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.

data(titanic)
str(titanic)
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.

summary(titanic)
 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.

BivariateAssoc(titanic$Survived, titanic[,-1])
$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.

catdesc(titanic$Survived, titanic[,-1], min.phi=0.1)
$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.

set.seed(1912)

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
plot(arbre)

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.

GetSplitStats(arbre)
$`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.

surro <- SurrogateTree(foret, maxdepth=3)

surro$r.squared %>% round(3)
[1] 0.96
plot(surro$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).

importance <- -varImpAUC(foret)
importance %>% round(3)
     Sex   Pclass      Age Embarked 
   0.238    0.073    0.030    0.010 
ggVarImp(importance)