### Abstract

We show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model. A smoothed plot of the martingale residuals was suggested for this purpose by Therneau, Grambsch, and Fleming (1990, Biometrika 77, 147160); however, its consistency required that the covariates be independent. They also noted that the plot could be biased for large covariate effects. We introduce two refinements which overcome these difficulties. The first is based on a ratio of scatter plot smooths, where the numerator is the smooth of the observed count plotted against the covariate, and the denominator is a smooth of the expected count. This is related to the Arias goodness-of-fit plot (1988, Journal of the American Statistical Association 83, 204-212). The second technique smooths the martingale residuals divided by the expected count, using expected count as a weight. This latter approach is related to a GLM partial residual plot, as well as to the iterative methods of Hastic and Tibshirani (1990, Biometrics 46, 1105-1016) and Gentleman and Crowley (1991, Biometrics 47, 1283-1296). Applications to survival data sets are given.

Original language | English (US) |
---|---|

Pages (from-to) | 1469-1482 |

Number of pages | 14 |

Journal | Biometrics |

Volume | 51 |

Issue number | 4 |

DOIs | |

State | Published - 1995 |

Externally published | Yes |

### Fingerprint

### Keywords

- Augmented partial residuals
- Cox model
- Partial residuals
- Proportional hazards model

### ASJC Scopus subject areas

- Agricultural and Biological Sciences(all)
- Public Health, Environmental and Occupational Health
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics
- Statistics and Probability

### Cite this

*Biometrics*,

*51*(4), 1469-1482. https://doi.org/10.2307/2533277

**Diagnostic plots to reveal functional form for covariates in multiplicative intensity models.** / Grambsch, P. M.; Therneau, Terry M; Fleming, T. R.

Research output: Contribution to journal › Article

*Biometrics*, vol. 51, no. 4, pp. 1469-1482. https://doi.org/10.2307/2533277

}

TY - JOUR

T1 - Diagnostic plots to reveal functional form for covariates in multiplicative intensity models

AU - Grambsch, P. M.

AU - Therneau, Terry M

AU - Fleming, T. R.

PY - 1995

Y1 - 1995

N2 - We show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model. A smoothed plot of the martingale residuals was suggested for this purpose by Therneau, Grambsch, and Fleming (1990, Biometrika 77, 147160); however, its consistency required that the covariates be independent. They also noted that the plot could be biased for large covariate effects. We introduce two refinements which overcome these difficulties. The first is based on a ratio of scatter plot smooths, where the numerator is the smooth of the observed count plotted against the covariate, and the denominator is a smooth of the expected count. This is related to the Arias goodness-of-fit plot (1988, Journal of the American Statistical Association 83, 204-212). The second technique smooths the martingale residuals divided by the expected count, using expected count as a weight. This latter approach is related to a GLM partial residual plot, as well as to the iterative methods of Hastic and Tibshirani (1990, Biometrics 46, 1105-1016) and Gentleman and Crowley (1991, Biometrics 47, 1283-1296). Applications to survival data sets are given.

AB - We show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model. A smoothed plot of the martingale residuals was suggested for this purpose by Therneau, Grambsch, and Fleming (1990, Biometrika 77, 147160); however, its consistency required that the covariates be independent. They also noted that the plot could be biased for large covariate effects. We introduce two refinements which overcome these difficulties. The first is based on a ratio of scatter plot smooths, where the numerator is the smooth of the observed count plotted against the covariate, and the denominator is a smooth of the expected count. This is related to the Arias goodness-of-fit plot (1988, Journal of the American Statistical Association 83, 204-212). The second technique smooths the martingale residuals divided by the expected count, using expected count as a weight. This latter approach is related to a GLM partial residual plot, as well as to the iterative methods of Hastic and Tibshirani (1990, Biometrics 46, 1105-1016) and Gentleman and Crowley (1991, Biometrics 47, 1283-1296). Applications to survival data sets are given.

KW - Augmented partial residuals

KW - Cox model

KW - Partial residuals

KW - Proportional hazards model

UR - http://www.scopus.com/inward/record.url?scp=0029621092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029621092&partnerID=8YFLogxK

U2 - 10.2307/2533277

DO - 10.2307/2533277

M3 - Article

C2 - 8589234

AN - SCOPUS:0029621092

VL - 51

SP - 1469

EP - 1482

JO - Biometrics

JF - Biometrics

SN - 0006-341X

IS - 4

ER -