What is the difference between rel error and x error in a rpart decision tree?

Question

I have a purely categorical dataframe from the UCI machine learning database https://archive.ics.uci.edu/ml/datasets/Diabetes+130-US+hospitals+for+years+1999-2008

I am using rpart to form a decision tree based on a new category on whether patients return before 30 days (a new failed category).

I am using the following parameters for my decision tree

    tree_model <- rpart(Failed ~ race + gender + age+ time_in_hospital+ medical_specialty + num_lab_procedures+ num_procedures+num_medications+number_outpatient+number_emergency+number_inpatient+number_diagnoses+max_glu_serum+ A1Cresult+metformin+glimepiride+glipizide+glyburide+pioglitazone+rosiglitazone+insulin+change,method="class", data=training_data, control=rpart.control(minsplit=2, cp=0.0001, maxdepth=20, xval = 10), parms = list(split = "gini"))

Printing the results yields:

       CP     nsplit rel error  xerror     xstd
1 0.00065883      0   1.00000  1.0000   0.018518
2 0.00057648      8   0.99424  1.0038   0.018549
3 0.00025621     10   0.99308  1.0031   0.018543
4 0.00020000     13   0.99231  1.0031   0.018543

I see that the relative error is going down as the decision tree branches off, but the xerror goes up - which I don't understand as I would have thought that the error would reduce the more branches there are and the more complex the tree is.

I take it that the xerror is most important, since most methods for tree pruning would cut the tree at the root.

Why is the xerror what is focused on when pruning the tree? And when we summarise what the error of the decision tree classifier is, is the error 0.99231 or 1.0031?

Solution 1

The x-error is the cross-validation error (rpart has built-in cross validation). You use the 3 columns, rel_error, xerror and xstd together to help you choose where to prune the tree.

Each row represents a different height of the tree. In general, more levels in the tree mean that it has lower classification error on the training. However, you run the risk of overfitting. Often, the cross-validation error will actually grow as the tree gets more levels (at least, after the 'optimal' level).

A rule of thumb is to choose the lowest level where the rel_error + xstd < xerror.

If you run plotcp on your output it will also show you the optimal place to prune the tree.

Also, see the SO thread How to compute error rate from a decision tree?

Solution 2

I would like to add some info to @Harold Ship's answer. Actually, there are three ways to select the optimal cp value for pruning:

1. Use the first level (i.e. least nsplit) with minimum xerror. The first level only kicks in when there are multiple level having same, minimum xerror. This is the most common used method.

2. Use the first level where xerror falls into the ±1 xstd range of min(xerror), i.e., xerror < min(xerror) + xstd, the level whose xerror is at or below horizontal line. This method takes into account the variability of xerror resulting from cross-validation.

   Note: rel_error should NOT be used in pruning.

3. (A rarely used method) Use the first level where xerror ± xstd overlaps with min(xerror) ± xstd. i.e., the level whose lower limit is at or below horizontal line.

Comments

[cloudscomputes]

Do you have the explicit formula to calculate the rel_error? Or if you have the link related to it, let me know