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When dealing with Survival Analysis, we create models that estimate two properties:

  • Survival: Survival Probabilities tend to be more straightforward to understand. Survival Probabilities estimate the probability of surviving some "event" past a certain time, given that you have survived up until that time.

  • Hazard: On the other hand, Hazard Rate is said to represent the "instantaneous rate of experiencing the event" at any given time.

I have the following question: When we work with on Survival Analysis data, what kinds of problems are better answered using "Survival Probabilities" and what kinds of problems are better answered using "Hazard Rates"?

For instance, I could use Survival Probabilities to estimate out which cohorts of patients are more likely to survive a certain "event" (e.g. some disease) - and I could then use Hazard Rates to estimate which of these same cohorts are more likely to experience this "event" at different times. I am not a biomedical researcher by background, but both of these quantities seem equally important to me. But I was hoping someone could help me better understand this question.

Usually in biomedical studies, which of these two quantities tends to be "more important" - are we ever able to interpret both of these quantities simultaneously? In short, which kinds of problems are better answered using "Survival Probabilities" and what kinds of problems are better answered using "Hazard Rates"?

Thanks!

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2 Answers 2

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The survival function and hazard function are directly related. In continuous time:

$$h(t) = - \frac{S'(t)}{S(t)} .$$

So "answering a problem" in terms of survival probabilities is equivalent to doing so in terms of hazard rates. Which you work with first depends on the type of model.

If a proportional hazards (PH) assumption holds, then you can work first in terms of hazards with a semi-parametric Cox PH survival model. Under the PH assumption only the relative hazards are affected by covariate values; the underlying shape of the baseline survival curve doesn't matter. A Cox PH regression model thus doesn't model the survival curve directly. It estimates the relative hazards as functions of covariates under a PH assumption. Hazard ratios then provide compact summaries of the associations of covariates with outcome, without considering the survival curve per se. My impression is that Cox models are predominant in clinical survival analysis. Once the Cox model is fit, however, you can estimate corresponding survival curves.

With a fully parametric survival model that involves censored or truncated survival-time data, you could say that model fitting is done more in terms of the survival function. This page shows the forms of the likelihoods associated with each type of data that are used to fit the model, expressed in terms of the survival function and the event density, $f(t)=-S'(t)$. If the parametric functional form isn't maintained under a PH assumption, then relative hazards aren't constant over time and interpretation of results in terms of hazards is less straightforward. Such models (e.g., log-normal models) can instead follow an accelerated failure time interpretation, in which covariates stretch or compress the time axis of a single underlying survival curve.

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  • $\begingroup$ Could you clarify what S' is? how does it differ from S? $\endgroup$
    – Wojty
    Nov 8, 2023 at 13:37
  • $\begingroup$ @Wojty the ' is a standard symbol for the first derivative of a function of a single variable. In this case, $S'(t)=dS(t)/dt$. $\endgroup$
    – EdM
    Nov 8, 2023 at 14:42
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Hazard models are highly flexible methods that produce summary measures of differences in the time to event.

Using the hazard function, we can create multivariable regression models with independent variables of any form. Such models can be extended to accommodate situations that include repeated measures, stratification, competing outcomes, or time-dependent covariates. The resulting models can be used to create plots of any form (eg, cumulative hazard, instantaneous hazard, or survival) or to generate probabilities or rates that can be used in an array of applications.

The cumulative hazard can exceed a value of 1 (it’s a rate), and some people find it unsettling or nonintuitive to see graphs that exceed a rate of 1.0 event/time.

Survival models depict the empiric time to event data and can provide a test of the difference in group survival.

Survival analysis (traditional Kaplan-Meier analysis) is a much more restrictive approach to modeling time to event data, but it models or displays the difference between distributions as they are (ie, does not “transform” the time to event data). Traditional Kaplan-Meier analysis can only compare groups and can only make a determination on the probability of seeing the observed data assuming all groups were sampled from the same distribution (ie, provides a p-value, but does not provide measures of relative risk). When all of the work to reduce bias and confounding is done with the experiment's design, as in a randomized trial, traditional survival analysis can provide an excellent measure of the differences between survival at some time point along with confidence intervals and a p-value.

Survival estimates are bounded by 1 and 0; most people find these measures intuitive and easy to interpret in graphical form.

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