Depending on your selection, the optimal tree is the tree that
yields the minimum misclassification cost or the smallest tree with a
misclassification cost within a multiple of standard errors of the minimum
misclassification cost. The determination of the optimal tree depends on the
validation method.

For more details on the model validation methods and complexity parameters,
see Breiman, Friedman, Olshen and Stone (1984)^{1}.

Model summary statistics, such as the deviance R^{2}, tend to be
optimistic when you calculate them with the same data that you use to fit a
model. Model validation methods leave a portion of the data out of the model
fitting process, then calculate statistics that evaluate the performance of the
model on the omitted data. Model validation techniques provide a better
estimate of how well models perform on new data. The misclassification cost
from the omitted data is the criterion for the selection of the optimal tree.
Minitab offers two validation methods for predictive analytics techniques:
k-fold cross-validation and validation with a separate test data set.

K-fold cross-validation is the default method in Minitab when the data have 5000 cases or less. With this method, Minitab portions the data into K subsets. The subsets are called folds. K-fold cross validation tends to work well with data sets that are relatively small compared to data sets that work well with a test data set. Because the process repeats K times, cross-validation is usually slower than validation with a test data set.

To complete k-fold cross-validation, Minitab produces 1 + k sequences of subtrees. One sequence of subtrees, the master sequence, uses the entire training data set. The other k sequences are for the k folds. For each fold, the sequence of subtrees uses (k – 1)/k of the cases in the training data set.

Each sequence consists of a finite sequence of nested subtrees. Each fold
has a finite sequence of complexity parameters
*α*_{d} ≤
*α* ≤
*α*_{d + 1} that correspond to the largest tree and the
subtrees in the sequence. The sequence that is for the full data set has
complexity parameters
*β*_{d
} ≤
*β* ≤
*β*_{d + 1}where
*d* = 0, 1, ... D, where
*β*_{0} is the parameter for the largest tree in the sequence.

For any subtree in the master sequence, assume the corresponding
complexity parameters are
*β*_{d
} and
*β*_{d + 1
}. Let .
Then, Minitab uses this alpha to find the k corresponding subtrees from the k
folds. For each fold, calculate the misclassification cost for the subtree
using the formula in
Methods and formulas for the model summary in CART® Classification.
The average misclassification cost across k folds is the estimated
misclassification cost for the subtree in the master sequence. Repeat the
calculation of the estimated misclassification cost for each subtree in the
master sequence. The procedure identifies the subtree with the minimum average
misclassification cost. The tree with the minimum misclassification cost or the
smallest tree with a misclassification cost within a multiple of standard
errors of the misclassification cost becomes the optimal tree in the results.

In validation with a test data set, a portion of the data is set aside for validation. This portion of the data is the training data set. First, Minitab fits all the trees for with the training data set. Then, Minitab calculates either the mean square error or the absolute deviation for the test data set for each tree. The tree with the optimal value of the criterion for the test data set is the optimal tree.

Without any validation, Minitab uses the entire data set to grow the sequence of subtrees. The subtree with the most terminal nodes has the minimum misclassification cost and is the optimal tree.