Recursive partitioning

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In statistics, recursive partitioning is a method for the multivariable analysis of diagnostic tests.^[1][2][3][4]. Recursive partitioning creates a decision tree that strives to correctly classify members of the population based on a dichotomous dependent variable. Compared to other multivariable:

• Advantages are:
  • Generates clinically more intuitive models that do not require the user to perform calculations.^[5]
  • Allows "adjustable misclassification penalties" or misclassification costs in order to create a decision rule that has more sensitivity or specificity. This has also been called the "diversity index" which is the sum of the false-negatives and false-positives. Either the false-negatives and false-positives can be weighted in order to preferentially reduce their occurrence.^[6][7]

• Disadvantages are:
  • Does not work well for continuous variables^[8]
  • May overfit data.

Studies comparing the predictive accuracy of recursive partitioning with logistic regression have reported no difference^[5], recursive partitioning to be more accurate^[9] and logistic regression to be more accurate.^[10]

A variation is 'Cox linear recursive partitioning'.^[6]

Examples are available of using recursive partitioning in research of diagnostic tests.^{[10][11][12][13][14][15]} Goldman used recursive partitioning to prioritize sensitivity in the diagnosis of myocardial infarction among patients with chest pain in the emergency room.^[14]

Additional alternatives to recursive partitioning include latent class analysis.^[16][17]

Calculations[edit]

Unweighted analysis[edit]

An 'Initial misclassification rate' is calculated by assigning all cases to a single class. Using the example in the CART Manual, in trying to diagnose 59 patients who have a disease of interest among a sample of 189 patients, arbitrarily classifying all patients as normal yields:^[18]
For the top node, also called node 1:

${\displaystyle {\mbox{Initial misclassification rate}}=\left({\frac {\mbox{Total with disease}}{\mbox{Total sample size}}}\right)=\left({\frac {\mbox{59}}{\mbox{189}}}\right)={\mbox{0.312}}}$

For the next nodes: The next best independent, classifier variable creates two nodes:

• Left node: 159 patients; 118 have disease and 41 are false positives who do not have the disease
• Right node: 30 patients; 12 have disease and false negatives

${\displaystyle {\mbox{Misclassification costs}}=\left({\frac {\mbox{41}}{\mbox{159}}}\right)+\left({\frac {\mbox{12}}{\mbox{30}}}\right)=\left({\mbox{0.258}}\right)+\left({\mbox{0.400}}\right)={\mbox{0.658}}}$

${\displaystyle {\mbox{Relative misclassification costs}}=\left({\frac {\mbox{0.658}}{\mbox{0.312}}}\right)={\mbox{2.109}}}$

Weighted analysis[edit]

In this analysis, the sensitivity is preferenced by weighted it by 2.2, which is the ratio of patients without disease over patients with disease.
For the top node:

${\displaystyle {\mbox{Initial misclassification rate}}=2.2\times \left({\frac {\mbox{Total with disease}}{\mbox{Total sample size}}}\right)=2.2\times \left({\frac {59}{189}}\right)=2.2\times 0.312=0.686}$

For the next nodes: The next best independent, classifier variable creates two nodes:

• Left node: 159 patients; 118 have disease and 41 are false positives who do not have the disease
• Right node: 30 patients; 12 have disease and false negatives

${\displaystyle {\mbox{Misclassification costs}}=2.2\times \left({\frac {41}{159}}\right)+\left({\frac {12}{30}}\right)=2.2\times 0.258+0.400=0.568+0.400=0.968}$

${\displaystyle {\mbox{Relative misclassification costs}}=\left({\frac {0.968}{0.686}}\right)=1.411}$

Stopping rules[edit]

Various methods have been proposed to avoid overfitting of data.

• One method is to grow the tree as far as possible, then to 'prune' the tree. Pruning is based on a balance between misclassification costs and tree complexity.^[2][19]

Another method is to grow the tree "until p<0.05 or until one or more of the marginal counts in the two-by-two table is less than 10."^[5]

Software for recursive partitioning[edit]

Software available for recursive partitioning includes:

• CART
• R programming language:
  • rpart package
  • HSAUR2 interactive package with a chapter containing sample demonstrations, "Recursive Partitioning: Predicting Body Fat and Glaucoma Diagnosis" in "A Handbook of Statistical Analyses Using R".^[20]

References[edit]

External links[edit]

• CART website