Structure Equation With Nonnormal Variables

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Structure Equation With Nonnormal Variables

• Presented in DHPR, NHRI
• 2004.5.

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Major Source of Inappropriate Use of SEM

• Fail to satisfy the scaling and normality
  assumption
• Many measurements are dichotomous or ordered
  categories, e.g. agree no preference
  disagree
• Some are continuous, but depart from normal
  dramatically, e.g. amount of cigarettes smoked by
  females per day
• In 1990, 72 articles published in personality and
  psychology journals used SEM, only 19
  acknowledged normality assumption, less than 10
  explicitly considered whether the assumption had
  been violated

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Review of Normal Theory Estimation

• Estimation minimize the difference between each
  element in S and the corresponding elements in
• S is the sample covariance matrix based on
  observed data
• is the covariance matrix implied by a
  set of parameters for the hypothesized model

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Most Commonly Used Estimation Techniques
??(maximize the function)

• Maximum likelihood (ML)

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• Generalized Least Squares(GLS)
• Normal Function (used in ML)
• Therefore, both ML and GLS are based on normal
  assumption

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More on GLS

• Minimize
• W -1 is the weight function, most common choice
  is S 1
• Sample covariance between xi and xj
• The large sample distribution of the elements of
  S is assumed to be multivariate normal

W-1
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Problems
Very Large Sample
Continuous
Assumptions
Multivariate Normal
Statistical Properties ?
Robustness of the Estimators?
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Effects and Detection

• The observed variables do not have multivariate
  normal
• The X2 goodness-of-fit test is an accurate
  assessment of fit, rejecting too many (gt5) true
  models
• Tests of all parameter estimates are expected to
  be biased, yielding too many significant results
• Categorical variables assumed continuous
• Correlation is stronger than it should be

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Studies on the Effect of Non-Normality

• Olsson, Foss, Troye and Howell (Structure
  Equation Modeling, 2000)

REALITY
TURE MODEL
POPULATION (TURE)
STATES
AND COVARIANCE ?
Mtrue and ? true
Theoretic fit
True Empirical fit
Empirical fit
COVARIANCE IMPLIED BY THEORETICAL MODEL ? (?)
SAMPLE COVARIANCE S
Mtheory and ?theory
THEORETICAL DOMAIN
EMPIRICAL DOMAIN
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• Theoretical fit the degree of isomorphism
  between structure and parameter values of a
  theoretical models and of the true model that
  generates the data.
• Empirical fit the discrepancy between the
  observed covariance structure and the one implied
  by a theoretical model.
• True empirical fit the correspondence between
  the population covariance matrix (?) and the
  covariance structure implied by the theoretical
  model (? (?))

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Comparisons Among ML, GLS, and WLS

• The performance in terms of empirical and
  theoretical fit of the three models is
  differentially affected by sample size,
  specification error, and kurtosis.
• ML is considerably more insensitive than the
  others to variation in sample size and kurtosis.
  Only empirical fit is affected by specification
  error. In general, ML tends to be more stable,
  high accuracy
• GLS requires well-specified models, but allows
  small sample sizes. Its appealing performance in
  terms of empirical fit can be misleading
• WLS requires well-specified models as well as
  large sample sizes.

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Detecting Departure From Normal

• Skewness and Kurtosis
• Skewness ? Kurtosis ( vs. -)
• SAS PROC UNIVARIATE
• Univariate vs. Multivariate
• When univariate normal is violated in each
  variable, then multivariate normal (joint
  distribution) cannot be true. But the converse is
  not true.
• Mardia (1970) measures
• Outliers
• Checking errors, leverage statistics, etc.

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Remedies for Multivariate Nonnormalilty

• Alternative Estimation Techniques
• Asymptotically Distribution Free Estimator (ADF)
• Optimal weight matrix consisting of a combination
  of second- and fourth- order terms
• It has many more elements than the normal theory
  GLS weight matrix (S-1)
• Computation demanding e.g. 15 measured
  variables, it has ½1516120 unique elements,
  the matrix has 12012014,400 elements. Inversing
  the matrix can be difficult.
• GLS only take the diagonal of the matrix (120
  elements).

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• SCALED ?2 statistic and standard errors (Satorra,
  1990)
• Corrected or rescaled the ?2
• The ?2 from ML or GLS is divided by a constant k,
  whose value is a function of the model-implied
  residual weight matrix, the multivariate
  kurtosis, and the degree of freedom for the
  model.
• k as kurtosis adjusted ?2
• Its available in EQS program

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Bootstrapping

• Taking repeated samples from a population of
  interest
• Calculate the parameter estimates of interests
  resulting in an empirical sampling distribution
  of the estimates.
• Repeated samples of the same sample size are
  taken from the original sample with replacement.
• For example, the original sample consists (1, 2,
  3, 4), possible bootstrap samples are (1,4, 1,1),
  (2,3,1,3), or (4,2,2,4).

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Re-expression of Variables

• Item Parcels sum or mean of several items that
  measure the same domain.
• Potential complication in the interpretation of
  relationships and structure.
• Use of too few measured variables as indicators
  of a domain yields less stringent tests of the
  proposed structure of confirmatory factor models
• Identification problems are more likely to occur

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• Transformation of variables
• Linear transformations (e.g. standardization)
  have no effect on either the distributions of
  variables or the results of simple structural
  equation models
• Non-linear transformations potentially alter the
  distribution of the measured variables as well as
  the relationships among measured variables,
  potentially eliminating some forms of curvilinear
  effects and interactions between variables.

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Selecting an Appropriate Transformation

• Power function
• Positively skewed generally, raising the scores
  on the measured variable to a power less than
  1.0, e.g. log, squared root, reciprocal
• Negatively skewed raising raw scores to a power
  greater than 1.0.
• Box-Cox transformation when scattered plots show
  a possible non-linear relationship between pairs
  of variables.

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About the Transformation

• Examine the univariate skewness and kurtosis of
  the transformed data
• Examine the multivariate skewness and kurtosis of
  the transformed data using Mardia measures
• y y, so the covariance the y should be
  computed, not the original
• Box-Cox transformation can result in considerable
  confusion in the interpretation of the variables.

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Choice Among Remedies

• In large samples (1000 to 5000), ADE and SCALED
  ?2 and standard errors for continuous nonnormal
  data perform well.
• In median samples (200 to 500) depend on the
  degree of nonnormality
• Small samples (nonnormality is not severe) SCALED
  ?2 begin to depart from normality (e.g
  skewness2 kurtosis7)
• Variable re-expression is recommended.