# Cox PH Regression: likelihood based on all subjects

The likelihood function for Cox PH regression when there are $$k$$ failures is

$$$$\mathcal{L} = \prod_{j=1}^k \frac{\exp ( \mathbf{x}_j^{\mathsf{T}} \boldsymbol{\beta}) } {\sum_{i\in R(t_{(j)})} \exp ( \mathbf{x}_i^{\mathsf{T}} \boldsymbol{\beta}) }.\\$$$$

However, the likelihood can be written to represent all $$n$$ subjects as

$$$$\mathcal{L} = \prod_{j=1}^n \left\{ \frac{w_j}{W_j} \right\}^{d_j},$$$$

where $$d_j$$ equals 1 for a failure and zero otherwise, and

$$$$W_j = \sum_{i\in R(t_{(j)})} \exp ( \mathbf{x}_i^{\mathsf{T}} \boldsymbol{\beta})$$$$

Taking the log of the likelihood above gives

$$$$\ell = \sum_j d_j \log(w_j) -\sum_j d_j \log(W_j).$$$$

Next, let $$Y_i(t_j)$$ be an at-risk indicator, equal to 1 if subject $$i$$ is at risk at time $$t_j$$, and 0 otherwise. If we let the relative probability of failure for subject $$i$$ be $$w_i$$, then the absolute probability of failure for subject $$i$$ at time $$t_j$$ is

$$$$\pi_{ij} =Y_i(t_j) \frac{w_i}{W_j}.$$$$

Looking at the log-likelihood, $$\ell$$, since we know that $$w_j=\exp ( \mathbf{x}_i^{\mathsf{T}} \boldsymbol{\beta})$$, taking the log of $$w_j$$ simply gives us $$\mathbf{x}_i^{\mathsf{T}} \boldsymbol{\beta}=\eta_j$$.

So we can now write the log-likelihood as

$$$$\ell = \sum_j d_j \eta_j - \sum_j d_j \log(W_j)$$$$

The question is: If we know there are many $$\eta$$ in $$W_j$$, how can the first partial derivative of the log-likelihood w.r.t. $$\eta_i$$ be

$$$$\frac{\partial \ell}{\partial \eta_i} = d_j - \sum_j \pi_{ij}d_j$$$$

It can't.

Your $$\ell$$ isn't the log of your $${\cal L}$$; there's a sum missing. If we call it $$\ell_j$$ instead, because it's got an unbound $$j$$ index, we can look at its derivatives

We have $$\frac{\partial \log(W_j)}{\partial\eta_i}=\frac{1}{W_j}\frac{\partial W_j}{\partial\eta_i}$$ Now $$W_j$$ is a sum of terms $$\exp\eta_k$$ for $$k$$ in the relevant risk set. The derivative of one of these terms with respect to $$\eta_i$$ is zero unless $$i=k$$, when it's $$\exp\eta_i=w_i$$. That is $$\frac{\partial \log(W_j)}{\partial\eta_i}=\frac{1}{W_j}\frac{\partial W_j}{\partial\eta_i}=\frac{w_i}{W_j}\mathbb{1}_{\textrm{something}}$$ where $$\mathbb{1}_{\textrm{something}}$$ is the indicator that $$\exp\eta_i$$ actually does appear in $$W_j$$. That turns out to be precisely $$Y_i(t_j)$$; the event that $$i$$ is alive when $$j$$ dies or is censored, so we get $$\frac{\partial \ell_j}{\partial\eta_i}=d_j-\pi_{ij}d_j$$ with $$i$$ and $$j$$ unbound on the right because they are on the left

For the full loglikelihood $$\ell=\sum_j \ell_j$$ we have $$\frac{\partial \ell}{\partial\eta_i}=\sum_j\frac{\partial \ell_j}{\partial\eta_i}=\sum_j(d_j-\pi_{ij}d_j)$$

That's written as a sum over observations.

The score contribution with respect to $$\eta_i$$ at time $$t_j$$, though, is not $$d_j-\pi_{ij}d_j$$ but $$d_j-\sum_{t_k>t_j} \pi_{ik}d_k$$ The sum is still there, because in this decomposition we keep together the score contributions of all observations at time $$t_j$$, rather than the score contributions of observation $$j$$ at all times. There are like $$n^/2$$ terms $$d_{k}\pi_{ik}$$ in the score vector, and we're just choosing how to split them into sums.

• Thanks, I forgot to list the sums in $\ell$, which is now fixed.
– user318288
Feb 4, 2022 at 6:08
• I like this approach to the likelihood for Cox PH regression because it is amenable to IRLS (non-GLM). IRLS models for non-linear, logistic, Poisson regression, etc., all have a score vector element values of $u_j=\sum_i x_{ij}e_i$, where $e_i$ is a residual, or at least a delta of some sort. So, in IRLS here, $e_i=d_j - \sum \pi_{ik}d_k$, and a single additional step using the chain rule will yield the $x$.
– user318288
Feb 4, 2022 at 16:20