Question Iso-risk decompression schedules

[TooCold]

"(...) , before a diver sets out to perform a repetitive series of three dives, each with 2.3 % PDCS , the risk of DCS on at least one of the dives is the binomial probability of no DCS on all dives and is (...) 0.067.

Something missing here.

The probability of DCS on at least one of the dives is (1 - the probability of no DCS on all dives); probably what the paper said, or meant.
This is 1 - (3 * 0.977), which is indeed 0.067, or 6.7%. But this analysis assumes statistical independence of the events, like you're going to toss a coin 3 times no matter what the outcome of the earlier tosses were. But is the probability of getting bent on the second dive really independent of whether you got bent on the first dive a few hours earlier?

 

[dmaziuk]

But is the probability of getting bent on the second dive really independent of whether you got bent on the first dive a few hours earlier?

I don't have a cite but statistically it's the other way around: most "incidents" happen on the first day of a dive trip and the first dive of the day.

 

[Alekseolsen]

I don't have a cite but statistically it's the other way around: most "incidents" happen on the first day of a dive trip and the first dive of the day.

Hmm, I seem to believe that TooVold meant that the second dive wouldn't happen if you get DCS on the first dive, thus the outcome of dive 1 and 2 affects the probability of DCS from dive 3, as dive 3 won't happen if DCS occurs in the two previous dives.

 

[dmaziuk]

Yes, it's called The Gambler's Fallacy.

 

[inquis]

the second dive wouldn't happen if you get DCS on the first dive

It works out the same. There's a 2.3% chance of DCS on dive 1, a 2.3% chance on dive 2 given that you make it, but only a 97.7% chance of making it, so a net 2.2% chance. Etc.: 2.3 + 97.7*2.3 + 97.7*97.7*2.3 = 6.7%

The 1-0.977^3 is an equivalent shortcut.

My take was that @TooCold meant the residual loading on dive 2 increases the risk. Which it might, but there are other factors that might override that; e.g., rusty skills on dive 1, but better control on 2 & still better on 3.

 

[Kendall Raine]

It works out the same. There's a 2.3% chance of DCS on dive 1, a 2.3% chance on dive 2 given that you make it, but only a 97.7% chance of making it, so a net 2.2% chance. Etc.: 2.3 + 97.7*2.3 + 97.7*97.7*2.3 = 6.7%

The 1-0.977^3 is an equivalent shortcut.

My take was that @TooCold meant the residual loading on dive 2 increases the risk. Which it might, but there are other factors that might override that; e.g., rusty skills on dive 1, but better control on 2 & still better on 3.

This illustrates the best reason to not dive to the schedule limit (Pdcs of 3.5%, 2.3%, 1.5% or whatever) but to back off to lower Pdcs considerably. Using SAUL for this is a way to have "fun with numbers."

I agree that DCS is likely not an independent event, even if the repets are correct(?!). In addition to your listed factors are things like: 1) potential complement system depletion (reduced "activation" from VGE) over successive dives resulting in "adaptation" or "acclimatization," 2) improved sleep/hydration after a long journey and 3) lower emotional stress as one gets back into the diving grove/familiar with conditions, etc. I have noticed my Doppler scores are generally highest/more volatile in the early part of a successive series of similar dives (multiple dives/day across multiple dive days).

Consistent with:

Acclimatization

Acclimatization is the adaptation to natural exposure, but could this apply and alter decompression stress? Read more about acclimatization and diving.

dan.org

 

[TooCold]

TLDR: this is all pretty deep in the weeds, so just skip it.

Note that the quote from the study carefully says "... series of three dives, each with 2.3 % PDCS ..." If we interpret this literally, that the dives in this scenario by definition all have 2.3% PDCS, then the dives are indeed statistically independent (by definition - since nothing at all can condition the PDCS of any of the dives, never mind what might have happened on an earlier dive). So given that assumption, the calculation (which requires independence) is appropriate.

But what I asked is "is the probability of getting bent on the second dive really independent of whether you got bent on the first dive a few hours earlier?" I think the actual probability of getting bent on a later dive does depend on whether or not you just got bent on an earlier dive the same day. If true, then the dives in a repetitive series are not statistically independent. That's all I was getting at in my earlier post.

For the record, I'm not criticising the study, the methodology, the research program, the investigators, any of it. My stats persona just hickupped at the implicit assumption of and requirement for independence when my gut says they're not.

It works out the same. There's a 2.3% chance of DCS on dive 1, a 2.3% chance on dive 2 given that you make it, but only a 97.7% chance of making it, so a net 2.2% chance. Etc.: 2.3 + 97.7*2.3 + 97.7*97.7*2.3 = 6.7%

The 1-0.977^3 is an equivalent shortcut.

Exactly. I was fleshing out the calculation as the quote described it.
And thanks for correcting my slip of the brain - I wrote (1 - 3 * 0.977). That's a ludicrous number for a probability.

My take was that @TooCold meant the residual loading on dive 2 increases the risk. Which it might, but there are other factors that might override that; e.g., rusty skills on dive 1, but better control on 2 & still better on 3.

I agree, lots of other factors. But I wasn't thinking of the residual loading, but the case where dive 1 results in DCS (since the described scenario models dives with a binary outcome: bent or not bent). I didn't make any assumptions about the direction of the influence, but not surprisingly I do indeed think that continuing to dive right after getting bent increases the risk.

 

[dmaziuk]

Residual loading only matters if you treat each subsequent dive as the 1st dive of the series (in which case you shall receive what you ask for). If you're running a multi-dive schedule, residual loading is accounted for, and the probability remains at 2.3%.