Understanding proportional hazards models
Let’s make two generalizations. First, we let the hazard depend on some covariates \(x_1,x_2, \dots, x_p\); we will indicate this dependence by extending our notation for hazard:
Definition 1 (conditional hazard) The conditional hazard of outcome \(T\) at value \(t\), given covariate vector \(\tilde{x}\), is the conditional density of the event \(T=t\), given \(T \ge t\) and \(\tilde{X}= \tilde{x}\):
\[{\lambda}(t|\tilde{x}) \stackrel{\text{def}}{=}\operatorname{p}(T=t|T\ge t, \tilde{X}= \tilde{x}) \tag{1}\]
Definition 2 (baseline hazard)
The baseline hazard, base hazard, or reference hazard, denoted \({\lambda}_0(t)\) or \(\lambda_0(t)\), is the hazard function for the subpopulation of individuals whose covariates are all equal to their reference levels:
\[{\lambda}_0(t) \stackrel{\text{def}}{=}{\lambda}(t | \tilde{X}= \tilde{0}) \tag{2}\]
The baseline hazard is somewhat analogous to the intercept term in linear regression, but it is not a mean.
Definition 3 (baseline cumulative hazard)
The baseline cumulative hazard, base cumulative hazard, or reference cumulative hazard, denoted \(H_0(t)\) or \(\Lambda_0(t)\), is the cumulative hazard function for the subpopulation of individuals whose covariates are all equal to their reference levels:
\[{\Lambda}_0(t) \stackrel{\text{def}}{=}{\Lambda}(t | \tilde{X}= \tilde{0}) \tag{3}\]
Definition 4 (Baseline survival function) The baseline survival function is the survival function for an individual whose covariates are all equal to their default values.
\[\operatorname{S}_0(t) \stackrel{\text{def}}{=}\operatorname{S}(t | \tilde{X}= \tilde{0})\]
Now, let’s define how the hazard function depends on covariates. We typically use a log link to model the relationship between the hazard function, \({\lambda}(t|\tilde{x})\), and the linear component, \(\eta(t|\tilde{x})\), as we did for Poisson models in models for count outcomes; that is:
Definition 5 (log-hazard)
The log-hazard function, denoted \(\eta(t)\), is the natural logarithm of the hazard function:
\[\eta(t) \stackrel{\text{def}}{=}\operatorname{log}\mathopen{}\left\{{\lambda}(t)\right\}\mathclose{}\]
Definition 6 (conditional log-hazard)
The conditional log-hazard function, denoted \(\eta(t|\tilde{x})\), is the natural logarithm of the conditional hazard function:
\[\eta(t | \tilde{x}) \stackrel{\text{def}}{=}\operatorname{log}\mathopen{}\left\{{\lambda}(t | \tilde{x})\right\}\mathclose{}\]
In contrast with Poisson regression, here \(\eta(t|\tilde{x})\) depends on both \(t\) and \(\tilde{x}\).
Definition 7 (baseline log-hazard)
The baseline log-hazard, denoted \(\eta_0(t)\), log-hazard function for the subpopulation of individuals whose covariates are all equal to their reference levels:
\[\eta_0(t) \stackrel{\text{def}}{=}\eta(t | \tilde{X}= \tilde{0})\]
Theorem 1 (Hazard from log-hazard) \[
\begin{aligned}
{\lambda}(t|\tilde{x}) &= \operatorname{exp}\mathopen{}\left\{\eta(t|\tilde{x})\right\}\mathclose{}
\end{aligned}
\]
Definition 8 (difference in log-hazards) The difference in log-hazards between covariate patterns \(\tilde{x}\) and \({\tilde{x}^*}\) at time \(t\) is:
\[\Delta\eta(t|\tilde{x}: {\tilde{x}^*}) \stackrel{\text{def}}{=}\eta(t|\tilde{x}) - \eta(t|{\tilde{x}^*})\]
Theorem 2 (Difference of log-hazards vs hazard ratio) If \(\Delta\eta(t|\tilde{x}: {\tilde{x}^*})\) is the difference in log-hazard between covariate patterns \(\tilde{x}\) and \({\tilde{x}^*}\) at time \(t\), and \(\theta_{{\lambda}}(t| \tilde{x}: {\tilde{x}^*})\) is corresponding hazard ratio, then:
\[\Delta\eta(t|\tilde{x}: {\tilde{x}^*})= \operatorname{log}\mathopen{}\left\{\theta_{{\lambda}}(t| \tilde{x}: {\tilde{x}^*})\right\}\mathclose{}\]
Proof. Using the hazard ratio definition:
\[
\begin{aligned}
\Delta\eta(t|\tilde{x}: {\tilde{x}^*})
&\stackrel{\text{def}}{=}\eta(t|\tilde{x}) - \eta(t|{\tilde{x}^*})
\\
&= \operatorname{log}\mathopen{}\left\{{\lambda}(t|\tilde{x})\right\}\mathclose{} - \operatorname{log}\mathopen{}\left\{{\lambda}(t|{\tilde{x}^*})\right\}\mathclose{}
\\
&= \operatorname{log}\mathopen{}\left\{\frac{{\lambda}(t|\tilde{x})}{{\lambda}(t|{\tilde{x}^*})}\right\}\mathclose{}
\\
&= \operatorname{log}\mathopen{}\left\{\theta_{{\lambda}}(t| \tilde{x}: {\tilde{x}^*})\right\}\mathclose{}
\end{aligned}
\]
Corollary 1 (Hazard ratio vs difference of log-hazards) \[\theta_{{\lambda}}(t| \tilde{x}: {\tilde{x}^*}) = \operatorname{exp}\mathopen{}\left\{\Delta\eta(t|\tilde{x}: {\tilde{x}^*})\right\}\mathclose{}\]
Definition 9 (difference in log-hazard from baseline)
The difference in log-hazard for covariate pattern \(\tilde{x}\) compared to the baseline covariate pattern \(\tilde{0}\) is:
\[
\begin{aligned}
\Delta\eta(t|\tilde{x})
&\stackrel{\text{def}}{=}\Delta\eta(t | \tilde{x}: \tilde{0})
\end{aligned}
\]
Theorem 3 (Decomposition of log-hazard) \[
\begin{aligned}
\eta(t|\tilde{x}) = \eta_0(t) + \Delta\eta(t|\tilde{x})
\end{aligned}
\]
Definition 10 (Hazard ratio versus baseline) \[\theta_{{\lambda}}(t|\tilde{x}) \stackrel{\text{def}}{=}\theta_{{\lambda}}(t| \tilde{x}: \tilde{0}) \tag{4}\]
Corollary 2 (Hazard factor from difference of log-hazard from baseline) \[\theta_{{\lambda}}(t|\tilde{x})= \operatorname{exp}\mathopen{}\left\{\Delta\eta(t|\tilde{x})\right\}\mathclose{}\]
Proof. \[
\begin{aligned}
\theta_{{\lambda}}(t|\tilde{x})
&\stackrel{\text{def}}{=}\theta_{{\lambda}}(t| \tilde{x}: \tilde{0})
\\
&= \operatorname{exp}\mathopen{}\left\{\Delta\eta(t|\tilde{x})\right\}\mathclose{}
\end{aligned}
\]
Corollary 3 (Difference of log-hazard from baseline equals log of the hazard factor) \[\Delta\eta(t|\tilde{x}) = \operatorname{log}\mathopen{}\left\{\theta_{{\lambda}}(t| \tilde{x})\right\}\mathclose{}\]
As the second generalization, we let the base hazard, cumulative hazard, and survival functions depend on \(t\), but not on any covariates (for now). We can do this using either parametric or semi-parametric approaches.
Definition 11 (Cox proportional hazards model) The Cox proportional hazards model (Cox 1972) for a time-to-event outcome \(T\) is a model where the difference in log-hazard from the baseline log-hazard is equal to a linear combination of the predictors:
\[\Delta\eta(t|\tilde{x}) = \tilde{x}\cdot \tilde{\beta} \tag{5}\]
Equivalently:
Lemma 1 (Log-hazard as baseline plus a linear combination) In a proportional hazards model (that is, if Equation 5 holds):
\[
\begin{aligned}
\eta(t|\tilde{x}) &= \eta_0(t) + \tilde{x}\cdot \tilde{\beta}
\\
&= \eta_0(t) + \beta_1 x_1+ \dots + \beta_p x_p
\end{aligned}
\tag{6}\]
In a proportional hazards model, the baseline log-hazard is analogous to the intercept term in a generalized linear model, except that the baseline log-hazard depends on time, \(t\).
Lemma 2 (Difference of log-hazards between two covariate patterns) If \(\eta(t|\tilde{x}) = \eta_0(t) + \tilde{x}\cdot \tilde{\beta}\), then:
\[
\begin{aligned}
\Delta\eta(t|\tilde{x}: {\tilde{x}^*})
&= (\tilde{x}- {\tilde{x}^*}) \cdot \beta
\end{aligned}
\]
Theorem 4 (Hazard ratio under proportional hazards) If \(\eta(t|\tilde{x}) = \eta_0(t) + \tilde{x}\cdot \tilde{\beta}\), then:
\[
\begin{aligned}
\theta_{{\lambda}}(t|\tilde{x}: {\tilde{x}^*})
&= \operatorname{exp}\mathopen{}\left\{\Delta\eta(t|\tilde{x}: {\tilde{x}^*})\right\}\mathclose{}
\\
&= \operatorname{exp}\mathopen{}\left\{(\tilde{x}- {\tilde{x}^*}) \cdot \beta\right\}\mathclose{}
\\
&= \operatorname{exp}\mathopen{}\left\{\Delta\tilde{x}\cdot \beta\right\}\mathclose{}
\end{aligned}
\]
where \(\Delta\tilde{x}\stackrel{\text{def}}{=}\tilde{x}- {\tilde{x}^*}\) is the difference in covariate vectors.
Proof. \[
\begin{aligned}
\theta_{{\lambda}}(t|\tilde{x}: {\tilde{x}^*})
&\stackrel{\text{def}}{=}\frac{{\lambda}(t|\tilde{x})}{{\lambda}(t|{\tilde{x}^*})}
\\
&= \operatorname{exp}\mathopen{}\left\{\eta(t|\tilde{x}) - \eta(t|{\tilde{x}^*})\right\}\mathclose{}
\\
&= \operatorname{exp}\mathopen{}\left\{\Delta\eta(t|\tilde{x}: {\tilde{x}^*})\right\}\mathclose{}
\\
&= \operatorname{exp}\mathopen{}\left\{(\tilde{x}- {\tilde{x}^*}) \cdot \beta\right\}\mathclose{}
\\
&= \operatorname{exp}\mathopen{}\left\{\Delta\tilde{x}\cdot \beta\right\}\mathclose{}
\end{aligned}
\]
Expanding the definition of \(\theta_{{\lambda}}\), the first equality writes the ratio as the exponential of a difference of logarithms; the second applies the definition of \(\Delta\eta\) as that difference; the third is Lemma 2; and the last substitutes \(\Delta\tilde{x}\) for \(\tilde{x}- {\tilde{x}^*}\).
So for proportional hazards models, we can write the hazard ratio using a shorthand notation:
\[\theta_{{\lambda}}(t| \tilde{x}: {\tilde{x}^*}) = \theta_{{\lambda}}(\tilde{x}: {\tilde{x}^*})\]
Lemma 3 (Difference of log-hazard from baseline) \[\Delta\eta(t|\tilde{x})= \tilde{x}\cdot \tilde{\beta} \tag{7}\]
Theorem 5 (Hazard ratio versus baseline under proportional hazards) If \(\eta(t|\tilde{x}) = \eta_0(t) + \tilde{x}\cdot \tilde{\beta}\), then:
\[\theta_{{\lambda}}(t|\tilde{x}) = \operatorname{exp}\mathopen{}\left\{\tilde{x}\cdot \tilde{\beta}\right\}\mathclose{}\]
Proof. \[
\begin{aligned}
\theta_{{\lambda}}(t|\tilde{x})
&\stackrel{\text{def}}{=}\theta_{{\lambda}}(t| \tilde{x}: \tilde{0})
\\
&= \operatorname{exp}\mathopen{}\left\{\Delta\eta(t|\tilde{x})\right\}\mathclose{}
\\
&= \operatorname{exp}\mathopen{}\left\{\tilde{x}\cdot \tilde{\beta}\right\}\mathclose{}
\end{aligned}
\]
Theorem 6 (Proportional-hazards decomposition of the hazard) \[{\lambda}(t|\tilde{x}) = {\lambda}_0(t)\theta_{{\lambda}}(\tilde{x})\]
Definition 12 (Risk Score) In a Cox proportional hazards model (see Definition 11) with coefficient vector \(\tilde{\beta}\), the risk score (also called the hazard multiplier or partial hazard) for a subject with covariate vector \(\tilde{x}\) is:
\[\theta_{{\lambda}}\mathopen{}\left(\tilde{x}\right)\mathclose{} = \operatorname{exp}\mathopen{}\left\{\tilde{x}\cdot \tilde{\beta}\right\}\mathclose{}\]
The risk score \(\theta_{{\lambda}}\mathopen{}\left(\tilde{x}\right)\mathclose{}\) is a theoretical quantity depending on the true (unknown) parameter \(\tilde{\beta}\).
Also:
Theorem 7 (Equivalent forms of the proportional-hazards model) \[
\begin{aligned}
\theta_{{\lambda}}(\tilde{x}) &= \operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde{x})\right\}\mathclose{}
\\
\operatorname{log}\mathopen{}\left\{{\lambda}(t|\tilde{x})\right\}\mathclose{}
&= \operatorname{log}\mathopen{}\left\{{\lambda}_0(t)\right\}\mathclose{} + \Delta\eta(\tilde{x})
\\
&= \eta_0(t) + \Delta\eta(\tilde{x})
\\
\Delta\eta(\tilde{x}) &= \tilde{x}\cdot \tilde{\beta}\\
&\stackrel{\text{def}}{=}\beta_1x_1+\cdots+\beta_px_p
\end{aligned}
\]
This model is semi-parametric, because the linear predictor depends on estimated parameters but the base hazard function is unspecified. There is no constant term in \(\eta(x)\), because it is absorbed in the base hazard.
Alternatively, we could define \(\beta_0(t) = \operatorname{log}\mathopen{}\left\{{\lambda}_0(t)\right\}\mathclose{}\), and then:
\[\eta(x,t) = \beta_0(t) + \beta_1x_1+\cdots+\beta_px_p\]
For two different individuals with covariate patterns \(\tilde{x}_1\) and \(\tilde{x}_2\), the ratio of the hazard functions (a.k.a. hazard ratio, a.k.a. relative hazard) is:
\[
\begin{aligned}
\frac{{\lambda}(t|\tilde{x}_1)}{{\lambda}(t|\tilde{x}_2)}
&=\frac{{\lambda}_0(t)\theta_{{\lambda}}(\tilde{x}_1)}{{\lambda}_0(t)\theta_{{\lambda}}(\tilde{x}_2)}\\
&=\frac{\theta_{{\lambda}}(\tilde{x}_1)}{\theta_{{\lambda}}(\tilde{x}_2)}\\
\end{aligned}
\]
Under the proportional hazards model, this ratio (a.k.a. proportion) does not depend on \(t\). This property is a structural limitation of the model; it is called the proportional hazards assumption.
Definition 13 (proportional hazards) A conditional probability distribution \(p(T|X)\) has proportional hazards if the hazard ratio \({\lambda}(t|\tilde{x}_1)/{\lambda}(t|\tilde{x}_2)\) does not depend on \(t\). Mathematically, it can be written as:
\[
\frac{{\lambda}(t|\tilde{x}_1)}{{\lambda}(t|\tilde{x}_2)}
= \theta_{{\lambda}}(\tilde{x}_1,\tilde{x}_2)
\]
As we saw above, Cox’s proportional hazards model has this property, with \(\theta_{{\lambda}}(\tilde{x}_1,\tilde{x}_2) = \frac{\theta_{{\lambda}}(\tilde{x}_1)}{\theta_{{\lambda}}(\tilde{x}_2)}\).
Theorem 8 (Relating the hazard-ratio and hazard-factor notations)
We are using two similar notations, \(\theta_{{\lambda}}(\tilde{x},{\tilde{x}^*})\) and \(\theta_{{\lambda}}(\tilde{x})\). We can link these notations: \[\theta_{{\lambda}}(\tilde{x}) \stackrel{\text{def}}{=}\theta_{{\lambda}}(\tilde{x}, \tilde{0})\]
Then:
\[\theta_{{\lambda}}(\tilde{x}, {\tilde{x}^*}) = \frac{\theta_{{\lambda}}(\tilde{x})}{\theta_{{\lambda}}({\tilde{x}^*})}\] \[\theta_{{\lambda}}(\tilde{0}) = \theta_{{\lambda}}(\tilde{0}, \tilde{0}) = 1\]
The proportional hazards model also has additional notable properties:
\[
\begin{aligned}
\frac{{\lambda}(t|\tilde{x}_1)}{{\lambda}(t|\tilde{x}_2)}
&=\frac{\theta_{{\lambda}}(\tilde{x}_1)}{\theta_{{\lambda}}(\tilde{x}_2)}\\
&=\frac{\operatorname{exp}\mathopen{}\left\{\eta(\tilde{x}_1)\right\}\mathclose{}}{\operatorname{exp}\mathopen{}\left\{\eta(\tilde{x}_2)\right\}\mathclose{}}\\
&=\operatorname{exp}\mathopen{}\left\{\eta(\tilde{x}_1)-\eta(\tilde{x}_2)\right\}\mathclose{}\\
&=\operatorname{exp}\mathopen{}\left\{\tilde{x}_1'\tilde{\beta}-\tilde{x}_2'\tilde{\beta}\right\}\mathclose{}\\
&=\operatorname{exp}\mathopen{}\left\{(\tilde{x}_1 - \tilde{x}_2)'\tilde{\beta}\right\}\mathclose{}\\
\end{aligned}
\]
Hence on the log scale, we have:
Theorem 9 (Difference of log-hazards is a linear combination) \[
\begin{aligned}
\operatorname{log}\mathopen{}\left\{\frac{{\lambda}(t|\tilde{x})}{{\lambda}(t|{\tilde{x}^*})}\right\}\mathclose{}
&= \Delta\eta(t|\tilde{x}: {\tilde{x}^*})
\\
&\stackrel{\text{def}}{=}\eta(t|\tilde{x}) - \eta(t|{\tilde{x}^*})
\\
&=\eta(\tilde{x})-\eta({\tilde{x}^*})\\
&= \tilde{x}'\tilde{\beta}-\mathopen{}\left({\tilde{x}^*}\right)\mathclose{}'\tilde{\beta}\\
&= (\tilde{x}- {\tilde{x}^*})'\tilde{\beta}
\end{aligned}
\]
Applying the general procedure for interpreting a regression coefficient to the log-hazard linear predictor \(\eta(\tilde{x}) = \Delta\eta(\tilde{x}) = \tilde{x}\cdot \tilde{\beta}\) (differentiating or differencing with respect to \(x_j\), depending on whether \(x_j\) is continuous or discrete, with all other covariates held fixed) yields the following interpretation of \(\beta_j\):
If only one covariate \(x_j\) is changing, then:
\[
\begin{aligned}
\operatorname{log}\mathopen{}\left\{\frac{{\lambda}(t|\tilde{x}_1)}{{\lambda}(t|\tilde{x}_2)}\right\}\mathclose{}
&= (x_{1j} - x_{2j}) \cdot \beta_j\\
&\propto (x_{1j} - x_{2j})
\end{aligned}
\]
That is, under Cox’s model \({\lambda}(t|\tilde{x}) = {\lambda}_0(t)\operatorname{exp}\mathopen{}\left\{\tilde{x}'\tilde{\beta}\right\}\mathclose{}\), the log of the hazard ratio is proportional to the difference in \(x_j\), with the proportionality coefficient equal to \(\beta_j\).
Further,
\[
\begin{aligned}
\operatorname{log}\mathopen{}\left\{{\lambda}(t|\tilde{x})\right\}\mathclose{}
&=\operatorname{log}\mathopen{}\left\{{\lambda}_0(t)\right\}\mathclose{} + \tilde{x}\cdot \tilde{\beta}
\end{aligned}
\]
Additional properties of the proportional hazards model
If \({\lambda}(t|\tilde{x})= {\lambda}_0(t)\theta_{{\lambda}}(\tilde{x})\), then:
Theorem 10 (Cumulative hazards are also proportional to \({\Lambda}_0(t)\)) \[
\begin{aligned}
{\Lambda}(t|\tilde{x})
&\stackrel{\text{def}}{=}\int_{u=0}^t {\lambda}(u)du\\
&= \int_{u=0}^t {\lambda}_0(u)\theta_{{\lambda}}(\tilde{x})du\\
&= \theta_{{\lambda}}(\tilde{x})\int_{u=0}^t {\lambda}_0(u)du\\
&= \theta_{{\lambda}}(\tilde{x}){\Lambda}_0(t)
\end{aligned}
\]
where \({\Lambda}_0(t) \stackrel{\text{def}}{=}{\Lambda}(t|\tilde{0}) = \int_{u=0}^t {\lambda}_0(u)du\).
Theorem 11 (The logarithms of cumulative hazard should be parallel) \[
\operatorname{log}\mathopen{}\left\{{\Lambda}(t|\tilde{x})\right\}\mathclose{} =\operatorname{log}\mathopen{}\left\{{\Lambda}_0(t)\right\}\mathclose{} + \tilde{x}\cdot \tilde{\beta}
\]
Corollary 4 (linear model for log-negative-log survival) \[
\operatorname{log}\mathopen{}\left\{-\operatorname{log}\mathopen{}\left\{\operatorname{S}(t|\tilde{x})\right\}\mathclose{}\right\}\mathclose{} =
\operatorname{log}\mathopen{}\left\{-\operatorname{log}\mathopen{}\left\{\operatorname{S}_0(t)\right\}\mathclose{}\right\}\mathclose{} + \tilde{x}\cdot \tilde{\beta}
\]
Theorem 12 (Survival functions are exponential multiples of \(\operatorname{S}_0(t)\)) \[\operatorname{S}(t|\tilde{x}) = \mathopen{}\left[\operatorname{S}_0(t)\right]\mathclose{}^{\theta_{{\lambda}}(\tilde{x})}\]
Proof. \[
\begin{aligned}
\operatorname{S}(t|\tilde{x})
&= \operatorname{exp}\mathopen{}\left\{-{\Lambda}(t|\tilde{x})\right\}\mathclose{}\\
&= \operatorname{exp}\mathopen{}\left\{-\theta_{{\lambda}}(\tilde{x})\cdot {\Lambda}_0(t)\right\}\mathclose{}\\
&= \mathopen{}\left(\operatorname{exp}\mathopen{}\left\{- {\Lambda}_0(t)\right\}\mathclose{}\right)\mathclose{}^{\theta_{{\lambda}}(\tilde{x})}\\
&= \mathopen{}\left[\operatorname{S}_0(t)\right]\mathclose{}^{\theta_{{\lambda}}(\tilde{x})}\\
\end{aligned}
\]
Summary of proportional hazards model structure and assumptions
Joint likelihood of data set: \(\mathcal{L}\stackrel{\text{def}}{=}\operatorname{p}(\tilde{Y}= \tilde{y}, \tilde{D}= \tilde{d}| \mathbf{X}= \mathbf{x})\)
Marginal likelihood contribution of obs. i : \(\mathcal{L}_i \stackrel{\text{def}}{=}\operatorname{p}(Y_i= y_i, D_i = d_i | \tilde{X}_i = \tilde{x}_i)\)
Independent Observations Assumption: \(\mathcal{L}= \prod_{i=1}^n \mathcal{L}_i\)
Non-Informative Censoring Assumption: \(T_i\perp\!\!\!\perp C_i | \tilde{X}_i\)
\[
\mathcal{L}_i \propto [\operatorname{f}_T(y_i|\tilde{x}_i)]^{d_i} [\operatorname{S}_T(y_i | \tilde{x}_i)]^{1-d_i}
= \operatorname{S}_T(y_i|\tilde{x}_i) \cdot [{\lambda}_T(y_i|\tilde{x}_i)]^{d_i}
\]
Survival function: \(\operatorname{S}(t | \tilde{x}) \stackrel{\text{def}}{=}\operatorname{P}(T > t|\tilde{X}= \tilde{x}) = \int_{u=t}^{\infty} \operatorname{f}(u|\tilde{x})du = \operatorname{exp}\mathopen{}\left\{-{\Lambda}(t|\tilde{x})\right\}\mathclose{}\)
Probability density function: \(\operatorname{f}(t| \tilde{x}) \stackrel{\text{def}}{=}\operatorname{p}(T=t|\tilde{X}= \tilde{x}) = -\operatorname{S}'(t|\tilde{x}) = {\lambda}(t| \tilde{x}) \operatorname{S}(t | \tilde{x})\)
Cumulative hazard function: \({\Lambda}(t | \tilde{x}) \stackrel{\text{def}}{=}\int_{u=0}^t {\lambda}(u|\tilde{x})du = -\operatorname{log}\mathopen{}\left\{\operatorname{S}(t|\tilde{x})\right\}\mathclose{}\)
Hazard function: \({\lambda}(t |\tilde{x}) \stackrel{\text{def}}{=}\operatorname{p}(T=t|T\ge t,\tilde{X}= \tilde{x}) = {\Lambda}'(t|\tilde{x}) = \frac{\operatorname{f}(t|\tilde{x})}{\operatorname{S}(t|\tilde{x})}\)
Hazard ratio: \(\theta_{{\lambda}}(t| \tilde{x}: {\tilde{x}^*}) \stackrel{\text{def}}{=}\frac{{\lambda}(t|\tilde{x})}{{\lambda}(t|{\tilde{x}^*})}\)
Log-Hazard function: \(\eta(t|\tilde{x}) \stackrel{\text{def}}{=}\operatorname{log}\mathopen{}\left\{\lambda(t|\tilde{x})\right\}\mathclose{} = \eta_0(t) + \Delta\eta(t|\tilde{x})\)
Proportional Hazards Assumption:
\[
\begin{aligned}
{\lambda}(t |\tilde{x}) &= {\color{blue}{{\lambda}_0(t)}} \cdot {\color{red}{\theta_{{\lambda}}(\tilde{x})}}
\\
{\Lambda}(t |\tilde{x}) &= {\color{blue}{{\Lambda}_0(t)}} \cdot {\color{red}{\theta_{{\lambda}}(\tilde{x})}}
\\
\eta(t|\tilde{x}) &= {\color{blue}{\eta_0(t)}} + {\color{red}{\Delta\eta(\tilde{x})}}
\end{aligned}
\]
Here and throughout this chapter, we color the \({\color{blue}{\text{baseline hazard}}}\) — the part shared by every covariate pattern — and the \({\color{red}{\text{covariate hazard ratio}}}\) — the part specific to \(\tilde{x}\) — so the eye can follow each piece through the algebra.
Logarithmic Link Function Assumption:
Link function: \[\operatorname{log}\mathopen{}\left\{{\lambda}(t|\tilde{x})\right\}\mathclose{} = \eta(t|\tilde{x})\] \[\operatorname{log}\mathopen{}\left\{\theta_{{\lambda}}(\tilde{x})\right\}\mathclose{} = \Delta\eta(\tilde{x})\]
Inverse link function: \[{\lambda}(t|\tilde{x}) = \operatorname{exp}\mathopen{}\left\{\eta(t|\tilde{x})\right\}\mathclose{}\] \[\theta_{{\lambda}}(\tilde{x}) = \operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde{x})\right\}\mathclose{}\]
Linear Predictor Component:
\[\eta(t|\tilde{x}) = \eta_0(t) + \Delta\eta(t|\tilde{x})\] \[\Delta\eta(t|\tilde{x}) = \tilde{x}\cdot \tilde{\beta}\]
Linear Predictor Component Functional Form Assumption:
\[\Delta\eta(t|\tilde{x}) = \tilde{x}\cdot \tilde{\beta}\stackrel{\text{def}}{=}\beta_1 x_1 + \cdots + \beta_p x_p\]
Fitting proportional hazards models to data
Fitting a proportional hazards model means estimating two pieces: the coefficients \(\tilde{\beta}= (\beta_1, \ldots, \beta_p)\) that govern the covariate effects, and the baseline hazard \({\lambda}_0(t)\). We estimate them in two stages:
- estimate \(\tilde{\beta}\) by maximizing a partial likelihood in which the unknown baseline \({\lambda}_0(t)\) cancels out, so the covariate effects can be fit without committing to a shape for the baseline;
- then, with \(\hat{\tilde{\beta}}\) in hand, estimate the baseline cumulative hazard \({\Lambda}_0(t)\) by the Breslow estimator.
Throughout, assume for simplicity that there are no tied event times; let \(t_1 < t_2 < \cdots < t_K\) be the distinct event times for the whole data set, let \(R(t_i)\) be the risk set at time \(t_i\), and let \((i)\) denote the subject who fails at \(t_i\) (unique under the no-ties assumption). Recall the model:
\[
\begin{aligned}
\Delta\eta(\tilde{x}) &\stackrel{\text{def}}{=}\beta_1x_{1}+\cdots+\beta_p x_{p}
\\
\theta_{{\lambda}}(\tilde{x}) &= \operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde{x})\right\}\mathclose{}
\\
{\lambda}(t|\tilde{x}) &= {\color{blue}{{\lambda}_0(t)}}\,{\color{red}{\operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde{x})\right\}\mathclose{}}}
\\
&= {\color{red}{\theta_{{\lambda}}(\tilde{x})}}\,{\color{blue}{{\lambda}_0(t)}}
\end{aligned}
\]
Stage 1: estimating the coefficients \(\tilde{\beta}\). The key idea is to condition on when the failures happen and ask only which at-risk subject failed at each event time. Over a short interval of width \(\Delta t\), the probability that an at-risk subject \(j\) fails at \(t_i\) is approximately its hazard times the interval width, \({\lambda}(t_i \mid \tilde{x}_j)\,\Delta t = {\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\). So, conditional on exactly one failure at \(t_i\) among the risk set \(R(t_i)\), the probability that the subject who failed is subject \((i)\) is that subject’s share of the total — and the shared baseline factor \({\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\) cancels:
\[
\begin{aligned}
\Pr(\text{subject } (i) \text{ fails} \mid \text{1 failure at } t_i,\ R(t_i))
&= \frac{{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{\sum_{k \in R(t_i)}{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_k)}}}
\\
&= \frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{\sum_{k \in R(t_i)} {\color{red}{\theta_{{\lambda}}(\tilde{x}_k)}}}
\end{aligned}
\]
Because the baseline factor cancels, this conditional probability depends only on \(\tilde{\beta}\). Multiplying these probabilities across the \(K\) event times gives the partial likelihood:
Definition 14 (Cox PH partial likelihood) \[
\begin{aligned}
\mathcal{L}^*_i(\tilde{\beta}) &\stackrel{\text{def}}{=}\frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{\sum_{k \in R(t_i)} {\color{red}{\theta_{{\lambda}}(\tilde{x}_k)}}}
\\
\mathcal{L}^*(\tilde{\beta}) &\stackrel{\text{def}}{=}
\prod_{\mathopen{}\left\{i:\ d_i = 1\right\}\mathclose{}} \mathcal{L}^*_i(\tilde{\beta})
\end{aligned}
\]
where, for the distinct event times \(t_1 < \cdots < t_K\) of the data set:
-
\((i)\) is the subject who fails at \(t_i\) (unique under the no-ties assumption);
-
\(R(t_i) \stackrel{\text{def}}{=}\mathopen{}\left\{j : Y_j \ge t_i\right\}\mathclose{}\) is the risk set at \(t_i\) (where \(Y_j = \min(T_j, C_j)\) is subject \(j\)’s observed follow-up time) — the subjects still under observation just before \(t_i\);
-
\(d_i\) is the number of events at \(t_i\) (so \(d_i = 1\) when there are no ties);
-
\(\theta_{{\lambda}}(\tilde{x}) \stackrel{\text{def}}{=}\operatorname{exp}\mathopen{}\left\{{\tilde{x}}^{\top}\tilde{\beta}\right\}\mathclose{}\) is the hazard ratio (risk score) for covariate pattern \(\tilde{x}\).
To prove that this partial likelihood is free of the baseline hazard, we first assemble a chain of general discretized-likelihood building blocks — used only by this proof, so we develop them here rather than in the Intro to Survival Analysis chapter. One piece carries over from that chapter: the interval failure probability \(q_j(\tau)\) (introduced ahead of that chapter’s Kaplan-Meier section, since the Kaplan-Meier derivation also relies on it). From it we build its first-order approximation, the censored-data likelihood, the exact discretized survival factor, the per-interval likelihood factor, and the likelihood regrouped by interval:
Definition 15 (First-order interval failure probability) To first order in \(\Delta t\), the interval failure probability \(q_j(\tau)\) is the hazard times the interval width. We give that approximation its own symbol, the first-order interval failure probability:
\[
q^*_j(\tau) \stackrel{\text{def}}{=}{\lambda}(\tau)\,\Delta t,
\]
so that
\[
q_j(\tau) \approx q^*_j(\tau).
\]
Example 1 (First-order approximation versus the exact value) Continue the setup of the interval failure probability example: subject \(j\) has constant hazard \({\lambda}(t) = 0.1\) per year, on the half-year interval \([4.0,\, 4.5)\) with \(\Delta t = 0.5\) years. The first-order approximation replaces the exact value with the hazard times the interval width:
\[
\begin{aligned}
q^*_j(4.0)
&= {\lambda}(4.0)\,\Delta t
&& \text{(definition of } q^*_j \text{)}
\\
&= 0.1 \times 0.5
&& \text{(substitute } {\lambda}= 0.1,\ \Delta t = 0.5\text{)}
\\
&= 0.05 .
&& \text{(arithmetic)}
\end{aligned}
\]
The first-order approximation \(q^*_j(4.0) = 0.05\) overstates the exact value \(q_j(4.0) \approx 0.0488\) by about \(0.0012\), a relative error of roughly \(2.5\%\). The discrepancy is exactly the leading term dropped by the linearization \(1 - \operatorname{exp}\mathopen{}\left\{-x\right\}\mathclose{} = x - \frac{x^2}{2} + \cdots\): here \(x = {\lambda}(4.0)\,\Delta t = 0.05\), so the dropped term is \(\frac{x^2}{2} = \frac{0.05^2}{2} \approx 0.00125\), which shrinks to zero as \(\Delta t \to 0\). This vanishing discrepancy is why proofs based on this discretization become exact in the continuous-time limit.
Lemma 4 (Exact discretization of the survival factor) Partition \([0, \max_j Y_j]\) into intervals of width \(\Delta t\) with left endpoints \(\tau\), and assume each observed time \(Y_j\) falls on an interval boundary (for an interior \(Y_j\) the identity instead holds in the \(\Delta t \to 0\) limit). Then each subject’s survival factor is the exact product of its per-interval conditional survival probabilities:
\[
\operatorname{S}(Y_j \mid \tilde{x}_j) = \prod_{\tau < Y_j} \mathopen{}\left(1 - q_j(\tau)\right)\mathclose{},
\tag{8}\]
where \(q_j(\tau)\) is the interval failure probability.
Proof. Surviving to \(Y_j\) is the intersection of the events “survive interval \(\tau\)”, over all grid intervals \(\tau < Y_j\), which tile \([0, Y_j)\) exactly because \(Y_j\) is a grid boundary. By the chain rule for survival over a partition, the probability of this intersection factors into the product, over those intervals, of the conditional probability of surviving each interval given survival to its left endpoint. That conditional survival probability is, by definition, \(1 - q_j(\tau)\). The factorization is exact: each factor is the exact conditional survival probability \(1 - q_j(\tau)\), not its first-order hazard approximation, so no error is incurred at this step.
Example 2 (Discretizing a constant hazard) Suppose subject \(j\) has constant hazard \({\lambda}(t \mid \tilde{x}_j) = 0.1\)/yr and is observed to \(Y_j = 1.5\) yr, discretized with \(\Delta t = 0.5\) yr (three intervals, \(\tau = 0,\, 0.5,\, 1.0\)). Each exact per-interval survival factor is
\[
\begin{aligned}
1 - q_j(\tau)
&= \operatorname{exp}\mathopen{}\left\{-\textstyle\int_{\tau}^{\tau + 0.5} 0.1 \, du\right\}\mathclose{}
&& \text{(exact conditional survival over the interval)}
\\
&= \operatorname{exp}\mathopen{}\left\{-0.05\right\}\mathclose{}
&& \text{(evaluate the integral: } 0.1 \times 0.5\text{)}
\\
&\approx 0.9512 ,
&& \text{(arithmetic)}
\end{aligned}
\]
so Equation 8 gives
\[
\begin{aligned}
\operatorname{S}(1.5 \mid \tilde{x}_j)
&= \prod_{\tau \in \mathopen{}\left\{0,\, 0.5,\, 1.0\right\}\mathclose{}} \mathopen{}\left(1 - q_j(\tau)\right)\mathclose{}
&& \text{(three intervals)}
\\
&\approx 0.9512^3
&& \text{(substitute the common per-interval factor)}
\\
&\approx 0.8607 ,
&& \text{(arithmetic)}
\end{aligned}
\]
which matches the closed form \(\operatorname{S}(1.5 \mid \tilde{x}_j) = \operatorname{exp}\mathopen{}\left\{-0.1 \times 1.5\right\}\mathclose{} = \operatorname{exp}\mathopen{}\left\{-0.15\right\}\mathclose{} \approx 0.8607\) exactly, confirming that the discretization introduces no error.
Lemma 5 (Censored-data likelihood) Under noninformative censoring (conditional on \(\tilde{x}\), censoring times are independent of event times), the likelihood of the observed data \(\mathopen{}\left\{(Y_j, D_j, \tilde{x}_j)\right\}\mathclose{}_{j=1}^n\) is
\[
\mathcal{L}
= \prod_{j=1}^{n}
\operatorname{S}(Y_j \mid \tilde{x}_j)\,
{\lambda}(Y_j \mid \tilde{x}_j)^{D_j},
\tag{9}\]
where \(Y_j = \min(T_j, C_j)\) is the observed follow-up time and \(D_j = \text{1}_{T_j \le C_j}\) the event indicator.
Proof. A subject observed to fail (\(D_j = 1\)) contributes the event-time density \({\lambda}(Y_j \mid \tilde{x}_j)\,\operatorname{S}(Y_j \mid \tilde{x}_j)\), while a subject still censored (\(D_j = 0\)) contributes the survival probability \(\operatorname{S}(Y_j \mid \tilde{x}_j)\). Under noninformative censoring the two cases combine into the single contribution \(\operatorname{S}(Y_j \mid \tilde{x}_j)\,{\lambda}(Y_j \mid \tilde{x}_j)^{D_j}\) (the likelihood with censoring section). Subjects are independent, so the full likelihood is the product of these contributions over \(j\).
Example 3 (A two-subject likelihood) Suppose subject \(1\) fails at \(Y_1 = 3\) (\(D_1 = 1\)) and subject \(2\) is still censored at \(Y_2 = 5\) (\(D_2 = 0\)). Then Equation 9 gives
\[
\mathcal{L}
= \underbrace{\operatorname{S}(3 \mid \tilde{x}_1)\,{\lambda}(3 \mid \tilde{x}_1)}_{\text{subject 1 fails}}
\;\times\;
\underbrace{\operatorname{S}(5 \mid \tilde{x}_2)}_{\text{subject 2 censored}} .
\]
The failing subject contributes both a survival factor and a hazard factor; the censored subject contributes only a survival factor.
Definition 16 (Per-interval likelihood factor) For a grid interval with left endpoint \(\tau\) and risk set \(R(\tau)\), the per-interval likelihood factor collects one factor from every member of \(R(\tau)\) — a failure factor for the subject who fails in the interval (if any) and a survival factor for each subject who does not:
\[
\mathcal{L}_\tau
\stackrel{\text{def}}{=}\prod_{m \in R(\tau)}
\begin{cases}
q_m(\tau), & m \text{ fails in interval } \tau,\\[2pt]
1 - q_m(\tau), & \text{otherwise,}
\end{cases}
\tag{10}\]
where \(q_m(\tau)\) is the interval failure probability.
Example 4 (Per-interval factor for a three-subject risk set) Suppose interval \(\tau\) has risk set \(R(\tau) = \mathopen{}\left\{1, 2, 3\right\}\mathclose{}\), with subject \(2\) failing in the interval and subjects \(1\) and \(3\) surviving it. Then the per-interval likelihood factor is
\[
\mathcal{L}_\tau
= \mathopen{}\left(1 - q_1(\tau)\right)\mathclose{}\,q_2(\tau)\,\mathopen{}\left(1 - q_3(\tau)\right)\mathclose{} ,
\]
a survival factor for each of subjects \(1\) and \(3\) and a failure factor for subject \(2\).
Lemma 6 (Likelihood regrouped by interval) Substituting Equation 8 into Equation 9 and regrouping the factors by interval, the likelihood is the product of the per-interval likelihood factors (Definition 16):
\[
\mathcal{L}
\approx
\prod_{\tau} \mathcal{L}_\tau ,
\tag{11}\]
so each interval \(\tau\) contributes one factor per member of its risk set \(R(\tau)\). Under the no-ties assumption each interval holds at most one failure.
Proof. By Lemma 5 and Lemma 4, substituting the discretized survival factor into the likelihood and replacing the failure subject’s hazard-density factor \({\lambda}(Y_j \mid \tilde{x}_j)^{D_j}\) by \(\mathopen{}\left(q_j(Y_j) / \Delta t\right)\mathclose{}^{D_j}\), via the first-order approximation \(q_j(Y_j) \approx {\lambda}(Y_j \mid \tilde{x}_j)\,\Delta t\) (Definition 15) rearranged to \({\lambda}(Y_j \mid \tilde{x}_j) \approx q_j(Y_j) / \Delta t\), then absorbing the constant \(\Delta t^{-D_j}\) (which is independent of all model parameters),
\[
\mathcal{L}
\approx
\prod_{j=1}^{n}
\mathopen{}\left[
q_j(Y_j)^{D_j}
\prod_{\tau < Y_j}\mathopen{}\left(1 - q_j(\tau)\right)\mathclose{}
\right]\mathclose{}.
\tag{12}\]
Each factor in Equation 12 is indexed by a (subject, interval) pair: subject \(j\) contributes one factor to every interval \(\tau\) with \(j \in R(\tau)\) — a failure factor \(q_j(\tau)\) if \(j\) fails in \(\tau\), and a survival factor \(1 - q_j(\tau)\) otherwise. Reindexing the double product by interval instead of by subject collects, for each \(\tau\), exactly one factor from every member \(m \in R(\tau)\) — the per-interval likelihood factor \(\mathcal{L}_\tau\) of Definition 16 — giving Equation 11. The \(\approx\) is inherited from the density approximation in Equation 12; the regrouping itself is an exact reindexing of the same factors.
Example 5 (Regrouping a two-subject likelihood) Take two subjects at risk over two intervals with left endpoints \(\tau = 0\) and \(\tau = 1\). Subject \(1\) survives interval \(0\) and fails in interval \(1\); subject \(2\) survives both intervals. Grouping the factors by subject (Equation 12) gives
\[
\underbrace{\mathopen{}\left(1 - q_1(0)\right)\mathclose{}\,q_1(1)}_{\text{subject 1}}
\;\times\;
\underbrace{\mathopen{}\left(1 - q_2(0)\right)\mathclose{}\mathopen{}\left(1 - q_2(1)\right)\mathclose{}}_{\text{subject 2}} ,
\]
while grouping the same factors by interval (Equation 11) gives
\[
\underbrace{\mathopen{}\left(1 - q_1(0)\right)\mathclose{}\mathopen{}\left(1 - q_2(0)\right)\mathclose{}}_{\tau = 0,\ \text{no failure}}
\;\times\;
\underbrace{q_1(1)\mathopen{}\left(1 - q_2(1)\right)\mathclose{}}_{\tau = 1,\ \text{subject 1 fails}} .
\]
The two products contain identical factors; regrouping only changes the order of multiplication, which is why it is exact.
Under the proportional-hazards model, the first-order interval failure probability splits into a shared baseline factor and a subject-specific hazard ratio:
Definition 17 (Proportional-hazards first-order interval failure probability) Under the proportional-hazards model (Theorem 6), the first-order interval failure probability (Definition 15) factors into a shared baseline term and a subject-specific hazard ratio:
\[
q^*_j(\tau) \stackrel{\text{def}}{=}{\color{blue}{{\lambda}_0(\tau)\,\Delta t}}\;{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}.
\]
Example 6 (First-order approximation under the PH model) Take a half-year interval \([4.0,\, 4.5)\) with \(\Delta t = 0.5\) yr, baseline hazard \({\color{blue}{{\lambda}_0(4.0) = 0.1}}\)/yr, and a subject with hazard ratio \({\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}} = 2\). The proportional-hazards factorization splits the first-order probability into the shared baseline factor and the subject’s hazard ratio:
\[
\begin{aligned}
q^*_j(4.0)
&= {\color{blue}{{\lambda}_0(4.0)\,\Delta t}}\;{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}
&& \text{(definition of } q^*_j \text{)}
\\
&= {\color{blue}{(0.1 \times 0.5)}}\;{\color{red}{2}}
&& \text{(substitute } {\lambda}_0 = 0.1,\ \Delta t = 0.5,\ \theta_{{\lambda}}= 2\text{)}
\\
&= {\color{blue}{0.05}}\;{\color{red}{2}}
&& \text{(arithmetic)}
\\
&= 0.10 .
&& \text{(arithmetic)}
\end{aligned}
\]
Equivalently, the subject’s total hazard is \({\lambda}(4.0 \mid \tilde{x}_j) = {\color{blue}{{\lambda}_0(4.0)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}} = 0.2\)/yr, so \(q^*_j(4.0) = {\lambda}(4.0 \mid \tilde{x}_j)\,\Delta t = 0.2 \times 0.5 = 0.10\).
We then build up to the partial likelihood through a chain of proportional-hazards lemmas: the single-subject failure probability and the two factors of each event (whether a failure occurs and which subject it is), and finally the assembly that cancels the baseline hazard.
Lemma 7 (Single-subject failure probability in a risk set) For any subject \(j \in R(t_i)\), under noninformative censoring, the probability that \(j\) fails in the interval at \(t_i\) given that \(j\) is at risk is the interval failure probability, and to first order in \(\Delta t\),
\[
\begin{aligned}
\Pr\mathopen{}\left[j \text{ fails at } t_i \mid j \in R(t_i)\right]\mathclose{}
&= q_j(t_i)
&& \text{(definition of } q_j(t_i) \text{)}
\\
&\approx q^*_j(t_i)
&& \text{(first-order hazard approximation)}
\\
&= {\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}} .
&& \text{(definition of } q^*_j(t_i)\text{)}
\end{aligned}
\tag{13}\]
Proof. Since \(j \in R(t_i)\) means \(Y_j = \min(T_j, C_j) \ge t_i\), we have both \(T_j \ge t_i\) and \(C_j \ge t_i\). Under noninformative censoring, the event \(C_j \ge t_i\) carries no information about \(T_j\) given \(\tilde{x}_j\), so conditioning on \(j \in R(t_i)\) is equivalent to conditioning on \(T_j \ge t_i\) alone. This conditional failure probability is exactly the interval failure probability \(q_j(t_i)\). The first-order interval failure probability (Definition 15) then gives \(q_j(t_i) \approx q^*_j(t_i)\), and the proportional-hazards decomposition (Definition 17) splits \(q^*_j(t_i)\) into the shared baseline factor \({\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\) and the subject-specific hazard ratio \({\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\).
Example 7 (Failure probabilities in a three-subject risk set) This example introduces a running scenario reused later in this section: the risk set \(R(t_i) = \mathopen{}\left\{1, 2, 3\right\}\mathclose{}\) at \(t_i = 4\) yr, with hazard ratios \({\color{red}{\theta_{{\lambda}}(\tilde{x}_1)}} = 1\), \({\color{red}{\theta_{{\lambda}}(\tilde{x}_2)}} = 2\), \({\color{red}{\theta_{{\lambda}}(\tilde{x}_3)}} = 3\), baseline \({\color{blue}{{\lambda}_0(4)}} = 0.1\)/yr, and grid width \(\Delta t = 0.5\) yr, so the shared factor is \({\color{blue}{{\lambda}_0(4)\,\Delta t}} = 0.05\). Subject \(2\) is the one who fails at \(t_i\). By Equation 13,
\[
q^*_1(4) = 0.05,
\qquad
q^*_2(4) = 0.10,
\qquad
q^*_3(4) = 0.15 ,
\]
each the shared \(0.05\) scaled by that subject’s hazard ratio \({\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\).
Definition 18 (Risk-set score total) The risk-set score total at event time \(t_i\) is the sum of the subject-specific hazard ratios over everyone at risk at \(t_i\):
\[
S_i \stackrel{\text{def}}{=}\sum_{k \in R(t_i)} {\color{red}{\theta_{{\lambda}}(\tilde{x}_k)}},
\tag{14}\]
where \(R(t_i)\) is the risk set at \(t_i\) and \({\color{red}{\theta_{{\lambda}}(\tilde{x})}} \stackrel{\text{def}}{=}\operatorname{exp}\mathopen{}\left\{{\tilde{x}}^{\top}\tilde{\beta}\right\}\mathclose{}\) is the hazard ratio (risk score) for covariate pattern \(\tilde{x}\).
Example 8 (Risk-set score total in the three-subject risk set) For the running scenario of Example 7 — \(R(4) = \mathopen{}\left\{1, 2, 3\right\}\mathclose{}\) with hazard ratios \({\color{red}{\theta_{{\lambda}}(\tilde{x}_1)}} = 1\), \({\color{red}{\theta_{{\lambda}}(\tilde{x}_2)}} = 2\), \({\color{red}{\theta_{{\lambda}}(\tilde{x}_3)}} = 3\) — the risk-set score total (Equation 14) is
\[
\begin{aligned}
{\color{red}{S_i}}
&= {\color{red}{\theta_{{\lambda}}(\tilde{x}_1)}} + {\color{red}{\theta_{{\lambda}}(\tilde{x}_2)}} + {\color{red}{\theta_{{\lambda}}(\tilde{x}_3)}}
&& \text{(definition of } S_i \text{)}
\\
&= 1 + 2 + 3
&& \text{(substitute the hazard ratios)}
\\
&= 6 .
&& \text{(arithmetic)}
\end{aligned}
\]
Lemma 8 (The whether-a-failure-occurs factor) To first order in \(\Delta t\), the probability of exactly one failure in the interval at \(t_i\) is the shared baseline factor times the risk-set score total \({\color{red}{S_i}}\) (Definition 18):
\[
\Pr\mathopen{}\left[1 \text{ failure at } t_i \mid R(t_i)\right]\mathclose{}
\approx
{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{S_i}}.
\tag{15}\]
Proof. Treating subjects’ failure events as conditionally independent given \(R(t_i)\) (the Cox-model assumption used throughout; see (Klein and Moeschberger 2003, sec. 8.3)), the probability of exactly one failure is, by inclusion-exclusion,
\[
\Pr\mathopen{}\left[1 \text{ failure at } t_i \mid R(t_i)\right]\mathclose{}
= \sum_{k \in R(t_i)}
\Pr\mathopen{}\left[k \text{ fails} \mid R(t_i)\right]\mathclose{}
\prod_{m \in R(t_i),\, m \ne k} \Pr\mathopen{}\left[m \text{ survives} \mid R(t_i)\right]\mathclose{}.
\]
By Lemma 7, \(\Pr\mathopen{}\left[m \text{ fails} \mid R(t_i)\right]\mathclose{} = q_m(t_i) = O(\Delta t)\), so each surviving product is \(\prod_{m \ne k}(1 - O(\Delta t)) = 1 - O(\Delta t)\), and dropping the \(O(\Delta t^2)\) cross-terms (each pair of co-failures has probability \(O(\Delta t^2)\)) leaves the sum of the single-subject pieces. Factoring out the shared baseline,
\[
\begin{aligned}
\Pr\mathopen{}\left[1 \text{ failure at } t_i \mid R(t_i)\right]\mathclose{}
&\approx
\sum_{k \in R(t_i)} q_k(t_i)
&& \text{(drop } O(\Delta t^2) \text{ cross-terms)}
\\
&\approx
\sum_{k \in R(t_i)} q^*_k(t_i)
&& \text{(first-order hazard approximation)}
\\
&=
\sum_{k \in R(t_i)} {\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{\theta_{{\lambda}}(\tilde{x}_k)}}
&& \text{(definition of } q^*_k(t_i) \text{)}
\\
&=
{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\sum_{k \in R(t_i)} {\color{red}{\theta_{{\lambda}}(\tilde{x}_k)}}
&& \text{(factor out the shared baseline)}
\\
&=
{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{S_i}}.
&& \text{(definition of } {\color{red}{S_i}}\text{)}
\end{aligned}
\]
Example 9 (One-failure probability in the three-subject risk set) Continue the running scenario of Example 7, with \({\color{blue}{{\lambda}_0(4)\,\Delta t}} = 0.05\) and risk-set score total \({\color{red}{S_i}} = 6\) (Example 8). By Equation 15, the probability that some one subject fails at \(t_i = 4\) is
\[
\Pr\mathopen{}\left[1 \text{ failure at } 4 \mid R(4)\right]\mathclose{}
\approx {\color{blue}{0.05}} \times {\color{red}{6}} = 0.30 .
\]
Lemma 9 (Chain-rule split of an event factor) Under the no-ties assumption, the probability that subject \((i)\) fails at \(t_i\), given the risk set \(R(t_i)\), factors into a which-subject-fails factor and a whether-a-failure-occurs factor:
\[
\Pr\mathopen{}\left[(i) \text{ fails at } t_i \mid R(t_i)\right]\mathclose{}
= \underbrace{\Pr\mathopen{}\left[(i) \text{ fails} \mid 1 \text{ failure at } t_i,\ R(t_i)\right]\mathclose{}}_{\text{which subject fails}}
\;\times\;
\underbrace{\Pr\mathopen{}\left[1 \text{ failure at } t_i \mid R(t_i)\right]\mathclose{}}_{\text{whether a failure occurs}} .
\tag{16}\]
Proof. Let
-
\(A \stackrel{\text{def}}{=}\) “subject \((i)\) fails at \(t_i\)”,
-
\(B \stackrel{\text{def}}{=}\) “exactly one failure occurs at \(t_i\)”,
-
\(C \stackrel{\text{def}}{=}\) “the risk set at \(t_i\) is \(R(t_i)\)”.
By the chain rule of probability, \(\Pr(A \cap B \mid C) = \Pr(A \mid B \cap C)\,\Pr(B \mid C)\). Under the no-ties assumption \(A \subset B\) (subject \((i)\) failing at \(t_i\) is automatically a one-failure event), so \(A \cap B = A\) and therefore \(\Pr(A \mid C) = \Pr(A \mid B \cap C)\,\Pr(B \mid C)\), which is Equation 16.
Example 10 (Splitting the event probability numerically) Continue the running scenario of Example 7, in which subject \(2\) is the one who fails at \(t_i = 4\). The left side of Equation 16 is subject \(2\)’s single-subject failure probability, \(q^*_2(4) = 0.10\) (Example 7), and the whether-a-failure factor is \(0.30\) (Example 9). The chain rule therefore pins the which-subject factor at
\[
\Pr\mathopen{}\left[2 \text{ fails} \mid 1 \text{ failure at } 4,\ R(4)\right]\mathclose{}
= \frac{\Pr\mathopen{}\left[2 \text{ fails at } 4 \mid R(4)\right]\mathclose{}}{\Pr\mathopen{}\left[1 \text{ failure at } 4 \mid R(4)\right]\mathclose{}}
\approx \frac{0.10}{0.30} = \frac{1}{3} ,
\]
and indeed \(0.10 \approx \tfrac{1}{3} \times 0.30\) reproduces the left side.
Lemma 10 (The which-subject-fails factor) The which-subject-fails factor of Equation 16 equals the partial-likelihood contribution \(\mathcal{L}^*_i(\tilde{\beta})\) of Definition 14: the shared baseline factor cancels, leaving
\[
\Pr\mathopen{}\left[(i) \text{ fails} \mid 1 \text{ failure at } t_i,\ R(t_i)\right]\mathclose{}
\approx
\frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{{\color{red}{S_i}}}
= \mathcal{L}^*_i(\tilde{\beta}).
\tag{17}\]
Proof. Solve Equation 16 for the which-subject factor and substitute the numerator \(q_{(i)}(t_i)\) from Lemma 7 and the denominator \({\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\,{\color{red}{S_i}}\) from Lemma 8; the shared \({\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\) cancels between numerator and denominator:
\[
\begin{aligned}
\Pr\mathopen{}\left[(i) \text{ fails} \mid 1 \text{ failure at } t_i,\ R(t_i)\right]\mathclose{}
&= \frac{\Pr\mathopen{}\left[(i) \text{ fails at } t_i \mid R(t_i)\right]\mathclose{}}
{\Pr\mathopen{}\left[1 \text{ failure at } t_i \mid R(t_i)\right]\mathclose{}}
&& \text{(rearrange)}
\\
&\approx \frac{q^*_{(i)}(t_i)}{{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{S_i}}}
&& \text{(numerator and denominator from the previous two lemmas)}
\\
&= \frac{{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}
{{\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{S_i}}}
&& \text{(definition of } q^*_{(i)}(t_i) \text{)}
\\
&= \frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{{\color{red}{S_i}}}
&& \text{(cancel the shared } {\color{blue}{{\lambda}_0(t_i)\,\Delta t}} \text{)}
\\
&= \mathcal{L}^*_i(\tilde{\beta}).
&& \text{(definition of } \mathcal{L}^*_i(\tilde{\beta})\text{)}
\end{aligned}
\]
The final \(=\) is exact: the \({\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\) factors cancel algebraically, leaving a ratio independent of \(\Delta t\), so the which-subject factor equals the partial-likelihood contribution \(\mathcal{L}^*_i(\tilde{\beta})\) free of both the baseline hazard and the grid width.
Example 11 (The which-subject factor in the three-subject risk set) Continue the running scenario of Example 7, with subject \(2\) failing at \(t_i = 4\) and \({\color{red}{S_i}} = 6\) (Example 9). By Equation 17,
\[
\begin{aligned}
\mathcal{L}^*_i(\tilde{\beta})
&= \frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_2)}}}{{\color{red}{S_i}}}
&& \text{(which-subject formula, } (i) = 2\text{)}
\\
&= \frac{2}{6}
&& \text{(substitute } \theta_{{\lambda}}(\tilde{x}_2) = 2,\ S_i = 6\text{)}
\\
&= \frac{1}{3} ,
&& \text{(arithmetic)}
\end{aligned}
\]
independent of the baseline \({\color{blue}{{\lambda}_0(4)}}\) and the grid width \(\Delta t\) — the cancellation that the partial likelihood relies on. This value \(\tfrac{1}{3}\) matches the one pinned by the chain rule in Example 10.
With the lemmas in hand, the proof assembles them and shows the baseline cancels — confirming that the product is free of \({\lambda}_0\):
Proof. Adapted from (Klein and Moeschberger 2003, sec. 8.3, Theoretical Note 1, p. 257). This proof draws on the discretized-likelihood building blocks assembled earlier in this section — Lemma 5, Lemma 4, and Lemma 6 — together with the proportional-hazards lemmas Lemma 7, Lemma 8, Lemma 9, and Lemma 10.
Assume noninformative censoring (conditional on \(\tilde{x}\), censoring times are independent of event times) and no tied event times. Let \(t_1 < \cdots < t_K\) be the distinct observed failure times, and let \((i)\) be the subject who fails at \(t_i\) (unique under the no-ties assumption).
By Lemma 6, the likelihood factors over intervals, \(\mathcal{L}\approx \prod_\tau \mathcal{L}_\tau\). Separate the \(K\) intervals that contain a failure from the rest. A no-failure interval contributes \(\mathcal{L}_\tau^{\text{no failure}} = \prod_{m \in R(\tau)}\mathopen{}\left(1 - q_m(\tau)\right)\mathclose{}\); the interval containing \(t_i\) contributes the event factor \(\Pr\mathopen{}\left[(i) \text{ fails at } t_i \mid R(t_i)\right]\mathclose{}\) times the survival of the other members, which to first order is just the event factor itself, since each other member survives with probability \(1 - O(\Delta t)\). Applying Lemma 9 to each event factor, and then Lemma 8 and Lemma 10 to its two pieces,
\[
\begin{aligned}
\mathcal{L}
&\approx
\mathopen{}\left(\prod_{i=1}^{K}
\Pr\mathopen{}\left[(i) \text{ fails} \mid 1 \text{ failure at } t_i,\ R(t_i)\right]\mathclose{}\,
\Pr\mathopen{}\left[1 \text{ failure at } t_i \mid R(t_i)\right]\mathclose{}\right)\mathclose{}
\mathopen{}\left(\prod_{\tau\,:\,\text{no failure}} \mathcal{L}_\tau^{\text{no failure}}\right)\mathclose{}
&& \text{(event chain rule)}
\\
&\approx
\underbrace{\mathopen{}\left(\prod_{i=1}^{K} \frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{{\color{red}{S_i}}}\right)\mathclose{}}
_{\mathcal{L}^*(\tilde{\beta})\ \text{(kept)}}
\;\times\;
\underbrace{
\mathopen{}\left[\prod_{i=1}^{K} {\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\;{\color{red}{S_i}}\right]\mathclose{}
\mathopen{}\left[\prod_{\tau\,:\,\text{no failure}} \prod_{m \in R(\tau)} \mathopen{}\left(1 - q_m(\tau)\right)\mathclose{}\right]\mathclose{}}
_{\mathcal{B}(\tilde{\beta},\,{\lambda}_0)\ \text{(discarded)}} .
&& \text{(whether-failure and which-subject factors)}
\end{aligned}
\tag{18}\]
Under the no-ties assumption each event time has \(d_i = 1\) (where \(d_i\) is the number of events at \(t_i\)), so the kept product \(\prod_{i=1}^{K}(\cdots) = \prod_{\mathopen{}\left\{i:\ d_i = 1\right\}\mathclose{}}(\cdots)\) matches the form of \(\mathcal{L}^*(\tilde{\beta})\) in Definition 14.
The partial likelihood keeps the first product, \(\mathcal{L}^*(\tilde{\beta})\), and drops the second, \(\mathcal{B}(\tilde{\beta}, {\lambda}_0)\). The discarded factor \(\mathcal{B}(\tilde{\beta}, {\lambda}_0)\) collects both the whether-a-failure-occurs factors and the entire between-event survival background, and it is the only place the baseline hazard \({\color{blue}{{\lambda}_0(\cdot)}}\) appears — equivalently, the shared \({\color{blue}{{\lambda}_0(t_i)\,\Delta t}}\) cancels from each kept factor in Equation 17 — which is why maximizing \(\mathcal{L}^*(\tilde{\beta})\) alone frees the estimator from \({\lambda}_0(\cdot)\). The discarded factor is not free of \(\tilde{\beta}\), however: it still depends on \(\tilde{\beta}\) through the risk-set score totals \({\color{red}{S_i}}\) and the survival background. Dropping it therefore sacrifices the information about \(\tilde{\beta}\) carried by when (and within how large a risk set) failures occur, retaining only the information in which subject fails at each event time.
The factorization in Equation 18 is a construction rather than an exact decomposition of the joint probability: the factors are not formally independent (the risk set \(R(t_{i+1})\) depends on who failed and was censored before \(t_{i+1}\), and the discretization is exact only in the \(\Delta t \to 0\) limit), but Cox showed that treating \(\mathcal{L}^*(\tilde{\beta})\) as a likelihood — i.e., maximizing it as a function of \(\tilde{\beta}\) — yields estimators with the usual large-sample properties (consistency, asymptotic normality, the standard score and information identities; see also (Klein and Moeschberger 2003, sec. 8.3, Theoretical Note 1)). Under noninformative censoring, each kept factor depends on the observed history only through \(R(t_i)\) and \(\mathopen{}\left\{\tilde{x}_j\right\}\mathclose{}_{j \in R(t_i)}\), and Equation 17 gives exactly the partial likelihood \(\mathcal{L}^*(\tilde{\beta})\) of Definition 14.
We maximize this partial likelihood numerically with respect to \(\tilde{\beta}\) (which determines \(\Delta\eta(\tilde{x}) = {\tilde{x}}^{\top}\tilde{\beta}\) and \(\theta_{{\lambda}}(\tilde{x}) = \operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde{x})\right\}\mathclose{}\)). When there are tied event times some adjustments are needed, but the likelihood is similar. Crucially, the baseline hazard never enters, so we obtain \(\hat{\tilde{\beta}}\) without estimating \({\lambda}_0\).
Stage 2: estimating the baseline hazard. Once we have the coefficient estimates \(\hat{\tilde{\beta}}= (\hat\beta_1, \ldots, \hat\beta_p)\) — and hence \(\hat\Delta\eta(\tilde{x})\) and \(\hat\theta_{{\lambda}}(\tilde{x})\) — the baseline cumulative hazard is estimated by the Breslow estimator:
Definition 19 (Breslow estimator of the baseline cumulative hazard) \[\hat {\Lambda}_0(t) \stackrel{\text{def}}{=}
\sum_{t_i \le t} \frac{d_i}{\sum_{k\in R(t_i)} \theta_{{\lambda}}(\tilde{x}_k)}\]
where, for the distinct event times \(t_1 < \cdots < t_K\) of the data set:
-
\(d_i\) is the number of events at \(t_i\);
-
\(R(t_i) \stackrel{\text{def}}{=}\mathopen{}\left\{j : Y_j \ge t_i\right\}\mathclose{}\) is the risk set at \(t_i\) (where \(Y_j = \min(T_j, C_j)\) is subject \(j\)’s observed follow-up time) — the subjects still under observation just before \(t_i\);
-
\(\theta_{{\lambda}}(\tilde{x}) \stackrel{\text{def}}{=}\operatorname{exp}\mathopen{}\left\{{\tilde{x}}^{\top}\tilde{\beta}\right\}\mathclose{}\) is the hazard ratio (risk score) for covariate pattern \(\tilde{x}\).
Example 12 (Breslow estimator with two event times) Suppose there are two distinct event times \(t_1 = 2\) and \(t_2 = 4\) (no ties, \(d_1 = d_2 = 1\)), with risk-set-weighted totals \(\sum_{k \in R(2)} \theta_{{\lambda}}(\tilde{x}_k) = 10\) and \(\sum_{k \in R(4)} \theta_{{\lambda}}(\tilde{x}_k) = 6\). Then
\[
\hat{\Lambda}_0(4) = \frac{1}{10} + \frac{1}{6} \approx 0.267 .
\]
This estimator reduces to the Nelson-Aalen estimator when there are no covariates; several other estimators have also been proposed. To derive it as a profile likelihood, substitute the proportional-hazards decomposition into the full censored-data likelihood (Lemma 5):
\[
\mathcal{L}\mathopen{}\left[\tilde{\beta},\,{\lambda}_0(\cdot)\right]\mathclose{}
= \prod_{j = 1}^{n}
\mathopen{}\left[{\color{blue}{{\lambda}_0(\tilde{T}_j)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\right]\mathclose{}^{\delta_j}\,
\operatorname{exp}\mathopen{}\left\{-{\color{blue}{{\Lambda}_0(\tilde{T}_j)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\right\}\mathclose{},
\tag{19}\]
where \(\delta_j\) is the event indicator and \(\tilde{T}_j\) the observed time for subject \(j\) — the same quantities denoted \(D_j\) and \(Y_j\) in the censored-data likelihood lemma, so \(\tilde{T}_j = Y_j = \min(T_j, C_j)\) and \(\delta_j = D_j\) — using \({\lambda}(t \mid \tilde{x}) = {\color{blue}{{\lambda}_0(t)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x})}}\) (Theorem 6) and \(\operatorname{S}(t \mid \tilde{x}) = \operatorname{exp}\mathopen{}\left\{-{\color{blue}{{\Lambda}_0(t)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x})}}\right\}\mathclose{}\) (by Theorem 10, using \(\operatorname{S}(t) = \operatorname{exp}\mathopen{}\left\{-{\Lambda}(t)\right\}\mathclose{}\) from the survival/cumulative hazard relationship). Fix \(\tilde{\beta}\) and maximize over \({\lambda}_0(\cdot)\). We build up to the Breslow estimator’s profile-likelihood justification through a chain of lemmas: the point-mass form of the maximizing hazard, the resulting profile likelihood, the maximizer of each point mass, and the assembly that recovers the partial likelihood.
Lemma 11 (Point-mass optimality of the profile baseline hazard) Fix \(\tilde{\beta}\) and maximize the censored-data likelihood Equation 19 over \({\lambda}_0(\cdot)\). The maximizing \({\lambda}_0(\cdot)\) places mass only at the \(K\) distinct event times \(t_1 < \cdots < t_K\), as a discrete hazard measure with point masses \({\color{blue}{h_{0i}}} \stackrel{\text{def}}{=}{\color{blue}{{\lambda}_0(t_i)}}\):
\[
{\color{blue}{{\lambda}_0(t)}} = \begin{cases}
{\color{blue}{h_{0i}}}, & t = t_i \text{ for some } i \in \mathopen{}\left\{1, \dots, K\right\}\mathclose{},\\
0, & \text{otherwise}.
\end{cases}
\tag{20}\]
Proof. Subjects with \(\delta_j = 0\) contribute only the survival term \(\operatorname{exp}\mathopen{}\left\{-{\color{blue}{{\Lambda}_0(\tilde{T}_j)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\right\}\mathclose{}\) to Equation 19, since \({\color{blue}{{\lambda}_0(\tilde{T}_j)}}^{\delta_j} = {\color{blue}{{\lambda}_0(\tilde{T}_j)}}^0 = 1\) regardless of \({\lambda}_0\). Adding mass to \({\lambda}_0\) at any non-event time \(t\) increases \({\color{blue}{{\Lambda}_0(\tilde{T}_j)}}\) for every subject with \(\tilde{T}_j \ge t\), penalizing their survival terms in Equation 19 without any compensating gain in a hazard-density factor — so no mass should be placed away from the \(K\) event times.
For an event subject \(j\) with \(\delta_j = 1\) whose failure occurs at \(t_i\), consider any allocation of a fixed total mass \({\color{blue}{h_{0i}}}\) to a neighborhood of \(t_i\). The cumulative hazard \({\color{blue}{{\Lambda}_0(\tilde{T}_j)}}\) depends only on that total mass, not on how it is distributed within the neighborhood, so the survival penalty \(\operatorname{exp}\mathopen{}\left\{-{\color{blue}{{\Lambda}_0(\tilde T_j)}}{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\right\}\mathclose{}\) is unchanged by the allocation. But the hazard-density factor \({\color{blue}{{\lambda}_0(t_i)}}^{\delta_j}\) in Equation 19 is maximized when all of that mass is concentrated as a single point at \(t_i\) (so \({\color{blue}{{\lambda}_0(t_i)}} = {\color{blue}{h_{0i}}}\)): spreading the same total mass across a wider neighborhood would leave less of it exactly at \(t_i\), lowering \({\color{blue}{{\lambda}_0(t_i)}}\) below \({\color{blue}{h_{0i}}}\), while leaving the survival penalty fixed. Since concentrating the mass strictly increases the hazard-density factor without changing the survival penalty, Equation 19 is maximized by Equation 20.
Example 13 (Concentrating versus spreading a point mass) Suppose a total mass of \(0.1\) is available to place near an event time \(t_1\). Concentrating it exactly at \(t_1\) gives \({\color{blue}{{\lambda}_0(t_1)}} = {\color{blue}{h_{01}}} = 0.1\), contributing a hazard-density factor of \(0.1\) to Equation 19 for the subject failing at \(t_1\). Splitting the same total mass into two halves — \(0.05\) exactly at \(t_1\) and \(0.05\) at a nearby non-event point — leaves only \({\color{blue}{{\lambda}_0(t_1)}} = 0.05\) at \(t_1\) itself, half the hazard-density factor, while the cumulative hazard \({\color{blue}{{\Lambda}_0(\tilde T_j)}}\) (and hence every survival term) is unchanged, since both allocations place the same total mass \(0.1\) below any \(\tilde T_j \ge t_1\). Splitting the mass therefore strictly decreases the likelihood, confirming that concentrating all of it at \(t_1\) is optimal.
Lemma 12 (Profile likelihood in terms of point masses) Substituting the point-mass form of \({\lambda}_0(\cdot)\) (Lemma 11) into the censored-data likelihood Equation 19 gives the profile likelihood as a function of \(\tilde{\beta}\) and the point masses \(h_{01},\ldots,h_{0K}\):
\[
\mathcal{L}\mathopen{}\left[\tilde{\beta},\,h_{01},\dots,h_{0K}\right]\mathclose{}
= \prod_{i = 1}^{K}
{\color{blue}{h_{0i}}}\;{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}\;
\operatorname{exp}\mathopen{}\left\{-{\color{blue}{h_{0i}}}\,{\color{red}{S_i}}\right\}\mathclose{},
\tag{21}\]
where \({\color{red}{S_i}} = \sum_{j \in R(t_i)} {\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\) is the risk-set score total (Definition 18).
Proof. By Equation 20, \({\color{blue}{{\Lambda}_0(\tilde{T}_j)}} = \sum_{i\,:\,t_i \le \tilde{T}_j} {\color{blue}{h_{0i}}}\). Expanding the survival exponent in Equation 19 and swapping the order of summation:
\[
\begin{aligned}
\sum_j {\color{blue}{{\Lambda}_0(\tilde{T}_j)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}
&= \sum_j {\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}} \sum_{i:\ t_i \le \tilde{T}_j} {\color{blue}{h_{0i}}}
&& \text{(expand } {\color{blue}{{\Lambda}_0(\tilde{T}_j)}} \text{ as point masses)}
\\
&= \sum_i {\color{blue}{h_{0i}}} \sum_{j:\ \tilde{T}_j \ge t_i} {\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}
&& \text{(swap the order of summation)}
\\
&= \sum_i {\color{blue}{h_{0i}}} \sum_{j \in R(t_i)} {\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}
&& \text{(}R(t_i) = \{j : \tilde{T}_j \ge t_i\}\text{)}
\\
&= \sum_i {\color{blue}{h_{0i}}}\,{\color{red}{S_i}} .
&& \text{(definition of } S_i \text{)}
\end{aligned}
\]
Here \(R(t_i)\) uses the “at risk at \(t_i\)” convention (the risk set definition) with the \(\ge\) boundary — a subject censored exactly at \(t_i\) is in \(R(t_i)\) and contributes \({\color{blue}{h_{0i}}}\) to \({\color{blue}{{\Lambda}_0(\tilde{T}_j)}}\). Reindexing the hazard-density product in Equation 19: subjects with \(\delta_j = 0\) contribute a factor of \({\color{blue}{{\lambda}_0(\tilde{T}_j)}}^0 = 1\) (no contribution), and for each subject with \(\delta_j = 1\) their event occurred at some \(t_i\), so \({\color{blue}{{\lambda}_0(\tilde{T}_j)}} = {\color{blue}{h_{0i}}}\):
\[
\begin{aligned}
\prod_{j=1}^n \mathopen{}\left[{\color{blue}{{\lambda}_0(\tilde{T}_j)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}\right]\mathclose{}^{\delta_j}
&= \prod_{j:\,\delta_j=1} {\color{blue}{{\lambda}_0(\tilde{T}_j)}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_j)}}
&& \text{(terms with } \delta_j = 0 \text{ equal } 1\text{, dropped)}
\\
&= \prod_{i=1}^K {\color{blue}{h_{0i}}}\,{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}} .
&& \text{(}{\color{blue}{{\lambda}_0(\tilde{T}_j)}} = {\color{blue}{h_{0i}}} \text{ for event } j \text{ at event time } t_i\text{)}
\end{aligned}
\]
Factoring \(\operatorname{exp}\mathopen{}\left\{-\sum_i {\color{blue}{h_{0i}}}\,{\color{red}{S_i}}\right\}\mathclose{} = \prod_i \operatorname{exp}\mathopen{}\left\{-{\color{blue}{h_{0i}}}\,{\color{red}{S_i}}\right\}\mathclose{}\) and combining factor-by-factor with the hazard-density product gives Equation 21.
Example 14 (Profile likelihood for two event times) Extend the running scenario of Example 7 with an earlier event time \(t_1 = 2\) at which subject \(4\) fails, with risk set \(R(2) = \mathopen{}\left\{1,2,3,4\right\}\mathclose{}\) and hazard ratios \({\color{red}{\theta_{{\lambda}}(\tilde{x}_1)}} = 1\), \({\color{red}{\theta_{{\lambda}}(\tilde{x}_2)}} = 2\), \({\color{red}{\theta_{{\lambda}}(\tilde{x}_3)}} = 3\), \({\color{red}{\theta_{{\lambda}}(\tilde{x}_4)}} = 4\), so \({\color{red}{S_1}} = 10\) (Definition 18). Subject \(4\) then leaves the risk set, so at \(t_2 = 4\) the risk set is \(R(4) = \mathopen{}\left\{1,2,3\right\}\mathclose{}\) as before, with \({\color{red}{S_2}} = 6\) (Example 8). By Equation 21, with \(K = 2\),
\[
\mathcal{L}\mathopen{}\left[\tilde{\beta},\,h_{01},\,h_{02}\right]\mathclose{}
= \mathopen{}\left[{\color{blue}{h_{01}}} \times {\color{red}{4}} \times \operatorname{exp}\mathopen{}\left\{-{\color{blue}{h_{01}}} \times {\color{red}{10}}\right\}\mathclose{}\right]\mathclose{}
\times
\mathopen{}\left[{\color{blue}{h_{02}}} \times {\color{red}{2}} \times \operatorname{exp}\mathopen{}\left\{-{\color{blue}{h_{02}}} \times {\color{red}{6}}\right\}\mathclose{}\right]\mathclose{} .
\]
Lemma 13 (Profile maximum likelihood estimate of each point mass) The value of \({\color{blue}{h_{0i}}}\) that maximizes the profile likelihood Equation 21 (Lemma 12) is
\[
{\color{blue}{\hat h_{0i}}} \stackrel{\text{def}}{=}\frac{1}{{\color{red}{S_i}}} .
\tag{22}\]
Proof. Taking logs of Equation 21, the log-likelihood is separable in the \({\color{blue}{h_{0i}}}\) — each summand depends on only one point mass:
\[
\ell\mathopen{}\left[\tilde{\beta},\,h_{01},\dots,h_{0K}\right]\mathclose{}
= \sum_{i = 1}^{K} \log{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}
+ \sum_{i = 1}^{K}
\mathopen{}\left\{\log {\color{blue}{h_{0i}}} - {\color{blue}{h_{0i}}}\,{\color{red}{S_i}}\right\}\mathclose{} .
\]
So each \({\color{blue}{h_{0i}}}\) can be maximized separately. Differentiate the \(i\)-th summand with respect to \({\color{blue}{h_{0i}}}\) and set the derivative to zero:
\[
\begin{aligned}
\frac{1}{{\color{blue}{h_{0i}}}} - {\color{red}{S_i}} &= 0
&& \text{(stationarity condition)}
\\
\Longrightarrow\quad {\color{blue}{\hat h_{0i}}} &= \frac{1}{{\color{red}{S_i}}} .
&& \text{(solve for } {\color{blue}{\hat h_{0i}}} \text{)}
\end{aligned}
\]
The second derivative \(-1/{\color{blue}{h_{0i}}}^2 < 0\) confirms this critical point is a maximum, giving Equation 22.
Example 15 (Maximum likelihood point masses for two event times) Continue the two-event scenario of Example 14, with \({\color{red}{S_1}} = 10\) at \(t_1 = 2\) and \({\color{red}{S_2}} = 6\) at \(t_2 = 4\). By Equation 22,
\[
{\color{blue}{\hat h_{01}}} = \frac{1}{{\color{red}{10}}} = 0.1 ,
\qquad
{\color{blue}{\hat h_{02}}} = \frac{1}{{\color{red}{6}}} \approx 0.167 .
\]
Summing gives the Breslow estimator at \(t = 4\): \({\color{blue}{\hat{\Lambda}_0(4)}} = {\color{blue}{\hat h_{01}}} + {\color{blue}{\hat h_{02}}} = 0.1 + 0.167 = 0.267\) (Definition 19).
Lemma 14 (The profile likelihood at the MLE recovers the partial likelihood) Substituting the maximum likelihood point masses \({\color{blue}{\hat h_{0i}}} = 1/{\color{red}{S_i}}\) (Lemma 13) back into the profile likelihood Equation 21 (Lemma 12) gives, up to a \(\tilde{\beta}\)-free constant, the partial likelihood of Definition 14:
\[
\mathcal{L}\mathopen{}\left[\tilde{\beta},\,\hat h_{01},\dots,\hat h_{0K}\right]\mathclose{} = e^{-K}\,\mathcal{L}^*(\tilde{\beta}) .
\tag{23}\]
Proof. Substitute \({\color{blue}{\hat h_{0i}}} = 1/{\color{red}{S_i}}\) into Equation 21 term by term:
\[
\begin{aligned}
\mathcal{L}\mathopen{}\left[\tilde{\beta},\,\hat h_{01},\dots,\hat h_{0K}\right]\mathclose{}
&= \prod_{i = 1}^{K}
{\color{blue}{\hat h_{0i}}}\;{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}\;
\operatorname{exp}\mathopen{}\left\{-{\color{blue}{\hat h_{0i}}}\,{\color{red}{S_i}}\right\}\mathclose{}
&& \text{(the profile likelihood)}
\\
&= \prod_{i = 1}^{K}
\frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{{\color{red}{S_i}}}\;
\operatorname{exp}\mathopen{}\left\{-{\color{red}{S_i}} / {\color{red}{S_i}}\right\}\mathclose{}
&& \text{(substitute } {\color{blue}{\hat h_{0i}}} = 1/{\color{red}{S_i}}\text{)}
\\
&= \prod_{i = 1}^{K}
\frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{{\color{red}{S_i}}}\;
e^{-1}
&& \text{(since } {\color{red}{S_i}}/{\color{red}{S_i}} = 1\text{)}
\\
&= e^{-K}
\cdot \prod_{i = 1}^{K} \frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{{\color{red}{S_i}}} .
&& \text{(collect the } K \text{ factors of } e^{-1}\text{)}
\end{aligned}
\]
Under the no-ties assumption each event time \(t_i\) has \(d_i = 1\), so \(\prod_{i=1}^K \frac{{\color{red}{\theta_{{\lambda}}(\tilde{x}_{(i)})}}}{{\color{red}{S_i}}}\) equals the partial likelihood \(\mathcal{L}^*(\tilde{\beta})\) of Definition 14, giving Equation 23.
Example 16 (Verifying the profile likelihood at the MLE) Continue the two-event scenario of Example 15, with \({\color{blue}{\hat h_{01}}} = 0.1\), \({\color{red}{S_1}} = 10\), \(\theta_{{\lambda}}(\tilde{x}_4) = 4\) at \(t_1 = 2\), and \({\color{blue}{\hat h_{02}}} = 1/6\), \({\color{red}{S_2}} = 6\), \(\theta_{{\lambda}}(\tilde{x}_2) = 2\) at \(t_2 = 4\). Substituting into Equation 21,
\[
\begin{aligned}
\mathcal{L}\mathopen{}\left[\tilde{\beta},\,\hat h_{01},\,\hat h_{02}\right]\mathclose{}
&= \mathopen{}\left[{\color{blue}{0.1}} \times {\color{red}{4}} \times \operatorname{exp}\mathopen{}\left\{-{\color{blue}{0.1}}\times {\color{red}{10}}\right\}\mathclose{}\right]\mathclose{}
\times
\mathopen{}\left[{\color{blue}{\tfrac{1}{6}}} \times {\color{red}{2}} \times \operatorname{exp}\mathopen{}\left\{-{\color{blue}{\tfrac{1}{6}}}\times {\color{red}{6}}\right\}\mathclose{}\right]\mathclose{}
&& \text{(substitute into the profile likelihood)}
\\
&= \mathopen{}\left[\frac{{\color{red}{4}}}{{\color{red}{10}}}\,e^{-1}\right]\mathclose{} \times \mathopen{}\left[\frac{{\color{red}{2}}}{{\color{red}{6}}}\,e^{-1}\right]\mathclose{}
&& \text{(}{\color{blue}{\hat h_{0i}}}\,{\color{red}{S_i}} = 1\text{)}
\\
&= e^{-2}\times\frac{{\color{red}{4}}}{{\color{red}{10}}}\times\frac{{\color{red}{2}}}{{\color{red}{6}}} ,
&& \text{(collect the two factors of } e^{-1}\text{)}
\end{aligned}
\]
matching Equation 23 with \(K = 2\) and \(\mathcal{L}^*(\tilde{\beta}) = \frac{\theta_{{\lambda}}(\tilde{x}_4)}{S_1}\times\frac{\theta_{{\lambda}}(\tilde{x}_2)}{S_2} = \frac{4}{10}\times\frac{2}{6}\).
With the lemmas in hand, the proof assembles them into the Breslow estimator and shows the profile likelihood recovers the partial likelihood:
Proof. Adapted from (Klein and Moeschberger 2003, sec. 8.3, Theoretical Note 2, p. 258); the original profile-likelihood argument is due to Johansen (1983). This proof assembles four building-block lemmas: Lemma 11, Lemma 12, Lemma 13, and Lemma 14.
Assume, as in the partial-likelihood proof, that there are no tied event times, so each ordered event time \(t_i\) corresponds to exactly one event, and let \(K\) denote the number of distinct event times \(t_1 < \cdots < t_K\).
Fix \(\tilde{\beta}\). By Lemma 11, the maximizing \({\lambda}_0(\cdot)\) places point masses \({\color{blue}{h_{0i}}} \stackrel{\text{def}}{=}{\color{blue}{{\lambda}_0(t_i)}}\) only at the \(K\) event times. By Lemma 12, substituting this form into Equation 19 gives the profile likelihood Equation 21 in terms of the point masses and the risk-set score totals \({\color{red}{S_i}}\) (Definition 18). By Lemma 13, each point mass is maximized at \({\color{blue}{\hat h_{0i}}} = 1/{\color{red}{S_i}}\). Summing the estimated point masses over event times \(t_i \le t\) gives the Breslow estimator of the baseline cumulative hazard,
\[
{\color{blue}{\hat {\Lambda}_0(t)}} = \sum_{t_i \le t} {\color{blue}{\hat h_{0i}}} = \sum_{t_i \le t} \frac{1}{{\color{red}{S_i}}} ,
\]
matching Definition 19 under the no-ties assumption (\(d_i = 1\) for every \(i\)). With tied event times, the numerator generalizes to \(d_i\), the number of events at \(t_i\) (see (Klein and Moeschberger 2003, sec. 8.8) for the tie-handling adjustments).
By Lemma 14, substituting \({\color{blue}{\hat h_{0i}}}\) back into the profile likelihood recovers the partial likelihood up to a \(\tilde{\beta}\)-free constant, \(\mathcal{L}\mathopen{}\left[\tilde{\beta},\,\hat h_{01},\dots,\hat h_{0K}\right]\mathclose{} = e^{-K}\,\mathcal{L}^*(\tilde{\beta})\), where \(\mathcal{L}^*(\tilde{\beta})\) is the partial likelihood of Definition 14. Since \(e^{-K}\) does not depend on \(\tilde{\beta}\), the profile likelihood is proportional to \(\mathcal{L}^*(\tilde{\beta})\). This proportionality justifies treating the partial likelihood as a profile likelihood for \(\tilde{\beta}\), with \({\lambda}_0(\cdot)\) concentrated out.
Adjustment for Ties (optional)
At each time \(t_i\) at which more than one of the subjects has an event, let \(d_i\) be the number of events at that time, \(D_i\) the set of subjects with events at that time, and let \(s_i\) be a covariate vector for an artificial subject obtained by adding up the covariate values for the subjects with an event at time \(t_i\). Let \[\bar\eta_i = \beta_1s_{i1}+\cdots+\beta_ps_{ip}\] and \(\bar\theta_i = \operatorname{exp}\mathopen{}\left\{\bar\eta_i\right\}\mathclose{}\).
Note that
\[
\begin{aligned}
\bar\eta_i &=\sum_{j \in D_i}\beta_1x_{j1}+\cdots+\beta_px_{jp}\\
&= \beta_1s_{i1}+\cdots+\beta_ps_{ip}\\
\bar\theta_i &= \operatorname{exp}\mathopen{}\left\{\bar\eta_i\right\}\mathclose{}\\
&= \prod_{j \in D_i}\theta_i
\end{aligned}
\]
Breslow’s method for ties
Breslow’s method estimates the partial likelihood as
\[
\begin{aligned}
L(\beta|T) &=
\prod_i \frac{\bar\theta_i}{[\sum_{k \in R(t_i)} \theta_k]^{d_i}}\\
&= \prod_i \prod_{j \in D_i}\frac{\theta_j}{\sum_{k \in R(t_i)} \theta_k}
\end{aligned}
\]
This method is equivalent to treating each event as distinct and using the non-ties formula. It works best when the number of ties is small. It is the default in many statistical packages, including PROC PHREG in SAS.
Efron’s method for ties
The other common method is Efron’s, which is the default in R.
\[L(\beta|T)=
\prod_i \frac{\bar\theta_i}{\prod_{j=1}^{d_i}[\sum_{k \in R(t_i)} \theta_k-\frac{j-1}{d_i}\sum_{k \in D_i} \theta_k]}\] This is closer to the exact discrete partial likelihood when there are many ties.
The third option in R (and an option also in SAS as discrete) is the “exact” method, which is the same one used for matched logistic regression.
Example: Breslow’s method
Suppose as an example we have a time \(t\) where there are 20 individuals at risk and three failures. Let the three individuals have risk parameters \(\theta_1, \theta_2, \theta_3\) and let the sum of the risk parameters of the remaining 17 individuals be \(\theta_R\). Then the factor in the partial likelihood at time \(t\) using Breslow’s method is
\[
\left(\frac{\theta_1}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_2}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_3}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\]
If on the other hand, they had died in the order 1,2, 3, then the contribution to the partial likelihood would be:
\[
\left(\frac{\theta_1}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_2}{\theta_R+\theta_2+\theta_3}\right)
\left(\frac{\theta_3}{\theta_R+\theta_3}\right)
\]
as the risk set got smaller with each failure. The exact method roughly averages the results for the six possible orderings of the failures.
Example: Efron’s method
But we don’t know the order they failed in, so instead of reducing the denominator by one risk coefficient each time, we reduce it by the same fraction. This is Efron’s method.
\[\left(\frac{\theta_1}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_2}{\theta_R+2(\theta_1+\theta_2+\theta_3)/3}\right)
\left(\frac{\theta_3}{\theta_R+(\theta_1+\theta_2+\theta_3)/3}\right)\]
