)-1(Gx z)
  (I-B)-1 Var(Gx z)(I-B)-1 T
  (I-B)-1 GFG T Y(I-B)-1 T

18
Voila! The Fitted Variance/Covariance Matrix Can
Be Written

• Once the form of the model is specified, and the
  parameters indicated, we can begin to fit the
  variance covariance matrix of the data.

19
Issues in Measurement Models

• Model identification
• Number of factors
• Second order factor models?
• Scaling of latent variables
• Influence of parts of model on overall fit
• Naming of latent variables
• Reification of latent variables
• Items, Item parcels
• Overly simple factor models