understanding stats and probability - discussion and tutorials

[Fundy]

That 20% figure I guess was actually based on people that were hospitalized, thus 20% of those hospitalized died. But if it were based on a per
person, I think your expression should be correct. But also for some reason I still believe 20% of 65 if it were a 100 person sample would have to hold true. Or those 1%,2% numbers would prove false.

Hey, try doing the math in that example for the next five year period of someone at age 70, using the 2% event rate with a 20% mortality.

(1-.996^5)=.0198 or almost 2 people in one hundred would die every year for that period.

What's up with that?

I think it violates the binomial nature required for use with the equation somehow.
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No, that doesn't express what I thought it would.
Probability of dying all five years is .004^5 which is basically zero. Not sure what I thought it would express. At the moment.

But the independence necessary is definitely violated. The years rates following each other depend on whether or not a mortality happened. so 2% in year two isn't 2% if year 1 was fatal. In roulette each of the spins is independent from the other, but its not when an event can have something like mortality associated with it that affect the other year probability rates.

I think the algorithm needs to be either more complicated or maybe just differ different altogether. My understanding of both stat and probability algorithms needs way too big of a refresher to tell what sort of adjustment needs done to adjust for bleed event probabilities lack of independence.

 

[newmitral]

Hi Fundy,

Hey, try doing the math in that example for the next five year period of someone at age 70, using the 2% event rate with a 20% mortality.
(1-.996^5)=.0198 or almost 2 people in one hundred would die every year for that period.
What's up with that?

The above calculation is the probability of ONE person NOT surviving for FIVE years.
That works out to about 1.98% with your assumptions of 2% event rate of which 20% are fatal.
It doesn't mean that 2 people will die each year.

On average, with a very large number of people, you would expect about 2 deaths out of 100 every 5 years, which if you take an average of 1.98%/5 which is about 0.4% (0.00396 probability) each year. This is, in fact, roughly the same probability of fatal event you started with (0.02 * 0.2 = 0.004). So, it basically agrees.

Probability of dying all five years is .004^5 which is basically zero. Not sure what I thought it would express.

The reason that you can't use the probability of dying each year every year for 5 years is that you only die once (unless you are a coward in which case it is claimed that you can die a thousand deaths, or a cat which is rumored to have 9 lives). So, that calculation is meaningless in this context. You have to use the probability of survival for each of the 5 years, which you get by subtracting the probability of dying from 1.00

So, for 5 years, if you go with the 2% event rate and 20% fatality, you are back to .004 probability of dying each year, or 0.996 probability of surviving each year. The combined probability of surviving 5 years is then the product of the probabilities of surviving each year, or
0.996^5 = 0.980159 or about 98.02%
so the probability of dying in 5 years is 0.01984 or about 1.98%

There is a difference between the average deaths per year for multiple years, and the probability of dying each year. For small probability numbers, the difference is small, but if the probability of death is very high, you can more clearly see that it is wrong to use the average number of deaths each year as an individual probability.
Do the calculation with a 50% fatality rate starting with 3200 people, and you'll see the difference very clearly.

Starting Population = 3200 people
After 1st year = 1600 people
After 2nd year = 800 people
After 3rd year = 400 people
After 4th year = 200 people
After 5th year = 100 people

So, at a 50%/year mortality rate, starting with 3200 people, you lose 3100 people in 5 years. If you simply average that over 5 years, you get 3100/5 = 620/year. For the first year, this is only 620/3200 = 19.3%
For Years 3 through 5 it is even more obvious that you can't just take the average deaths because you have more deaths than you have people. So, hopefully this illustrates that the average number of deaths/year is quite different from the probability of death each year, which was defined to be a constant 50% in the above example.

If you read a medical paper that gives you a probability of an adverse event occurring to one person, this is NOT the same as an average number of events per year among a group of 100 people.for several years.

 

[ElectLive]

Wow...this is a fun thread (in a crazy sort of way of course)! It kind of reminds me of college, though, desperately searching every brain cell for the way too complicated formula on the hardest question on the exam, feeling the amazing satisfaction of actually remembering it, only to not be sure what to plug in where once I had it in hand! :biggrin2:

I just posted on the sister thread to this and some of that applies here, particularly the part about study flaws and sample size. I mentioned the FDA 800 patient year rule, and often as small as a 400 patient year sample is used to "sell" valves in various manufacturer literature, as strange as that seems. Of course, manufacturer studies also introduces bias, either overt or hidden, and that is equally problematic.

So, for that matter, in some ways, I think the "internal" PROACT study is actually much more worthwhile for a lot of these issues you all bring up, since it is comparing results in a much more controlled sample with a single valve type. In other words, you get to see what happens to stroke and hemorrhage at various INR levels, as well as evaluate the degree to which protocols can really be maintained and what happens when not met. That issue was raised a time or two above. So, anyway, I'm unfortunately way too short on time to give this any justice, but here's a link: http://my.americanheart.org/idc/groups/ahamah-public/@wcm/@sop/@scon/documents/downloadable/ucm_425310.pdf.

It has some very illustrative results of complications rates at different INR levels, within and outside of range, etc. More importantly, it has lots of numbers for you all to plug and play... :wink2:

Seriously, though, I do think this a great discussion, and sadly, many doctors have the same failings of understanding on a lot of these issues. So, keep up the good work folks! Now, here's a little known stat for the day:

A clinical study compared moderate warfarin therapy (INR 2.65) to high intensity warfarin therapy (INR 9.0 - wow!) in a set of 258 mechanical valve patients. Thromboembolism rate was virtually identical - 4.0% for moderate and 3.7% for high intensity while hemorrhage was 0.95% for moderate and 2.1% for high intensity. Pretty interesting, I thought, not nearly as significant a difference as I might have guessed.