Marginal Structural Models and Causal Inference in Epidemiology Modeling Treatment, Censoring and Mi

1
Marginal Structural Models and Causal Inference
in Epidemiology
Modeling Treatment, Censoring and Missing Values

• Kooperberg, et al., 1997, JASA 92(437) 117.
• Robin, et al., 2000, Epidemiology 11(5) 550
• Hernan, et al., 2002, Stat in Med. 211689.

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Review of Set-up of Data

• Measurements made at regular times, 1,2,...,K.
• Ak is the AZT does on the kth day.
• Yk is dichotomous outcome measured (maybe
  repeatedly) on kth day.
• is the history
  of treatment as measured at time k.
• is the history
  of the potential confounders measured at time k.

3
Equivalence (algrothmically) of repeated binary
outcome data and survival data.

• In Survival data, the outcome of interest is T
  the time at which an event occurs relative to an
  index time.
• Example time of recurrence of cancer measured
  from removal of original tumor.
• Or, death from dx of AIDS.
• In order to practically implement the MSM
  estimator with survival data, must discretize the
  time interval into (equal) blocks of time (e.g.,
  months).

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• If we break up the time scale into blocks, then
  one variable, T, is converted to set of binary
  variables, Yk, that are 0 (not dead yet) or 1
  (died) and blank either after the event or
  censoring (o-censor, xfailure).

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Treatment Weights are now the same for
longitudinal data with repeated measures

• Now, the k represents the time-interval.
• Estimation works by treating each interval as a
  unique observation and just using the same old
  weighting.
• If not fitting a stratified MSM, then ignore the
  V in the numerator.

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Censoring Weights

• Same as the missigness weights.
• In this case, missingness has more predictable
  pattern once missing (censored), always
  missing.
• Use this to make slightly less general censoring
  weight.

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Censoring Weights

• Let C(k)0 if the subject is censored in that
  interval, C(k)1 if still in risk set opposite
  of syntax in Hernan et al.,1999 paper.
• Informally, the denominator is the probability of
  being observed in interval k (not censored),
  given the past (not censored yet, covariates and
  treatment).

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Estimating Stabilized Missingness Weights

• Need a model for

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Total Weights

• Then the weight for every non-censored interval
  is (just like repeated outcome data)
• It would be nice, lovely, marvelous, wonderful
  ... to have an automatic procedure to select the
  models.
• Good news they exist!!

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What a good method would look like

• Want a procedure that looks among a large list of
  possible models and choses the best model based
  on a sensible criterion.
• The procedure should try, automatically, to
  balance the competing goals of minimizing bias
  and variance.
• It should, thus, try to fit the data as closely
  as possible without over-fitting the data.
• The model should accomodate, in a flexible
  manner, both non-linear dose responses and
  possible statistical interactions.

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What would the optimal procedure try to optimize?

• Let the following were our MSM of interest
• Then, the optimal procedure would choose the
  model for the weights that minimized, given the
  data

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What would the optimal procedure try to optimize?

• However, procedures to do so are not currently
  available (perhaps soon using recently developed
  x-validation theory).
• Thus, we concentrate on a routine that still
  optimizes the fit of the censoring and treatment
  models and has the qualities we desire
  POLYCLASS.

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POLYCLASS

• Procedure used to model categorical outcomes.
• Thus, can be used for categorical treatment, (the
  Ak) and binary missing or censoring (the Ck)
• Developed by Kooperberg, et al., 1997.
• Works by building progressively more complicated
  models based on interactions and linear splines.

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Linear Splines

• A linear spline allows the function to bend at a
  point called a knot.
• Linear splines are based on basis functions
  that have the following form
• Usually written as x(x-c).

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Example of Linear Spline

• E(YXx)3x-1.5(x-5)

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Description of the Polyclass Model(assume binary
outcome)

• Where gi are (basis) functions of the relevant
  covariates (linear terms, spline terms,
  interactions, etc.
• ?i are the coefficients in front of these basis
  functions.

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How POLYCLASS build models up

• Starts with the constant model (just ?0),and at
  each step records a fit statistic default is
  BICp-2log(like)log(n)(p1), where n is the
  sample size and p is the number of non-constant
  basis functions in the model
• Among all potential covariates, finds the one
  that gives the smallest p-value (Rao statistic).
• At next step considers either adding 1) a new
  covariate, 2) a new knot to existing covariates,
  or 3) an interaction between basis functions
  already in model.

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How POLYCLASS pair models down.

• Removes basis functions one at a time, again at
  each step recording the fit statistic.
• Removes terms hierarchically so only removes
  main effect if variable not involved in a spline
  term nor interaction term.
• Among all candidate variables for removal, picks
  one with highest p-value (Wald statistic).

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The End Result

• Among all models (both in the building up and
  pairing down procedures), choses the model with
  the best (smallest) fit statistic.
• One frequently gets a complicated and somewhat
  hard to interpret model involving splines,
  interactions and main effects.
• Good news is you dont really care just need
  probability of observed treatment, not-missing or
  not censored and follow-up prediction procedures
  are available.
• Implemented in both R and Splus.

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POLYCLASS function in R

• polyclass(data, cov, weight, penalty, maxdim,
  exclude, include, additive FALSE, linear,
  delete 2, fit, silent TRUE, normweight
  TRUE, tdata, tcov, tweight, cv, select, loss,
  seed)
• data vector of outcomes
• cov matrix of covariates

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Example with Diarrhea Data

• Diarrhea
• Read in the data from a comma delimited file
  with header
• diarrhea1lt-read.csv("c/hubbard/causal2003/dataset
  s/diarboil.csv",headerT)
• Omit rows with missing observations
• diarrhea1lt-na.omit(diarrhea1)

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Diarrhea/Bioling Example

• Point Treatment Study with boiling water
  (categorical) as explanatory variable (0none,
  1sometimes, 2always)
• Many potential confounders
• Diarrhea (outcome) is 0 (no) 1 (yes)
• MSM

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Treatment Model - multinomial logistic regression

• j 0, 1 or 2.
• ?i00

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Fitting with Polyclass

• boillt-diarrhea1,"boilcat"
• covarlt-as.matrix(diarrhea1,321)
• Results
• Fit model (use cross-validation to choose
  penatly to minimize miss-classification)
• poly.boillt-polyclass(boil,covar,cv3,select0)
• dim1 knot1 dim2 knot2 Class 0 Class 1 Class 2
• 1 NA NA NA NA -10.878 -15.019 0
• 2 1 NA NA NA -5.515 -2.434 0
• 3 1 1 NA NA -5.183 -7.281 0
• 4 19 NA NA NA 21.701 23.556 0
• 5 15 NA NA NA 10.685 11.580 0
• 6 1 NA 15 NA 12.203 11.348 0
• 7 5 NA NA NA -11.551 -11.345 0
• 8 18 NA NA NA 4.290 3.840 0
• 9 18 NA 19 NA -1.989 -2.338 0
• 10 18 1 NA NA -3.166 -2.300 0
• 11 1 NA 19 NA 14.985 14.899 0

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Fitting with Polyclass, cont.

• The importance-anova decomposition is
• Cov-1 Cov-2 Percentage
• NA NA 53.60
• 1 NA 12.72
• 5 NA 5.07
• 15 NA 10.41
• 18 NA 1.24
• 19 NA 2.88
• 1 15 2.67
• 1 19 4.70
• 18 19 6.71
• Getting Probabilities using ppolyclass
• Description
• Classify new cases (cpolyclass'), compute
  class probabilities for
• new cases (ppolyclass'), and generate
  random multinomials for new
• cases (rpolyclass') for a polyclass'
  model.

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Probabilities and weights of Observed Treatment

• ppolyclt-ppolyclass(boil,covar,poly.boil)
• wghtslt-1/ppolyc

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Stabilized Weights

• boilwlt-table(boil)/sum(table(boil))
• swlt-wghts
• swboil0lt-boilw1swboil0
• swboil1lt-boilw2swboil1
• swboil2lt-boilw3swboil2