Simpson's Paradox (1 Viewer)

[McLake]

I was shown this the other day, I think it is really cool:

Table 1. Contingency Table for Males Treated/Not Treated With a New Drug

<table border = "4"><TR><TD>&nbsp;</TD><TD><P>Treated</TD><TD><P>Not treated</TD></TR><TR><TD>Recovered</TD><TD ALIGN="CENTER">700</TD><TD ALIGN="CENTER">80</TD></TR><TR><TD>Not recovered</TD><TD ALIGN="CENTER">800</TD><TD ALIGN="CENTER">130</TD></TR></TABLE>
Table 2. Contingency Table for Females Treated/Not Treated With a New Drug

<table border = "4"><TR><TD>&nbsp;</TD><TD><P>Treated</TD><TD><P>Not treated</TD></TR><TR><TD>Recovered</TD><TD ALIGN="CENTER">150</TD><TD ALIGN="CENTER">400</TD></TR><TR><TD>Not recovered</TD><TD ALIGN="CENTER">70</TD><TD ALIGN="CENTER">280</TD></TR></TABLE>
Table 3. Contingency Table (Aggregated) for Persons Treated/Not Treated With a New Drug

<table border = "4"><TR><TD>&nbsp;</TD><TD><P>Treated</TD><TD><P>Not treated</TD></TR><TR><TD>Recovered</TD><TD ALIGN="CENTER">850</TD><TD ALIGN="CENTER">480</TD></TR><TR><TD>Not recovered</TD><TD ALIGN="CENTER">870</TD><TD ALIGN="CENTER">410</TD></TR></TABLE>
Statistically, the 1st and 2nd tables suggest the drug is more effective.

Table 3, however, suggests the drug is less effective.

[Data blatently stolen from http://www.amstat.org/publications/jse/v9n1/sowey.html]

 

[Affinity]

what's the paradox here?

46.7 % males treated recovers
38.1 % who aren't treated recovers

68.2 % females with treatment recovers
58.8 % females without treatment recovers.

and always being subtly sexist we SNEAKILY decided to give different weightings to male and females, and those who are treated and those who aren't.
So we can stack the statistics:
since the male "treated" results are not as good as the females, we include 1500 of them and 220 females.

this way we get
49.4% treated recovers
53.9%who aren't treated recovers

but if we give "treated" and "not treated" , male and female equal weightings, we get:

on average 57% who are treated recover
and 48% who are not treated recover.

 

[xiao1985]

well i guess the paradox mainly states that if:
a/b > c/d, e/f > g/h are true the following may not necessarily be true:
a/b + e/f > c/d + g/h

hehe, the 850 bucks summer school did pay off at the end.... lolz

 

[maniacguy]

Originally posted by xiao1985
well i guess the paradox mainly states that if:
a/b > c/d, e/f > g/h are true the following may not necessarily be true:
a/b + e/f > c/d + g/h

hehe, the 850 bucks summer school did pay off at the end.... lolz

No, it says that (a+e)/(b+f) is not necessarily greater than (c+g)/(d+h)

(Incidentally, if you meant NMSS by the $850 summer school, it does pay off. For example, the UNSW subject MATH2400 Finite Mathematics (required for Software engineers) is pretty similar to the number theory stuff done at NMSS, just over a longer time span).

Simpson's paradox is that the marginal results (i.e. overall) differ to the conditional results. The main explanation, as Affinity said, is to consider tables of treatment and recovery for the genders:
TREATED NOT TREATED
MALE 1500 210
FEMALE 220 680

RECOVER NOT RECOVER
MALE 780 930
FEMALE 550 350

Thus males are more likely to be treated, but less likely to recover, and so it appears that overall people being treated are less likely to recover. In fact treatment does not harm recovery.

(Gender is called a confounder in the relationship between treatment and recovery. If you want to know more about this, you can do the subject MATH2810/2910 at UNSW on Categorical data analysis. I should note though that it's not a hugely interesting or challenging subject, so you might even be better off just turning up to the odd lecture here and there).

 

[xiao1985]

lolz, hey, how did u noe dat?? u r in da same comp aren't u??

lolz, sori for the retarded mem..... me <-- old , mem <-- goin down

 

[maniacguy]

I'm in second year @ UNSW, but I was at NMSS a few years back. (The number theory is the same each year - he even trots out the same jokes But it's all fun, even if it doesn't have a huge amount of relevance to the HSC)

 

[Newbie]

this stuff isnt in hsc right?

 

[Affinity]

Yes, ofcourse it is in the HSC
topic: harder 4 unit

 

[McLake]

Originally posted by Affinity
Yes, ofcourse it is in the HSC
topic: harder 4 unit

That's a 5U maths topic, right?

 

[Affinity]

hmm don't know....

 

[Newbie]

wtf is harder 4u? NOOOOOOOOOOOO WTF hehe