Confounding adjustment: Ideas in Action a case study

1
Confounding adjustment Ideas in Action -a case
study

• Xiaochun Li, Ph.D. Associate Professor Division
  of Biostatistics Indiana University School of
  Medicine

2
Outline

• Description of the data set
• Quantity to be estimated
• Summary of baseline characteristics
• Approaches to data analyses
• Results
• Discussion

3
Simulation Setup

• Linder Center data described and analyzed in
  Kereiakes et al. (2000)
• 6 month follow-up data on 996 patients who
• underwent an initial Percutaneous Coronary
  Intervention (PCI)
• were treated with usual care alone or usual
  care plus a relatively expensive blood thinner
  (IIB/IIIA cascade blocker
• has10 variables
• Y 2 outcomes, mort6mo (efficacy) and cardcost
  (cost)
• X 1 treatment variable, and 7 baseline
  covariates, stent, height, female, diabetic,
  acutemi, ejecfrac and ves1proc

4
Baseline characteristics
5
The LSIM10K dataset

• Simulation data set was based on the Linder
  Center data
• 17 copies of the clustered Lindner data, with
  fudge factors added to ejfract and hgt, and some
  clipping
• same correlation among covariates, same
  clustering patterns
• Contains the values of 10 simulated variables for
  10,325 hypothetical patients
• To simplify analyses, the data contain no missing
  values.
• Details and dataset available from Bobs website

6
What do we want to estimate?

• The population average treatment effect (ATE),
  i.e.,
• E(Y1) - E(Y0)
• Y1 and Y0 are conterfactual outcomes
• In plain words what if scenarios
• The expected response if treatment had been
  assigned to the entire study population minus the
  expected response if control had been assigned to
  the entire study population

7
Baseline covariate balanceassessment
8
Visualizing overall imbalance
Deep blue high values
C
T
9
Analytical Methodsfor confounding adjustment

• The following methods were applied to lsim10k
• Outcome regression adjustment (OR)
• Propensity score (PS) stratification
• Inverse-probability-treatment-weighted (IPTW)
• Doubly robust estimation
• Matching by
• Mahalonobis distance
• PS only

10
Analysis of mort6mo

• OR model for mort6mo
• treatment indicator (trtm)
• main effect terms for all seven covariates
• quadratic terms for both height and ejfract
• Residual deviance 2410.4 on 10323 degrees
  of freedom
• PS model
• saturated model for the five categorical
  covariates (main effects and interaction terms up
  to fifth-order)
• main effects and quadratic terms for height and
  ejfract

11
Covariates Balance Evaluations based on PS
Quintiles
12
Stent
13
Female
14
Diabetic
15
Acutemi
16
Ves1proc
17
Heightstrata 2 (0.95 cm) and 3 (-1.50cm)
18
Height

• Existence of residual confounding after adjusting
  for PS quintiles
• The within-stratum between-group height
  difference
• mean s.d. p
• Stratum 2 0.949 0.44 0.032
• Stratum 3 -1.497 0.43 0.0005

19
Ejfractstrata 1 (0.81), 2 (-1.32) and 3 (-0.72)
20
Ejfract

• Existence of residual confounding after
  adjusting for PS quintiles
• The within-strata between-group height
  difference
• mean s.d. p-value
• Stratum 1 0.812 0.41 0.0475
• Stratum 2 -1.322 0.33 7.38e-5
• Stratum 3 -0.721 0.32 0.025

21
PS Stratification

• Residual confounding within strata
• In PS stratification method, height and ejfract
  are further adjusted
• stratum specific
• Treatment effect
• Height, ejfract main effects and their quadratic
  terms

22
Results mort6mo
True ?-0.036

Results of all methods are consistent, providing
evidence of treatment effectiveness at
preventing death at 6 months.
23
Analysis of cardcost
ps model same as before

• cardcost model
• treatment indicator (trtm)
• main effect terms for all seven covariates
• quadratic terms for both height and ejfract

• cardcost model of CA with PS stratification
• stratum specific
• Treatment effect
• Height, ejfract main effects and their quadratic
  terms

24
Model checking OR Adjusted R-squared 0.0386
25
Model checking OR (log transformed) Adjusted
R-squared 0.0693
26
Results cardcost

27
IPTW 1 vs 2
28
Discussion

• All methods give consistent results on the 2
  outcomes
• All PS based results have similar variance except
  IPTW1
• IPTWs depend on approx. correct PS model
• OR depends on approx. correct outcome model
• DR is a fortuitous combination of OR and IPTW
  depends on one of models being right
• Nonparametric models of either models may be an
  alternative to parametric models

29
Double Robustness

• wrong PS model adjust for one covariate
  acutemi only
• wrong OR model for card cost adjust for the
  treatment indicator trtm and the acutemi
  covariate

By right, we mean approximately.
30
Propensity score estimation

• The majority applications in literature use a
  parametric logistic regression model that assume
  covariates are linear and additive on the log
  odds scale
• May include selected interactions and polynomial
  terms
• Accurate PS estimation is impeded by
• High dimensional covariates which ones should
  we de-confound?
• Unknown functional form how do they relate to
  the treatment selection
• PS model misspecification can substantially bias
  the estimated treatment effect
• Nonparametric approach is flexible to accommodate
  nonlinear/non-additive relationship of covariates
  to treatment assignment, e.g., trees

31
Nonparametric regression techniques

• Generalized Boosted Models (GBM) to estimate the
  propensity score function
• Friedman, 2001 Madigan and Ridgeway, 2004
  McCaffrey, Ridgeway, and Morral, 2004
• R package twang
• Regression tree model to predict cardcost
• Ripley, 1996 Therneau and Atkinson, 1997
• R package rpart

32
Generalized Boosted Models (GBM)

• A multivariate nonparametric regression technique
• Sum of a large set of simple regression trees
  modelling log-odds
• gbm finds mle of g(x)log(p(x)/(1-p(x)),
  p(x)P(T1x)
• Predict treatment assignment from a large number
  of pretreatment covariates adaptively choose
  them
• Nonlinear
• No need to select variables
• Can model complex interactions
• Invariant to monotone transformations of x
• E.g, same PS estimates whether use age, log(age)
  or age2
• Outperforms alternative methods in prediction
  error

33
Results cardcostnonparametric approach

34
Future research

• People try quintiles, deciles for propensity
  score stratification need data driven approach
  (based on bias-variance tradeoff) for number of
  strata
• Model selection PS model, and outcome model
• Nonparametric estimation of models may be
  intuitive, but not clear about the properties of
  the causal estimates
• Nonparametric caveat still need to define a set
  of confounders based on knowledge of causal
  relationship among treatment, outcome and
  covariates rather than conditioning
  indiscriminatly on all covariates that have
  associations with treatment and outcome