A shared random effects transition model for longitudinal count data with informative missingness

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A shared random effects transition model for
longitudinal count data with informative
missingness

• Jinhui Li
• Joint work with
• Yingnian Wu, Xiaowei Yang

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Outline

• Informative missingness
• SPMTM (shared-parameter Markov transition models)
• Bayesian Inference
• Gibbs Sampling methods
• Application to smoking cessation data
• Future work

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Data structure

• Drug abuse research and longitudinal design the
  effects of treatments or interventions are
  expected to change behavior over time.
• Often missing data involved. Participants with
  drug dependence frequently miss their scheduled
  clinic visits or drop out of studies prematurely
• The data contains the following
• ----the repeated measures Y
• ---- the missingness patterns R
• ---- the covariates X

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Missingness

• ---- Missing values on covariates
  (design matrix)
• ---- Intermittent missing values
  (repeated measures)
• ---- Missing values due to dropout
  (repeated measures)

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Missingness(2)

• We adopt the commonly accepted concepts about
  missingness(Rubin,1976 Little Rubin, 2002)
• Y .
• MCAR(Missing completely at random)
• MAR( Missing at random )
• NMAR (Missing not at random)
• Above conditions violated.

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Missingness(3)

• The possible relationships between the repeated
  measures Y and the missingness patterns R in the
  case of NMAR
• While the case A and C are nonignorable
  missingness and the informative missingness
  refers to the case of shared-parameter
  missingness which is our focus.

(A) Outcome-Dependent
(C) Pattern-Mixture
(B) Shared-Parameter
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Shared parameter Markov transition model

• Order-1 Markov Chain --- Modeling Repeated
  Measures
• 3-Category Logit Regression --- Modeling
  Missingness Indicators

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Note on SPMTM

• Here the shared parameter is the random effect.
• Y and R are conditional independent, given
  , which can be assumed following normal
  distribution .
• There are some minor constraints on the
  missingness model.

(B) Shared-Parameter
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Bayesian Inference

• Denote the parameters as
  which follow certain prior distribution ,
  and denote the data as
  
• 
  with conditional distribution
• we can apply Bayes rule to get the posterior
  distribution

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Bayesian Inference(2)

• Then, how to get the statistics of the posterior
  distribution?
• By sampling. We can sample parameters which
  follow the posterior distribution
  and estimate the statistics with sample
  statistics .

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Sampling strategies

• Gibbs sampling is adopted. Remember that we have
  missing data involved. Starting from some initial
  values, we have two iterative steps
• 1.Imputation step, draw one intermittent
  missing value from the conditional predictive
  distribution
• 2.Probability step,sample one single parameter
  conditional on all the others at a time
• Where - means exclusion. The above conditional
  distributions are got by fixing other things in
  and only
  leaving our target free.

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Sampling strategies(2)

• Sampling from the conditional distributions
• 1. Log-concave or log-concave after certain
  simple transformation case Adaptive Rejection
  Sampling method ( Gilks and Wild(1992)).
• gtgt The basic idea is setting the upper hull
  and lower hull for the target distribution and
  update them with the rejected sampled point if
  there is any.
• 2. General case Griddy Gibbs method ( Ritter and
  Tanner (1992)).
• gtgt The basic idea is estimating the support of
  the target distribution, discretize it to get a
  grid and sample from the grid points.
• In both cases, we can take the part not related
  to our target as constants to save the effort of
  integrating.

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Simulation

• We are mostly concerned about how the estimation
  of the treatment effect change as the missingness
  pattern parameters change. We fix the treatment
  effect at 0.5 and set others parameters,
  simulate the data according to the model and
  estimate the parameters starting from different
  random initial values 10 times

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Application to smoking cessation data

• The dataset contains the number of cigarettes
  smoked reported once a week by 172 smoking
  persons during a 12 weeks clinical trial. There
  are few intermittent missing and some dropouts.
• We are mostly concerned about the treatment
  effect . To evaluate the performance of our
  algorithm, we randomly generated some percent of
  intermittent missingness in the data 20 times and
  estimate the parameters.

5 missing 10 missing 15 missing 20 missing
-0.9388 -1.0484 -0.9975 -1.0298
0 -0.1096 -0.0587 -0.0910
Var( ) 0.0713 0.1122 0.1228 0.1426
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Future work

• Extend the model to the cases when the repeated
  measure follows continuous distribution, e.g.
  normal distribution.

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References

• Gilks, W. R. and Wild, P. (1992) Adaptive
  rejection sampling for Gibbs sampling. Applied
  Statistics 41, pp 337-348.
• Ritter, C., and Tanner, M. A. (1992).
  Facilitating the Gibbs Sampler The Gibbs Stopper
  and the Griddy-Gibbs Sampler. Journal of the
  American Statistical Society, 87, 861--868.
• Rubin, D. B. (1976). Inference and missing data,
  Biometrika 63. 581-582.
• Little, R. J. A. and Rubin, D. B. (2002).
  Statistical Analysis with Missing Data, 2nd
  edition, New York John Wiley.