Suppose we wish to estimate an unknown function, e.g. a survival function. We can take several approaches:
- Non-parametric using [[kaplan-meier]]
- Advantage: no assumption regarding type of distribution.
- Disadvantage: requires i.i.d. observations
- Parametric
- Advantage: (1) We only need to estimate a few parameters that completely characterize distribution (e.g. Exponential or Weibull); (2) gives low variance of estimates; (3) can be extended to non-iid observations using regression on covariates.
- Disadvantages: assumed model class may be (or always is) incorrect leading to model error or in other words, bias.
- Semi-Parametric
- Cox’s Proportional Hazards Model
Cox proposed the following model for the hazard function given covariates \(z\in \mathbb{R}^p\) \[ h(t;z) = h_{0}(t)\exp(z^T \beta ),\quad \beta \in \mathbb{R}^p. \] Here \(h_{0}\) completely unspecified function except that it must be non-negative.
Thus model combines great flexibility via non-parametric
References - [[lecture-notes-waagepetersen-2022-pdf]] - [[stat331-stanford-lecture-notes-pdf]]
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