Cox partial likelihood re-derived as normal equation of GLM

Bypassing risk-set construction and event conditioning arguments

Biostatistics
Generalized Linear Models
Survival Analysis
Partial Likelihood
Published

February 16, 2025

The Cox proportional hazards model is a popular semiparametric regression model in survival analysis. Its partial likelihood offers a clever way to filter out the nonparametric baseline function while keeping focus on the regression coefficients (log-hazard ratios).

The partial likelihood was originally derived through careful construction of risk sets and conditioning arguments specific to the survival context. However, with a bit of handwaving, I’ll show that it can be re-derived, mechanically, as the “normal equation” of a generalized linear model (GLM).1

Normal equations of GLMs

Consider a GLM for response \(Y\) against covariates \(Z\): \[\begin{equation}\label{eq:mean_mod} E(Y\mid Z) = \mu(\beta_0 + \beta^\mathrm{T} Z) \tag{1} \end{equation}\] through some mean function \(\mu(\cdot)\). The normal equation is defined as \[\begin{equation}\label{eq:normal} \sum_{i=1}^n \left(\begin{array}{c} 1\\ Z_i\\ \end{array}\right) \left\{Y_i - \mu(\beta_0 + \beta^\mathrm{T} Z_i)\right\} =0. \tag{2} \end{equation}\] This is also the score equation for a GLM with a canonical link function. Solving \(\eqref{eq:normal}\) gives estimates of \(\beta_0\) and \(\beta\).

  • Linear regression: \(\mu(x) = x\), with \(\eqref{eq:normal}\) leading to standard least squares.
  • Logistic regression: \(\mu(x) = \exp(x)/\{1+\exp(x)\}\), with \(\eqref{eq:normal}\) corresponding to score equation for MLE.

Cox model as a GLM

Model specification

Let \(T\) be the survival time of interest. The Cox model specifies \[\begin{equation}\label{eq:cox} \lambda(t\mid Z) = \lambda_0(t)\exp(\beta^\mathrm{T} Z), \tag{3} \end{equation}\] where \(\lambda(t\mid Z)\) is the conditional hazard of \(T\) given \(Z\). However, this seems a bit far from the mean model formulation of \(\eqref{eq:mean_mod}\).

Reformulation as a mean model

Let’s find a mean model implied by \(\eqref{eq:cox}\). With censoring time \(C\), we observe \(X = \min(T, C)\) and \(\delta = I(T\leq C)\). Write \(N(t) = I(T\leq t, \delta = 1)\).

Consider \(\mathrm{d} N(t)=N(t)-N(t-)\) as a binary response indicating an event observed at \(t\). Then \(\eqref{eq:cox}\) implies \[\begin{align*} E\{\mathrm{d} N(t)\mid X\geq t, Z\} &= I(X\geq t)\exp(\beta^\mathrm{T} Z)\lambda_0(t)\mathrm{d} t\\ &=I(X\geq t)\exp\{\beta_0(t)+\beta^\mathrm{T} Z\}. \end{align*}\] where \(\exp\{\beta_0(t)\}= \lambda_0(t)\mathrm{d} t\).

Now, we have a formulation similar to \(\eqref{eq:mean_mod}\) with \(\mu(x) = I(X\geq t)\exp(x)\), though the intercept \(\beta_0(t)\) is time-varying.

Normal equation with time-varying intercept

Therefore, \(\eqref{eq:normal}\) becomes \[\begin{equation}\label{eq:normal_cox} \sum_{i=1}^n\left(\begin{array}{c} 1\\ Z_i\\ \end{array}\right) \big[\mathrm{d} N_i(t) - I(X_i\geq t)\exp\{\beta_0(t)+\beta^\mathrm{T} Z_i\}\big] =0. \tag{4} \end{equation}\]

Use the first-line equation in \(\eqref{eq:normal_cox}\) to solve for the “intercept” \[ \exp\{\hat\beta_0(t)\} = \frac{\sum_{i=1}^n \mathrm{d} N_i(t)}{\sum_{i=1}^n I(X_i\geq t)\exp(\beta^\mathrm{T} Z_i)}. \] Plugging it back to the left-hand side of \(\eqref{eq:normal_cox}\):

\[\begin{align*} &\sum_{i=1}^n Z_i\mathrm{d} N_i(t) - \exp\{\hat\beta_0(t)\}\sum_{i=1}^n Z_iI(X_i\geq t)\exp(\beta^\mathrm{T} Z_i) \notag\\ =&\sum_{i=1}^n Z_i\mathrm{d} N_i(t) - \frac{\sum_{i=1}^n \mathrm{d} N_i(t)}{\sum_{i=1}^n I(X_i\geq t)\exp(\beta^\mathrm{T} Z_i)}\sum_{i=1}^n Z_iI(X_i\geq t)\exp(\beta^\mathrm{T} Z_i) \notag\\ =& \sum_{i=1}^n \left\{Z_i - \frac{\sum_{j=1}^n I(X_j\geq t)\exp(\beta^\mathrm{T} Z_j)Z_j}{\sum_{j=1}^n I(X_j\geq t)\exp(\beta^\mathrm{T} Z_j)}\right\}\mathrm{d} N_i(t) \end{align*}\] Integrating this over \(t\) yields precisely the partial-likelihood score function.

Conclusion

The Cox model can be reframed as a GLM for a binary event-indicator against an exponentially linked linear predictor with a time-dependent intercept. In this view, the partial likelihood score function aligns exactly with the normal equation of the GLM. This connection offers a new perspective on the Cox model and its estimation.

Footnotes

  1. Whitehead (1980) presented a similar derivation (Whitehead, J. “Fitting Cox’s Regression Model to Survival Data Using GLIM.” Journal of the Royal Statistical Society Series C: Applied Statistics 29 (3): 268. https://doi.org/10.2307/2346901.)↩︎