Partially missing at random and ignorable inferences for parameter subsets with missing data

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Partially missing at random and ignorable
inferences for parameter subsets with missing data

• Roderick Little

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Outline

• Survey Bayesics in three slides
• Inference with missing data Rubin's (1976) paper
  on conditions for ignoring the missing-data
  mechanism
• Rubins standard conditions are sufficient but
  not necessary example
• Propose definitions of MAR, ignorability for
  likelihood (and Bayes) inference for subsets of
  parameters
• Examples
• Joint work with Sahar Zanganeh

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Calibrated Bayes

• Frequentists should be Bayesian
• Bayes is optimal under assumed model
• Bayesians should be frequentist
• We never know the model (and all models are
  wrong)
• Inferences should have good repeated sampling
  characteristics
• Calibrated Bayes (e.g. Box 1980, Rubin 1984,
  Little 2012)
• Inference based on a Bayesian model
• Model chosen to yield inferences that are
  well-calibrated in a frequentist sense
• Aim for posterior probability intervals that have
  (approximately) nominal frequentist coverage

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Calibrated Bayes models for surveys should
incorporate sample design features

• All models are wrong, some models are useful
• Design-assisted make the estimator more robust
• Calibrated Bayes make the model more robust
  many models yield design-consistent estimates
• Models that ignore features like survey weights
  are vulnerable to misspecification
• But models can be successfully applied in survey
  setting, with attention to design features
• Weighting, stratification, clustering
• Capture design weights as covariates in the
  prediction model (e.g. Gelman 2007)

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Benefits of Bayes

• Unified approach to all problems
• Avoids current approach -- inferential
  schizophrenia
• Not asymptotic
• Propagates errors in estimating parameters
• Avoids frequentist pitfalls
• Conditions on ancillaries
• Obeys likelihood principle

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v
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There are those who predict
and those who weight
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Rubin (1976 Biometrika)

• Landmark paper (3700 citations, after being
  rejected by many journals!)
• RL wrote his first (11 page) referee report, and
  an obscure discussion
• Modeled the missing data mechanism by treating
  missingness indicators as random variables,
  assigning them a distribution
• Sufficient conditions under which missing data
  mechanism can be ignored for likelihood and
  frequentist inference about parameters
• Focus here on likelihood, Bayes

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Ignoring the mechanism

• Full likelihood
• Likelihood ignoring mechanism
• Missing data mechanism can be ignored for
  likelihood inference when

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Rubins sufficient conditions for ignoring the
mechanism

• Missing data mechanism can be ignored for
  likelihood inference when
• (a) the missing data are missing at random (MAR)
• (b) distinctness of the parameters of the data
  model and the missing-data mechanism
• MAR is the key condition without (b), inferences
  are valid but not fully efficient

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Sufficient for ignorable is not the same as
ignorable

• These definitions have come to define
  ignorability (e.g. Little and Rubin 2002)
• However, Rubin (1976) described (a) and (b) as
  the "weakest simple and general conditions under
  which it is always appropriate to ignore the
  process that causes missing data".
• These conditions are not necessary for ignoring
  the mechanism in all situations.

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Example 1 Nonresponse with auxiliary data
Or whole population N
0 0 0 1 1
? ?
? ?
Not linked
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MAR, ignorability for parameter subsets

• MAR and ignorability are defined in terms of the
  complete set of parameters in the data model for
  D
• It would be useful to have a definition of MAR
  that applies to subsets of parameters, including
  parameters of substantive interest.
• A trivial example It seems plausible that a
  nonignorable mechanism would be MAR for the
  parameters of distributions of variables that are
  not missing.

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MAR, ignorability for parameter subsets
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MAR, ignorability for parameter subsets
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Partial MAR given a function of mechanism
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Example 1 Auxiliary Survey Data
0 0 0 1 1
? ?
? ?
Not linked
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Ex. 2 MNAR Monotone Bivariate Data

• Paper presents more interesting case with Y1, Y2
  blocks of variables and missing data in each
  block

0 0 0 1 1
? ?
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More generally
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Ex. 3 Complete Case Analysis in Regression
0 0 0 0 1 1
? ?
? ?
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Ex. 4A normal pattern-mixture model
0 0 0 1 1
? ?
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Ex. 5 Subsample ignorable likelihood
Little and Zhang (2011) Columns could be
vectors v fully observed ? observed or missing
Pattern Z W X Y
P1 v v ? ?
P2 v ? ? ?

• Interest concerns parameters of regression
  of Y on (Z,X,W)
• Z complete, W and (X,Y) incomplete. W complete in
  P1.
• Division of covariates into W, X is based on
  following MNAR assumptions about the missing data
  mechanism
• Pr(W complete) fn(W,X,Z) (not Y)
• (X,Y) MAR in subsample with W fully
  observed (that is, P1)

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Ex. 6 Auxiliary data, survey nonresponse
1 . . r . . n . . N
? ?
? ?
Not linked
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Simulation Study
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Simulation Study methods

• CC Complete Case estimates based on the
  responding units
• M1 ML based on a logistic regression with
  interaction for Y3
• M2 ML based on an additive logistic regression
  for Y3
• NR Weighting class estimates where nonresponse
  weights are obtained based on Y1
• PS Post-stratification weighted estimates (PS)
  based on Y2
• NRPS Adjust weights using both Y1 and Y2. For
  the case of
• categorical variable, this method is equivalent
  to Linear Calibration regression, or Generalized
  Raking estimates

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Simulation summary findings

• When response depends on Y1 Y2 interaction, all
  methods do poorly
• When data are MCAR, all methods do similarly well
• Model-based methods remove almost all the bias
  and perform better when response doesnt depend
  on Y1 Y2 interaction
• Qualitative patterns hold for different sample
  sizes

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Frequentist inference

• Rubins (1976) sufficient conditions for
  ignorability for frequentist inference were even
  stronger (essentially MCAR)
• These can be weakened too for example
  asymptotic frequentist inference based on ML and
  observed information matrix works under
  conditions given here
• Small sample inference seems more problematic

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Frequentist inference

• Rubins (1976) sufficient conditions for
  ignorability for frequentist inference were even
  stronger (essentially MCAR)
• These can be weakened too for example
  asymptotic frequentist inference based on ML and
  observed information matrix works under
  conditions given here
• Small sample inference is more complex

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Summary

• Proposed definitions of partial MAR, ignorability
  for subsets of parameters
• Expands range of situations where missing data
  mechanism can be ignored
• Though, in some cases, MAR analysis entails a
  loss of information
• How much is lost is an interesting question,
  varies by context

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References

• Harel, O. and Schafer, J.L. (2009). Partial and
  Latent Ignorability in missing data problems.
  Biometrika, 2009, 1-14
• Little, R.J.A. (1993). Pattern-Mixture Models for
  Multivariate Incomplete Data. JASA, 88, 125-134.
• Little, R. J. A., and Rubin, D. B. (2002).
  Statistical Analysis with Missing Data (2nd ed.)
  Wiley.
• Little, R.J. and Zangeneh, S.Z. (2013). Missing
  at random and ignorability for inferences about
  subsets of parameters with missing data.
  University of Michigan Biostatistics Working
  Paper Series.
• Little, R. J. and Zhang, N. (2011). Subsample
  ignorable likelihood for regression analysis with
  missing data. JRSSC, 60, 4, 591605.
• Rubin, D. B. (1976). Inference and Missing Data.
  Biometrika 63, 581-592.