Survival analysis for different diseases on same patients

I want to apply survival analysis on UFC-fights. Each fighter represents a "disease" and each knock-out is a "death". Each UFC fight consists of a number of rounds and the number of rounds corresponds to the time alive. There are basically three different possible outcomes per fight:

• Fighter KO’s opponent in round r - event is observed in round r
• Opponent KO’s fighter in round r - event is right-censored in round r
• Fighter or opponent wins by jury decision - event is right-censored in round r = 3 ( normal fight) or r = 5 ( title fight)

This looks for a single fighter as follows:

The goal is to determine the conditional probability of opponent o getting KO’ed when fighting against fighter f in round r give that he/she has survived until that round (a.k.a. the hazard rate):

The person period data looks as follows:

I assume that because fighters are fighting against each other within their weight classes, there is a lot of information that can borrowed between the opponents. Therefore, I intend to use (bayesian) discrete survival analysis. Singer and Willet (1993) show the likelihood function of this is the same as logistic regression / likelihood function for N independent Bernoulli trials with parameter λ (the hazard rate).

Is anyone aware of similar papers where the methodology consist of different diseases and more or less same patient pool? There are no competing risks between the fighters / diseases because there is enough rest between the fights (sometimes months). Although, I understand that it is difficult to die more than once, can please someone point me out how to determine the "deadliest" fighter?

• What is UFC? care to explain? Commented May 29, 2021 at 3:23
• It's the name of the competition and stands for ultimate fighting championship. Commented May 31, 2021 at 6:37

You could consider an approach similar to this modeling of football (soccer) matches. There, a simple additive model for the number of goals scored includes the team (as one predictor), whether the team is home or away, and the opponent (another predictor). Then sampling from the model predictions gives the probabilities of wins or draws.

This approach should move fairly easily into the logistic regression model you envision for your discrete-time survival analysis. You would include the fighter, the opponent, the round, and probably the type of match (3-round versus 5-round) as predictors, plus any other covariates that might be relevant. For example: Is there some equivalent of "home turf" for a fighter? Might weight within the weight class matter? Might the time elapsed since a previous fight matter? Do some fighters tire faster than others, so that you need to include an interaction of fighter with round in the model?

It seems most straightforward to code the event as the fighter in question losing rather than winning. It's the terminal events, not the censored values, that provide information. For completeness, you might want to add a virtual 4th or 6th round representing the result of a decision. Then the "deadliest" fighter would be the one with the lowest probability of losing.

There might be a problem with modeling and predicting 5-round title fights, as presumably not all fighters have been in such fights. That will make predictions for first-time title fighters unavailable directly from the above type of model. You'd have to apply your knowledge of the subject matter to decide how to extend results on 3-round fights to potential outcomes of 5-round fights, for fighters who haven't been in one.