Introduction to DynForest methodology

The tree building

The tree building process, summarized in Figure 1, aims to recursively partition the subjects into groups/nodes that are the most homogeneous regarding the outcome \(Y\).

For each tree \(b\) \((b = 1, ..., B)\), we first draw a bootstrap sample from the original dataset of \(N\) subjects (\(N\) draws among the \(N\) subjects with replacement). The subjects excluded by the bootstrap constitute the out-of-bag (OOB) sample, noted \(OOB^b\) for tree \(b\). At each node \(d \in \mathcal{D}^b\) of the tree, we recursively repeat the following steps using the \(N^{(d)}\) subjects located at node \(d\):

1. An ensemble of \(\mathcal{M}^{(d)}=\{\mathcal{M}_x^{(d)},\mathcal{M}_z^{(d)}\}\) candidate predictors are randomly selected among \(\{\mathcal{M}_x,\mathcal{M}_z\}\) (see Figure 1B). The size of \(\mathcal{M}^{(d)}\) is defined by the hyperparameter \(mtry\).

2. For each time-dependent predictor in \(\mathcal{M}_z^{(d)}\):

   1. We independently model the trajectory of the predictor using a flexible linear mixed model (Laird and Ware 1982) according to time (the specification of the model is defined by the user). It is defined as: \(Z_{ij} = X1_{ij}(t_{ij})\beta + X2_{ij}(t_{ij})b_i + \epsilon_{ij}\) where \(Z_{ij}\) is the value of predictor \(Z\) for subject \(i\) at occasion \(j\), \(X1_{ij}\) and \(X2_{ij}\) are vectors of time functions and covariates to be specified by the user and \(t_{ij}\) is the time at occasion \(j\) for subject \(i\). Beta are population effects, and \(b_i\) individual random effects which follow a multivariate normal distribution. \(\epsilon_{ij}\) are the zero-mean independent Gaussian errors of measurement.
   2. We predict the vector of random-effects \(\hat{b}_i\) using the available information of individual \(i = 1, ..., N^{(d)}\). \(\hat{b}_i\) constitute time-independent features summarizing each time-dependent predictors. We thus derive the ensemble \(\mathcal{M}_{z \star}^{(d)}\) for all the variables in \(\mathcal{M}_z^{(d)}\).

3. We define \(\mathcal{M}_\star^{(d)}=\{\mathcal{M}_x^{(d)},\mathcal{M}_{z \star}^{(d)}\}\) our new ensemble of candidate features.

4. For each candidate feature \(W \in \mathcal{M}_\star^{(d)}\):

   1. We build a series of splits \(c_W^{(d)}\) according to the feature values if continuous, or subsets of categories otherwise (see Figure 1C), leading each time to two groups.
   2. We quantify the distance between the two groups according to the nature of \(Y\):
      • If \(Y\) continuous: we compute the weighted within-group variance with the proportion of subjects in each group as weights
      • If \(Y\) categorical: we compute the weighted within-group Shannon entropy (Shannon 1948) (i.e., the amount of uncertainty) with the proportion of subjects in each group as weights
      • If \(Y\) survival competing events: we compute the log-rank statistic test (Peto and Peto 1972)
      • If \(Y\) survival competing events: we compute the Fine & Gray statistic test (Gray 1988)

5. We split the subjects into the two groups that minimize (for continuous and categorical outcome) or maximize (for survival outcome) the quantity defined previously. We denote \(\{W_0^d,c_0^d\}\) the optimal couple used to split the subjects and assign them to the left and right daughter nodes \(2d\) and \(2d + 1\), respectively (see Figure 1D and A).

6. Step 1 to 5 are iterated on the daughter nodes until stopping criteria are met.

We define two stopping criteria: nodesize the minimal number of subjects in a node required to reiterate the split and minsplit the minimal number of events required to split the node. minsplit is only defined with survival outcome. In the following, we call leaves the terminal nodes.

In each leaf \(h \in \mathcal{H}^b\) of tree \(b\), a summary \(\pi^{h^b}\) is computed using the individuals belonging to the leaf. The leaf summary is defined according to the outcome:

• the mean, for \(Y\) continuous
• the category with the highest probability, for \(Y\) categorical
• the cumulative incidence function over time computed using the Nelson-Aalen cumulative hazard function estimator (Nelson 1969; O. Aalen 1976), for \(Y\) single cause time-to-event
• the cumulative incidence function over time computed using the non-parametric Aalen-Johansen estimator (O. O. Aalen and Johansen 1978), for \(Y\) time-to-event with multiple causes

NOTE: Step 2 involves the estimation of a parametric mixed model for each time-dependent predictor. The specification of this model should be carefully determined in preliminary analyses by the user, keeping in mind that there should be a trade-off between goodness-of-fit and parameter parsimony. The critical point is to adequately specify the trajectory shape over time at the population and individual levels. Different bases of time functions can be considered (e.g., fractional polynomials, splines, adhoc) and compared in terms of fit.

Individual prediction of the outcome

Out-Of-Bag prediction error

Using the OOB individual predictions, an OOB prediction error can be internally assessed. The OOB prediction error quantifies the difference between the observed and the predicted values. It is defined according to the nature of \(Y\) as:

• for \(Y\) continuous, the mean square error (MSE) defined by:

\[\begin{equation} errOOB = \frac{1}{N} \sum_{i=1}^N ( \hat{\pi}_i - \pi_i^0 )^2 \end{equation}\]

• for \(Y\) categorical, the missclassification error defined by:

\[\begin{equation} errOOB = \frac{1}{N} \sum_{i=1}^N {1}_{( \hat{\pi}_i \neq \pi_i^0 )} \end{equation}\]

• for \(Y\) survival, the Integrated Brier Score (IBS) (Sène et al. 2016) between \(\tau_1\) and \(\tau_2\) defined by:

\[\begin{equation} errOOB = \int_{\tau_1}^{\tau_2} \frac{1}{N} \sum_{i=1}^{N} \hat{\omega}_i(t) \Big\{ I(T_i \leq t, \delta_i = k) - \hat{\pi}_{ik}(t) \Big) \Big\}^2 dt \end{equation}\]

with \(T\) the time-to-event, \(k\) the cause of interest and \(\hat{\omega}(t)\) the estimated weights using Inverse Probability of Censoring Weights (IPCW) technique that accounts for censoring (Gerds and Schumacher 2006).

The OOB error of prediction is used to particular to tune the random forest by determining the hyperparameters (i.e., mtry, nodesize and minsplit) which give the smallest OOB prediction error.

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