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C-index

Introduction

The concordance index or C-index is a generalization of the area under the ROC curve (AUC) that can take into account censored data. It represents the global assessment of the model discrimination power: this is the model’s ability to correctly provide a reliable ranking of the survival times based on the individual risk scores. It can be computed with the following formula: $$\begin{equation} \text{C-index} = \frac{ \sum_{i, j} \mathbb{1}_{T_j < T_i} \cdot \mathbb{1}_{\eta_j > \eta_i} \cdot \delta_j }{\sum_{i, j} \mathbb{1}_{T_j < T_i}\cdot \delta_j } \end{equation}$$

with:

• $\eta_i$, the risk score of a unit $i$
• $\mathbb{1}_{ T_j < T_i } = 1$ if $T_j < T_i$ else $0$
• $\mathbb{1}_{ \eta_j > \eta_i } = 1$ if $\eta_j > \eta_i$ else $0$

Similarly to the AUC, $\text{C-index}= 1$ corresponds to the best model prediction, and $\text{C-index} = 0.5$ represents a random prediction.

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Location

The function can be found at pysurvival.utils.metrics.concordance_index.

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API

concordance_index - Concordance Index computations

    concordance_index(model, X, T, E, include_ties = True, additional_results=False)

Parameters:

• model : Pysurvival object -- Pysurvival model

• X : array-like -- input samples; where the rows correspond to an individual sample and the columns represent the features (shape=[n_samples, n_features]).

• T : array-like -- target values describing the time when the event of interest or censoring occurred.

• E : array-like -- values that indicate if the event of interest occurred i.e.: E[i]=1 corresponds to an event, and E[i] = 0 means censoring, for all i.

• include_ties: bool (default=True) -- Specifies whether ties in risk score are included in calculations

• additional_results: bool (default=False) -- Specifies whether only the c-index should be returned (False) or if a dict of values should returned. In that case, the values are:

  • c_index
  • nb_pairs
  • nb_concordant_pairs

Returns:

• c_index: float or dict -- Result of the function

  • if additional_results = False then c_index is float.
  • if additional_results = True then c_index is dict, such that c_index = {'c_index': ., 'nb_pairs': ., 'nb_concordant_pairs': .}

References:

• Uno, Hajime et al. “On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data” Statistics in medicine vol. 30,10 (2011): 1105-17.