#StackBounty: #statistical-significance #mathematical-statistics #predictive-models #survival #proportional-hazards How to calculate co…

Bounty: 50

I am looking for a statistical method to calculate the conditional hazard before the diagnoses or study has started; in other words, time is not a variable.

For example, here is a hypothetical graph:

enter image description here

where

x is a continuous predictor/covariate i.e.: humidity

damage is analogous to death i.e.: a banana going completely brown

h(damage|x) is the instantaneous hazard rate given $x$ is at a specific value i.e.: the hazard rate that a banana goes fully brown given that humidity = 6 g/m^3

Here is a motivating image

enter image description here

So, the goal is to statistically quantify the force that humidity exerts on the banana to make it turn brown. A specific level might make the banana go immediately brown (instant death) and some other level has no effect or even preserving effect. Once again, this is to be determined before we begin the survival study (in time) for one or many bananas. This is very important.

All my online searches online take me to Cox Proportional Model which is not what I am looking for.

One promising formula is the hazard ratio from probability as shown in this wiki

$$HR=frac{p}{1-p}$$

However, $p$ is left undefined. Is it some kind of conditional probability? Does it apply to other models other than Cox’s..? I don’t know but I like the formula since it gives me hope that I can use basic probability methods like the maximum likelihood, some parametric model, etc.

Edit:
I found a formula in a paper which they also called odds ratio and this makes more sense

$$O(t)=frac{F(t)}{S(t)}$$

where $F(t)$ and $S(t)$ are the the cumulative and survival function respectively.

Also, I believe at time zero there is no conditioning; hence, the hazard reduces to the probability of dying in the first $delta t$

Another thought is to let $HR$ itself be a random variable, take enough measurements (not sure if it is possible to measure an HR observation) to fit a distribution, then derive the the continuous conditional hazard at each level.

Any help would be greatly appreciated.


Get this bounty!!!

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