October 23, 2024
In terms of the Cox proportional hazards model, adding in a lasso feature for variable selection or other penalized method was typically done in the partial likelihood. Their method has sought to add the lasso penalty into the full likelihood. As the authors have stated, despite the
predominance of the partial likelihood in existing R routines, there aresome advantages when using the full likelihood which they pointed out. The first is that the baseline hazard can be modeled explicitly, for example, using a basis function approach such as B-splines (see, e.g., Eilers and Marx 1996), The second is that the full likelihood modelcan easily be extended by a wide class of frailty distributions including random intercepts and random slopes, The third is that time-varying covariates can be naturally incorporated.
The authors stated that the partial likelihood ignores thenonfailure intervals which might influence survival outcomes so information could be lost and estimates could be biased. They have developed a function in the R software, coxFL, which implements theunregularized Cox full likelihood approach, allowing for changing covariates, frailties, and time-varying coefficients but their main contribution is a function, coxlasso, which usesthe classical lasso penalization and can adapt the same items as the other function. They wrote their likelihood interms of an individual predictor for the jth individual belonging to theith cluster. The random effects part of this individual predictor, ηij,followed a Gaussian distribution. It was maximized via a penalized quasi-likelihood
approach proposed by Breslow and Clayton (1993), which applied a Laplace
approximation to the penalty term. They modeled the baseline hazard as a smooth
function using B-splines. The authors used the spline coefficients corresponding to the baseline hazard and time-varying effects. They used frailty on the grouping structurewith a log-normal distribution. Also they used a lasso penalty. The estimation was done by an iterativeNewton-Raphson algorithm.
They conducted simulations to compare the performance oftheir method to predominant survival packages. They compared to coxph, coxme, penalized,and glmnet. They could really only compare to coxph and coxme in terms of the random effects, but these do notperform variable selection and also coxmedoes not allow to extract the baseline hazard so they couldn’t compare that
part to their package, coxlassoso they could only compare that part to the glmnet and penalized. In all respects, their package performed the best overall in terms of providing the most robust estimates. Also, in a real dataset genetic lung cancer as well as a breastfeeding data application, it
also performed well.
Their work essentially proposed a flexible regularized Coxfrailty model based on the full likelihood. Using that framework allowed them to directly estimate the smoothbaseline hazard via p-splines and also include time-varying covariates and effects. The smoothing was carried out via a mixed model representation of the spline coefficient and the covariates were regularized using a lasso penalty with adaptive weights where categorical variables were penalized using a group
lasso. All of this was done in their function, coxlasso. They recommend using their function in a high dimensional setting with time-varying covariates and/or cluster structure
and/or also important variables of interest.
Written by,
Keywords: survival,Cox model, full likelihood, frailty, time-varying, lasso, penalty, splines
References
Eilers, P. H. C., and B. D. Marx. 1996. “Flexible Smoothing With B-Splines and Penalties.” Statistical Science 11, no. 2: 89–121.
HohbergM and Groll A (2024). “A Flexible Adaptive Lasso Cox Frailty Model Based on the Full Likelihood” BiometricalJournal. https://doi.org/10.1002/bimj.202300020
