Simpson's paradox

WickedCossack · Post by **WickedCossack** » 06 Sep 2015, 20:10

Saw this interesting paradox today, and no it's not to do with the cartoon.

So suppose you have a kidney stone which can be either a small stone or a large stone. Now there are two treatments for kidney stones, treatment A &' treatment B.

You go to a doctor and he gives you this information on treatment success:

Treatment A vs. small stones: 93% (81/87)
Treatment B vs. small stones: 87% (234/270)
Treatment A vs. large stones: 73% (192/263)
Treatment B vs. large stones: 69% (55/80)

Supposing you don't know if you have a small or large kidney stone ? Surely you'd pick treatment A whatever right ?

But not so fast !

Since you don't know whether you have a small or large kidney stone then you may as well combine the stones for each treatment ...

Treatment A: 78% (273/350)
Treatment B: 83% (289/350)

Same sample size out of 350 and it turns out treatment B is actually better. Or is it ?

What would you choose if you had a kidney stone ?

momuuu · Post by **momuuu** » 06 Sep 2015, 20:26

Thats because the weights of the averages are wrong. Here, the worse variant (large stones) weighs more heavily for the A average than for the B average. Its just unfair statistics. If you draw the chance tree for any random chance of having large kidney stones, A will have a significantly larger succesrate for any chance of large kidney stones.

WickedCossack · Post by **WickedCossack** » 06 Sep 2015, 20:32

jerom wrote:Thats because the weights of the averages are wrong. Here, the worse variant (large stones) weighs more heavily for the A average than for the B average. Its just unfair statistics. If you draw the chance tree for any random chance of having large kidney stones, A will have a significantly larger succesrate for any chance of large kidney stones.

Well the weighting is what creates the paradox. It''s skewed but it''s not what you can call "wrong."

350 patients were given treatment A, another 350 treatment B and if those are the results then you''ve got a choice on your hands.

For sure you can ask them to do a better experiment as well but they don''t always have the time

_venox_ · Post by **_venox_** » 06 Sep 2015, 20:52

It's not really a paradox is it?

WickedCossack · Post by **WickedCossack** » 06 Sep 2015, 20:55

It''s been around a long time: [url=https://en.wikipedia.org/wiki/Simpson''s_paradox]linky[/url]

I thought it was interesting anyhows. :happytomato:

momuuu · Post by **momuuu** » 06 Sep 2015, 20:59

If you like paradoxes, look up the olbers paradox. Thats my favorite paradox

_venox_ · Post by **_venox_** » 06 Sep 2015, 20:59

It surely is interesting and probably a bit confusing if you have to choose but depending on your mathmatical and logical skills different people will come to different conclusions or will just throw a dice. The really intelligent people can't throw a dice though since they will wonder if the throw of dice really gives an equal chance for each number.

iNcog · Post by **iNcog** » 06 Sep 2015, 21:00

-- deleted post --

_venox_ · Post by **_venox_** » 06 Sep 2015, 21:02

Maybe the light passed so many different really heavy stars and black holes that the light will be slightly tilted that it doesn't reach us or the stars are so far away that their light hasn't reached us yet, or asteroids trash and planets all absorb the light on the way to earth.

momuuu · Post by **momuuu** » 06 Sep 2015, 21:38

The actual answer is something with a big bang. Youre theory about absorption has been proven wrong actually.

_venox_ · Post by **_venox_** » 07 Sep 2015, 04:50

I just read it and it sounds really cool, or should I say hot because of the big bang?

purplesquid · Post by **purplesquid** » 07 Sep 2015, 05:18

There really is no paradox here. In terms of treatment effectiveness, all that really matters is the percents. More people with the harder to treat condition (larger stones) were given treatment A. This is why more people with treatment B were cured even though treatment A is clearly the better treatment.

It comes down to the fact that the group given treatment B was "healthier" than the group that was given treatment A.

gibson · Post by **gibson** » 07 Sep 2015, 05:55

purplesquid wrote:There really is no paradox here. In terms of treatment effectiveness, all that really matters is the percents. More people with the harder to treat condition (larger stones) were given treatment A. This is why more people with treatment B were cured even though treatment A is clearly the better treatment.

It comes down to the fact that the group given treatment B was "healthier" than the group that was given treatment A.

I agree I see no paradox

deleted_user0 · Post by **deleted_user0** » 07 Sep 2015, 06:22

mathers like to call everything a paradox, paradoxally they dont have a clue what paradox means because they arent linguists =_= unless they are logical linguists in which case they are simply clueless, full stop.

momuuu · Post by **momuuu** » 07 Sep 2015, 08:13

This would definitely be a paradox to a linguist. Without some mathematical knowledge this is seemingly a contradiction, even though it actually isnt. Which is exactly what a paradox is (it appears to me some of you might think a paradox is actually a contradiction, which is wrong).

deleted_user0 · Post by **deleted_user0** » 07 Sep 2015, 08:19

it would appear that way, but you are making a logical mistake somewhere, jerom!

Metis · Post by **Metis** » 08 Sep 2015, 22:26

This is a paradox that you will probably discuss in your an introductory logic course. As with most paradoxes, it's not so much of something in the universe that doesn't make sense as that it is a mistake in the way we analyze and present information. If I recollect correctly, the paradox occurs when one variable has an overwhelming effect when included in the whole. Basic multivariate analysis will easily find such a variable in a lumped data set.

yemshi · Post by **yemshi** » 13 Sep 2015, 17:22

wickedcossack wrote:Saw this interesting paradox today, and no it''s not to do with the cartoon.

So suppose you have a kidney stone which can be either a small stone or a large stone. Now there are two treatments for kidney stones, treatment A &' treatment B.

You go to a doctor and he gives you this information on treatment success:

Treatment A vs. small stones: 93% (81/87)
Treatment B vs. small stones: 87% (234/270)
Treatment A vs. large stones: 73% (192/263)
Treatment B vs. large stones: 69% (55/80)

Supposing you don''t know if you have a small or large kidney stone ? Surely you''d pick treatment A whatever right ?

But not so fast !

Since you don''t know whether you have a small or large kidney stone then you may as well combine the stones for each treatment ...

Treatment A: 78% (273/350)
Treatment B: 83% (289/350)

Same sample size out of 350 and it turns out treatment B is actually better. Or is it ?

What would you choose if you had a kidney stone ?

If I''d only understood half of the post, I might have been able to give a serious answer.

Kaiserklein · Post by **Kaiserklein** » 13 Sep 2015, 18:38

wickedcossack wrote:Saw this interesting paradox today, and no it''s not to do with the cartoon.

So suppose you have a kidney stone which can be either a small stone or a large stone. Now there are two treatments for kidney stones, treatment A &' treatment B.

You go to a doctor and he gives you this information on treatment success:

Treatment A vs. small stones: 93% (81/87)
Treatment B vs. small stones: 87% (234/270)
Treatment A vs. large stones: 73% (192/263)
Treatment B vs. large stones: 69% (55/80)

Supposing you don''t know if you have a small or large kidney stone ? Surely you''d pick treatment A whatever right ?

But not so fast !

Since you don''t know whether you have a small or large kidney stone then you may as well combine the stones for each treatment ...

Treatment A: 78% (273/350)
Treatment B: 83% (289/350)

Same sample size out of 350 and it turns out treatment B is actually better. Or is it ?

What would you choose if you had a kidney stone ?

It''s basically a math problem about conditional probabilities. Here, it means you need to consider the probability you have to get small or large stones before you can apply the probabilities of success for each treatment. Said in another way, the thing is treatment A has his best success when it matters less : 93% but on only 87 stones... While B has his best success on the biggest part : 87% on 270, which matters a lot. And ofc A sucks when the sample is greater (73% on 263 stones) while B sucks when it doesn''t really matter (69% on only 80 stones).

In fact it''s just the law of total probability :

Pr("A works") = Pr("you have small stones") x Pr("A works for small stones") + Pr("you have large stones") x Pr("A works for large stones")
So according to what you said, we get Pr("A works") = 87/350 x 93% + 263/350 x 73% ~ 78%.

You can make exactly the same for treatment B, and it will work. This is not a paradox, just maths

cerebellum · Post by **cerebellum** » 26 Sep 2015, 19:23

@kaiserklein: you made a mistake when calculating Pr("you have small stones"). It should be (87+270)/(87+270+263+80). When you do this, you get the correct answer of Pr("A works") ~= .83. As mentioned previously, the paradox arises out of the inappropriate weighting of the subgroups. Also, quoting from Wikipedia, "When the less effective treatment (B) is applied more frequently to easier cases, it can appear to be a more effective treatment"

Kaiserklein · Post by **Kaiserklein** » 26 Sep 2015, 22:11

cerebellum wrote:@kaiserklein: you made a mistake when calculating Pr("you have small stones"). It should be (87+270)/(87+270+263+80). When you do this, you get the correct answer of Pr("A works") ~= .83. As mentioned previously, the paradox arises out of the inappropriate weighting of the subgroups. Also, quoting from Wikipedia, "When the less effective treatment (B) is applied more frequently to easier cases, it can appear to be a more effective treatment"

Why would you take into account the small stones on which treatment A hasn''t been tested to calculate the probability that A works ?

momuuu · Post by **momuuu** » 27 Sep 2015, 01:10

In the end this paradox is just disappointing. It's like the proof that 1=2, it's just the result of poor maths:

a = b
a[sup]2 [/sup]= ab (multiply by a)
2a[sup]2 [/sup]= a[sup]2 [/sup]+ ab (add a[sup]2[/sup])
2a[sup]2[/sup]-2ab = a[sup]2[/sup] - ab (substract 2ab)
2(a[sup]2[/sup]-ab) = a[sup]2[/sup] - ab (rewrite)
2 = 1 (divide by a[sup]2 [/sup]- ab)

[span]This one is really funny, wonder how many of you see what's wrong ^_^ [/span]

Kaiserklein · Post by **Kaiserklein** » 27 Sep 2015, 01:40

21
Nicely found... Divisions are tricky

rsy · Post by **rsy** » 27 Sep 2015, 02:09

kaiserklein wrote:21
Nicely found... Divisions are tricky

oh my God took me like half an hour of rewriting again and again until I finally realized that division by 0 thing. Damn disappointing indeed :/

momuuu · Post by **momuuu** » 27 Sep 2015, 02:15

rsy wrote:
kaiserklein wrote:21
Nicely found... Divisions are tricky
oh my God took me like half an hour of rewriting again and again until I finally realized that division by 0 thing. Damn disappointing indeed :/

Rewrite b=a and notice that from step 4 onward, it just says 0 = 0.

You have a[sup]2 [/sup]and substract ab which equals [span style="font-size:13.3333px'"]a[sup]2 [/sup]aswell :p[/span]

ESOC

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