Case Study in R: Recruitment agency

Camille Mondon

camille.mondon@tse-fr.eu

Toulouse School of Economics

Last updated on January 19, 2026

Table of contents

• 1 About trees
  • 1.1 Introduction
  • 1.2 Full binary trees
  • 1.3 When to use decision trees?
• 2 Procedure
  • 2.1 Overview
  • 2.2 Recursive partitioning
  • 2.3 Growing the tree
  • 2.4 Stopping the tree
  • 2.5 Pruning the tree
• 3 Appendix
• References

1 About trees

1.1 Introduction

Applications

1. Predict the land use of a given area (agriculture, forest, etc.) given satellite information, meteorological data, socio-economic information, prices information, etc.
2. Determine how computer performance is related to a number of variables which describe the features of a PC (the size of the cache, the cycle time of the computer, the memory size and the number of channels. Both the last two were not measured but minimum and maximum values obtained).

Risk of heart attack

Predict high risk for heart attack:

• University of California: a study into patients after admission for a heart attack.
• 19 variables collected during the first 24 hours for 215 patients (for those who survived the 24 hours)
• Question: Can the high risk (will not survive 30 days) patients be identified?

Figure 1: A binary tree to identify patients at high risk of heart attack. Source: (Breiman et al. 1984, fig. 1.1).

A recruiting agency

# Data
candidates <- read.csv("data/recruitment.csv", row.names = 1)
# Transformation of the variable to predict into a factor
candidates$Res <- as.factor(candidates$Res)
candidates

Table 1: Data set on candidates for a job.

Id	Dip	Test	Exp	Res
A	1	5	4	0
B	2	3	3	0
C	1	4	5	1
D	2	3	4	0
E	1	4	4	0
F	4	3	4	1
G	3	4	4	1
H	1	1	5	0
I	3	2	5	1
J	5	4	4	1

# Classification tree
library(rpart)
library(rpart.plot)

# Building the tree (prune is not useful here)
candidates_fit <- rpart(Res ~ Dip + Test + Exp,
  data = candidates, method = "class",
  parms = list(split = "gini"), control = rpart.control(minsplit = 2)
)
rpart.plot(candidates_fit, extra = 101)

1.2 Full binary trees

Figure 2: A (reversed) true tree.

1.3 When to use decision trees?

2 Procedure

2.1 Overview

A synthetic 2D example

library(ggplot2)
set.seed(0)

a <- data.frame(X_1 = rnorm(800), X_2 = rnorm(800), Y = as.factor("A"))
b <- data.frame(X_1 = rnorm(200, 2), X_2 = rnorm(200, 2), Y = as.factor("B"))
ab <- rbind(a, b)

ggplot(ab, aes(X_1, X_2, color = Y, label = Y)) +
  geom_point()

Figure 3: A two-classes variable to predict with two quantitative predictors \(X_1\) and \(X_2\).

2.2 Recursive partitioning

Depth 0

library(rpart)
library(rpart.plot)
library(parttree)

ab_fit <- rpart(Y ~ ., data = ab, minbucket = nrow(ab), cp = 0)
rpart.plot(ab_fit, box.palette = "RdBu")

ggplot(ab, aes(X_1, X_2, color = Y, label = Y)) +
  geom_point()

(a) Depth 0: a root node.

(b) Depth 0: the feature space.

Figure 4: Building splits recursively.

Depth 1

depth = 1
ab_fit <- rpart(Y ~ ., data = ab, maxdepth = depth, cp = -Inf)

rpart.plot(ab_fit, box.palette = "RdBu")

p <- ggplot(ab, aes(X_1, X_2, color = Y, fill = Y)) +
  geom_parttree(data = ab_fit, alpha = 0.1, flip = depth == 1) +
  geom_point()
print(p)

(a) Depth 1: the first split.

(b) Depth 1: partitioning the feature space.

Figure 5: Building splits recursively.

Depth 2

depth = 2
ab_fit <- rpart(Y ~ ., data = ab, maxdepth = depth, cp = -Inf)

rpart.plot(ab_fit, box.palette = "RdBu")

p <- ggplot(ab, aes(X_1, X_2, color = Y, fill = Y)) +
  geom_parttree(data = ab_fit, alpha = 0.1, flip = depth == 1) +
  geom_point()
print(p)

(a) Depth 2: a taller binary tree.

(b) Depth 2: partitioning the feature space again.

Figure 6: Building splits recursively.

Depth 3

depth = 3
ab_fit <- rpart(Y ~ ., data = ab, maxdepth = depth, cp = -Inf)

rpart.plot(ab_fit, box.palette = "RdBu")

p <- ggplot(ab, aes(X_1, X_2, color = Y, fill = Y)) +
  geom_parttree(data = ab_fit, alpha = 0.1, flip = depth == 1) +
  geom_point()
print(p)

(a) Depth 3: an even taller binary tree.

(b) Depth 3: partitioning the feature space even further.

Figure 7: Building splits recursively.

2.3 Growing the tree

The need of an impurity criterion

Chosing an impurity criterion

We consider the split: \(X_1 <\) 1.64.

ab$split <- ifelse(ab$X_1 < ab_fit$splits[1, 4], "Left", "Right")
dplyr::sample_n(ab, 10)

X_1	X_2	Y	split
-1.5017872	1.6803619	A	Left
-0.9233743	-0.8534551	A	Left
1.5462761	3.3774228	B	Left
-0.8320433	1.1174336	A	Left
-1.0854224	2.2061932	A	Left
0.9922884	0.0171323	A	Left
2.6586580	0.3255937	A	Right
-0.1726686	-0.2422814	A	Left
-0.8874201	-0.9109120	A	Left
-1.1729836	-1.6521118	A	Left

ab_ct <- table(ab$split, ab$Y) |>
  addmargins() |>
  as.data.frame.matrix()
ab_ct

	A	B	Sum
Left	766	63	829
Right	34	137	171
Sum	800	200	1000

ab_freqs <- ab_ct$Sum / ab_ct$Sum[3]

(ab_probs <- ab_ct / ab_ct$Sum)

	A	B	Sum
Left	0.9240048	0.0759952	1
Right	0.1988304	0.8011696	1
Sum	0.8000000	0.2000000	1

ab_probs$Freq <- ab_freqs
ab_probs$Gini <- 1 - ab_probs$A^2 - ab_probs$B^2
ab_probs

	A	B	Sum	Freq	Gini
Left	0.9240048	0.0759952	1	0.829	0.1404398
Right	0.1988304	0.8011696	1	0.171	0.3185938
Sum	0.8000000	0.2000000	1	1.000	0.3200000

The total difference in Gini index of this split is:

with(ab_probs, Freq[3] * Gini[3] - Freq[2] * Gini[2] - Freq[1] * Gini[1])

[1] 0.1490959

2.4 Stopping the tree

2.5 Pruning the tree

Why and how pruning?

library(rpart)
library(rpart.plot)

cars_fit <- rpart(Price ~ ., data = cu.summary, minbucket = 1, cp = 0)

rpart.plot(cars_fit, tweak = 2)

Figure 8: This tree is too complex, it needs to be pruned!

Figure 9: Why pruning?

A penalized criterion

printcp(cars_fit)

Regression tree:
rpart(formula = Price ~ ., data = cu.summary, minbucket = 1, 
    cp = 0)

Variables actually used in tree construction:
[1] Country     Mileage     Reliability Type       

Root node error: 7407472615/117 = 63311732

n= 117 

           CP nsplit rel error  xerror    xstd
1  2.5052e-01      0   1.00000 1.01330 0.15932
2  1.4836e-01      1   0.74948 0.86105 0.16322
3  8.7654e-02      2   0.60112 0.70608 0.14694
4  6.2818e-02      3   0.51347 0.58381 0.10526
5  3.4875e-02      4   0.45065 0.61862 0.12371
6  2.4396e-02      5   0.41577 0.61587 0.12360
7  1.1966e-02      8   0.34259 0.63034 0.12155
8  1.0640e-02     14   0.27079 0.68456 0.14321
9  9.9092e-03     15   0.26015 0.71913 0.14533
10 8.8587e-03     16   0.25024 0.72795 0.14614
11 7.3572e-03     20   0.21480 0.74085 0.15294
12 7.2574e-03     22   0.20009 0.73494 0.15289
13 3.8972e-03     28   0.15655 0.77054 0.15609
14 1.9968e-03     31   0.14334 0.78644 0.15729
15 1.9131e-03     33   0.13935 0.78699 0.15818
16 1.6070e-03     34   0.13744 0.78804 0.15816
17 1.1151e-03     35   0.13583 0.78647 0.15815
18 9.0617e-04     36   0.13471 0.79763 0.15711
19 4.7736e-04     42   0.12928 0.79600 0.15718
20 1.4084e-04     43   0.12880 0.79752 0.15718
21 1.0325e-04     45   0.12852 0.80091 0.15732
22 1.0187e-04     46   0.12841 0.80038 0.15734
23 8.0922e-05     47   0.12831 0.80051 0.15734
24 6.9751e-05     48   0.12823 0.80065 0.15733
25 6.0368e-05     49   0.12816 0.80068 0.15733
26 5.2584e-05     50   0.12810 0.80058 0.15733
27 5.1761e-05     52   0.12800 0.80058 0.15733
28 2.5252e-05     53   0.12794 0.79930 0.15735
29 1.2155e-05     54   0.12792 0.79968 0.15735
30 7.9655e-06     55   0.12791 0.79972 0.15734
31 1.5920e-06     56   0.12790 0.79958 0.15735
32 4.1013e-07     57   0.12790 0.79956 0.15735
33 0.0000e+00     58   0.12790 0.79958 0.15735

Iterative pruning

plotcp(cars_fit)

1-SE method.

Choosing the optimal subtree

cars_fit <- prune(cars_fit, cp = 0.074)
rpart.plot(cars_fit)

The resulting pruned tree.

3 Appendix

# Plot
summary(candidates_fit)

Call:
rpart(formula = Res ~ Dip + Test + Exp, data = candidates, method = "class", 
    parms = list(split = "gini"), control = rpart.control(minsplit = 2))
  n= 10 

    CP nsplit rel error xerror      xstd
1 0.80      0       1.0    2.0 0.0000000
2 0.10      1       0.2    0.2 0.1897367
3 0.01      3       0.0    0.6 0.2898275

Variable importance
 Dip Test  Exp 
  67   20   13 

Node number 1: 10 observations,    complexity param=0.8
  predicted class=0  expected loss=0.5  P(node) =1
    class counts:     5     5
   probabilities: 0.500 0.500 
  left son=2 (6 obs) right son=3 (4 obs)
  Primary splits:
      Dip  < 2.5 to the left,  improve=3.3333330, (0 missing)
      Test < 1.5 to the left,  improve=0.5555556, (0 missing)
      Exp  < 3.5 to the left,  improve=0.5555556, (0 missing)

Node number 2: 6 observations,    complexity param=0.1
  predicted class=0  expected loss=0.1666667  P(node) =0.6
    class counts:     5     1
   probabilities: 0.833 0.167 
  left son=4 (4 obs) right son=5 (2 obs)
  Primary splits:
      Exp  < 4.5 to the left,  improve=0.6666667, (0 missing)
      Test < 3.5 to the left,  improve=0.3333333, (0 missing)
      Dip  < 1.5 to the right, improve=0.1666667, (0 missing)

Node number 3: 4 observations
  predicted class=1  expected loss=0  P(node) =0.4
    class counts:     0     4
   probabilities: 0.000 1.000 

Node number 4: 4 observations
  predicted class=0  expected loss=0  P(node) =0.4
    class counts:     4     0
   probabilities: 1.000 0.000 

Node number 5: 2 observations,    complexity param=0.1
  predicted class=0  expected loss=0.5  P(node) =0.2
    class counts:     1     1
   probabilities: 0.500 0.500 
  left son=10 (1 obs) right son=11 (1 obs)
  Primary splits:
      Test < 2.5 to the left,  improve=1, (0 missing)

Node number 10: 1 observations
  predicted class=0  expected loss=0  P(node) =0.1
    class counts:     1     0
   probabilities: 1.000 0.000 

Node number 11: 1 observations
  predicted class=1  expected loss=0  P(node) =0.1
    class counts:     0     1
   probabilities: 0.000 1.000 

# Rules
rpart.rules(candidates_fit)

	Res
4	0.00	when	Dip	<	3	&	Exp	<	5
10	0.00	when	Dip	<	3	&	Exp	>=	5	&	Test	<	3
11	1.00	when	Dip	<	3	&	Exp	>=	5	&	Test	>=	3
3	1.00	when	Dip	>=	3

# Prediction
predict(candidates_fit, candidates, type = "class")

 1  2  3  4  5  6  7  8  9 10 
 0  0  1  0  0  1  1  0  1  1 
Levels: 0 1

predict(candidates_fit, candidates, type = "prob")

   0 1
1  1 0
2  1 0
3  0 1
4  1 0
5  1 0
6  0 1
7  0 1
8  1 0
9  0 1
10 0 1

predict(candidates_fit, candidates, type = "matrix")

   [,1] [,2] [,3] [,4] [,5] [,6]
1     1    4    0    1    0  0.4
2     1    4    0    1    0  0.4
3     2    0    1    0    1  0.1
4     1    4    0    1    0  0.4
5     1    4    0    1    0  0.4
6     2    0    4    0    1  0.4
7     2    0    4    0    1  0.4
8     1    1    0    1    0  0.1
9     2    0    4    0    1  0.4
10    2    0    4    0    1  0.4

# confusion table
candidates_predict <- predict(candidates_fit, candidates, type = "vector")
table(candidates_predict, candidates$Res)

                  
candidates_predict 0 1
                 1 5 0
                 2 0 5

# prediction of a new observations
predict(candidates_fit, newdata = data.frame(Dip = c(1, 2, 2), Exp = c(5, 5, 3), Test = c(3, 1, 5)), type = "vector")

1 2 3 
2 1 1 

# Pruning
# print and plot the complexity parameter (cp)
printcp(candidates_fit)

Classification tree:
rpart(formula = Res ~ Dip + Test + Exp, data = candidates, method = "class", 
    parms = list(split = "gini"), control = rpart.control(minsplit = 2))

Variables actually used in tree construction:
[1] Dip  Exp  Test

Root node error: 5/10 = 0.5

n= 10 

    CP nsplit rel error xerror    xstd
1 0.80      0       1.0    2.0 0.00000
2 0.10      1       0.2    0.2 0.18974
3 0.01      3       0.0    0.6 0.28983

plotcp(candidates_fit)

candidates_fit_prune <- prune(candidates_fit, cp = 0.28)
rpart.plot(candidates_fit_prune)

# Prediction on the pruned data set
predict(candidates_fit_prune, candidates, type = "matrix")

   [,1] [,2] [,3]      [,4]      [,5] [,6]
1     1    5    1 0.8333333 0.1666667  0.6
2     1    5    1 0.8333333 0.1666667  0.6
3     1    5    1 0.8333333 0.1666667  0.6
4     1    5    1 0.8333333 0.1666667  0.6
5     1    5    1 0.8333333 0.1666667  0.6
6     2    0    4 0.0000000 1.0000000  0.4
7     2    0    4 0.0000000 1.0000000  0.4
8     1    5    1 0.8333333 0.1666667  0.6
9     2    0    4 0.0000000 1.0000000  0.4
10    2    0    4 0.0000000 1.0000000  0.4

References

Breiman, Leo, Jerome H. Friedman, Richard A. Olshen, and Charles J. Stone. 1984. Classification And Regression Trees. 1st ed. Routledge. https://doi.org/10.1201/9781315139470.