# Methods and Formulas for the Receiver Operating Characteristic (ROC) curve chart for Random Forests® Classification

###### Note

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.

## Out-of-bag validation

For a given tree in the forest, a class vote for a row in the out-of-bag data is the predicted class for the row from the single tree. The predicted class for a row in out-of-bag data is the class with the highest vote across all trees in the forest. The predicted class probability for a row in the out-of-bag data is the ratio of the number of votes for the class and the total votes for the row.

For the curve for the out-of-bag data, each point on the chart represents a distinct predicted class probability. The highest event probability is the first point on the chart and appears leftmost. The other probabilities are in decreasing order.

Use the following process to find the x- and y-coordinates for the chart.

1. Use every distinct 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 event probability. 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 simplistic model with two, 2-level categorical predictors. These predictors give four distinct event probabilities, which are rounded to 2 decimal places:

A: Order B: Predictor 1 C: Predictor 2 D: Number of events E: Number of nonevents F: Number of trials G: Threshold (Fitted event probability)
1 1 1 18 12 30 0.60
2 1 2 25 42 67 0.37
3 2 1 12 44 56 0.21
4 2 2 4 32 36 0.11
Totals 59 130 189

The following are the corresponding four tables with their respective false positive rates and true positive rates rounded to 2 decimal places:

Table 1. Threshold = 0.60.

False positive rate = 12 / (12 + 118) = 0.09

True positive rate = 18 / (18 + 41) = 0.31

Predicted
event nonevent
Observed event 18 41
nonevent 12 118
Table 2. Threshold = 0.37.

False positive rate = (12 + 42) / 130 = 0.42

True positive rate = (18 + 25) / 59 = 0.73

Predicted
event nonevent
Observed event 43 16
nonevent 54 76
Table 3. Threshold = 0.21.

False positive rate = (12 + 42 + 44) / 130 = 0.75

True positive rate = (18 + 25 + 12) / 59 = 0.93

Predicted
event nonevent
Observed event 55 4
nonevent 98 32
Table 4. Threshold = 0.11.

False positive rate = (12 + 42 + 44 + 32) / 130 = 1

True positive rate = (18 + 25 + 12 + 4) / 59 = 1

Predicted
event nonevent
Observed event 59 0
nonevent 130 0

## Separate test set

Use the same steps as the out-of-bag procedure, but calculate the event probabilities from the cases in the test set.

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