IT method

The IT method works in a similar way to the CART procedure but instead uses a

splitting criterion that detects treatment-covariate interaction. Suppose we split a node

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we have a continuous response Y and a treatment indicator, say Trt (control=0 and intervention=1). We can display this split as follows

𝝉_𝑳 𝝉_𝑹

Trt=0 𝑌̅₀𝐿 _𝑠

12 𝑛1 𝑌̅0𝑅 𝑠22 𝑛2

Trt=1 𝑌̅₁𝐿 _𝑠

32 𝑛3 𝑌̅1𝑅 𝑠42 𝑛4

The terms in the first quadrant above (𝑌̅₀𝐿 _𝑠

12 𝑛1) represent the sample mean, sample

variance and sample size respectively in the left child node 𝜏_𝐿 for when Trt=0. The other quadrants are interpreted in the same manner. Now that two nodes have been formed, all that is required is to determine the treatment effect heterogeneity between both nodes. This requires that we compare the treatment effect in the left node (𝑌̅₁𝐿₋

𝑌̅₀𝐿_{) with the effect in the right node (𝑌̅}

1𝑅− 𝑌̅0𝑅). In other words, we are interested in the

treatment-covariate interaction. The authors therefore proposed a splitting criterion for evaluating the interaction effect for each potential split in the tree growing process. The proposed splitting criterion is given by

𝐺(𝑠) = ( (𝑌̅₁𝐿_{− 𝑌̅} 0𝐿) − (𝑌̅1𝑅− 𝑌̅0𝑅) 𝜎̂ ∙ √_𝑛1 1+ 1 𝑛₂+𝑛1₃+𝑛1₄) 2 (6.8) where 𝜎̂2_{= ∑ 𝑤} 𝑖 4

𝑖=1 𝑠𝑖2 is a pooled estimator of the constant variance and where 𝑤𝑖 = (𝑛𝑖−1)

∑4𝑗=1(𝑛𝑗−1) (136). Thus when growing the tree, the splitting criterion G(s) is evaluated

for every single potential split and the split with the maximum G(s) is chosen as the best split. In other words, the split that gives the largest interaction effect is chosen as the best split.

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Growing a tree – parameter specifications

A fully grown tree, say 𝑇0, is grown using the aforementioned split function until either

some stopping criterion is met or the tree cannot split nodes any further. The code for the IT approach was very kindly provided by the author Dr Xiaogang Su. When

applying the IT procedure, it requires a couple of user-defined parameters to be specified to aid the tree growing process. These include the minimum node size i.e. the minimum number of individuals in any given node, and the maximum depth of a tree i.e. how many levels the tree has (the complexity of a tree).

Pruning

The IT method uses weakest link pruning, similar to CART (see equation 6.4), to determine the complexity parameter values and the associated subtrees of the fully grown tree 𝑇₀. Thus the function

𝛼(𝜏) =∑𝜏∈𝜏−𝜏̃𝐺(𝜏) |𝜏 − 𝜏̃|

is evaluated for every internal node 𝜏 where the numerator ∑𝜏∈𝜏−𝜏̃𝐺(𝜏) is the overall

amount of interaction of the internal nodes in the branch connected to the internal node 𝜏 and the denominator |𝜏 − 𝜏̃| is the number of internal nodes in the branch connected to 𝜏. The internal node with the smallest value of 𝛼(𝜏) is pruned i.e. the branch connected to the internal node is removed and the internal node itself becomes a terminal node to form the first subtree 𝑇₁. In the same way as CART, this process continues to form a sequence of subtrees until we are left with just the root node.

Selecting the best tree

The quality or performance of an interaction tree, say T, is assessed using an

interaction complexity measure; similar to the CART procedure (see equation (6.3)). The interaction- complexity measure is given by

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𝐺_𝛼(𝑇) = 𝐺(𝑇) − 𝛼 ∙ |𝑇 − 𝑇̃|

where 𝐺(𝑇) = ∑_{𝜏∈𝜏−𝜏̃}𝐺(𝜏) is the sum of the interaction in the internal nodes of the tree

T and |𝑇 − 𝑇̃| is the number of internal nodes in tree T. The above interaction-

complexity measure can be evaluated for every subtree and the one with the maximum value is chosen as the best tree. What remains is the selection of the complexity

parameter value 𝛼, i.e. the penalty for additional splits. This can be done using V-fold cross-validation as described in section 6.2.5, using the 1-SE rule to determine the best tree size and thus the complexity parameter value. The authors also recommend a bootstrapping method, used by LeBlanc et al, as an alternative option for validating the trees (136, 143).

Interpretation of the final tree

Once a final tree has been selected, the interpretation is rather straightforward. The outcome of interest, which in this case is the treatment effect, is computed for each of the terminal nodes of the final selected tree. The conclusions are then based on the comparison of the terminal nodes summaries. For example, say that a single one-way interaction effect exists in a dataset, then the final chosen interaction tree should consist of a single split of the root node i.e. two terminal nodes. Thus, the difference in the treatment effect between the two terminal nodes should be equivalent to the size of the interaction effect.

6.3.2 STIMA method