### Abstract

Discrimination slope, defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status, has recently gained popularity as a measure of model performance. It is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared. Several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis-calibrated models, whereas others have raised concerns about the ability of improving discrimination slope without adding new information. In this paper, we show that under certain assumptions the discrimination slope is asymptotically related to two other R-squared measures, one of which is a rescaled version of the Brier score, known to be proper. Furthermore, we illustrate how a simple recalibration makes the slope equal to the rescaled Brier R-squared metric. We also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the Gini-Brier regret function. Using theoretical and practical examples, we illustrate how all of these metrics are affected by different levels of model mis-calibration. In particular, we demonstrate that simple recalibration ascertaining calibration in-the-large and calibration slope equal to 1 are not sufficient to correct for some forms of mis-calibration. We conclude that R-squared metrics, including the discrimination slope, offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration.

Language | English (US) |
---|---|

Pages | 4482-4490 |

Number of pages | 9 |

Journal | Statistics in Medicine |

Volume | 36 |

Issue number | 28 |

DOIs | |

State | Published - Dec 10 2017 |

### Fingerprint

### Keywords

- IDI
- model
- proper
- R-squared
- risk

### ASJC Scopus subject areas

- Epidemiology
- Statistics and Probability

### Cite this

*Statistics in Medicine*,

*36*(28), 4482-4490. DOI: 10.1002/sim.7139

**Discrimination slope and integrated discrimination improvement – properties, relationships and impact of calibration.** / Pencina, Michael J.; Fine, Jason P.; D'Agostino, Ralph B.

Research output: Research - peer-review › Article

*Statistics in Medicine*, vol 36, no. 28, pp. 4482-4490. DOI: 10.1002/sim.7139

}

TY - JOUR

T1 - Discrimination slope and integrated discrimination improvement – properties, relationships and impact of calibration

AU - Pencina,Michael J.

AU - Fine,Jason P.

AU - D'Agostino,Ralph B.

PY - 2017/12/10

Y1 - 2017/12/10

N2 - Discrimination slope, defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status, has recently gained popularity as a measure of model performance. It is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared. Several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis-calibrated models, whereas others have raised concerns about the ability of improving discrimination slope without adding new information. In this paper, we show that under certain assumptions the discrimination slope is asymptotically related to two other R-squared measures, one of which is a rescaled version of the Brier score, known to be proper. Furthermore, we illustrate how a simple recalibration makes the slope equal to the rescaled Brier R-squared metric. We also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the Gini-Brier regret function. Using theoretical and practical examples, we illustrate how all of these metrics are affected by different levels of model mis-calibration. In particular, we demonstrate that simple recalibration ascertaining calibration in-the-large and calibration slope equal to 1 are not sufficient to correct for some forms of mis-calibration. We conclude that R-squared metrics, including the discrimination slope, offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration.

AB - Discrimination slope, defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status, has recently gained popularity as a measure of model performance. It is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared. Several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis-calibrated models, whereas others have raised concerns about the ability of improving discrimination slope without adding new information. In this paper, we show that under certain assumptions the discrimination slope is asymptotically related to two other R-squared measures, one of which is a rescaled version of the Brier score, known to be proper. Furthermore, we illustrate how a simple recalibration makes the slope equal to the rescaled Brier R-squared metric. We also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the Gini-Brier regret function. Using theoretical and practical examples, we illustrate how all of these metrics are affected by different levels of model mis-calibration. In particular, we demonstrate that simple recalibration ascertaining calibration in-the-large and calibration slope equal to 1 are not sufficient to correct for some forms of mis-calibration. We conclude that R-squared metrics, including the discrimination slope, offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration.

KW - IDI

KW - model

KW - proper

KW - R-squared

KW - risk

UR - http://www.scopus.com/inward/record.url?scp=84994890073&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994890073&partnerID=8YFLogxK

U2 - 10.1002/sim.7139

DO - 10.1002/sim.7139

M3 - Article

VL - 36

SP - 4482

EP - 4490

JO - Statistics in Medicine

T2 - Statistics in Medicine

JF - Statistics in Medicine

SN - 0277-6715

IS - 28

ER -