The procedure for the points on the gain chart depends on the
validation method. For a multinomial response variable, Minitab displays
multiple charts that treat each class as the event in turn.

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
*n*is the number of cases in the event class in the_{1,k}*k*^{th}node*N*is the number of cases in the_{k}*k*^{th}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. Multiply these true positive rates by 100 to find
the y-coordinates for the chart.
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/D) 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:

Table 1. Threshold = 0.60. True positive rate = 18 / 59 = 0.31 Predicted **event****nonevent****Observed****event**18 41 **nonevent**12 118 Table 2. Threshold = 0.37. True positive rate = (18 + 25) / 59 = 0.73 Predicted **event****nonevent****Observed****event**43 16 **nonevent**54 76 Table 3. Threshold = 0.21. True positive rate = (18 + 25 + 12) / 59 = 0.93 Predicted **event****nonevent****Observed****event**55 4 **nonevent**98 32 Table 4. Threshold = 0.11. True positive rate = (18 + 25 + 12 + 4) / 59 = 1 Predicted **event****nonevent****Observed****event**59 0 **nonevent**130 0 - From the sorted terminal
nodes, find the percentage of the population in the terminal nodes:
where
*N*is the number of cases in the_{k}*k*^{th}node*N*is the number of cases in the training data set

- For the sorted terminal nodes,
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 predicted probability has 0.35 of the data, then the cumulative percentage of the data for the first terminal node is 0.16 and the cumulative percentage of the data for the second terminal node is 0.16 + 0.35 = 0.51.

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 (y-coordinate) | F: Percent in data (C/ sum of C) | G: Cumulative percent in data, x-coordinate |
---|---|---|---|---|---|---|

4 | 18 | 30 | 0.60 | 0.31 | 0.16 | 0.16 |

1 | 25 | 67 | 0.37 | 0.73 | 0.35 | 0.51 |

3 | 12 | 56 | 0.21 | 0.93 | 0.30 | 0.81 |

2 | 4 | 36 | 0.11 | 1 | 0.19 | 1.00 |

Use the same steps as the training data set case, but calculate the event probability from the cases for the test data set.

The procedure to define the x- and y-coordinates on the gain 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 i^{th} 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.