Cox partial likelihood re-derived as normal equation of GLM

Bypassing risk-set construction and event conditioning arguments

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 $Z$: $$\begin{array}{}\text{(1)}& E(Y\mid Z)=\mu ({\beta }_0+{\beta }^{\mathrm{T}}Z)\end{array}$$ through some mean function $\mu (\cdot )$. The normal equation is defined as $$\begin{array}{}\text{(2)}& \sum _{i=1}^n\left(\begin{array}{c}1\\ Z_i\end{array}\right)\{Y_i-\mu ({\beta }_0+{\beta }^{\mathrm{T}}{Z}_i)\}=0.\end{array}$$ This is also the score equation for a GLM with a canonical link function. Solving $\text{(2)}$ gives estimates of ${\beta }_0$ and $\beta$.

• Linear regression: $\mu (x)=x$, with $\text{(2)}$ leading to standard least squares.
• Logistic regression: $\mu (x)=\exp (x)/\{1+\exp (x)\}$, with $\text{(2)}$ 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 $$\begin{array}{}\text{(3)}& \lambda (t\mid Z)={\lambda }_0(t)\exp ({\beta }^{\mathrm{T}}Z),\end{array}$$ 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 $\text{(1)}$.

Reformulation as a mean model

Let’s find a mean model implied by $\text{(3)}$. With censoring time $C$, we observe $X=min(T,C)$ and $\delta =I(T\le C)$. Write $N(t)=I(T\le t,\delta =1)$.

Consider $\mathrm{d}N(t)=N(t)-N(t-)$ as a binary response indicating an event observed at $t$. Then $\text{(3)}$ implies $$\begin{array}{rl}E\{\mathrm{d}N(t)\mid X\ge t,Z\}& =I(X\ge t)\exp ({\beta }^{\mathrm{T}}Z){\lambda }_0(t)\mathrm{d}t\\ & =I(X\ge t)\exp \{{\beta }_0(t)+{\beta }^{\mathrm{T}}Z\}.\end{array}$$ where $\exp \{{\beta }_0(t)\}={\lambda }_0(t)\mathrm{d}t$.

Now, we have a formulation similar to $\text{(1)}$ with $\mu (x)=I(X\ge t)\exp (x)$, though the intercept ${\beta }_0(t)$ is time-varying.

Normal equation with time-varying intercept

Therefore, $\text{(2)}$ becomes $$\begin{array}{}\text{(4)}& \sum _{i=1}^n\left(\begin{array}{c}1\\ Z_i\end{array}\right)[\mathrm{d}{N}_i(t)-I(X_i\ge t)\exp \{{\beta }_0(t)+{\beta }^{\mathrm{T}}{Z}_i\}]=0.\end{array}$$

Use the first-line equation in $\text{(4)}$ to solve for the “intercept” $$\exp \{{\hat{\beta }}_0(t)\}=\frac{\sum _{i=1}^n\mathrm{d}{N}_i(t)}{\sum _{i=1}^{n}I(X_i\ge t)\exp ({\beta }^{\mathrm{T}}{Z}_i)}.$$ Plugging it back to the left-hand side of $\text{(4)}$:

$$\begin{array}{rl}& \sum _{i=1}^n{Z}_i\mathrm{d}{N}_i(t)-\exp \{{\hat{\beta }}_0(t)\}\sum _{i=1}^n{Z}_{i}I(X_i\ge t)\exp ({\beta }^{\mathrm{T}}{Z}_i)\\ =& \sum _{i=1}^n{Z}_i\mathrm{d}{N}_i(t)-\frac{\sum _{i=1}^n\mathrm{d}{N}_i(t)}{\sum _{i=1}^{n}I(X_i\ge t)\exp ({\beta }^{\mathrm{T}}{Z}_i)}\sum _{i=1}^n{Z}_{i}I(X_i\ge t)\exp ({\beta }^{\mathrm{T}}{Z}_i)\\ =& \sum _{i=1}^n\{Z_i-\frac{\sum _{j=1}^{n}I(X_j\ge t)\exp ({\beta }^{\mathrm{T}}{Z}_j)Z_j}{\sum _{j=1}^{n}I(X_j\ge t)\exp ({\beta }^{\mathrm{T}}{Z}_j)}\}\mathrm{d}{N}_i(t)\end{array}$$ 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.)