Model Selection Techniques for Repeated Measures Covariance Structures

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Model Selection Techniques for Repeated Measures
Covariance Structures

• E. Barry MoserLouisiana State University
• Raúl E. Macchiavelli
• University of Puerto Rico

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Outline

• Introduction
• Covariance and Inverse Covariance Matrices and
  Structures
• Likelihood and Likelihood Ratio Test
  Decomposition
• Graphical Fingerprints
• Examples

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Covariance Structures

• Used in repeated measures to model the
  dependencies among observations taken on the same
  unit.
• The choice of structure affects both power and
  validity of test procedures about the mean.
• In general, if the structure is simpler than
  necessary, tests may be invalid.
• On the other hand, unnecessarily complex
  structures may decrease the power.

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Consider n independent normal observations, each
of them representing T repeated measures
Define
Assume a linear model
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We are usually interested in making inferences
about ?, while ? contains the nuisance
parameters.
Define the concentration matrix as ??-1
We can see that the concentration matrix is in
the canonical parameterization for the
likelihood
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Compound Symmetry

• Equi-correlated times

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Structures with zeroes in

• First order autoregressive, AR(1)

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Structures with zeroes in

• First order heteroscedastic autoregressive,
  ARH(1), same as AR(1) but

• First order antedependence, ANTE(1)

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Structures with zeroes in

• Banded Toeplitz, TOEP(q)

• Heteroscedastic banded Toeplitz, TOEPH(q)

• Banded general covariance, UN(q)

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Importance of Zeros

• Structural zeroes -gt covariance matrix
• Marginal independence
• Structural zeroes -gt concentration matrix
• Conditional independence

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Making inferences about the mean (Diggle, 1988)

• Fit a mean model (overfit is preferable to
  underfit avoid spurious correlations).
• Obtain initial covariance structures
• Relevant theory
• Graphics
• Choose an appropriate structure using formal
  statistical techniques.
• Make inferences about the mean.

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Methods for choosing covariances

• Sequence of likelihood ratio tests
• Penalized likelihood criteria
• Graphical models
• Methods used in linear structural equations
  (Jöreskögs LISREL)
• Goodness of fit indices
• Covariance residuals

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Likelihood ratio test

• Consider a likelihood ratio test comparing a
  proposed structure with the general unstructured
  alternative

where
and
are residuals of structured and unstructured
models, respectively.
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• Applying the spectral decomposition to
• the LRT we can write

and
where
is a diagonal matrix of diagonal elements
of
and
and
are the jth eigen-
value and vector, respectively, of
the correlation matrix.
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This results in the decomposition of the LRT as
where
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Decomposition emphasis

• Provides contributions to the LRT according to
  each subject
• Provides contributions to the LRT according to
  the components of time
• Since the LRT has an asymptotic expected value of
  df (difference in number of parameters), divide
  each component dij by df to spot large individual
  contributions, or sum up by subject or dimension.

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Alternative decomposition of the LRT
Let
then
which can be further decomposed into
where
and
are variances from
and
respectively, and
and
are eigenvalues from
and
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Decomposition emphasis

• Variances at each time are directly compared
• Correlation structures are compared by comparing
  the variances of the principal components of the
  correlation matrix estimates
• Contributions by the residuals are kept separate

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Graphical fingerprints

• Principal components of several of the
  correlation structures have characteristic
  patterns
• Consider Compound Symmetry

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Graphical fingerprints

• AR(1) - eigenvalues

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Graphical fingerprints

• AR(1) - eigenvectors

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Graphical fingerprints

• Plot the time vectors in the space of the
  principal components from the UN correlation
  matrix and examine the resulting pattern.
• Rotate the UN correlation pattern to best match
  (Procrustes rotation) the fitted structured
  correlation model and overlay the vector plots.

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AR(1) fingerprint
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AR(1) fingerprint
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Examples

• Simulated CS structure
• Simulated ARH(1) structure
• Pasture forage experiment

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Simulated CS structure
Likelihood ratio test (df19) decomposition for
the compound symmetry data fitted with a compound
symmetry model.
 
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Simulated CS structure
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Simulated CS structure
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Simulated ARH(1) structure
Likelihood ratio test (df19) decomposition for
the ARH(1) data fitted with a homogeneous
compound symmetry (CS) model .
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Simulated ARH(1) structure
Likelihood ratio test (df14) decomposition for
the ARH(1) data fitted with a heterogeneous
compound symmetry (CSH) model .
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Simulated ARH(1) structure
Likelihood ratio test (df19) decomposition for
the ARH(1) data fitted with a homogeneous
first-order autoregressive (AR(1)) model.
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Simulated ARH(1) structure
Likelihood ratio test (df14) decomposition for
the ARH(1) data fitted with a heterogeneous
first-order autoregressive (ARH(1)) model.
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Endpoints of the 6 time vectors for the ARH(1)
data fit with a heterogeneous compound symmetry
(CSH) model plotted in the space of the first two
principal axes.
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Endpoints of the 6 time vectors for the ARH(1)
data fit with a homogeneous first-order
autoregressive (AR(1)) model plotted in the space
of the first two principal axes.
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Endpoints of the 6 time vectors for the ARH(1)
data fit with a homogeneous first-order
autoregressive (AR(1)) model plotted in the space
of the last two principal axes.
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Pasture forage experiment

• Comparison of minimum and non-tillage with
  conventional methods of signal-grass pasture
  establishment
• Randomized complete block design with 3
  replicates of 5 treatments
• 5 repeated measurements of average plot coverage
  taken at monthly intervals

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Pasture forage experiment

• Treatments
• Minimum tillage then seeded
• Minimum tillageherbicideminimum tillage again
  at 45 days then seeded
• Minimum tillageherbicidedisking at 45 days then
  seeded
• Non-tillageherbicide then seeded at 45 days
• Conventional planting using disking and harrowing
  then seeded

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Proc MIXED Code - UN

• Proc Mixed DataForage MethodML
• Class Block Treatment Time
• Model Coverage Block Treatment Time /
  OutPredResidUN
• Repeated Time / SubjectPlot TypeUN R RCorr
• Run

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Unstructured Correlation Matrix
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Inverse of UnstructuredCorrelation Matrix
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Proc MIXED Code ARH(1)

• Proc Mixed DataForage MethodML
• Class Block Treatment Time
• Model Coverage Block Treatment Time /
  OutPredResidARH1
• Repeated Time / SubjectPlot TypeARH(1) R
  RCorr
• Run

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Likelihood ratio test (df9) decomposition for
the pasture forage data fitted with a
heterogeneous first-order autoregressive (ARH(1))
model.
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Likelihood ratio test (df9) decomposition of
variances (V) for the pasture forage data fitted
with a heterogeneous first-order autoregressive
(ARH(1)) model.
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Likelihood ratio test (df9) decomposition of
eigenvalues (l) of correlation matrix for the
pasture forage data fitted with a heterogeneous
first-order autoregressive (ARH(1)) model.
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Endpoints of the 6 time vectors for the pasture
forage data fit with a heterogeneous first-order
autoregressive (ARH(1)) model plotted in the
space of the first two principal axes.
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Likelihood ratio test (df3) decomposition for
the pasture forage data fitted with a two-factor
factor analytic (FA(2)) model.
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Likelihood ratio test (df3) decomposition of
variances (V) for the pasture forage data fitted
with a two-factor factor analytic (FA(2)) model.
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Likelihood ratio test (df3) decomposition of
eigenvalues (l) of correlation matrix for the
pasture forage data fitted with a two-factor
factor analytic (FA(2)) model.
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Endpoints of the 6 time vectors for the pasture
forage data fit with a two-factor factor analytic
(FA(2)) model plotted in the space of the first
two principal axes.
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Conclusions

• Decomposition of the LRT focuses attention
  separately upon the variance structure,
  correlation structure, and residuals.
• Graphical fingerprints yield a visual measure of
  fit indicating problems with the fitted
  correlation structure, and may help identify
  outlier time points.

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Future

• Missing data likelihood
• Subject and time-point outliers/diagnostics
• Fixed-effect tests
• Mixed models
• Categorical repeated measures data models

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Selected references

• Diggle, P. (1988) An approach to the analysis of
  repeated measurements. Biometrics 44, 959-71.
• Wolfinger, R. (1996) Heterogeneous
  variance-covariance structures for repeated
  measures. Journal of Agricultural, Biological,
  and Environmental Statistics 1, 205-230.

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Selected references (cont)

• Macchiavelli, R. and E.B. Moser (1997) Analysis
  of repeated measures with ante-dependence models.
  Biometrical Journal 39 (3), 339-350.