Proportional hazards model

Sir David Cox observed that if the proportional hazards assumption holds (or, is assumed to hold) then it is possible to estimate the effect parameter(s), denoted ${\displaystyle \beta _{i}}$ below, without any consideration of the full hazard function. This approach to survival data is called application of the Cox proportional hazards model,[2] sometimes abbreviated to Cox model or to proportional hazards model.[3] However, Cox also noted that biological interpretation of the proportional hazards assumption can be quite tricky.[4][5]

Let X_i = (X_i1, … , X_ip) be the realized values of the p covariates for subject i. The hazard function for the Cox proportional hazards model has the form

${\displaystyle {\begin{aligned}\lambda (t|X_{i})&=\lambda _{0}(t)\exp(\beta _{1}X_{i1}+\cdots +\beta _{p}X_{ip})\\&=\lambda _{0}(t)\exp(X_{i}\cdot \beta )\end{aligned}}}$

This expression gives the hazard function at time t for subject i with covariate vector (explanatory variables) X_i. Note that between subjects, the baseline hazard ${\displaystyle \lambda _{0}(t)}$ is identical (has no dependency on i). The only difference between subjects' hazards comes from the baseline scaling factor ${\displaystyle \exp(X_{i}\cdot \beta )}$.

To start, suppose we only have a single covariate, ${\displaystyle x}$, and therefore a single coefficient, ${\displaystyle \beta _{1}}$. Consider the effect of increasing ${\displaystyle x}$ by 1:

${\displaystyle {\begin{aligned}\lambda (t|x+1)&=\lambda _{0}(t)\exp(\beta _{1}(x+1))\\&=\lambda _{0}(t)\exp(\beta _{1}x+\beta _{1})\\&={\Bigl (}\lambda _{0}(t)\exp(\beta _{1}x){\Bigr )}\exp(\beta _{1})\\&=\lambda (t|x)\exp(\beta _{1})\end{aligned}}}$

We can see that increasing a covariate by 1 scales the original hazard by the constant ${\displaystyle \exp(\beta _{1})}$. Rearranging things slightly, we see that:

${\displaystyle {\frac {\lambda (t|x+1)}{\lambda (t|x)}}=\exp(\beta _{1})}$

The right-hand-side is constant over time (no term has a ${\displaystyle t}$ in it). This relationship, ${\displaystyle x/y={\text{constant}}}$, is called a proportional relationship.

More generally, consider two subjects, i and j, with covariates ${\displaystyle X_{i}}$ and ${\displaystyle X_{j}}$ respectively. Consider the ratio of their hazards:

${\displaystyle {\begin{aligned}{\frac {\lambda (t|X_{i})}{\lambda (t|X_{j})}}&={\frac {\lambda _{0}(t)\exp(X_{i}\cdot \beta )}{\lambda _{0}(t)\exp(X_{j}\cdot \beta )}}\\&={\frac {{\cancel {\lambda _{0}(t)}}\exp(X_{i}\cdot \beta )}{{\cancel {\lambda _{0}(t)}}\exp(X_{j}\cdot \beta )}}\\&=\exp((X_{i}-X_{j})\cdot \beta )\end{aligned}}}$

The right-hand-side isn't dependent on time, as the only time-dependent factor, ${\displaystyle \lambda _{0}(t)}$, was cancelled out. Thus the ratio of hazards of two subjects is a constant, i.e. the hazards are proportional.

Often there is an intercept term (also called a constant term or bias term) used in regression models. The Cox model lacks one because the baseline hazard, ${\displaystyle \lambda _{0}(t)}$, takes the place of it. Let's see what would happen if we did include an intercept term anyways, denoted ${\displaystyle \beta _{0}}$:

${\displaystyle {\begin{aligned}\lambda (t|X_{i})&=\lambda _{0}(t)\exp(\beta _{1}X_{i1}+\cdots +\beta _{p}X_{ip}+\beta _{0})\\&=\lambda _{0}(t)\exp(X_{i}\cdot \beta )\exp(\beta _{0})\\&=\left(\exp(\beta _{0})\lambda _{0}(t)\right)\exp(X_{i}\cdot \beta )\\&=\lambda _{0}^{*}(t)\exp(X_{i}\cdot \beta )\end{aligned}}}$

where we've redefined ${\displaystyle \exp(\beta _{0})\lambda _{0}(t)}$ to be a new baseline hazard, ${\displaystyle \lambda _{0}^{*}(t)}$. Thus, the baseline hazard incorporates all parts of the hazard that are not dependent on the subjects' covariates, which includes any intercept term (which is constant for all subjects, by definition).

The Cox partial likelihood, shown below, is obtained by using Breslow's estimate of the baseline hazard function, plugging it into the full likelihood and then observing that the result is a product of two factors. The first factor is the partial likelihood shown below, in which the baseline hazard has "canceled out". The second factor is free of the regression coefficients and depends on the data only through the censoring pattern. The effect of covariates estimated by any proportional hazards model can thus be reported as hazard ratios.

The likelihood of the event to be observed occurring for subject i at time Y_i can be written as:

${\displaystyle L_{i}(\beta )={\frac {\lambda (Y_{i}\mid X_{i})}{\sum _{j:Y_{j}\geq Y_{i}}\lambda (Y_{i}\mid X_{j})}}={\frac {\lambda _{0}(Y_{i})\theta _{i}}{\sum _{j:Y_{j}\geq Y_{i}}\lambda _{0}(Y_{i})\theta _{j}}}={\frac {\theta _{i}}{\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}}},}$

where θ_j = exp(X_j ⋅ β) and the summation is over the set of subjects j where the event has not occurred before time Y_i (including subject i itself). Obviously 0 < L_i(β) ≤ 1. This is a partial likelihood: the effect of the covariates can be estimated without the need to model the change of the hazard over time.

Treating the subjects as if they were statistically independent of each other, the joint probability of all realized events[6] is the following partial likelihood, where the occurrence of the event is indicated by C_i = 1:

${\displaystyle L(\beta )=\prod _{i:C_{i}=1}L_{i}(\beta ).}$

The corresponding log partial likelihood is

${\displaystyle \ell (\beta )=\sum _{i:C_{i}=1}\left(X_{i}\cdot \beta -\log \sum _{j:Y_{j}\geq Y_{i}}\theta _{j}\right).}$

This function can be maximized over β to produce maximum partial likelihood estimates of the model parameters.

The partial score function is

${\displaystyle \ell ^{\prime }(\beta )=\sum _{i:C_{i}=1}\left(X_{i}-{\frac {\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}X_{j}}{\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}}}\right),}$

and the Hessian matrix of the partial log likelihood is

${\displaystyle \ell ^{\prime \prime }(\beta )=-\sum _{i:C_{i}=1}\left({\frac {\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}X_{j}X_{j}^{\prime }}{\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}}}-{\frac {\left[\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}X_{j}\right]\left[\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}X_{j}^{\prime }\right]}{\left[\sum _{j:Y_{j}\geq Y_{i}}\theta _{j}\right]^{2}}}\right).}$

Using this score function and Hessian matrix, the partial likelihood can be maximized using the Newton-Raphson algorithm. The inverse of the Hessian matrix, evaluated at the estimate of β, can be used as an approximate variance-covariance matrix for the estimate, and used to produce approximate standard errors for the regression coefficients.

Several approaches have been proposed to handle situations in which there are ties in the time data. Breslow's method describes the approach in which the procedure described above is used unmodified, even when ties are present. An alternative approach that is considered to give better results is Efron's method.[7] Let t_j denote the unique times, let H_j denote the set of indices i such that Y_i = t_j and C_i = 1, and let m_j = |H_j|. Efron's approach maximizes the following partial likelihood.

${\displaystyle L(\beta )=\prod _{j}{\frac {\prod _{i\in H_{j}}\theta _{i}}{\prod _{\ell =0}^{m_{j}-1}\left[\sum _{i:Y_{i}\geq t_{j}}\theta _{i}-{\frac {\ell }{m_{j}}}\sum _{i\in H_{j}}\theta _{i}\right]}}.}$

The corresponding log partial likelihood is

${\displaystyle \ell (\beta )=\sum _{j}\left(\sum _{i\in H_{j}}X_{i}\cdot \beta -\sum _{\ell =0}^{m_{j}-1}\log \left(\sum _{i:Y_{i}\geq t_{j}}\theta _{i}-{\frac {\ell }{m_{j}}}\sum _{i\in H_{j}}\theta _{i}\right)\right),}$

the score function is

${\displaystyle \ell ^{\prime }(\beta )=\sum _{j}\left(\sum _{i\in H_{j}}X_{i}-\sum _{\ell =0}^{m_{j}-1}{\frac {\sum _{i:Y_{i}\geq t_{j}}\theta _{i}X_{i}-{\frac {\ell }{m_{j}}}\sum _{i\in H_{j}}\theta _{i}X_{i}}{\sum _{i:Y_{i}\geq t_{j}}\theta _{i}-{\frac {\ell }{m_{j}}}\sum _{i\in H_{j}}\theta _{i}}}\right),}$

and the Hessian matrix is

${\displaystyle \ell ^{\prime \prime }(\beta )=-\sum _{j}\sum _{\ell =0}^{m_{j}-1}\left({\frac {\sum _{i:Y_{i}\geq t_{j}}\theta _{i}X_{i}X_{i}^{\prime }-{\frac {\ell }{m_{j}}}\sum _{i\in H_{j}}\theta _{i}X_{i}X_{i}^{\prime }}{\phi _{j,\ell ,m_{j}}}}-{\frac {Z_{j,\ell ,m_{j}}Z_{j,\ell ,m_{j}}^{\prime }}{\phi _{j,\ell ,m_{j}}^{2}}}\right),}$

where

${\displaystyle \phi _{j,\ell ,m_{j}}=\sum _{i:Y_{i}\geq t_{j}}\theta _{i}-{\frac {\ell }{m_{j}}}\sum _{i\in H_{j}}\theta _{i}}$

${\displaystyle Z_{j,\ell ,m_{j}}=\sum _{i:Y_{i}\geq t_{j}}\theta _{i}X_{i}-{\frac {\ell }{m_{j}}}\sum _{i\in H_{j}}\theta _{i}X_{i}.}$

Note that when H_j is empty (all observations with time t_j are censored), the summands in these expressions are treated as zero.

Below are some worked examples of the Cox model in practice.

Suppose the endpoint we are interested in is patient survival during a 5-year observation period after a surgery. Patients can die within the 5-year period, and we record when they died, or patients can live past 5 years, and we only record that they lived past 5 years. The surgery was performed at one of two hospitals, A or B, and we would like to know if the hospital location is associated with 5-year survival. Specifically, we would like to know the relative increase (or decrease) in hazard from a surgery performed at hospital A compared to hospital B. Provided is some (fake) data, where each row represents a patient: T is how long the patient was observed for before death or 5 years (measured in months), and C denotes if the patient died in the 5-year period. We have encoded the hospital as a binary variable denoted X: 1 if from hospital A, 0 from hospital B.

Our single-covariate Cox proportional model looks like the following, with ${\displaystyle \beta _{1}}$ representing the hospital's effect, and i indexing each patient:

${\displaystyle \overbrace {\lambda (t|X_{i})} ^{\text{hazard for i}}=\underbrace {\lambda _{0}(t)} _{{\text{baseline}} \atop {\text{hazard}}}\cdot \overbrace {\exp(\beta _{1}X_{i})} ^{\text{scaling factor for i}}}$

Using statistical software, we can estimate ${\displaystyle \beta _{1}}$ to be 2.12. The hazard ratio is the exponential of this value, ${\displaystyle \exp(\beta _{1})=\exp(2.12)}$. To see why, consider the ratio of hazards, specifically:

${\displaystyle {\frac {\lambda (t|X=1)}{\lambda (t|X=0)}}={\frac {{\cancel {\lambda _{0}(t)}}\exp(\beta _{1}\cdot 1)}{{\cancel {\lambda _{0}(t)}}\exp(\beta _{1}\cdot 0)}}=\exp(\beta _{1})}$

Thus, the hazard ratio of hospital A to hospital B is ${\displaystyle \exp(2.12)=8.32}$. Putting aside statistical significance for a moment, we can make a statement saying that patients in hospital A are associated with a 8.3x higher risk of death occurring in any short period of time compared to hospital B.

There are important caveats to mention about the interpretation:

1. a 8.3x higher risk of death does not mean that 8.3x more patients will die in hospital B: survival analysis examines how quickly events occur, not simply whether they occur.
2. More specifically, "risk of death" is a measure of a rate. A rate has units, like meters per second. However, a relative rate does not: a bicycle can go two times faster than another bicycle (the reference bicycle), without specifying any units. Likewise, the risk of death (rate of death) in hospital A is 8.3 times higher (faster) than the risk of death in hospital B (the reference group).
3. the inverse quantity, ${\displaystyle 1/8.32={\frac {1}{\exp(2.12)}}=\exp(-2.12)=0.12}$ is the hazard ratio of hospital B relative to hospital A.
4. We haven't made any inferences about probabilities of survival between the hospitals. This is because we would need an estimate of the baseline hazard rate, ${\displaystyle \lambda _{0}(t)}$, as well as our ${\displaystyle \beta _{1}}$ estimate. However, standard estimation of the Cox proportional hazard model does not directly estimate the baseline hazard rate.
5. Because we have ignored the only time varying component of the model, the baseline hazard rate, our estimate is timescale-invariant. For example, if we had measured time in years instead of months, we would get the same estimate.
6. It is tempting to say that the hospital caused the difference in hazards between the two groups, but since our study is not causal (that is, we do not know how the data was generated), we stick with terminology like "associated".

To demonstrate a less traditional use case of survival analysis, the next example will be an economics question: what is the relationship between a companies' price-to-earnings ratio (P/E) on their 1-year IPO anniversary and their future survival? More specifically, if we consider a company's "birth event" to be their 1-year IPO anniversary, and any bankruptcy, sale, going private, etc. as a "death" event the company, we'd like to know the influence of the companies' P/E ratio at their "birth" (1-year IPO anniversary) on their survival.

Provided is a (fake) dataset with survival data from 12 companies: T represents the number of days between 1-year IPO anniversary and death (or an end date of 2022-01-01, if did not die). C represents if the company died before 2022-01-01 or not. P/E represents the companies price-to-earnings ratio at their 1-year IPO anniversary.

Unlike the previous example where there was a binary variable, this dataset has a continuous variable, P/E. However, the model looks similar:

${\displaystyle \lambda (t|P_{i})=\lambda _{0}(t)\cdot \exp(\beta _{1}P_{i})}$

where ${\displaystyle P_{i}}$ represents a company's P/E ratio. Running this dataset through a Cox model produces an estimate of the value of the unknown ${\displaystyle \beta _{1}}$, which is -0.34. Therefore, an estimate of the entire hazard is:

${\displaystyle \lambda (t|P_{i})=\lambda _{0}(t)\cdot \exp(-0.34P_{i})}$

Since the baseline hazard, ${\displaystyle \lambda _{0}(t)}$, was not estimated, the entire hazard is not able to be calculated. However, consider the ratio of the companies i and j's hazards:

${\displaystyle {\begin{aligned}{\frac {\lambda (t|P_{i})}{\lambda (t|P_{j})}}&={\frac {{\cancel {\lambda _{0}(t)}}\cdot \exp(-0.34P_{i})}{{\cancel {\lambda _{0}(t)}}\cdot \exp(-0.34P_{j})}}\\&=\exp(-0.34(P_{i}-P_{j}))\end{aligned}}}$

All terms on the right are known, so calculating the ratio of hazards between companies is possible. Since there is no time-dependent term on the right (all terms are constant), the hazards are proportional to each other. For example, the hazard ratio of company 5 to company 2 is ${\displaystyle \exp(-0.34(6.3-3.0))=0.33}$. This means that, within the interval of study, company 5's risk of "death" is 0.33 ≈ 1/3 as large as company 2's risk of death.

There are important caveats to mention about the interpretation:

1. The hazard ratio is the quantity ${\displaystyle \exp(\beta _{1})}$, which is ${\displaystyle \exp(-0.34)=0.71}$ in the above example. From the last calculation above, an interpretation of this is as the ratio of hazards between two "subjects" that have their variables differ by one unit: if ${\displaystyle P_{i}=P_{j}+1}$, then ${\displaystyle \exp(\beta _{1}(P_{i}-P_{j})=\exp(\beta _{1}(1))}$. The choice of "differ by one unit" is convenience, as it communicates precisely the value of ${\displaystyle \beta _{1}}$.
2. The baseline hazard can be represented when the scaling factor is 1, i.e. ${\displaystyle P=0}$.

${\displaystyle \lambda (t|P_{i}=0)=\lambda _{0}(t)\cdot \exp(-0.34\cdot 0)=\lambda _{0}(t)}$

Can we interpret the baseline hazard as the hazard of a "baseline" company whose P/E happens to be 0? This interpretation of the baseline hazard as "hazard of a baseline subject" is imperfect, as it is possible that the covariate being 0 is impossible. In this application, a P/E of 0 is meaningless (it means the company's stock price is 0, i.e., they are "dead"). A more appropriate interpretation would be "the hazard when all variables are nil".

1. It is tempting to want to understand and interpret a value like ${\displaystyle \exp(\beta _{1}P_{i})}$ to represent the hazard of a company. However, consider what this is actually representing: ${\displaystyle \exp(\beta _{1}P_{i})=\exp(\beta _{1}(P_{i}-0))={\frac {\exp(\beta _{1}P_{i})}{\exp(\beta _{1}0)}}={\frac {\lambda (t|P_{i})}{\lambda (t|0)}}}$. There is implicitly a ratio of hazards here, comparing company i's hazard to an imaginary baseline company with 0 P/E. However, as explained above, a P/E of 0 is impossible in this application, so ${\displaystyle \exp(\beta _{1}P_{i})}$ is meaningless in this example. Ratios between plausible hazards are meaningful, however.