gms | German Medical Science

Introduction: The mean can be used to describe the central location of a distribution. If the distribution is skewed (e.g. for event-time distributions), the median is commonly estimated instead. Another alternative is the mode, which is preferable if the distribution is heavily skewed or has multiple local modes. Just as least squares regression estimates the influence of covariates on the conditional mean, mode regression attempts to analyze changes in the mode. We show the benefits of mode regression by analyzing observational data from pancreatic cancer patients treated at a regional high volume cancer center.

Methods: We focus on mode regression based on kernel density estimates. There are two basic approaches: Local mode regression is done fully nonparametrically by first estimating the multivariate density and then finding local maxima with iterative algorithms such as the expectation-maximization or meanshift algorithm [1]. Global mode regression interprets a univariate kernel density estimate as a loss function which allows the inclusion of a linear predictor [2]. Local mode regression can detect multiple modes and functional forms of effects do not need to be specified. In contrast, global mode regression has computational advantages, avoids the curse of dimensionality and covariate effects can be summarized in simple point estimates. We show how global mode regression can be extended with semiparametric predictors to have similar flexibility as local mode regression.

Mode regression is then applied to the survival times of pancreatic cancer patients. The logarithm of the event times were regressed on multiple covariates, including surgical therapy (yes/no), UICC stage and age. To account for right-censoring we extended mode regression by using the kernel smoothed Kaplan-Meier estimator. Observations were weighted with the inverse probability of censoring. As a comparison, estimates for the effects on the mean and the median were calculated. We used the Buckley-James estimator [3] and the censored quantile regression estimator of Peng and Huang, which is based on the Nelson-Aalen estimator [4].

Results: The data consisted of 309 patients with a median follow-up of 8.2 months and 10.4% censoring. Surgical therapy was associated with an increase of 0.65 log days (95% CI: 0.18 to 1.12) for the mean survival time, 0.61 (CI: 0.05 to 1.19) for the median and 0.52 (CI: 0.02 to 1.03) for the mode. Higher age and more advanced cancer stages were negatively associated with overall survival for all three location measures.

Discussion: The mode of the survival time had a weaker association with surgical therapy than the median and the mean. Mode regression adds a different perspective that may be more appropriate than other location measures. The choice depends on what we consider a representative patient. Our future research on mode regression will include the optimal choice of the bandwidth and kernel for censored data.

The authors declare that they have no competing interests.

The authors declare that an ethics committee vote is not required.