Generating Simpson's Paradox with Z3

https://kevinlynagh.com/z3-simpsons-paradox/

by surprisetalk • 2 years ago

173 57 2 years ago

kevinlynagh.com

incognito124• 2 years ago

I just love the napkin equation in the middle of [1], it really made it clear to me

[1] https://robertheaton.com/2019/02/24/making-peace-with-simpso...

joe_the_user• 2 years ago

The wikipedia gives a reasonable take:

"Mathematician Jordan Ellenberg argues that Simpson's paradox is misnamed as 'there's no contradiction involved, just two different ways to think about the same data' and suggests that its lesson 'isn't really to tell us which viewpoint to take but to insist that we keep both the parts and the whole in mind at once.'"

My own take is that any statistic has a value and a strength (in the case of averages, strength can be the number of instances averaged, for instance). You can have to keep in mind both.

https://en.wikipedia.org/wiki/Simpson%27s_paradox

kgwgk3• 2 years ago

its lesson 'isn't really to tell us which viewpoint to take

When the analysis is intended to inform a decision it’s important to take the appropriate viewpoint.

https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf

https://www.unige.ch/cisa/files/9616/7501/5107/CISA_BM_stats...

TheMrZZ3• 2 years ago

Biggest trap of Simpson's paradox is the results can change with every level of granularity.

If you take the example of Treatment A vs Treatment B for tumors, you can get infinite layers of seemingly contradicting statemens: - Overall, Treatment A has better average results - But if you add tumor size, Treatment B is always better - But if you add gender to size, Treatment B is always better - But if you add age category to gender and size, Treatment A is always better - etc...

It totally contradicts our instincts, and shows statistics can be profoundly misleading (intentionally or not).

Narhem• 2 years ago

Like Tumors are higher dimensional objects than what human brains are trained to perceive.

oh_my_goodness• 2 years ago

If the previous comment is right, then this one is plainly also true in some sense. I'm disappointed to see downvotes.

gradschoolfail• 2 years ago

For pedagogues and practitioners alike: there is a subtle connection between Simpson’s paradox and the wild geometry of relative entropy. This might be partly why effect sizes are also contentious.

Besides Ellenberg’s mind-altering discussion of that link[1], see hints on the second page of:

https://www.qeios.com/read/XB1N2A/pdf

[1] "[the point of Simpson’s paradox] isn't really to tell us which viewpoint to take but to insist that we keep both the parts and the whole in mind at once."

Ellenberg, from Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else (2021)

mb77332• 2 years ago

I think the real-world resolution to this problem is straightforward though. You should look at the finest level of granularity available, and pick the best treatment in the relevant subpopulation for the patient.

jefftk• 2 years ago

Unfortunately our level of certainty generally falls off as we increase the granularity. For example, imagine the patient is a 77yo Polish-American man, and we're lucky enough to have one historical result for 77yo Polish-American men. That man got treatment A and did better than expected. But say if we go out to 70-79y white men we have 1,000 people, of which 500 got treatment A and generally did significantly worse than the 500 who got treatment B. While the more granular category gives us a little information, the sample size is so small that we would be foolish to discard the less granular information.

throwawaystress• 2 years ago

The smaller the subpopulation, the higher the variance, and the less significant the result.

TheMrZZ3• 2 years ago

To add some proofs to my answer, I actually coded a Z3 program to prove it! The 3-variables version takes too long to resolve, but I got results for the 2-variables version (tumor size + gender):

Results can be found in this GSheet: https://docs.google.com/spreadsheets/d/1tsBhElTgXjVTeas8quar...

Code is here: https://gist.github.com/TheMrZZ/c33927ca2cc917997a67d7f84b82...

I'm currently running the 3-variables version, hopefully I'll get results this afternoon.

We can clearly see the same problems that arise in the 1-variable Simpson's paradox (widely different population sizes).

tombert2• 2 years ago

Z3 is kind of my new favorite thing right now. I have a problem that lends itself quite well to constraints-based reasoning, and I need it to be optimized. I'm sure I could have hacked something together using any number of programming languages, but after playing with Z3 for a bit, I realized that this could be easily done in around ~100 lines of an SMT2 file, and probably be considerably faster.

Tools like this make me feel a lot better about all the time I wasted playing with predicate logic.

samstave• 2 years ago

Curious, would your data constraints be related to http://equationofhappiness.com/

Would like to know how you us Z3 to evaluate emotional happy Q. Could you apply that to evaluate a piece of content (like thread, feed, comment) and then evaluate the energy of the thing for its happiness quotient value whatever? then you can just Z3 the thread and determine the psychological predicted impact...

Maybe you could then design casula games that provide the positive happy Q vibes.

I was in a program from when I was a baby with UCSD that was a life tracking project and they would check in with you every so often to see where you were on that trajectory - and where you were happy in life etc.

Problem is that it was also tied to Morehouse University, MK, Vic Baranco, and a bunch of other Stanford thingy's from the 70s that we all know abou these days.

tombert2• 2 years ago

Nah, the core math for Equation of Happiness is my dad's thing, and it's using a genetic algorithm for its optimization stuff, all I did was port some of his code to Julia and write a basic web frontend. If I had known about Z3's optimization tools when he was writing the book, I might have tried to use it, but I'm not sure how well Z3 would actually work with the differential equation stuff he was doing, since it's not really discrete.

I'm working on something that is trying to utilize Z3 for some financial market stuff that I hack on in my free time.

darkteflon• 2 years ago

Same. I’d been circling SMT solvers for a while but a recent HN post on knuckledragger[1] (built on Z3) made me finally take a closer look at Z3 itself. It kind of feels like stealing fire.

[1] https://news.ycombinator.com/item?id=41161455

User23• 2 years ago

If you’re not already familiar with it, have a look at Dafny. It’s an imperative programming language built using Boogie and Z3 that allows extremely interesting compile time assertions.

clircle• 2 years ago

Seems like an inappropriate use of the word "paradox". How about Simpson's intuitive situation?

travisjungroth• 2 years ago

“Going against common expectation” is well established as one of the definitions of “paradox”. Check any dictionary you choose.

reallybees• 2 years ago

Paradoxical iff data sets are isolated; explains the Gemini effect. Preferences tend to model future outcomes if sample sizes of surveyors are combined.

eadler• 2 years ago

It is a veridical paradox, not a falsidical paradox.

A falsidical paradox is what most people think of as a formal paradox: from the assertions you derive a conclusion which is false (either deductively or inductively).

A veridical paradox is where you derive a true result which appears absurd. it is an informal classification based on intuition. https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox is a classic veridical paradox.

staplung2• 2 years ago

Simpson's Paradox keeps experimenters up at night because it embodies the idea that although your data might say one thing, it's always possible that slicing the data via some unknown axis of finer granularity might paint a very different picture. It's hard to know if there is such an axis lurking there in your data, let alone, what it might be.

If you get paranoid about its presence it can lead you to second guess pretty much every statistic. "I know that 4 out of 5 dentists recommend chewing X Brand gum but what if I slice the dentists by number of eyes? Maybe both one-eyed dentists and two-eyed dentists aren't so enthusiastic."

grobbyy• 2 years ago

It does, but only experiments who don't have a strong grounding in research methods.

Simpson's paradox is part of a broader problem: correlation does not imply causation. In practice, it's one of many problems with making decisions based on correlations.

In a randomized control trial, with a large random sample, the odds of Simpson's Paradox coming up are astronomically low.

Good statisticians WILL second-guess ANY conclusions based on post-hoc data analysis. To frame this in scientific jargon, exploratory data analysis and correlations are great at generating hypotheses, but those need to be confirmed with methods appropriate for confirmatory analysis.

zekrioca• 2 years ago

I dislike the baseball example, because it is too context specific to the other several Bi people who don’t follow the sport.

0xDEADFED5• 2 years ago

Does Bi = Baseball ignorant? I'm definitely Bi

staplung2• 2 years ago

I know essentially nothing about Z3 but it seems like there's a potential problem in the code. There's a section headed by the comment "All hits and miss counts must be positive" followed by a bunch of assertions that those numbers are greater than 0. Isn't it possible that you have 0 hits or misses? I mean, what if your season is short and you only face a couple of lefty pitchers and strike out both times?

In any case, I would explain the paradox differently than the author. The author says: "The key to understanding the paradox is that the players did not bat against the same set of pitchers. A batted against 5 lefties and 12 righties; B against 2 and 11."

I would say instead that the key to understanding the paradox is to observe that both players are much better when batting against lefties and that player A batted against lefties much more often, both in absolute and relative terms. In other words, A is not as good against lefties as B but he faced a lot more of these comparatively easy pitchers.

cortesoft• 2 years ago

That would be possible for a hitter, but it would be impossible to be an example of Simpson's paradox in that case.

Izkata2• 2 years ago

This visualization on Wikipedia was what I needed to understand Simpson's Paradox, the descriptions never made a whole lot of sense to me until seeing it like this: https://en.wikipedia.org/wiki/Simpson%27s_paradox#/media/Fil...

Along the same lines of "visualize your data to see what's really going on" is Anscombe's Quartet: https://en.wikipedia.org/w/index.php?title=Anscombe%27s_quar...

And then there's the Datasaurus [Dozen], which has some fun with the idea behind Anscombe's Quartet: https://en.wikipedia.org/wiki/Datasaurus_dozen (you can see it animated here: https://blog.revolutionanalytics.com/2017/05/the-datasaurus-... )

mjburgess6• 2 years ago

A more general lesson is that correlation and causation are unrelated: the former doesnt imply the latter, and the latter does not imply the former. Just because one thing causes another does not mean it will be correlated with it.

There is no contradiction in subsets having different correlations that the parent set. The apparent "paradox" arises from reading the data causally. The purpose of this lesson is to expose these assumptions in interpretation of data. Few seem to get the message though.

weinzierl• 2 years ago

If anyone has doubts about the second claim, think about a hash function. The input certainly causes the output, but they are not correlated in a statistical sense.

kqr3• 2 years ago

A more general lesson is that correlation and causation are unrelated

This is a bit extreme. The tongue-in-cheek variant I like (which I first read about in the book referenced by TFA) is "no correlation without causation". In order for two things to truly co-vary (and not just by accident, or as a consequence of poor data collection/manipulation), there needs to be some causal connection between the two – although it can be quite distant.

godelski4• 2 years ago

  > correlation and causation are unrelated

This is incorrect (but what followed is correct).

You have extended the meaning of the phrase "correlation does not imply causation" to a stronger case[0]. The correct way to say this is that "correlation are not necessarily related."

The other way you might determine this was wrong is that ,,association''[2] always occurs when there is causation. So we have the classic A ⇒ B ⇏ B ⇒ A (A implies B does not imply B implies A), where ordering matters.

Last, we should reference Judea Pearl's Ladder of Causality[1].

[0] Another similar example was given to us by Rumsfield with respect to the Iraq WMD search. Where the error was changing "the absence of proof is not proof of absence" to the much stronger "the absence of evidence is not evidence of absence". It also illustrates why we might want to "nitpick" here https://archive.is/20140823194745/http://logbase2.blogspot.c...

[1] https://web.cs.ucla.edu/~kaoru/3-layer-causal-hierarchy.pdf

[2] Edit for clarity: The reason I (and Pearl) use the word "association" rather than "correlation" is because in statistics "correlation" often refers to linear relationship. So association clarifies that there is mutual information. There might be masked relationships, so non-linear. But if we are to use the standard vernacular of "correlation" (what most people think) then we could correctly say "causation implies correlation" (or more accurately, "causation implies correlation, but not necessarily linear correlation."). And of course, causation implies high mutual information, but high mutual information does not imply causation :) https://stats.stackexchange.com/questions/26300/does-causati...

wodenokoto• 2 years ago

I'm not sure I understand how that would work if x is time, unless each category is sampled from different time spans.

Leszek• 2 years ago

I've always been partial to this visual: https://analyticsindiamag.com/wp-content/uploads/2018/12/sim...

hackandthink3• 2 years ago

"The key to understanding the paradox is that the players did not bat against the same set of pitchers."

This is misleading. I am baseball ignorant but I feel this is a contrived and bad example for Simpson's paradox.

(UC Berkeley gender bias example is much better)

kgwgk3• 2 years ago

There you have something like "The key to understanding the paradox is that the schools did not have the same set of applicants.” It seems similar.

hackandthink3• 2 years ago

My take:

The key to understanding the paradox is that women apply mostly to popular departments therefore women are declined more often than man who apply more to unpopular departments.

kqr3• 2 years ago

Popular wisdom regarding experimentation has always been to "vary just one thing at a time, keeping the others as constant as possible". Fisher argued to the contrary, that we should (systematically) try as many variations as possible simultaneously. Simpson's paradox (and perhaps the similarly counter-intuitive Berkson's paradox) are the reason why: when analysing just one variate at a time, we risk seeing relationships that aren't there, or run counter to what we are trying.

Proper multifactor analysis that accounts for all variations simultaneously is required to learn about complex phenomena.

cubefox2• 2 years ago

A basic question I always had about Simpson's paradox: If X is positively correlated with Y, but X is also negatively correlated with its parts when Y is broken down (Simpson's paradox) – is it then more likely that X causes Y or that X causes not-Y?

This seems to be a pretty fundamental question but I have never seen it addressed.

taeric2• 2 years ago

My understanding is that the answer is more that there are more important/causal elements than X acting on Y. That is it.

For the college admissions example, the more important factor was what department you applied to. That had far more of a meaningful contribution to whether you were admitted than what your sex was. So, if folks wanted to increase admissions, you wouldn't focus on sex, you would focus on expanding departments.

That is the general trend with all of the examples. The paradox is you think focusing on X would be the important thing to focus on, but the data had hidden that there was a Z that you should instead focus on.

im3w1l• 2 years ago

Well it depends on context I would say. Like first you have to consider sample size. The more you break something down the lower the sample size becomes, and the more breakdowns you try, the more likely you will find some kind of spurious pattern.

But let's say the effect is real. Then you have to start considering what is causing why, which is highly context dependant.

To take the example of gender bias in the grad school admissions, men were more likely to be admitted but the effect reversed when breaking down by department.

Hypotheses come to mind:

A department being easy to get into causes a higher male ratio. (Subhypotheses: Maybe men want to take it easy? Maybe women seek out prestige?)

High male ratio of applicants to a department causes it to become easier to get into. (Maybe funding flows to male departments?)

There is some kind of unknown factor that both causes a high male ratio and a high rate of admissions (Maybe a booming sector attracts men and leads to easy admissions?)

With all these possibilities I think it should become clear that there can't be a general solution to your question, you have to consider the context and dig deeper.

cubefox2• 2 years ago

Well, I think the standard explanation for this case was something this:

Being male (X) causes you psychologically to prefer STEM departments, which are more economically useful, which means they are better funded, which means they are larger, which means they can admit more people, which means you are more likely to be admitted (Y). So X is positively correlated with Y.

However, being male (X) also causes you to be less likely to be admitted in each individual department (parts of Y), e.g. because people in admissions offices have some degree of anti-male bias, known in psychology as women-are-wonderful effect [1]. So X is negatively correlated with the parts of Y (being admitted to individual departments).

So in this case, X does cause both Y, via the male psychological preference for STEM, and not-y_n (for {y_1, y_2, ..., y_n} = Y, for an n-partition of Y), via anti-male admissions bias like women-are-wonderful. So the previously most plausible initial explanation for why X causes Y (university being biased again women) is indeed false.

I guess the lesson is that detecting a case of Simpson's paradox (when partitions of a variable like Y have different direction of correlation with some other variable X than the entire variable Y) can point to causal explanations being different than what they would naively seem to be.

[1] https://en.wikipedia.org/wiki/Women-are-wonderful_effect

taeric2• 2 years ago

I am not sure I agree with that conclusion. In this particular example, it is almost certainly more important to consider frequency and sample size? Could have been the exact same pitchers, per se.

Generating Simpson's Paradox with Z3

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