## First steps

First, the necessary packages are loaded into memory.

```
library(tidyverse) # data management
library(caret) # confusion matrix
library(party) # conditional inference random forests and trees
library(partykit) # conditional inference trees
library(pROC) # ROC curves
library(measures) # performance measures
library(varImp) # variable importance
library(pdp) # partial dependence
library(vip) # measure of interactions
library(moreparty) # surrogate trees, accumulated local effects, etc.
library(RColorBrewer) # color palettes
library(GDAtools) # bivariate analysis
```

Now, we then import `titanic`

data set from `moreparty`

.

```
tibble [1,309 × 5] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
$ Survived: Factor w/ 2 levels "No","Yes": 2 2 1 1 1 2 2 1 2 1 ...
$ Sex : Factor w/ 2 levels "female","male": 1 2 1 2 1 2 1 2 1 2 ...
$ Pclass : Factor w/ 3 levels "1st","2nd","3rd": 1 1 1 1 1 1 1 1 1 1 ...
$ Age : num [1:1309] 29 0.917 2 30 25 ...
$ Embarked: Factor w/ 3 levels "Cherbourg","Queenstown",..: 3 3 3 3 3 3 3 3 3 1 ...
```

We have 1309 cases, one categorical explanained variable, `Survived`

, which codes whether or not an individual survived the shipwreck, and four explanatory variables (three categorical and one continuous): gender, age, passenger class, and port of embarkation. The distribution of the variables is examined.

```
Survived Sex Pclass Age Embarked
No :809 female:466 1st:323 Min. : 0.1667 Cherbourg :270
Yes:500 male :843 2nd:277 1st Qu.:21.0000 Queenstown :123
3rd:709 Median :28.0000 Southampton:914
Mean :29.8811 NA's : 2
3rd Qu.:39.0000
Max. :80.0000
NA's :263
```

The distribution of the explained variable is not balanced, as survival is largely in the minority. In addition, some explanatory variables have missing values, in particular `Age`

.

We examine the bivariate statistical relationships between the variables.

```
$YX
variable measure assoc p.value criterion
1 Sex cramer 0.527 0.00000 0.000000000
2 Pclass cramer 0.313 0.00000 0.000000000
3 Embarked cramer 0.184 0.00000 0.000000001
4 Age eta2 0.002 0.26069 0.302040642
$XX
variable1 variable2 measure assoc p.value criterion
1 Pclass Age eta2 0.170 0.00000 0.0000000000
2 Pclass Embarked cramer 0.280 0.00000 0.0000000000
3 Sex Pclass cramer 0.125 0.00004 0.0000378611
4 Sex Embarked cramer 0.122 0.00006 0.0000563134
5 Age Embarked eta2 0.006 0.01789 0.0180491352
6 Sex Age eta2 0.003 0.03964 0.0404512887
```

Survival is primarily associated with gender, secondarily with the passenger class. The explanatory variables are weakly related to each other.

```
$variables
variable measure assoc p.value criterion
1 Sex cramer 0.527 0.00000 0.000000000
2 Pclass cramer 0.313 0.00000 0.000000000
3 Embarked cramer 0.184 0.00000 0.000000001
4 Age eta2 0.002 0.26069 0.302040642
$bylevel
$bylevel$No
$bylevel$No$categories
categories pct.ycat.in.xcat pct.xcat.in.ycat pct.xcat.global phi
2 Sex.male 0.809 0.843 0.644 0.527
5 Pclass.3rd 0.745 0.653 0.542 0.282
8 Embarked.Southampton 0.667 0.754 0.699 0.150
6 Embarked.Cherbourg 0.444 0.148 0.207 -0.181
3 Pclass.1st 0.381 0.152 0.247 -0.278
1 Sex.female 0.273 0.157 0.356 -0.527
$bylevel$No$continuous.var
variables median.x.in.ycat median.x.global sd.x.in.ycat sd.x.global
1 Age 28 28 13.92254 14.4135
corr.coef
1 0.05551252
$bylevel$Yes
$bylevel$Yes$categories
categories pct.ycat.in.xcat pct.xcat.in.ycat pct.xcat.global phi
1 Sex.female 0.727 0.678 0.356 0.527
3 Pclass.1st 0.619 0.400 0.247 0.278
6 Embarked.Cherbourg 0.556 0.301 0.207 0.181
8 Embarked.Southampton 0.333 0.610 0.699 -0.150
5 Pclass.3rd 0.255 0.362 0.542 -0.282
2 Sex.male 0.191 0.322 0.644 -0.527
$bylevel$Yes$continuous.var
variables median.x.in.ycat median.x.global sd.x.in.ycat sd.x.global
1 Age 28 28 15.06148 14.4135
corr.coef
1 -0.05551252
```

Women, first class passengers and those who boarded at Cherbourg are over-represented among the survivors. Men, 3rd class passengers and those who boarded at Southampton are over-represented among the non-survivors.

Random forests imply a share of randomness (via resampling and drawing of splitting variables), as well as some interpretation tools (via variable permutations). From one program run to the next, the results may therefore differ slightly. If you wish to obtain the same results systematically and to ensure reproducibility, use the `set.seed`

function.

## Classification tree

In order to build a classification tree with CTree conditional inference algorithm, we use `partykit`

package, which allows more flexibility than `party`

package, in particular to deal with missing values.

The tree can be displayed in textual or graphical form.

```
arbre <- partykit::ctree(Survived~., data=titanic, control=partykit::ctree_control(minbucket=30, maxsurrogate=Inf, maxdepth=3))
print(arbre)
```

```
Model formula:
Survived ~ Sex + Pclass + Age + Embarked
Fitted party:
[1] root
| [2] Sex in female
| | [3] Pclass in 1st, 2nd
| | | [4] Pclass in 1st: Yes (n = 144, err = 3.5%)
| | | [5] Pclass in 2nd: Yes (n = 106, err = 11.3%)
| | [6] Pclass in 3rd
| | | [7] Embarked in Cherbourg, Queenstown: Yes (n = 87, err = 36.8%)
| | | [8] Embarked in Southampton: No (n = 129, err = 39.5%)
| [9] Sex in male
| | [10] Pclass in 1st
| | | [11] Age <= 53: No (n = 148, err = 38.5%)
| | | [12] Age > 53: No (n = 31, err = 12.9%)
| | [13] Pclass in 2nd, 3rd
| | | [14] Age <= 9: No (n = 77, err = 35.1%)
| | | [15] Age > 9: No (n = 587, err = 12.4%)
Number of inner nodes: 7
Number of terminal nodes: 8
```

`Sex`

is the first splitting variable, `Pclass`

is the second and all the explanatory variables are used in the tree.

The proportion of survivors varies greatly from one terminal node to another.

`nodeapply(as.simpleparty(arbre), ids = nodeids(arbre, terminal = TRUE), FUN = function(x) round(prop.table(info_node(x)$distribution),3))`

```
$`4`
No Yes
0.035 0.965
$`5`
No Yes
0.113 0.887
$`7`
No Yes
0.368 0.632
$`8`
No Yes
0.605 0.395
$`11`
No Yes
0.615 0.385
$`12`
No Yes
0.871 0.129
$`14`
No Yes
0.649 0.351
$`15`
No Yes
0.876 0.124
```

Thus, 96.5% of women travelling in 1st class survive, compared with only 12.4% of men over 9 years of age travelling in 2nd or 3rd class.

The graphical representation can be parameterized to obtain a simpler and more readable tree.

`plot(arbre, inner_panel=node_inner(arbre,id=FALSE,pval=FALSE), terminal_panel=node_barplot(arbre,id=FALSE), gp=gpar(cex=0.6), ep_args=list(justmin=15))`

Note that the `ggparty`

package graphically represents Ctree trees with the `ggplot2`

grammar.

To measure the performance of the tree, the AUC is calculated, by comparing predicted and observed survival.

```
pred_arbre <- predict(arbre, type='prob')[,'Yes']
auc_arbre <- AUC(pred_arbre, titanic$Survived, positive='Yes')
auc_arbre %>% round(3)
```

`[1] 0.838`

AUC is 0.838, which is a relatively high performance.

To plot the ROC curve of the model :

```
pROC::roc(titanic$Survived, pred_arbre) %>%
ggroc(legacy.axes=TRUE) +
geom_segment(aes(x=0,xend=1,y=0,yend=1), color="darkgrey", linetype="dashed") +
theme_bw() +
xlab("TFP") +
ylab("TVP")
```

Other performance measures are based on the confusion matrix.

```
ifelse(pred_arbre > .5, "Yes", "No") %>%
factor %>%
caret::confusionMatrix(titanic$Survived, positive='Yes')
```

```
Confusion Matrix and Statistics
Reference
Prediction No Yes
No 760 212
Yes 49 288
Accuracy : 0.8006
95% CI : (0.7779, 0.8219)
No Information Rate : 0.618
P-Value [Acc > NIR] : < 2.2e-16
Kappa : 0.5497
Mcnemar's Test P-Value : < 2.2e-16
Sensitivity : 0.5760
Specificity : 0.9394
Pos Pred Value : 0.8546
Neg Pred Value : 0.7819
Prevalence : 0.3820
Detection Rate : 0.2200
Detection Prevalence : 0.2574
Balanced Accuracy : 0.7577
'Positive' Class : Yes
```

The `GetSplitStats`

function allows to examine the result of the competition between covariates in the choice of splitting variables. If, for a given node, the splitting variable is significantly more associated with the explained variable than the other explanatory variables, we can think that this split is stable.

```
$`1`
Sex Pclass Age Embarked
statistic 3.656074e+02 1.277615e+02 3.2203137 4.420789e+01
p.value 6.771232e-81 7.227818e-28 0.2606920 1.005629e-09
criterion -6.771232e-81 -7.227818e-28 -0.3020406 -1.005629e-09
$`2`
Pclass Age Embarked
statistic 1.154544e+02 7.07039686 2.433650e+01
p.value 2.549845e-25 0.02332660 1.557812e-05
criterion -2.549845e-25 -0.02360298 -1.557824e-05
$`3`
Pclass Age Embarked
statistic 5.91071179 0.2003411 4.2482494
p.value 0.04447125 0.9587381 0.3174531
criterion -0.04549043 -3.1878160 -0.3819240
$`6`
Age Embarked
statistic 1.7248154 12.759465580
p.value 0.3423996 0.003388277
criterion -0.4191579 -0.003394030
$`9`
Pclass Age Embarked
statistic 3.299374e+01 10.838488444 17.7107774921
p.value 2.054102e-07 0.002979392 0.0004277725
criterion -2.054103e-07 -0.002983839 -0.0004278640
$`10`
Age Embarked
statistic 9.079224404 2.1938826
p.value 0.005163910 0.5562985
criterion -0.005177289 -0.8126033
$`13`
Pclass Age Embarked
statistic 0.03486032 2.540589e+01 5.2715095
p.value 0.99675089 1.393491e-06 0.1999551
criterion -5.72937273 -1.393492e-06 -0.2230874
```

In this case, for each of the nodes, the result of the competition is final (see `criterion`

). The tree therefore appears to be stable.

## Random forest

Then we build a random forest with conditional inference algorithm, `mtry`

=2 and `ntree`

=500.

`foret <- party::cforest(Survived~., data=titanic, controls=party::cforest_unbiased(mtry=2,ntree=500))`

To compare the performance of the forest to that of the tree, the predictions and then the AUC are calculated.

```
pred_foret <- predict(foret, type='prob') %>%
do.call('rbind.data.frame',.) %>%
select(2) %>%
unlist
auc_foret <- AUC(pred_foret, titanic$Survived, positive='Yes')
auc_foret %>% round(3)
```

`[1] 0.879`

The performance of the forest is 0.879, therefore slightly better than that of the tree.

The `OOB=TRUE`

option allows predictions to be made from out-of-bag observations, thus avoiding optimism bias.

```
pred_oob <- predict(foret, type='prob', OOB=TRUE) %>%
do.call('rbind.data.frame',.) %>%
select(2) %>%
unlist
auc_oob <- AUC(pred_oob, titanic$Survived, positive='Yes')
auc_oob %>% round(3)
```

`[1] 0.847`

Calculated in this way, the performance is indeed slightly lower (0.847).

## Surrogate tree

The so-called “surrogate tree” can be a way to synthesize a complex model.

`[1] 0.96`

`plot(surro$tree, inner_panel=node_inner(surro$tree,id=FALSE,pval=FALSE), terminal_panel=node_boxplot(surro$tree,id=FALSE), gp=gpar(cex=0.6), ep_args=list(justmin=15))`

The surrogate tree reproduces here very faithfully the predictions of the random forest (R2 = 0.96). It is also very similar to the initial classification tree.

## Variable importance

To go further in the interpretation of the results, permutation variable importance is calculated (using AUC as a performance measure). Since the explanatory variables have little correlation with each other, it is not necessary to use the “conditional permutation scheme” (available with the `conditional=TRUE`

option in the `varImpAUC`

function).

```
Sex Pclass Age Embarked
0.238 0.073 0.030 0.010
```