Minzhao Liu

notes for mike book

Notes for Missing Data in Longitudinal Studies by Mike Daniels

MTM

  • $logit(\mu_{ij}) = x\beta$
  • $\log(\phi) = \delta + y\gamma_{j}$

Misc

  • missing assumption can not be verified
  • when posterior is not closed form, use sample from posterior
  • use diffuse prior
  • prior can be improper, but posterior should be proper or just proper
  • in complete data, variance parameter is ancillary but important for efficiency. In missing data scenario, variance parameter is important for consistency.
  • For missing data, Bayesian allow assumptions about nonidentified parameters and uncertainty about the assumption. While frequentist uses fixed value or constraints.

Some Recommendation

  • $\sigma$: truncated uniform or folded normal
  • $\tau$ : uniform shrinkage
  • $\Sigma$ : flat prior , reference prior, just proper wishart. Since improper prior can not be used in Winbugs, so use just proper priors.

Posterior Consistency

$p(\theta y) \to \theta_{0}$ which is a point mass (ture value) as $n \to \infty$ the sample size goes to infinity

Sampling

Gibbs

  • each can be conjugate
  • If not random walk, optimal acceptance rate could be 100%
  • Choose normal approximation or Laplace approximation for full conditional distribution

Recommendation:

  1. random walk for 1 parameter and which is not expensive
  2. when not expensive, can be applied on full distribution
  3. works on heavy tailed well

Data Augmentation

What we want is $p(\theta y)$, use auxiliary variable z:
  1. $p(z \theta, y)$
  2. $p(\theta z, y)$
  • If there is latent variable, it suit better (probit, random effect)
  • often used for multivariate t distribution

Inference

After getting samples, the problem is how to use samples to get inference

  1. discard K (burn-in)
  2. multiple chain
  3. autocorrelated (lag k) (lag 10 is fine)
  4. block gibbs or non-random walk used to speed mixing by reducing autocorrelation
  5. thinning, but inefficient
  6. batching is more efficient

Model Selection

DIC

Drawbacks:

  1. likelihood based
  2. not invariant
  3. no closed likelihood

PPL

Nonparametric

  • $p(y \theta)$
  • $\theta \sim G(\theta)$
  • $G \sim DP(G_{0}, \alpha)$

$\alpha$ controls how similar nonparametric G is close to $G_{0}$. $\alpha \to \infty \rightarrow G \to G_{0}$.

  • can check random effect distribution

Missing

  • Full data : $p(y, r \theta, \omega)$
  • Full data response model : $p(y x, \theta)$

MAR

  • For monotone dropout, MAR is equivalent to $P(U=j Y) = P(U=j Y_{j})$, which means the hazard is only relevant to previous observed one.

  • MAR + Ignorability goes to simplification

  • By $p(y_{2} y_{1}, r= 0, \theta) = P(y_{2} y_{1}, r = 1, \theta)$, it is easy to impute data and then model regression from completed data
  • MTM not suitable
  • Example 5.9: MAR can not assure ignorability

MNAR (Non ignorable)##

  1. use unvarified parametric assumption
  2. use informative prior

Selection Model

$y, r = p(y)p(r y)$

Pros:

  1. $y x$
  2. $r y$
  3. when monotone dropout, $r y$ is hazard function

Cons:

  1. sensitive to model specification
  2. sometimes identification problem

Example:

$g(\pi_{i}) = \phi_{0} + \phi_{1}y_{1} + \phi_{2}y_{2}$

if $\phi_{2} = 0$ is equivalent to MAR

cons:

  1. assumptions can not be verified
  2. potential sensitivity problem

Mixture Model

$p(y, r) = p(y r,x,\omega)p(r x, \omega)$

cons:

  • $y r$ is sensible
$p(y_{2},y_{1} r, \alpha) = p(y_{2} y_{1}, r, \alpha_{E})p(y_{1} r, \alpha_{o})$

In general, $\alpha_{E}$ can not fully identified from the data alone.

$\alpha_{E}, \alpha_{O}$ may overlap. So assume

$\alpha_{E} = (\alpha_{EI},\alpha_{NI}$

where $\alpha_{NI}$ can be used to do sensitivity analysis, or use informative prior

Example: bivariate normal

cons:

  • when dimension of observations goes up, number of $\alpha_{NI}$ goes up too

Shared Parameter Model

Ignorability

  1. $p(ymis yobs, \theta)$ using Data augmentation
  2. $p(\theta y)$ using gibbs sampler
Strongly rely on the assumption of $p(y \theta)$

Misspecification leads to inconsistency.

GARP/IV

  • for ignorable data, only need to specify $p(y \omega)$
  • for non-ignorable: $p(y, r \omega)$ is mandatory

NONIGNORABLE

  • sensitivity parameter and sensitivity analysis
  • informative prior
  • not identifiable from observed data but when they are fixed, remainder is identified.

Selection Model

parametric model are not suitable to sensitivity analysis

Semiparametric Selection Model

  • parametric form for $r y$
  • non or semi-parametric form for $y \omega$ or $y,r \omega$
  • informative prior on $\phi_{2}$

pros:

  1. specify $r y, y \omega$, and $\beta$ explicit

cons:

  1. sensitivity analysis

Mixture Model

$p(y \omega,x) = \sum_{r}(y,r \omega, x)$

Identification strategy:

Usually mixture model is used to deal with MNAR.

For monotone dropout, MAR is equivalent to ACMV constraints.

Table here.

Interior Family Constraints (IFC)

  1. complete case missing value (CCMV)
  2. nearest-neighbour (NN)
  3. available case (ACMV) , for monotone dropout, MAR == ACMV
  4. non future dependence missing value restrictions
  5. identification via extrapolation: $\Sigma$ is common across pattern , then identified

Mixture Model with discrete time dropout

Pattern Mixture Model with Covariates

  • linear link: $\beta = \sum \phi_s\beta^{s}$
  • nonlinear link: $\beta = \sum_{s} \partial{1}{x} g^{-1}(x\beta^{s}) \phi_{s}(x)$, when $\phi_{s}(x)$ does not depend on x.

So when dealing with mixture model with covariates, be sure to check if

  1. mean is linear in covariates (identical link or something)
  2. missingness depends on covariates: $r x$
  3. covariates effects are time varying

Shared Parameter Model

$y b$ is independent with $r b$
  • $y x,z \sim N(x\beta+z,\sigma^2)$
  • $h_u(t z) = h_{0}(t)\exp(z\gamma)$

if $\gamma = 0 $ , then MAR, otherwise, MNAR

pros:

  1. good at hazard
  2. complex data structure
  3. latent variable
  4. good at jointly analyzing repeated measure and event time

cons:

  1. $y u$ is not explicit ($p(u y)$ or p(y u))
  2. $h_{u}$ depend on future observations
  3. hard to separate p(ymis yobs, u), ie, hard to embed MAR
  4. rely on $b$ distribution

Model Selection in Nonignorable Models

based on p(y,r $\omega$) , instead of p(y $\theta$). While for
ignorable data, based on p(y $\theta$)  
  • DIC : can be obtained in winbugs
  • PPL: posterior predictive checks

Ch9: Informative Priors and Sensitivity Analysis

  • sensitivity for UN-verifiable missing assumption
  • incorporate subjective belief

Pattern Mixture Model

$p(y, r \alpha) = p(y r,)p(r) = p(ymis yobs, r)p(yobs r) p(r)$

Sensitivity Analysis

  • fix value constant
  • exam inference across range
  • assign appropriate prior
  • prefer mixture model than selection model
  • freq: fix $\delta$ and see how inference change with $\delta$
  • Bayesian: specify priors on $\delta$ based on the belief of missing mechanism.

Examples

Spcify Priors

  • MAR with no uncertainty
  • MAR with uncertainty
  • MNAR with no uncertainty
  • MNAR with uncertainty

Examples

Published

12 February 2013

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