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Tree Structure, Advanced Techniques
The basic decision tree is a graphical representation of a decision model consisting of nodes, branches, variables, and expressions. The standard node types are decision, chance, and terminal nodes. Since decision analysis first became widely used in healthcare, analysts have gradually extended the basic representation to include a number of advanced features that make models more flexible and versatile. This entry describes these advances and their use in decision modeling.
Branches and Bindings
A branch connects two nodes in the tree (Figure 1) and specifies a specific context. Each branch, or collective path of branches, between nodes in a tree represents a different clinical context, having potentially different probabilities of subsequent events, costs, and utilities. When such values are expressed in terms of symbolic expressions, any parameters may be affected by the corresponding series of events. Bindings are mathematical expressions that reassign the value of parameters in a tree context. For example, in Figure 1, the parameter pCure should be different in the contexts of “Empiric therapy” and “Observation.” This can be implemented by applying a binding of the form
on the branch “Empiric therapy.” This expression uses the assignment operator “: =” to indicate that the variable on the left of the operator is assigned to the value of the expression on the right. The assignment of the new value applies to all tree contexts downstream from the binding expression but does not affect other tree contexts at any other points of the tree. While the previously bound value of the variable may be used in the binding expression, the new value overrides the previous value in the subsequent contexts.
Local versus Global Variable Values
The binding mechanism creates local values for variables. The values apply only to tree contexts downstream from the binding. In some cases, it is useful to create global values, which apply in all parts of a tree. An example is precalculating parameters such as mortality rates and incidence of disease that depend on other variables, such as age. The calculated variables can be used in all parts of the decision model and not only downstream from where they are defined. However, the same effect can often be accomplished by setting a binding at the root of the tree.
Applying Bayes's Rule in a Decision Tree
A common application of bindings in decision models is application of Bayes's rule to calculate the posttest probability of a test from the pretest probability and the test characteristics (sensitivity and specificity). Figure 2 shows a tree with an additional decision node branch labeled “Test.”
Figure 1 Decision tree
Figure 2 Tree with diagnostic test and bindings
The Test branch leads to a chance node with two branches, “Test positive” and “Test negative.”
The probability of the “Test positive” branch (pPOS) is equal to pDIS × SENS + (1 – pDIS) × (1 – SPEC), where SENS is the test sensitivity and SPEC is the test specificity, and pDIS is the probability of disease. Note that it is common to use short mnemonics for variable names, using the first character to indicate type, such as “p” for probability variables. This formula can be applied as a binding on the “Test” branch to convey an updated, posterior probability of positive test results in contexts in which disease is present. Given the values for pDIS in Figure 4, SENS = .95 and SPEC = .99, pPOS = .19. It is higher than pDIS because of false-positive tests.
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