Penalized maximum likelihood inference under the mixture cure model in sparse data

Stat Med. 2023 Jun 15;42(13):2134-2161. doi: 10.1002/sim.9715. Epub 2023 Mar 25.


Introduction: When a study sample includes a large proportion of long-term survivors, mixture cure (MC) models that separately assess biomarker associations with long-term recurrence-free survival and time to disease recurrence are preferred to proportional-hazards models. However, in samples with few recurrences, standard maximum likelihood can be biased.

Objective and methods: We extend Firth-type penalized likelihood (FT-PL) developed for bias reduction in the exponential family to the Weibull-logistic MC, using the Jeffreys invariant prior. Via simulation studies based on a motivating cohort study, we compare parameter estimates of the FT-PL method to those by ML, as well as type 1 error (T1E) and power obtained using likelihood ratio statistics.

Results: In samples with relatively few events, the Firth-type penalized likelihood estimates (FT-PLEs) have mean bias closer to zero and smaller mean squared error than maximum likelihood estimates (MLEs), and can be obtained in samples where the MLEs are infinite. Under similar T1E rates, FT-PL consistently exhibits higher statistical power than ML in samples with few events. In addition, we compare FT-PL estimation with two other penalization methods (a log-F prior method and a modified Firth-type method) based on the same simulations.

Discussion: Consistent with findings for logistic and Cox regressions, FT-PL under MC regression yields finite estimates under stringent conditions, and better bias-and-variance balance than the other two penalizations. The practicality and strength of FT-PL for MC analysis is illustrated in a cohort study of breast cancer prognosis with long-term follow-up for recurrence-free survival.

Keywords: Firth-type penalized likelihood; Newton-type algorithm; likelihood ratio test; maximum likelihood; mixture cure; nested deviance method.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Cohort Studies
  • Computer Simulation
  • Humans
  • Likelihood Functions
  • Neoplasm Recurrence, Local*
  • Proportional Hazards Models

Grant support