# Visualizing a marker’s degrees of necessity and of sufficiency in the predictiveness curve

## TL;DR

This paper introduces a new way to visualize how much a factor is necessary or sufficient for predicting an event using a predictiveness curve.

## Contribution

The paper shows that degrees of necessity and sufficiency can be interpreted as areas in the predictiveness curve plot.

## Key findings

- Degrees of necessity and sufficiency are connected to areas in the predictiveness curve.
- Explained variation is closely related to these areas in the curve.
- The extended predictiveness curve offers a comprehensive evaluation of markers for prediction.

## Abstract

The degrees to which a factor is necessary or sufficient for an event have been proposed as generalizations of attributable risk based on simple functions of unconditional and conditional event probabilities. Predictiveness curves show the risk for an event, as derived by a model with one or more predictors, depending on risk percentiles that represent the predictors’ distribution in the underlying population.

Connections between the degrees of necessity and of sufficiency and explained variation on the one hand and the predictiveness curve on the other hand are mathematically proved and exemplified using data of in-hospital death of Covid- 19 patients.

We show that the degrees of necessity and of sufficiency can be represented as proportions of areas easily identifiable in the plot of the predictiveness curve. In addition, we show that the proportion of explained variation, a common measure of predictiveness and relative importance of prognostic factors, is also closely connected to these areas.

Our investigations demonstrate that the predictiveness curve extended by these new interpretations of areas provides a comprehensive evaluation of markers or sets of markers for prediction.

Austrian Coronavirus Adaptive Clinical Trial (ACOVACT); ClinicalTrials.gov, identifier NCT04351724.