The goal of this vignette is to show most of all possibilies with *aVT* (for *aVirtualTwins* meaning *a*daptation of *Virtual Twins* method) package.

*VT* method (Jared Foster and al, 2011) has been created to find subgroup of patients with enhanced treatment effect, if it exists. Theorically, this method can be used for binary and continous outcome. This package only deals with binary outcome in a two arms clinical trial.

*VT* method is based on random forests and regression/classification trees.

I decided to use a simulated dataset called *sepsis* in order to show how *aVT* package can be used. Type `?sepsis`

to know more about this dataset. Anyway, the true subgroup is `PRAPACHE <= 26 & AGE <= 49.80`

.

**NOTE:** This true subgroup is defined with the *lower* event rate (`survival = 1`

) in treatement arm. Therefore in following examples we’ll search the subgroup with the *highest* event rate, and we know it is `PRAPACHE > 26 & AGE > 49.80`

.

Data used in *VT* are modelized by \(\left\{Y, T, X_1, \ldots, X_{p-2}\right\}\). \(p\) is the number of variables.

- \(Y\) is a binary outcome. In R, \(Y\) is a
`factor`

. Second level of this factor will be the desirable event. (\(Y=1\)) - \(T\) is treatment variable, \(T=1\) means
*active treatement*, \(T=0\) means*control treatment*. In R, \(T\) is numeric. - \(X_i\) is covariables, \(X_i\) can be categorical, continous, binary.

**NOTE:** if you run *VT* with interactions, categorical covariables must be transformed into binary variables.

Type `?formatRCTDataset`

for details.

Related functions/classes in aVirtualTwins package : `VT.object()`

, `vt.data()`

, `formatRCTDataset`

.

*VT* is a two steps method but with many possibilities

let \(\hat{P_{1i}} = P(Y_i = 1|T_i = 1, X_i)\)

let \(\hat{P_{0i}} = P(Y_i = 1|T_i = 0, X_i)\)

let \(X = \left\{X_1, \ldots, X_{p-2}\right\}\)

- Grow a random forest with data \(\left\{Y, T, X \right\}\).

- Grow a random forest with interaction treatement / covariable, i.e. \(\left\{Y, T, X, XI(T_i=0), XI(T_i=1)\right\}\)
- Grow two random forests, one for each treatement:
- The first with data \(\left\{Y, X \right\}\) where \(T_i = 0\)

- The second with data \(\left\{Y, X \right\}\) where \(T_i = 1\)

- The first with data \(\left\{Y, X \right\}\) where \(T_i = 0\)
- Build your own model

From one of these methods you can estimate \(\hat{P_{1i}}\) and \(\hat{P_{0i}}\).

Related functions/classes in aVirtualTwins package : `VT.difft()`

, `vt.forest()`

.

Define \(Z_i = \hat{P_{1i}} - \hat{P_{0i}}\)

- Use regression tree to explain \(Z\) by covariables \(X\). Then subjects with predicted \(Z_i\) greater than some threshold \(c\) are considered to define a subgroup.
- Use classification tree on new variable \(Z^{*}\) defined by \(Z^{*}_i=1\) if \(Z_i > c\) and \(Z^{*}_i=0\) otherwise.

The idea is to identify which covariable from \(X\) described variation of \(Z\).

Related function in aVirtualTwins package : `vt.tree()`

.

See **Introduction**.

*Sepsis* dataset is a simulated clinical trial with two groups treatment about sepsis desease. See details. This dataset is taken from SIDES method

*Sepsis* contains simulated data on 470 subjects with a binary outcome survival, that stores survival status for patient after 28 days of treatment, value of 1 for subjects who died after 28 days and 0 otherwise. There are 11 covariates, listed below, all of which are numerical variables.

Note that contrary to the original dataset used in SIDES, missing values have been imputed by random forest `randomForest::rfImpute()`

. See file *data-raw/sepsis.R* for more details.

True subgroup is `PRAPACHE <= 26 & AGE <= 49.80`

. **NOTE:** This subgroup is defined with the *lower* event rate (survival = 1) in treatement arm.

470 patients and 13 variables:

`survival`

: binary outcome`THERAPY`

: 1 for active treatment, 0 for control treatment`TIMFIRST`

: Time from first sepsis-organ fail to start drug`AGE`

: Patient age in years`BLLPLAT`

: Baseline local platelets`blSOFA`

: Sum of baselin sofa (cardiovascular, hematology, hepaticrenal, and respiration scores)`BLLCREAT`

: Base creatinine`ORGANNUM`

: Number of baseline organ failures`PRAPACHE`

: Pre-infusion apache-ii score`BLGCS`

: Base GLASGOW coma scale score`BLIL6`

: Baseline serum IL-6 concentration`BLADL`

: Baseline activity of daily living score`BLLBILI`

: Baseline local bilirubin

**Source:** http://biopharmnet.com/subgroup-analysis-software/

In order to begin the two steps of *VT* method, aVirtualTwins package needs to be initialized with `vt.data()`

function. type `?vt.data`

for more details.

**NOTE:** if running VT with interactions between \(T\) and \(X\), set `interactions = TRUE`

.

Code of `vt.data()`

:

```
vt.data <- function(dataset, outcome.field, treatment.field, interactions = TRUE, ...){
data <- formatRCTDataset(dataset, outcome.field, treatment.field, interactions = TRUE)
VT.object(data = data, ...)
}
```

**Example with Sepsis**

```
# load library VT
library(aVirtualTwins)
# load data sepsis
data(sepsis)
# initialize VT.object
vt.o <- vt.data(sepsis, "survival", "THERAPY", TRUE)
## "1" will be the favorable outcome
```

1 will be the favorable outcome because 1 is the second level of `"survival"`

column. It means that \(P(Y=1)\) is the probability of interest. Anyway, it’s still possible to compute \(P(Y=0)\).

**Quick example**

*Sepsis* does not have any categorical variable, following example show how `vt.data`

deals with categorical values depending on `interactions`

parameter

```
# Creation of categorical variable
cat.x <- rep(1:5, (nrow(sepsis))/5)
cat.x <- as.factor(cat.x)
sepsis.tmp <- cbind(sepsis, cat.x)
vt.o.tmp <- vt.data(sepsis.tmp, "survival", "THERAPY", TRUE)
## "1" will be the favorable outcome
## Creation of dummy variables for cat.x
## Dummy variable cat.x_1 created
## Dummy variable cat.x_2 created
## Dummy variable cat.x_3 created
## Dummy variable cat.x_4 created
## Dummy variable cat.x_5 created
```

Dummies variables are created for each category of `cat.x`

variable. And `cat.x`

is removed from dataset.

As described earlier, step 1 can be done via differents ways

Following example used *sepsis* data created in previous part.

To perform simple random forest on `VT.object`

, `randomForest`

, `caret`

and `party`

package can be used.

Class `vt.forest("one", ...)`

is used. It takes in arguments :

`forest.type`

: you have to set it to`"one"`

`vt.data`

: return of`vt.data()`

function`model`

: a random forest model`interactions`

: logical,`TRUE`

is default value`...`

: options to`randomForest()`

function

**with randomForest**

```
# use randomForest::randomForest()
library(randomForest, verbose = F)
## randomForest 4.6-12
## Type rfNews() to see new features/changes/bug fixes.
# Reproducibility
set.seed(123)
# Fit rf model
# default params
# set interactions to TRUE if using interaction between T and X
model.rf <- randomForest(x = vt.o$getX(interactions = T),
y = vt.o$getY(),
ntree = 500)
# initialize VT.forest.one
vt.f.rf <- vt.forest("one", vt.data = vt.o, model = model.rf, interactions = T)
### or you can use randomForest inside vt.forest()
vt.f.rf <- vt.forest("one", vt.data = vt.o, interactions = T, ntree = 500)
```

**with party**

`cforest()`

can be usefull however computing time is really long. I think there is an issue when giving *cforest object* in Reference Class parameter. Need to fix it.

```
# # use randomForest::randomForest()
# library(party, verbose = F)
# # Reproducibility
# set.seed(123)
# # Fit cforest model
# # default params
# # set interactions to TRUE if using interaction between T and X
# model.cf <- cforest(formula = vt.o$getFormula(), data = vt.o$getData(interactions = T))
# # initialize VT.forest.one
# vt.f.cf <- vt.forest("one", vt.data = vt.o, model = model.cf)
```

**with caret**

Using `caret`

can be usefull to deal with parallel computing for example.

**NOTE:** For `caret`

levels of outcome can’t be 0, so i’ll change levels name into “n”/“y”

```
# Copy new object
vt.o.tr <- vt.o$copy()
# Change levels
tmp <- ifelse(vt.o.tr$data$survival == 1, "y", "n")
vt.o.tr$data$survival <- as.factor(tmp)
rm(tmp)
# Check new data to be sure
formatRCTDataset(vt.o.tr$data, "survival", "THERAPY")
## "y" will be the favorable outcome
# use caret::train()
library(caret, verbose = F)
## Loading required package: lattice
## Loading required package: ggplot2
##
## Attaching package: 'ggplot2'
## The following object is masked from 'package:randomForest':
##
## margin
# Reproducibility
set.seed(123)
# fit train model
fitControl <- trainControl(classProbs = T, method = "none")
model.tr <- train(x = vt.o.tr$getX(interactions = T),
y = vt.o.tr$getY(),
method = "rf",
tuneGrid = data.frame(mtry = 5),
trControl = fitControl)
# initialize VT.forest.one
vt.f.tr <- vt.forest("one", vt.o.tr, model = model.tr)
```

To perform double random forest on `VT.object`

, same packages as simple random forest can be used.

Function `vt.forest("double", ...)`

is used. It takes in arguments :

`forest.type`

: You have to set is to`"double"`

`vt.data`

: return of`vt.data()`

function`model_trt1`

: a random forest model for \(T=1\) (this argument has to be specified)`model_trt0`

: a random forest model for \(T=0\) (this argument has to be specified)

**NOTE:** use `trt`

parameter in `VT.object::getX()`

or `VT.object::getY()`

methods to obtain part of data depending on treatment. See following example.

**with randomForest**

```
# grow RF for T = 1
model.rf.trt1 <- randomForest(x = vt.o$getX(trt = 1),
y = vt.o$getY(trt = 1))
# grow RF for T = 0
model.rf.trt0 <- randomForest(x = vt.o$getX(trt = 0),
y = vt.o$getY(trt = 0))
# initialize VT.forest.double()
vt.doublef.rf <- vt.forest("double",
vt.data = vt.o,
model_trt1 = model.rf.trt1,
model_trt0 = model.rf.trt0)
### Or you can use randomForest() inside
vt.doublef.rf <- vt.forest("double",
vt.data = vt.o,
ntree = 200)
```

Follow the same structure for `caret`

or `cforest`

models.

This idea is taken from *method 3* of Jared Foster paper :

A modification of [previous methods] is to obtain \(\hat{P_{1i}}\) and \(\hat{P_{0i}}\) via cross-validation. In this méthod the specific data for subject \(i\) is not used to obtain \(\hat{P_{1i}}\) and \(\hat{P_{0i}}\). Using k-fold cross-validation, we apply random forest regression approach to \(\frac{k-1}{k}\) of the data and use the resulting predictor to obtain estimates of \(P_{1i}\) and \(P_{0i}\) for the remaining \(\frac{1}{k}\) of the observations. This is repeated \(k\) times.

To use this approach, use `vt.forest("fold", ...)`

. This class takes in argument :

`forest.type`

: it has to be set to`"fold"`

`vt.data`

: return of`vt.data()`

function`fold`

: number of fold (e.g. \(5\))`ratio`

: Control of sampsize balance.`ratio`

of \(2\) means that there 2 times le highest level compared to the other. “Highest” means the level with larger observations. It’s in test.`interactions`

: Logical. If`TRUE`

, interactions between covariables and treatments are used.`FALSE`

otherwise.`...`

:`randomForest()`

function options

**NOTE:** This function use only `randomForest`

package.

```
# initialize k-fold RF
# you can use randomForest options
model.fold <- vt.forest("fold", vt.data = vt.o, fold = 5, ratio = 1, interactions = T, ntree = 200)
```

Random Forests are not the only models you can use to compute \(\hat{P_{1i}}\) and \(\hat{P_{0i}}\). Any prediction model can be used, as logitic regression, boosting …

Anyway, aVirtualTwins package can be used. To do so, you can use `VT.difft()`

class. It is important to note this the parent class of all “forests” classes. It takes in argument :

`vt.object`

: return of`vt.data()`

function`twin1`

: estimate of \(P(Y_{i} = 1 | T = T_{i})\) : meaning response probability under the correct treatment.`twin2`

: estimate of \(P(Y_{i} = 1 | T = 1-T_{i})\) : meaning response probability under the other treatment.`method`

:*absolute*(default),*relative*or*logit*. See`?VT.difft`

for details.

```
# you get twin1 and twin2 by your own method
# here, i'll use random number between 0 and 1 :
twin1_random <- runif(470)
twin2_random <- runif(470)
# then you can initialize VT.difft class :
model.difft <- VT.difft(vt.o, twin1 = twin1_random, twin2 = twin2_random, "absolute")
# compute difference of twins :
model.difft$computeDifft()
# See results
head(model.difft$difft)
## [1] 0.4883771 0.7632997 -0.2555044 -0.1270976 -0.4710701 -0.1426843
# Graph :
# hist(model.difft$difft)
```

**NOTE: Also, you can clone repository, write your own child class of VT.difft() AND submit it !**

As described in the method, we define \(Z_i = \hat{P_{1i}} - \hat{P_{0i}}\). It’s the difference in term of response of the active treatments compared to the control treatment. The idea is to try to explain this difference by few covariables.

We define a new variable \(Z^{*}\), \(Z^{*}_i=1\) if \(Z_i > c\) and \(Z^{*}_i=0\) otherwise. Classification tree’s goal is to explain the value \(Z^*=1\). \(c\) is a threshold given by the user. It’s the threshold for which the difference is “interesting”. One idea is to use quantiles of the *difft* distribution.

To compute a classifiction tree, `vt.tree("class", ...)`

is used. Internally, `rpart::rpart()`

is computed. It takes in argument:

`tree.type`

: it has to be set to`"class"`

`vt.difft`

:`VT.difft`

object (return of`vt.forest()`

function)`sens`

:`c(">", "<")`

.`sens`

corresponds to the way \(Z^{*}\) is defined.`">"`

(default) : \(Z^{*}\), \(Z^{*}_i=1\) if \(Z_i > c\) and \(Z^{*}_i=0\) otherwise.`"<"`

: \(Z^{*}\), \(Z^{*}_i=1\) if \(Z_i < c\) and \(Z^{*}_i=0\) otherwise.

`threshold`

: corresponds to \(c\), it can be a vector. \(seq(.5, .8, .1)\) by default.`screening`

:`NULL`

is default value. If`TRUE`

only covariables in`varimp`

`vt.data`

’s field is used.

See `?VT.tree`

for details.

```
# initialize classification tree
tr.class <- vt.tree("class",
vt.difft = vt.f.rf,
sens = ">",
threshold = quantile(vt.f.rf$difft, seq(.5, .8, .1)),
maxdepth = 3,
cp = 0,
maxcompete = 2)
# tr.class is a list if threshold is a vectoor
class(tr.class)
## [1] "list"
# acce trees with treeXX
class(tr.class$tree1)
## [1] "VT.tree.class"
## attr(,"package")
## [1] "aVirtualTwins"
```

Use regression tree to explain \(Z\) by covariables \(X\). Then some leafs have predicted \(Z_i\) greater than the threshold \(c\) (if \(sens\) is “>”), and it defines which covariables explain \(Z\).

The function to use is `vt.tree("reg", ...)`

. It takes same parameters than classification mehod.

```
# initialize regression tree
tr.reg <- vt.tree("reg",
vt.difft = vt.f.rf,
sens = ">",
threshold = quantile(vt.f.rf$difft, seq(.5, .8, .1)))
# tr.class is a list if threshold is a vectoor
class(tr.reg)
## [1] "list"
# access trees with treeXX
class(tr.reg$tree1)
## [1] "VT.tree.reg"
## attr(,"package")
## [1] "aVirtualTwins"
```

Once trees have been computed, you surely want to see what are the subgroups. This package provides a wrapper function of intern methods of `VT.tree`

class : `vt.subgroups()`

.

This function takes in argument :

`vt.tree`

: object or list of class`VT.tree`

. Return of the`vt.tree()`

function.`only.leaf`

: logical. Set`TRUE`

(default) to visualize only terminal nodes.`only.fav`

: logical. Set`TRUE`

(default) to visualize only class 1 nodes. (\(\hat{A}\))`tables`

: logical. Set`FALSE`

(default) to prevent tables of incidences from being printed.`verbose`

: logical. Set`FALSE`

(default) to prevent detailed stuffs from being printed.`compete`

: logical. Set`TRUE`

to print competitors rules thanks to competitors split.`FALSE`

is default value.

If `vt.tree`

is a list, unique subgroups are printed.

```
# use tr.class computed previously
vt.sbgrps <- vt.subgroups(tr.class)
# print tables with knitr package
library(knitr)
knitr::kable(vt.sbgrps)
```

Subgroup | Subgroup size | Treatement event rate | Control event rate | Treatment sample size | Control sample size | RR (resub) | RR (snd) | |
---|---|---|---|---|---|---|---|---|

tree1.11 | PRAPACHE< 26.5 & AGE>=51.74 & BLGCS< 11.5 | 49 | 0.485 | 0.375 | 33 | 16 | 1.293 | 1.043 |

tree1.3 | PRAPACHE>=26.5 | 157 | 0.752 | 0.327 | 105 | 52 | 2.300 | 1.865 |

tree2.11 | PRAPACHE< 26.5 & BLGCS< 10.5 & AGE>=64.68 | 14 | 0.7 | 0.5 | 10 | 4 | 1.400 | 1.200 |

tree2.13 | PRAPACHE>=26.5 & AGE< 51.74 & BLLCREAT< 2.3 | 21 | 0.375 | 0.6 | 16 | 5 | 0.625 | 1.506 |

tree2.7 | PRAPACHE>=26.5 & AGE>=51.74 | 120 | 0.897 | 0.31 | 78 | 42 | 2.894 | 2.007 |

tree3.13 | PRAPACHE>=26.5 & AGE< 51.74 & BLLBILI>=2.95 | 13 | 0.333 | 0.25 | 9 | 4 | 1.332 | 1.564 |

tree4 | PRAPACHE>=26.5 & AGE>=51.74 & PRAPACHE>=28.5 | 89 | 0.915 | 0.333 | 59 | 30 | 2.748 | 2.087 |

You can plot one tree with package `rpart.plot`

`## Loading required package: rpart`

If you want to see competitors split :

```
tr.class$tree2$createCompetitors()
head(tr.class$tree2$competitors)
## count ncat improve index var path string
## 1 470 -1 119.987009 26.500 PRAPACHE 2 PRAPACHE < 26.5
## 2 470 1 37.018261 10.500 BLGCS 2 BLGCS >= 10.5
## 3 470 -1 30.491058 61.671 AGE 2 AGE < 61.671
## 9 313 1 6.349646 10.500 BLGCS 4 BLGCS >= 10.5
## 10 313 -1 5.075061 61.671 AGE 4 AGE < 61.671
## 11 313 -1 2.027870 19.500 PRAPACHE 4 PRAPACHE < 19.5
```

If you want to print incidence of a subgroup :

```
vt.o$getIncidences("PRAPACHE >= 26 & AGE >= 52")
## $table.selected
## $table.selected$table
## trt
## resp 0 1 sum
## 0 30.000 18.000 48.000
## 1 15.000 72.000 87.000
## sum 45.000 90.000 135.000
## Incidence 0.333 0.800 0.644
##
## $table.selected$rr
## [1] 2.402402
##
##
## $table.not.selected
## $table.not.selected$table
## trt
## resp 0 1 sum
## 0 71.000 170.000 241.000
## 1 37.000 57.000 94.000
## sum 108.000 227.000 335.000
## Incidence 0.343 0.251 0.281
##
## $table.not.selected$rr
## [1] 0.7317784
# or
# tr.class$tree2$getIncidences("PRAPACHE >= 26 & AGE >= 52")
```

If you want to get infos about the tree

```
tr.class$tree2$getInfos()
##
## Threshold = 0.0675
## Delta = 0.0671
## Sens : >
## Size of Ahat : 155
# access Ahat
# tr.class$tree2$Ahat
```

You can re-run rpart computation:

`tr.class$tree2$run(maxdepth = 2)`

Type `?VT.tree`

for details.

This vignette is a bit messy right now, therefore feel free to ask anything to the repository issue reports :

- https://github.com/prise6/aVirtualTwins
- read documentation with
`?aVirtualTwins`