The procedure for the points on the ROC curve 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 chart.
Calculate the event
probability of each terminal node:
where
n1,k is the number of events 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 false
positive rate and the true positive rate for each terminal node. The false
positive rates are the x-coordinates for the chart The true positive rates are
the y-coordinates.
For example, suppose the following table summarizes a tree with 4
terminal nodes:
A: Terminal node
B: Number of events
C: Number of nonevents
D: Number of cases
E: Threshold (B/D)
4
18
12
30
0.60
1
25
42
67
0.37
3
12
44
56
0.21
2
4
32
36
0.11
Totals
59
130
189
Then the following are the corresponding 4 tables with their
respective false positive rates and true positive rates to 2 decimal places:
Use the same steps as the training data set procedure, 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 ROC curve 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.
