Estimating DRIPTW Population Intervention Models

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Estimating DR-IPTW Population Intervention Models

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General Causal Graph For a Point Treatment Study
V? W
confounders
A
treatment
Y
outcome
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Data (full and observed) for point Treatment Study

• A denotes a treatment or exposure of interest
  assume categorical.
• W is a vector (set) of confounders
• Y is an outcome
• Define X (Y,W)
• Ya are the counterfactual outcomes of interest
• The full data is

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Relevant Assumptions / Model

• Consistency Assumption (CA)
• O(A,XA)(A,W,YA)
• Randomization Assumptions
• Experimental Treatment Assignment all treatments
  are possible for all members of the target
  population, or
• for
  all W,a.
• We have a random sample from the target
  population of Oi, i1,..,n.

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Population Intervention Models of Interestand
IPTW Estimating Function

• m(a,Vb)?EYa-YV
• Note that
• where ?(V) is a new nuisance parameter
• Now, we can define an IPTW estimator for EYaV
  which is a function of our model of interest,
  m(a,Vb), and ?(V).

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IPTW-DR Estimating Function

• A Specific DR estimating function robust against
  misspecification of ?(V) , and 1 out of 2 of
  EYA,W, g(AX).

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IPTW-DR continued

• In this case, Q(A,W)EYA,W and I choose
• Algorithm
• Estimate EYA,W
• Estimate EYV
• Estimate g(AW)
• Estimate
• Let bn be the solution of

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Example of R Implementation of the IPTW-DR
Estimating Function

• Note that
• where the dimension is the same as the number
  of parameters, b, equal to p1

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Example of R Implementation of the IPTW-DR
Estimating Function

• To solve this using minimization algorithms (such
  as optim in R) calculate the Euclidean norm of
  the vector of sums
• optim has quasi-Newton algorithm that uses
  numerical derivatives or you can use
  Newton-Rhapton if you are willing to figure out
  the vector-function for the gradient.
• Inference yes, you guessed it, bootstrapping.

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Implementation of IPTW-DR on Simulated Data

• Data Generating Model
• nlt-100
• sigmaWlt-0.5
• sigmaVlt-0.5
• Generate random W and V
• Wlt-rnorm(n,0,sigmaW)
• Vlt-rnorm(n,0,sigmaV) W,VN(0,0.52)
• Generate random A given W,V
• alpha0lt-0
• alpha1lt-1
• alpha2lt--1
• alpha3lt-0.5
• PA.1givenWVlt-1/(1exp(-(alpha0alpha1Walpha2Va
  lpha3WV)))
• Alt-rbinom(n,size1,PA.1givenWV) Abin(p(W,V))
• Generate random Y given A,W,V
• beta0lt--2
• beta1lt-1
• beta2lt-1
• beta3lt-0.5

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Finding the True E(Y-YaV)

• Generate Y0 and Y1
• nlt-1000000
• Alt-rep(0,n)
• PY.1givenAWVlt-1/(1exp(-(beta0beta1Wbeta2V2b
  eta3Abeta4AV2)))
• Y0lt-rbinom(n,size1,PY.1givenAWV)
• Alt-rep(1,n)
• PY.1givenAWVlt-1/(1exp(-(beta0beta1Wbeta2V2b
  eta3Abeta4AV2)))
• Y1lt-rbinom(n,size1,PY.1givenAWV)
• Generate Y Y-Ya
• yslt-Y-Y0
• yslt-c(ys,Y-Y1)
• Alt-c(rep(0,n),rep(1,n))
• Vlt-rep(V,2)
• V2lt-V2
• AVlt-AV
• Get Coefficients from Intervention Model
  m(a,V)b0b1ab2V b3V2 b4aV
• glm.tlt-lm(ysAVV2AV)
• summary(glm.t)

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Finding the True E(Y-YaV)

• Results
• (Intercept) 0.047141 b0
• A -0.089942 b1
• V -0.025187 b2
• V2 0.001019 b3
• AV 0.002208 b4
• Thus, the effect of intervening such that the
  entire population has a0 (at V0) is a 4.74 (b0
  0.047) reduction in the prevalence of the
  disease.
• The effect of intervening such that the entire
  population has a0 (at V-1) is a 7.2 (b0 - b2
  b3)

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Estimating E(Y-YaV) using IPTW estimator

• Estimating weights 1/P(AW,V)
• glm.Alt-glm(AWV,familybinomial)
• pred.Alt-predict(glm.A,type"response")
• wt.Alt-A/pred.A(1-A)/(1-pred.A)
• Estimating EYV (incorrectly) as
• glm.YVlt-glm(YV,familybinomial)
• Getting residuals to regress against A,V
• res.Ylt-predict(glm.YV,type"response")-Y
• Fitting the IPTW population intervention model
• inter.lmlt-lm(res.YVV2AV,weightswt.A)

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Estimates

• Ignore SEs
• gt msm.lmlt-lm(res.YAVV2AV,weightswt.A)
• gt summary(msm.lm)
• Call
• lm(formula res.Y A V V2 AV, weights
  wt.A)
• Residuals
• Min 1Q Median 3Q Max
• -1.5152 -0.7906 0.3235 0.3940 0.7868
• Coefficients
• Estimate Std. Error t value Pr(gtt)
• (Intercept) 0.08643 0.06845 1.263 0.210
• A -0.10627 0.08809 -1.206 0.231
• V -0.12945 0.11721 -1.104 0.272
• V2 -0.03653 0.10038 -0.364 0.717
• AV 0.19294 0.15948 1.210 0.229

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Misspecification of EYV
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Function to return Euclidean norm
vector-estimation function

• driptwlt-function(beta,A,W,V,Y)
• talt-table(A)
• alt-as.numeric(names(ta))
• nalt-length(a)
• V2lt-V2
• AVlt-AV
• m(A,Vb)b0b1Ab2V b3AV
• mavlt-beta1beta2Abeta3Vbeta4V2beta5
  AV
• h(A,Vb)
• dmlt-cbind(1,A,V,V2,AV)
• 1/P(AW,V)
• glm.Alt-glm(AWV,familybinomial)
• pred.Alt-predict(glm.A,type"response")
• wt.Alt-A/pred.A(1-A)/(1-pred.A)
• Get EYV
• EYV
• glm.YVlt-glm(YV,familybinomial)
• pred.YVlt-predict(glm.YV,type"response")

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Estimating E(Y-YaV) using IPTW-DR estimator

• term1lt-dm(wt.A(Y-pred.YVmav))
• term2lt-dm(wt.A(pred.YAW-pred.YVmav))
• term3lt-matrix(0,dim(term2)1,dim(term2)2)
• term4lt-matrix(0,dim(term2)1,dim(term2)2)
• term5lt-matrix(0,dim(term2)1,dim(term2)2)
• Summing over the a
• for(i in 1na)
• aalt-rep(ai,length(V))
• aVlt-aaV
• dm2lt-cbind(1,aa,V,V2,aV)
• mav2lt-beta1beta2aabeta3Vbeta4V2bet
  a5aV

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Estimating E(Y-YaV) using IPTW-DR estimator

• totallt-term1-term2term3-term4sumt5
• norm.crtlt-apply(total,2,sum)
• sum(norm.crt2)

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Estimating E(Y-YaV) using IPTW-DR estimator

• term4lt-term4dm2Y
• term5lt-term5dm2pred.YV
• sumt5lt-t(matrix(rep(apply(term5,2,mean),length(V))
  ,dim(term2)2,dim(term2)1))

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Finding solution to IPTW-DR in R

• optim(par, fn, gr NULL,method
  c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B",
  "SANN"),lower -Inf, upper Inf,control
  list(), hessian FALSE, ...)
• Application
• iptw.dr.optimlt-
• optim(coef(glm.t),driptw,method"BFGS",AA,WW,VV
  ,YY)
• Results
• par
• (Intercept) A V V2
  AV
• 0.08519943 -0.14117898 -0.09240759 0.02160015
  0.06513375
• value
• 1 1.241466e-30
• counts
• function gradient
• 44 10
• convergence
• 1 0
• message

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Comparing Results

• Relative EfficiencyMSE(IPTW)/MSE(IPTW-DR) by
  coefficient
• cbind(coef(glm.t),coef(msm.lm),iptw.dr.optimpar)
• TRUE IPTW IPTW-DR
• (Intercept) 0.050865198 0.1168423 0.08519943
• A -0.090643429 -0.1672510 -0.14117898
• V -0.032297676 -0.2077673 -0.09240759
• V2 -0.002590980 0.2292895 0.02160015
• AV 0.002681724 0.1168423 0.06513375

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Comparing SimulationsRelative Efficiency

• Relative EfficiencyMSE(IPTW)/MSE(IPTW-DR) by
  coefficient
• (Intercept) A V V2
  AV
• 2.0232989 1.0090188 1.0868551 6.3401207
  0.8951305