To demonstrate how to estimate flexible individualized treatment
rules using `xgboost`

in the `personalized`

package, we simulate data with a binary treatment and a complex
relationship between covariates and the effect of the treatment. The
treatment assignments are based on covariates and thus mimic an
observational setting with no unmeasured confounders.

In this simulation, the treatment assignment depends on covariates and hence we must model the propensity score \(\pi(x) = Pr(T = 1 | X = x)\). In this simulation we will assume that larger values of the outcome are better.

```
library(personalized)
set.seed(1)
<- 500
n.obs <- 10
n.vars <- matrix(rnorm(n.obs * n.vars, sd = 1), n.obs, n.vars)
x
# simulate non-randomized treatment
<- 0.5 + 0.25 * x[,1] - 0.25 * x[,5]
xbetat <- exp(xbetat) / (1 + exp(xbetat))
trt.prob <- rbinom(n.obs, 1, prob = trt.prob)
trt
# simulate delta (CATE) as a complex function of the covariates
<- 2*(0.25 + x[,1] * x[,2] - x[,3] ^ {-2} * (x[,3] > 0.35) +
delta 1] < x[,3]) - (x[,1] < x[,2]))
(x[,
# simulate main effects g(X)
<- x[,1] + x[,2] + x[,4] - 0.2 * x[,4]^2 + x[,5] + 0.2 * x[,9] ^ 2
xbeta <- xbeta + delta * (2 * trt - 1) * 0.5
xbeta
# simulate continuous outcomes
<- drop(xbeta) + rnorm(n.obs) y
```

To estimate any ITR, we first must construct a propensity score function. We also optionally (and highly recommended for performance) can define an augmentation function that estimates main effects of covariates.

The `personalized`

package has functionality for doing so
using cross-fitting (see the vignette for augmentation):

```
# arguments can be passed to cv.glmnet via `cv.glmnet.args`.
# normally we would set the number of crossfit folds and internal folds to be larger,
# but have reduced it to make computation time shorter
<- create.propensity.function(crossfit = TRUE,
prop.func nfolds.crossfit = 4,
cv.glmnet.args = list(type.measure = "auc", nfolds = 3))
```

```
<- create.augmentation.function(family = "gaussian",
aug.func crossfit = TRUE,
nfolds.crossfit = 4,
cv.glmnet.args = list(type.measure = "mse", nfolds = 3))
```

For the sake of comparison, first fit a linear ITR. We have set
`nfolds`

to 3 for computational reasons; it should generally
be higher, such as 10.

```
<- fit.subgroup(x = x, y = y,
subgrp.model.linear trt = trt,
propensity.func = prop.func,
augment.func = aug.func,
loss = "sq_loss_lasso",
nfolds = 3) # option for cv.glmnet (for ITR estimation)
summary(subgrp.model.linear)
```

```
## family: gaussian
## loss: sq_loss_lasso
## method: weighting
## cutpoint: 0
## augmentation
## function: augment.func
## propensity
## function: propensity.func
##
## benefit score: f(x),
## Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
##
## Average Outcomes:
## Recommended 0 Recommended 1
## Received 0 0.4082 (n = 192) -1.5707 (n = 2)
## Received 1 -0.4157 (n = 305) -2.2139 (n = 1)
##
## Treatment effects conditional on subgroups:
## Est of E[Y|T=0,Recom=0]-E[Y|T=/=0,Recom=0]
## 0.8239 (n = 497)
## Est of E[Y|T=1,Recom=1]-E[Y|T=/=1,Recom=1]
## -0.6432 (n = 3)
##
## NOTE: The above average outcomes are biased estimates of
## the expected outcomes conditional on subgroups.
## Use 'validate.subgroup()' to obtain unbiased estimates.
##
## ---------------------------------------------------
##
## Benefit score quantiles (f(X) for 1 vs 0):
## 0% 25% 50% 75% 100%
## -1.10799 -0.57674 -0.44738 -0.33098 0.07272
##
## ---------------------------------------------------
##
## Summary of individual treatment effects:
## E[Y|T=1, X] - E[Y|T=0, X]
##
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -2.2160 -1.1535 -0.8948 -0.9094 -0.6620 0.1454
##
## ---------------------------------------------------
##
## 5 out of 10 interactions selected in total by the lasso (cross validation criterion).
##
## The first estimate is the treatment main effect, which is always selected.
## Any other variables selected represent treatment-covariate interactions.
##
## Trt1 V1 V3 V4 V5 V9
## Estimate -0.45 -0.134 -0.1052 0.0699 0.0118 0.01
```

The `{personalized}`

package allows use of
`{xgboost}`

routines for direct estimation of the CATE
(conditional average treatment effect) based on gradient boosted
decision trees. This allows for highly flexible estimates of the CATE
and thus benefit scores.

Several arguments used by the `xgb.train()`

and
`xgb.cv()`

functions from `{xgboost}`

must be
specified; they are:

`params`

: the list of parameters for the underlying`xgboost`

model (see help file for`xgb.train()`

: this includes`eta`

,`max_depth`

,`nthread`

,`subsample`

,`colsample_bytree`

, etc). However, note that`objective`

and`eval_metric`

will be overwritten, as they need to be set to custom values to work within`{personalized}`

.`nfold`

: number of cross validation folds to be used by`xgb.cv()`

for tuning`nrounds`

: the number of boosting iterations`early_stopping_rounds`

: can optionally be set. If set to an integer`k`

, training will stop if the performance doesnâ€™t improve for`k`

rounds.

We have set `nfolds`

to 3 for computational reasons; it
should generally be higher, such as 10.

```
## xgboost tuning parameters to use:
<- list(max_depth = 5, eta = 0.01, nthread = 1,
param booster = "gbtree", subsample = 0.623, colsample_bytree = 1)
<- fit.subgroup(x = x, y = y,
subgrp.model.xgb trt = trt,
propensity.func = prop.func,
augment.func = aug.func,
## specify xgboost via the 'loss' argument
loss = "sq_loss_xgboost",
nfold = 3,
params = param, verbose = 0,
nrounds = 5000, early_stopping_rounds = 50)
subgrp.model.xgb
```

```
## family: gaussian
## loss: sq_loss_xgboost
## method: weighting
## cutpoint: 0
## augmentation
## function: augment.func
## propensity
## function: propensity.func
##
## benefit score: f(x),
## Trt recom = 1*I(f(x)>c)+0*I(f(x)<=c) where c is 'cutpoint'
##
## Average Outcomes:
## Recommended 0 Recommended 1
## Received 0 1.7095 (n = 107) -1.3048 (n = 87)
## Received 1 -1.5548 (n = 177) 1.1216 (n = 129)
##
## Treatment effects conditional on subgroups:
## Est of E[Y|T=0,Recom=0]-E[Y|T=/=0,Recom=0]
## 3.2643 (n = 284)
## Est of E[Y|T=1,Recom=1]-E[Y|T=/=1,Recom=1]
## 2.4264 (n = 216)
##
## NOTE: The above average outcomes are biased estimates of
## the expected outcomes conditional on subgroups.
## Use 'validate.subgroup()' to obtain unbiased estimates.
##
## ---------------------------------------------------
##
## Benefit score quantiles (f(X) for 1 vs 0):
## 0% 25% 50% 75% 100%
## -8.4688 -0.9660 -0.2056 0.3896 3.3069
##
## ---------------------------------------------------
##
## Summary of individual treatment effects:
## E[Y|T=1, X] - E[Y|T=0, X]
##
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -16.9377 -1.9319 -0.4113 -0.8605 0.7792 6.6139
```

We first run the training/testing procedure to assess the performance
of the non-augmented estimator (ideally, with the number of replications
larger than `B=100`

). Note we do not run this part due to
computation time.

```
<- validate.subgroup(subgrp.model.linear, B = 100,
valmod.lin method = "training_test",
train.fraction = 0.75)
valmod.lin
```

Then we compare with the augmented estimator. Although this is based on just 5 replications, we can see that the augmented estimator is much better at discriminating between benefitting and non-benefitting patients, as evidenced by the large treatment effect among those predicted to benefit by the augmented estimator versus the smaller conditional treatment effect above.

```
<- validate.subgroup(subgrp.model.xgb, B = 100,
valmod.xgb method = "training_test",
train.fraction = 0.75)
valmod.xgb
```

We also plot the estimated CATE versus the true CATE for each approach:

```
## RMSE (note: this is still on the in-sample data so
## out-of-sample RMSE is preferred to evaluate methods)
sqrt(mean((delta - treatment.effects(subgrp.model.linear)$delta) ^ 2))
```

`## [1] 3.195867`

`sqrt(mean((delta - treatment.effects(subgrp.model.xgb)$delta) ^ 2))`

`## [1] 1.439912`

```
par(mfrow = c(2,1))
plot(delta ~ treatment.effects(subgrp.model.linear)$delta,
ylab = "True CATE", xlab = "Estimated Linear CATE")
abline(a=0,b=1,col="blue")
plot(delta ~ treatment.effects(subgrp.model.xgb)$delta,
ylab = "True CATE", xlab = "CATE via xgboost")
abline(a=0,b=1,col="blue")
```