Missing Data and Surrogate Splitters

Ideally, every row in a dataset would have values for every variable. Unfortunately, in the real world, missing values are encountered often: People being surveyed refuse or forget to answer questions, some questions may not apply to all people, some medical tests may not be performed on all patients, etc.

Some simple programs discard rows that have any missing values. But this is a waste of valuable information that may be available on other variables.

DTREG uses a sophisticated technique involving surrogate splitters to estimate the values of predictor variables with missing values.

Surrogate splitters are predictor variables that are not as good at splitting a group as the primary splitter but which yield similar splitting results; they mimic the splits produced by the primary splitter.

DTREG compares which rows are sent to the left and right child groups by the primary splitter with the rows sent to the corresponding child groups by each other predictor variable. The association between the primary splitter and each alternate predictor is computed as a function of how closely the alternate predictor matches the primary splitter. (This roughly corresponds to a count of how many rows each predictor sends left and right, but the actual calculation is more complex.) The alternate predictor variables are then ranked in decreasing order of association. The largest possible association value is 1.0 which means the surrogate sends exactly the same set of rows to the left and right groups as the primary splitter. An association value of 0.0 means that the surrogate does no better at assigning rows than simply putting them in the most probable group.

Surrogate splitters are similar to competitor splitters in the sense that they both yield splits of benefit but are not as good as the primary splitter. Often, the same variable will be listed as both a competitor and a surrogate. However, there is a significant difference between the way variables are ranked as competitors and as surrogates. Competitor splits are runners-up to the primary split: they are judged the same way the primary splitter is judged by how much improvement they make in reducing node impurity. Surrogate splitters are not ranked by the amount of improvement they produce but rather by how closely they mimic the split selected for the primary splitter. The optimal split point for a surrogate maximizes the association between the surrogate and the primary splitter; it does not necessarily maximize the improvement. If you compare entries for the same variable in the competitor and surrogate lists, you may see different split points selected and different values for the improvement from the splits.

Surrogate splitters are used to classify rows that have missing values in the primary splitter. They function both when the tree is being built and later when the tree is used to score additional datasets.

When a row is encountered that has a missing value on the primary splitter, DTREG searches the list of surrogate splitters and uses the one with the highest association to the primary splitter that has a non-missing value for the row.

Surrogate splitters provide the most accurate classification of rows with missing values. This is the default and recommended method.

In addition to their function in classifying rows with missing predictor values, the association between the primary splitter and surrogate splitters is used in the calculation of the overall importance of variables.

To understand why this is done, consider two variables that are very similar and highly correlated, for example height and weight. At some split point, weight may be selected as the primary splitter because it is slightly better than height. If this preference for weight prevails at many split points, weight would appear to be extremely important and height as unimportant. However, if you removed weight as a predictor variable and reran the analysis, an identical tree very well might be built using height as the splitting variable wherever weight was used before. Hence, height is nearly as important as weight. When one variable hides the importance of another variable, it is known as masking. By considering not only which variables are used as primary splitters but also the association of the surrogates, DTREG is able to provide a more accurate evaluation of variable importance.