The procedure for the calculation of cumulative lift depends on the
validation method. For a multinomial response variable, Minitab displays
multiple charts that treat each class as the event in turn.
Training data set or no validation
For the chart for a training data set, each point on the chart represents
a terminal node from the tree. The terminal node with the highest event
probability is the first point on the chart and appears leftmost. The other
terminal nodes are in order of decreasing event probability.
Use the following process to find the x- and y-coordinates for the
points.
Calculate the event
probability of each terminal node:
where
n1,k is the number of cases in the event class
in the
kth node
Nk is the number of cases in the
kth node
Rank the terminal nodes from
highest to lowest event probability.
Use every event probability
as a threshold. For a specific threshold, cases with estimated event
probability greater than or equal to the threshold get 1 as the predicted
class, 0 otherwise. Then, you can form a 2x2 table for all cases with observed
classes as rows and predicted classes as columns to calculate the true positive
rate for each terminal node.
For example, suppose the following table summarizes a tree with 4
terminal nodes:
A: Terminal node
B: Number of events
C: Number of cases
D: Threshold (B/C)
4
18
30
0.60
1
25
67
0.37
3
12
56
0.21
2
4
36
0.11
Totals
59
189
Then the following are the corresponding four tables with their
respective true positive rates to 2 decimal places:
From the sorted terminal
nodes, find the percentage of the population in the terminal nodes:
where
Nk is the number of cases in the
kth node
N is the number of cases in the training data set
From the sorted list,
calculate the cumulative percentage of the data in each terminal node. These
cumulative values are the x-coordinates on the chart.
For example, if the terminal node with the highest predicted probability
contains 0.16 of the data and the terminal node with the second-highest event
probability has 0.35 of the population, then the cumulative percentage of the
data for the first terminal node is 0.16 and the cumulative percentage of the
population for the second terminal node is 0.16 + 0.35 = 0.51.
To find the cumulative lift
for the y-coordinate, divide the true positive rate and the cumulative
percentage of the population:
The following table shows an example of the computations for a small tree.
The values are to 2 decimal places.
A: Terminal node
B: Number of events
C: Number of cases
D: Event probability for sorting (B/C)
E: True positive rate
F: Percent in data (C/ sum of C)
G: Cumulative percent in data, x-coordinate
H: Cumulative lift (E/G), y-coordinate
4
18
30
0.60
0.31
0.16
0.16
1.92
1
25
67
0.37
0.73
0.35
0.51
1.42
3
12
56
0.21
0.93
0.30
0.81
1.15
2
4
36
0.11
1
0.19
1.00
1
Separate test data set
Use the same steps as the training data set case but calculate the event
probability from the cases for the test data set.
Test with k-fold cross-validation
The procedure to define the x- and y-coordinates on the cumulative lift
chart with k-fold cross-validation has an additional step. This step creates
many distinct event probabilities. For example, suppose the tree diagram
contains 4 terminal nodes. We have 10-fold cross-validation. Then, for the
ith fold, you use 9/10 portion of the data to estimate the event
probabilities for cases in fold i. When this process repeats for each fold, the
maximum number of distinct event probabilities is 4 *10 = 40. After that, sort
all the distinct event probabilities in decreasing order. Use the event
probabilities as each of the threshold values to assign predicted classes for
cases in the entire data set. After this step, steps from 3 to the end for the
training data set procedure apply to find the x- and y-coordinates.
