I’ve got a ‘puzzle’ so to speak, but it’s pretty open ended.
I’ve got a data set: I’ve tested drug A and drug B on both women and men. For the sake of simplicity, we’ll just say that each subject either died or healed (just a binary measurement of the effect of the drug on each individual).
The percentage of men who were healed by drug A was higher than the percentage of men who were healed by drug B.
The percentage of women who were healed by drug A was higher than the percentage of women who were healed by drug B.
But, the percentage of people overall who were healed by drug B was higher than the percentage of people healed by drug A.
How is this possible? What does a data set have to look like to produce this result?
And no, it’s not a trick question – the answer is not anything like “Many of the participants were hermaphrodites or transvestites.” Just keep it simple: all participants are men or women.
Bonus points if you know the name of this mathematical anomaly.
I spent a while on this before I gave up and googled it, and learned that I had added some assumptions that were getting in my way.
This first entabment will only discuss assumptions, but I tab them because talking about them can sometimes suggest the answer; purists may want to avoid:
[tab]I had assumed that the treatment group sizes were the same, so that the number of individuals given each drug were the same, and thus that the number of men given drug A was the same as the number of men given drug B, and ditto for women. The trial sizes and compositions can be different.[/tab]
This next entabment will discuss the answer:
[tab]I’d never heard of Simpson’s paradox before this, so thanks for pointing it out. They have some interesting real-world examples in that article. Particularly interesting was the example of gender-biased admissions.
I’m having difficulty succinctly expressing what must be true of the data set for this to occur… I know that the compositions need to be reversed, so for example, more women participate in the drug A trial and more men participate in the drug B trial. And I intuitively grasp the “vector interpretation” given in the wikipedia article, especially the image:
But I need to think more on this to distil what’s going on. Every time I try to express the criteria that are necessary, it sounds like a tautological restatement of the question itself. I know the two trial groups must be of different sizes and ratios of men to women, and I think it is expressible as some relationship between the success group of one drug for one gender to the failure group of the other drug for the other gender.[/tab]
I would explain it like this:
It’s about weighting (think weighted averages).
Women have a higher healing rate than men. A has a higher healing rate than B.
If the healing rate of Women taking B is higher than the healing rate of Men taking A, and A is disproportionately taken by men while B is disproportionately taken by women, this (with some more mathematical constraints) can allow for the Simpson’s Paradox to occur.
I’ve taken a screenshot of a journal article about it that you might find enlightening:
[tab]I’m curious why you said it has to be June 18 or July 16. I eliminated May and June first and went from there, I’d be interested to know your reasoning.[/tab]
Like most of these types of things it is poorly worded.
Fact is that either having the knowledge but not the other, alone is simply not enough to choose between the the dates and the 3 line conversation is of no help.
Unless we know the reasons why the 3 line conversation occurs - such as how they know then the problem is insoluble given the information.
There was a similar problem in this site a couple of years ago, about Monks that was equally stupid.
[tab]I’m wasn’t looking very carefully or focusing on it much because of work, so yeah, June 18 makes no sense
Albert: there are only 2 daya that don’t repeat, and they don’t match the month I know, so Bernard can’t know the answer.
Bernard: if Albert knows that I don’t know the date, then it can’t be May or June, because those months have the only numbers that don’t repeat. Thus, since I know the day, and Albert ruled out May, it is July 16.
But now that I think of it, why not August 15?
Edit: because Albert then says that he knows it too. The criteria above eliminate all days except for July 16, August 15, and August 17. If Albert now knows it too, it can’t be August, since there are still 2 dates to choose from.[/tab]
[tab]However. B knowing that answer can ONLY conclude that June the 18th is the day, as he knows it is the 18th and from the list he can see only one date with that number. On hearing that B knows, A can also conclude that the birthday has to be on the only date that has an exclusive day number. As of all the dates, only one month has a day number not shared with any other month.[/tab]
[tab]I thought I was well on my way to solving it–I had just reasoned out what I considered the final key to the solution–, when I suddenly realised a very basic fact had escaped my attention. The fact that the numbers are between 2 and 99 means their product cannot be a prime number. But then it’s completely superfluous for Peter to say he doesn’t know the numbers (which he could, from the information given him, only know if the product was a prime number). But then it’s just as superfluous for Sarah to say she knew he didn’t know the numbers. For then Peter could have known Sarah could have known he didn’t know the numbers! So their saying those things need not give either of them any new information. So if Peter knows the numbers after those things have been said, he could have known them as soon as he was informed of the product. In fact, he would have known them, considering that he’s great at math. So the problem has now become completely obscure to me, and I suspect that it doesn’t add up.[/tab]
