"Survival" vs. "Hazard" : When to Use Which? When dealing with Survival Analysis, we create models that estimate two properties:

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*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!

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.
