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Fitting survival data with penalized Poisson regression

机译：用惩罚性Poisson回归拟合生存数据

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Cox's proportional hazards model is the most common way to analyze survival data. The model can be extended in the presence of collinearity to include a ridge penalty, or in cases where a very large number of coefficients (e.g. with microarray data) has to be estimated. To maximize the penalized likelihood, optimal weights of the ridge penalty have to be obtained. However, there is no definite rule for choosing the penalty weight. One approach suggests maximization of the weights by maximizing the leave-one-out cross validated partial likelihood, however this is time consuming and computationally expensive, especially in large datasets. We suggest modelling survival data through a Poisson model. Using this approach, the log-likelihood of a Poisson model is maximized by standard iterative weighted least squares. We will illustrate this simple approach, which includes smoothing of the hazard function and move on to include a ridge term in the likelihood. We will then maximize the likelihood by considering tools from generalized mixed linear models. We will show that the optimal value of the penalty is found simply by computing the hat matrix of the system of linear equations and dividing its trace by a product of the estimated coefficients.
机译：考克斯的比例风险模型是分析生存数据的最常用方法。可以在存在共线性的情况下扩展该模型以包括山脊罚值，或者在必须估计非常大量的系数（例如具有微阵列数据）的情况下。为了最大化惩罚的可能性，必须获得最佳的岭惩罚权重。但是，没有选择罚款权重的明确规则。一种方法建议通过最大化留一法交叉验证的部分可能性来最大化权重，但是，这非常耗时且计算量大，尤其是在大型数据集中。我们建议通过泊松模型对生存数据进行建模。使用这种方法，可以通过标准的迭代加权最小二乘最大化Poisson模型的对数似然性。我们将说明这种简单的方法，其中包括对危险函数进行平滑处理，然后继续在可能性中包括一个岭项。然后，通过考虑广义混合线性模型中的工具，我们将使可能性最大化。我们将表明，仅通过计算线性方程组的hat矩阵并将其迹线除以估计系数的乘积即可找到惩罚的最佳值。

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