Linearized v. Actuarial

[StretchL]

I received the following statement from a cardiothoracic surgeon the other day. I'd appreciate any analysis that y'all who are well versed in statistics could provide. My questions- Are this guy's assertions about the linearization of yearly stats valid, and what's the diff between linearized and actuarial. Thanks:

"Note that there is a difference between the actuarial freedom from bleeding or embolism and the linearized incidence of these. The actuarial freedoms are each in the 80% range at 20 years. (Look at the tables carefully to see how many patients are at risk after 20 years - not very many. And at 25 it's almost 0!) However, the annualized linear incidence of anticoagulant-related hemorrhage (ARH) is 2.7%/pt-yr and that for thrombo-embolic phenomena is 1.9%/pt-yr. The simplistic translation of this means there is a 4.6% chance each year for a patient to have one problem or the other. That extrapolates out to over 100% at 25 years, so you can see how realistic is the number of only 25% who have had neither problem by the time they get to 25 years."

This bud's for you, Andyrdj...

 

[Ross]

Looks and sounds to me like he is doing what others have mistaken before and using it as a cumulative total over the years and that's just not the way it works.

 

[Bryan B]

I don't know squat about statistics (even though my dad had a masters in statistics ), but I would have to agree with Ross. IMO each year starts fresh. If you have a 4% chance of incurring one or the other in one year and neither occurs you have the same 4% chance the next year. My uneducated guess is that as you get older there may be some other health factors that may influence the statistics, but as someone who had a successfull Ross and had selected a Carpentier Edwards bovine pericardial as a backup...I would change my backup to a mechanical after having experienced OHS first hand. If I had ended up with the CE bovine at age 43 I can tell you that the thought of another surgery in 10-15 years would always be on the back of my mind. Now if I was 10-15 years older when I had the surgery my choice may be entirely different, but having gone through the surgery once (which BTW went VERY smoothly), in hindsight I would be making my decision on minimizing how many times I had to be cracked open like a crab.

 

[StretchL]

...having gone through the surgery once (which BTW went VERY smoothly), in hindsight I would be making my decision on minimizing how many times I had to be cracked open like a crab.

If, in hindsight, you would've chosen the mechvalve as a backup for this reason, why would you not, also in hindsight, have chosen it as the first choice. Even though Rosses have the possibility of lasting a lifetime, the mechs will nearly definitely do so.

 

[LaughClown]

Think of it like flipping a coin. If you flip a coin 100 times there is a very big chance of getting heads around 50 times. This doesnt mean at flip #73 if you happened to get tails everytime that the next flip has a better chance of getting heads. Its still 50%. However at flip #23 there is a very good chance (better than 50%) that by flip 100 you will get at least one head.
4% a year means that however long its been since you started, the next year is four percent. But it also means you have greater than 4% if you have 20 years ahead of you.
Basically accumulated years behind you dont count, in front and they add up.

 

[Ross]

The trick to this question is patient compliance in taking their meds as precribed, then is the managing facility worthy of the job. We've seen plenty that are not. If you have non compliant patients, morons managing, then your going to have problems.

 

[Andyrdj]

Simple case

Simple case

We first need to ask whether the two types of events are independant. For example, your odds of catching a bus each day are mostly independant of the odds of you wearing red.

However, if it's winter the odds of getting a nosebleed each day are clearly not independant of whether you get a snowball in the face.

In the case of two independant probabilities, simply use the old method of calculating the probability that neither happens. So , for odds of 2.7% and 1.9% per year, the individual odds of not happening are 97.3% and 98.1%, so yearly probability of being free from event is 0.973 x 0.981 = 0.954513, or 95.45% per year.

(n.b. you can only use the formula "probability of (a and b)= prob(a) x prob(b) if the two events are independant.)
here a and b are considered to be FREEDOM from adverse events

For 20 years, the freedom from event is (0.954513)^20 =39.41%, or just over 60% of getting one or the other.

However - cheer up a bit, as these figures are for independant probabilities, which, if you think about it, is not the best case scenario.

If the two affect each other, then being free from one could mean you're likely to be free from the other too. A bleeding patient might well be a thromboembolic patient too from the same event, in which case the Maths I used effectively counts the same event twice. So the simple multiplication of probs is too pessimistic.

On the other hand, if bleeding and embolism were a trade off, then the more you protect yourself from one, the more you're vulnerable to the other. Really, to make an honest judgement I would need some data on proportions of patients with both, neither, or one complication.

This has sort of been of interest to me recently. My personal preference for a tissue valve has long been made clear, and recently something seemed to validate that judgement. I've been suffering from gastro reflux disease (and God alone knows what I've done to my stomach over the years). The medication I'm on (proton pump inhibitors) affects how warfarin works, and I suspect I might have a higher likelihood of gastrointestinal bleeding than the average person.

Which backs up a point I've made before - you need to consider all of your health issues when making a valve choice, not just your heart.

 

[StretchL]

However - cheer up a bit, as these figures are for independant probabilities, which, if you think about it, is not the best case scenario.

If the two affect each other, then being free from one could mean you're likely to be free from the other too. A bleeding patient might well be a thromboembolic patient too from the same event, in which case the Maths I used effectively counts the same event twice. So the simple multiplication of probs is too pessimistic.

On the other hand, if bleeding and embolism were a trade off, then the more you protect yourself from one, the more you're vulnerable to the other.

It seems to me that bleeding and emolism *are* a trade off, and that's the problem of Coumidin. Reducing the risk of one increases the risk of the other.

So how does the math (or, for you Brits, "the maths"!) work out if they are indeed a trade off?

 

[Andyrdj]

tricky

tricky

Well, strictly speaking we need to work out the following

To use the following symbol conventions - which will be a bit tricky to the uninitiated, but are necessary to prevent each equation being represented by pages of writing.

P(a n b) = probability of a AND b
P(a u b) = probability of a OR b
P(a|b) = probability of a given that b has already happened

You can therefore work out

P(a n b) = p(a) x p(b|a) - working out the probability of freedom from both

In the case of independant events, p(b|a) =p(b) as a never affects b, and we are left with the formula I used in my last post.

In one sense, the two are sort of independant here, in that on a given regime having a bleeding event wouldn't cause an embolism. (though you might be reduced on your anticoagulant medicine and be at a higher risk)

However, in another sense they aren't because, in a trade off between the two, the success of the regime depends on how you work it.

Imagine the bleeding and embolism are inversely related.
Imagine on one regime they are 4% and 4% each, so annual freedom =0.96^2
= 0.9216

I adjust to another regime, which adjusts one up to 16%, and the other down to 1%.

So the annual freedom prob is 0.99 x0.84=0.8316

I adjust to a third regime, with 2% and 8%,
annual freedom prob =0.92 x0.98= 0.9016

You can see from this that there is an optimum in balancing between the two.
I can't swear, though, that the risks work exactly this way - one might increase far faster than the other as you move along the "slider of warfarin".

It is probably fair to say, though, that there is an optimum for "freedom from events" in reality. this might not equal the "optimum for survival" though, as a doctor might feel that it's worth risking a few blleding episodes to avoid strokes because strokes are more dangerous (don't quote me on that one, it's for example only)



Aside
Some of you might wonder why I tend to work out the probabilities of something not happening, rather than saying "what's the probability of one event or another?"
Well, just to work out one year's probabilities in the example above, one has to use

p(a u b) =p(a) + p(b) - p(a n b).

Why subtract the p(a n b) off the end? Well, it's like saying "count the swimmers and tennis players in the classroom".

Adding "swimmers" + "tennis players" won't give quite the right answer
Some people will swim and play tennis both, and this select group is counted twice. so we subtract one lot of those who do both.

To calculate probabilities of an event at two years, you would have to say "well, they could have been free at one year, and had one prob, or had an event at one year, survived, but suffered in the second year, or dies at one year, or been free both of these years..... you can see how messy it gets!

Here's a diagram of a permutation tree to help visualise it- https://nrich.maths.org/discus/messages/67613/67821.jpg

the enclosing discussion is here https://nrich.maths.org/discus/messages/67613/67590.html?1134598273

N.B I haven't been through it all to check whether it's codswallop or not!

Far better to focus on the freedom figure!

 

[Mary]

Thank you Andy.
(I'm not certain for what, but I'm certain that it's probably something!)
Bravo!

 

[StretchL]

Is there a spinning head smiley that I can post here???

Thanks, Andy... I gotta read this when I have more time...

 

[Ross]

I like my formula better. It's simple and states that there are too many variables that come into play between individuals, that it makes statistics and numbers irrelevant.