Bayesian dynamic modeling of latent trait distributions

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Bayesian dynamic modeling of latent trait
distributions
Paper by David B. Dunson, Biostatistics, 2006
Duke University Machine Learning Group Presented
by Kai Ni Jan. 25, 2007
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Outline

• Introduction
• Measurement model
• Dynamic mixture of Dirichlet processes
• Inference
• Results Conclusion

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Motivation

• The general problem
• The primary response variable of interest cannot
  be measured directly and one must rely on
  multiple surrogates.
• The different measured outcomes are assumed to e
  manifestations of a latent variable, which may
  depend on covaraites.
• Example Cannot measure the frequency of DNA
  strand break but can use gel electrophoresis to
  get surrogates. The distribution of DNA damage
  across cells may have different shapes depending
  on the level of oxidative stress.

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Motivation (2)

• The paper focus on developing an approach for
  assessing dynamic changes in the latent response
  distribution across levels of a predictor.
• Dynamic mixture of Dirichlet processes (DMDP)
  The latent response distribution in group h is
  represented as a mixture of the distribution in
  group h-1 and an unknown innovation distribution,
  which is assigned a DP prior.

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Measurement Model

• Let yhi (yhi1,,yhip) denote a p x 1 vector of
  surrogate measurements for the latent response of
  the ith (i 1,,nh) subject in group h (h
  1,,d).
• For example, in the DNA damage study, yhi denotes
  surrogates of DNA damage for the ith cell in dose
  group h.
• The yhi has both continuous and categorical
  elements. Use some mapping function to get an
  underlying continuous variables yhi.

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Measurement Model (2)

• Relate the underlying continuous variables to the
  latent response through a measurement model
• Latent variable
• Intercept parameters
• Factor loadings
• Measurement errors
• A scale mixture of normal distribution is
  assumed for the residual distribution.
• The primary goal is to assess how the latent
  response distribution changes between groups.

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Dynamic mixture of Dirichlet process

• First the latent response distribution for group
  1 is assumed to be drawn from a DP
• and the predictive density of latent response
  for group 1 is
• Assume the distribution G2 for group 2 shares
  features with G1 but that innovation may have
  occurred. So G2
• G2 is randomly modified from G1 by (1) reducing
  the probabilities allocated to the atoms in G1 by
  a factor (1- ) and (2) incorporating new atoms
  drawn from the base

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DMDP

• The difference between G1 and G2 has mean and
  variance
• The hyperparameters control
  the magnitude of the expected changed from G1 to
  G2.

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DMDP
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Correlation

• For the special case in which
  for all l, so that the same base distribution is
  chosen for each component in the mixture. The
  correlation between consecutive Gs is
• The prior probability of clustering together two
  subjects h, i and h, i in the same or different
  groups is
• For the hyperparameters, beta distribution is
  chosen for and gamma distribution is chosen
  for

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Sampling in the latent response model
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Sampling in the measurement model
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Inference on the latent response distribution

• Collecting draws from the conditional predictive
  distribution for a future subject in dose groups
  
• After convergence, the samples of nh,nh1
  represent draws from the predictive density of
  the latent response in group h, and inferences
  can be based on comparing these densities between
  groups.

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DNA damage study

• The study assessed the effect of oxidative stress
  on the frequency of DNA strand breaks using
  single-cell gel electrophoresis.
• 500 human lymphoblastoid cells drawn from an
  immortalized cell line were randomized to one of
  the the five dose groups (0, 5, 20, 50, or 100
  micromoles H2O2).
• There are p5 surrogate measures of DNA damage,
  including (1) tail DNA, (2) tail extent divided
  by head extent, (3) extent tail moment, (4) Olive
  tail moment, and (5) tail extent.

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Conclusion

• The author proposed a Bayesian semiparametric
  latent response model in which the latent
  variable density can shift dynamically across
  groups.
• Use linear regression model to infer the latent
  variables may fail in many applications while the
  measurement model proposed by the author is quite
  flexible.
• The DMDP should prove useful when interest
  focuses on clustering of observations within and
  across groups.