# Difficulty calculating probability of a complicated event

#### Pharmastat

##### New Member
Trying to calculate this statistical issue for work, not sure if I am over complicating or what - but certainly need some help in solving. For whoever can help, if it can be done on Excel, that would be extremely appreciated.

I am looking at the clinical risk of a particular drug being taken off the market by the FDA. The purpose of this exercise is to apply it to a financial model and get a risk adjusted NPV. Specifically, I am looking at the chances of the drug being taken off market each year, applying that chance to the net present value of that year, and summing those up to get the value of the drug.

There are two actions the FDA can take to remove the drug from the market. The first is a "summary judgment" which would remove the drug immediately. For simplicity, this action can only happen at the end of the year. The second action they can take is request a hearing. Again, for simplicity, the hearing would be requested at the end of the year. When a hearing is requested, there are multiple possibilities of when the hearing would conclude, each with their own probability. Finally, when the hearing does conclude, there is a chance that the conclusion would be to remove the drug. If a hearing is requested, no summary judgment can be made. If a hearing is concluded positively, the drug will stay on until the end. By the last period (19) the drug will definitely be taken off the market regardless.

So these are the actual probabilities I am currently using, for each period (1-19), respectively:
Summary judgment: 5% (every year)
Hearing: 10%, 20%, 30%, 20%, 10% (from period 5 to 19)
Conclusion being reached after x years: 0% (after 1 year), 25%, 50%, 25%, 0% (I left the zeroes in so I could move these percentages around)
Conclusion removing drug from market: 90%

So for a hearing, the first year the drug can possibly be taken off market due to the conclusion of a hearing is period 3 (hearing requested in year 1, 0% 1 year after that, and 25%*90% chance 2 years after that - in period 3).

Note that these probabilities are independent, but should be made dependent (at least in my mind). Meaning, since you cannot remove the drug more than once, then the chance of the drug being removed in the second year, is the chance it is not removed in the first times the chance it is removed in the second. So again, these are only the chances of it being removed each year.

I think the complication for me is that the chance of a summary judgment being made is the chance a hearing is not requested cumulatively prior (as opposed to concluded) time the chance a summary judgment hasn't been made cumulatively prior. I may be wrong on this, but I feel pretty confident.

So one last example for illustrative purposes:
Period 1: total chance of being taken off market is 5% = Summary judgment
Period 2: total chance of being taken off market is (1-10%)*(1-5%)*5% = summary judgment times the chance there was not a hearing requested and no summary judgment the period prior
Period 3: total chance of being taken off market is (1-10%)*(1-20%)*(1-5%)^2*5%+10%*25%*90% = summary judgment in period 3 times the chance there was no hearing requested in first 2 years and no summary judgment in first 2 years plus the chance the drug is taken off market due to a hearing being requested in year one times the chance the hearing was concluded in 2 periods times the chance the conclusion was negative
Period 4: verbally ... the chance nothing has happened (hearings nor summary judgments in 3 periods prior) times the chance of summary judgment in period 4 plus the chance there was a hearing in period one but concluded in 3 periods plus the chance there was a hearing requested in period 2 and concluded in 2 periods both times the chance the hearing is negative and an adjusted chance (basically 1- (5%*25%)) that a summary judgment wasn't reached in the first year (the adjustment is to compensate for the fact that it only effects the second hearing period but not the first, but that the hearings can still be ongoing ...)

so these first 4 percentages come out to ... chance of being removed in that year:
1: 5%
2: 4.28%
3: 5.50%
4: 10.85%

I have attached a censored excel for the work I've done on this so far for anyone who wants to give it a shot!

#### ted00

##### New Member
complicated, indeed
I'm not sure to what extent some of the events are statistically independent, e.g. are the events {not taken off market year k} and {summary judgement in year k+1} truly independent?

have you considered the possibility of couching this in terms of a survival model or life table analysis context?

#### Pharmastat

##### New Member
complicated, indeed
I'm not sure to what extent some of the events are statistically independent, e.g. are the events {not taken off market year k} and {summary judgement in year k+1} truly independent?

have you considered the possibility of couching this in terms of a survival model or life table analysis context?
I think a large part of the complication is really thinking that through (in regards to independence). I also think a part of that problem is because I am saying both events happen at the end of the year - and also because I can't really read the minds of the FDA. What I mean is, realistically, these decisions (hearing or judgment) can be made any day of the year. When they are made - there is nothing else to be done (except in the case of the hearing of waiting for the result). But, from today's perspective, we don't know what will come first or at all - and so from that perspective, I feel you just need to take the average chance these things will happen (taken off market) or not happen (continue to stay on market).

As for a survival analysis and life table analysis, I am not too familiar with them, but after looking them up I can say I sort of already have that in a way, unless I am misunderstanding. You can see the bottom part - I have the individual cash flows (as opposed to total NPV) and I multiply those by the chances you will make it to that year. So, basically, the result is instead of saying that the cash flows have an X% chance of ending that year, it is kind of saying there is a (1-X%) chance of making it to this year and this is the additional cash you will get from that year. The result ends up being the same - I am not sure if that is a good thing or a bad thing (as there may be a flaw in my line of thought).

Again, I think I may be over-thinking things, but there is a chance I am not and was actually correct in my analysis - I just hope someone can correct or verify that (or even, recreate it).

I am going to attach a "decision tree" here, maybe this will help everyone understand what the breakoff points are.

#### ted00

##### New Member
I agree with you
I did a life table analysis once, not saying I'm world's leading expert in it or anything, but when I read your description it did kind of seem to line up
Especially if you're taking the route that these events happen -- for practical purposes -- at the end of a year
There could be justification for this, e.g. we're dealing with yearly finances

Can I try to summarize your tree, and you correct where/if I have something wrong? That's a nice figure, it helps. Starting at any time period k there are 5 possibilities given there's "survival" at time k:
(A) survive to period k+1
(B) death in period k+1
(C) death in period k+3
(D) death in period k+4
(E) death in period k+5
There's a probability number on each of the (C), (D), and (E) events in your figure, I'm still confused why

#### Pharmastat

##### New Member
I agree with you
I did a life table analysis once, not saying I'm world's leading expert in it or anything, but when I read your description it did kind of seem to line up
Especially if you're taking the route that these events happen -- for practical purposes -- at the end of a year
There could be justification for this, e.g. we're dealing with yearly finances

Can I try to summarize your tree, and you correct where/if I have something wrong? That's a nice figure, it helps. Starting at any time period k there are 5 possibilities given there's "survival" at time k:
(A) survive to period k+1
(B) death in period k+1
(C) death in period k+3
(D) death in period k+4
(E) death in period k+5
There's a probability number on each of the (C), (D), and (E) events in your figure, I'm still confused why
If I understand the question - you are asking - when there is a hearing, why are there probabilities that it will take different lengths of time to conclude? Maybe restating it like that helps you see the answer, but more directly, we do not know how long the hearing will take, but we do know it will take 2-4 years after the hearing is requested, with the 3rd year being the most probable - the 25%, 50%, 25% is somewhat arbitrary, but fitting for my current purposes. Just to go a step further - at the end of each of those probabilities, there is a 90% chance that the drug will be taken off market. The reason I have the tree drawn in this manner is - if a hearing is requested, then in the years between that request and the distribution of the possible years the hearing is concluded, there cannot be a summary judgment.

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#### Pharmastat

##### New Member
Just looking at it again and trying a different approach, I saw an error in my drawing - there aren't 3 branches after a hearing request, but rather, 1 branch with 3 points where it might end. Not sure if that makes a difference - I can't redraw it right now.

#### Pharmastat

##### New Member
I just wanted to update with my latest model for probabilities. I actually feel really good about this one, the logic seems to flow a bit better - and I did a few extra steps just to help with the display of logic.

So what I have are the base probabilities per year
The chance that neither of the two primary events happen
The chance the two primary events do happen in any given year effected by previous events (essentially the events not happening in years prior)
The chance the hearing happens and how that would take it off market
The chance the hearing happens and how it staying on market would effect the value (the biggest change)
The total chance in any given year the drug is taken off market
The total chance it makes it to the end and achieves full value (mostly as a result of the positive hearing)

Just to restate one more time how the two events effect one another:
1. A hearing is requested OR a summary judgment can be given - the inverse of the combined probabilities of these events is the chance nothing would happen
2. A hearing request after the first year would only happen if there were no hearings requested in years prior and no summary judgments given
3. A summary judgment would happen for the same reasons - no previous summary judgments and no hearing requested (since one is never requested, one cannot be active either)
4. No matter what happens, the drug cannot proceed past the last period (now 18.5)