Estimating DRIPTW Population Intervention Models - PowerPoint PPT Presentation

1 / 22

About This Presentation

Title:

Estimating DRIPTW Population Intervention Models

Description:

Data (full and observed) for point Treatment Study ... 'L-BFGS-B', 'SANN'),lower = -Inf, upper = Inf,control = list(), hessian = FALSE, ... – PowerPoint PPT presentation

Number of Views:25

Avg rating:3.0/5.0

Slides: 23

Provided by: alanhu

Category:

more less

Transcript and Presenter's Notes

Title: Estimating DRIPTW Population Intervention Models

1
Estimating DR-IPTW Population Intervention Models

2
General Causal Graph For a Point Treatment Study
V? W
confounders
A
treatment
Y
outcome
3
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

4
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.

5
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).

6
IPTW-DR Estimating Function

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

7
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

8
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

9
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.

10
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

11
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)

12
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)

13
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)

14
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

15
Misspecification of EYV
16
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")

17
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

18
Estimating E(Y-YaV) using IPTW-DR estimator

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

19
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))

20
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