Analysing partially observed datasets by embedding in multilevel models

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Analysing partially observed datasets by
embedding in multilevel models

• James Carpenter1, Michael Spratt2, Jonathan
  Sterne2 Mike Kenward1
• Medical Statistics Unit, London School of Hygiene
  Tropical Medicine
• Department of Social Medicine, University of
  Bristol

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Overview

• Motivating example from ALSPAC study
• Brief review of issues raised by missing data
• Handling missing data by embedding the analysis
  in a multilevel model
• Simulated example
• Analysis of ALSPAC blood pressure data
• Discussion
• References.

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Blood Pressure in ALSPAC

• ALSPAC Avon Longitudinal Study of Parents and
  Children
• General interest in the extent to which adverse
  circumstances in early life have a long term
  effect
• Here, take an illustrative example after
  adjusting for systolic blood pressure at 7 years,
  sex, teenage pregnancy and mothers educational
  level, is systolic blood pressure at 9 years
  still affected by defective housing at birth?

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Blood Pressure in ALSPAC

• Social deprivation variables fully observed
• Some missing blood pressure readings at both 7
  and 9 years
• 10.5 missing at age 7
• 15.5 missing at age 9

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Recap on missing data

• When we have missing observations, we need to
  consider the reason they are unseen the
  missingness mechanism
• We say they are Missing Completely at Random
  (MCAR) when each individuals missingness
  mechanism is orthogonal to anything we wish to
  draw inference about
• We say they are Missing At Random (MAR) when,
  given an individuals observed data, their
  missingness mechanism does not depend on their
  unseen values.
• We say they are Missing Not At Random (MNAR) if
  the missingness mechanism is neither of the
  above.
• Note that, strictly, different individuals can
  have different mechanisms within the MAR class

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Options for analysis

• Use complete cases
• If data are not MCAR, results will be biased if
  they are MCAR, we still lose power
• Assume unseen data are MAR and
• Use multiple imputation, or weighting, OR
• Embed the analysis in a multilevel model which
  takes care of the imputation step

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Is blood pressure MCAR?

• Logistic regression of the chance of observing 9
  year blood pressure on the other predictors

• Data not MCAR

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MAR analysis by embedding model of interest in a
multilevel model

• When data are MAR, we get valid inference by
  regressing the partially observed data on the
  fully observed data
• This is essentially what multiple imputation does
• Therefore we can avoid MI if we can set up a
  model where
• Partially observed data are responses and
• the model parameterisation includes the
  coefficients we are interested in

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When can we do this?

• We can do this when we wish to regress a
  partially observed response at some follow-up
  time on say birth data.
• we bring in intermediate data as additional
  responses
• A property of the normal distribution means we
  can also do this if we wish to adjust for a
  partially observed variable provided we set up
  the model for the mean and variance correctly

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Illustration

• Response y and covariate x both partially
  observed. Additional fully observed categorical
  variable (two levels)
• Model
• Exact numerical equivalence if no missing data
  consistent otherwise.
• Assumption of multivariate normality required
  for many MI packages too.
• Theoretical arguments in paper.

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Simulated example

• y is outcome variable, normally distributed
• z is normal variable correlated with y and also
  predictive of missingness of y
• z can be though of as a baseline, or earlier
  measurement

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Other covariates

• Generate binary x1 and x2 predictive of y
• Make some values of y and z MAR
• Estimate the association of y with x1, adjusted
  for x2 and z

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Missing data mechanism

• Missingness in y artificially created, with
  missingness mechanism dependent on x1, x2 and z
• Missingness in z created, with missingness
  mechanism dependent on y, x1 and x2

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Simulated Examples

• 5 analyses carried out
• Analysis using original data
• Then after observations made missing
• Analysis of remaining data (complete cases)
• MI analysis using Stata inorm 2
• Analysis using Stata ice 4
• Analysis by embedding in a multilevel models (SAS
  proc MIXED MLwiN)

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Results
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Analysis of ALSPAC Blood Pressure Example

• Blood pressure example analysed in similar way to
  the simulations
• Y sbp at 9 years
• X1 defective housing (main predictor of
  interest)
• X2 gender
• X3 teenage pregnancy
• X4 basic educational attainment
• Z sbp at 7 years

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Missing data assumption

• For each individual, their missing blood pressure
  (either at 7 or 9 years) is MAR given their
  observed covariates
• blood pressure
• defective housing
• gender,
• teenage pregnancy and
• mothers educational level

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Results

• Effect of defective housing at birth, adjusting
  for blood pressure at 7, gender, and teenage
  pregnancy

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Discussion blood pressure analysis

• Weve seen unseen data are not MCAR our results
  assume unseen data MAR
• Results consistent with an ongoing effect of
  defective housing at birth on blood pressure
• Complete case analysis overestimates the effect,
  and also looses information (by ignoring
  information from partially observed cases)
• Multiple imputation methods agree with multilevel
  model approach in line with theory

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Discussion implications

• Illustrated how missing data in a single-level
  analysis can be handled by embedding in a
  multilevel model
• Key assumption is multivariate normality
• MI is a stochastic estimation process so cannot
  be more efficient that direct estimation, which
  also has the advantage of easy repeatability
• Modelling can be done in your favourite package
  MLwiN, stata, SAS proc MIXED
• However, within Stata xtmixed, we currently
  cannot define the residual correlation matrix as
  unstructured

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References

• 1 James R Carpenter and Michael G Kenward.
  Missing data in randomised controlled trials a
  practical guide. Birmingham NCCRM (2007)
• 2 R.J.A. Little and D. B. Rubin. Statistical
  analysis with missing data (2nd edition). Wiley,
  (2002).
• 3 J. L.Schafer. Analysis of incomplete
  multivariate data. CRC Press (1997).
• 4 S Van Buuren, H. C. Boshuitzen and D.L.
  Knook. Multiple imputation of missing blood
  pressure covariates in survival analysis. SIM 18.
  681-694 (1999).

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Appendix SAS code for first example

• proc mixed data from_stata_x1
  
  
• title 'Reg condit on baseline introduction of
  mytreat to enforce similar means for treat at
  bl'
• class id time mytreat
  
  
• model newy mytreat/ s htype 2 noint ddfm
  kr
  
• repeated time/subject id type un
  
  
• lsmeans mytreat / diff
  
  
• run
  
  
• 
  
  

• Data in long format newy is z when time1 and
  is y when time2
• In order to adjust for z (i.e, baseline) a
  3-level categorical variable mytreat is defined
• 0 if time is 1
• 1 if time is 2 and x is 0
• 2 if time is 2 and x is 1