Moontower #103


There is a positive correlation between a high school student’s grades and standardized test scores.

Yet… high standardized test scores before entering university do not predict university grades.

I’ll let you think a bit about possible reasons for that.

Some more statements:

  1. “Smart students are less athletic.”
  2. “Good books make bad movies.”
  3. “Height does not correlate with performance in the NBA”
  4. “People winning engineering contests are not that good at their job.”
  5. “In a good restaurant, the least appetizing sounding items likely taste better.”

You may have nodded with 1 and 2. Too bad they are illusions.

You may have been surprised by 3,4 and 5.

These are all examples of Berkson’s Paradox. What the heck is that?

Via Wikipedia:

The most common example of Berkson’s paradox is a false observation of a negative correlation between two positive traits, i.e., that members of a population which have some positive trait tend to lack a second. Berkson’s paradox occurs when this observation appears true when in reality the two properties are unrelated—or even positively correlated—because members of the population where both are absent are not equally observed.

Here Wikipedia summarizes the same example I highlighted from Jordan Ellenberg’s book How Not To Be Wrong around the idea that attractive men are rude.

Suppose Alex will only date a man if his niceness plus his handsomeness exceeds some threshold. Then nicer men do not have to be as handsome to qualify for Alex’s dating pool. So, among the men that Alex dates, Alex may observe that the nicer ones are less handsome on average (and vice versa), even if these traits are uncorrelated in the general population. Note that this does not mean that men in the dating pool compare unfavorably with men in the population. On the contrary, Alex’s selection criterion means that Alex has high standards. The average nice man that Alex dates is actually more handsome than the average man in the population (since even among nice men, the ugliest portion of the population is skipped). Berkson’s negative correlation is an effect that arises within the dating pool: the rude men that Alex dates must have been even more handsome to qualify. 

The key to understanding all of these examples is the correlations you expect break down when the sample is narrowed. This is commonly referred to as “range restriction”.

Let’s go back to the surprising statement I opened with. This time I’ll boldface the restrictor.

High standardized test scores before entering university do not predict university grades. created a handy visual for understanding what’s happening. Even though SATs and GPA are positively correlated at large, at any particular university you may see a negative correlation.

They explain:

The admissions committee accepts students who have either a sufficiently high GPA, a sufficiently high SAT score, or some combination of the two. However, applicants who have both high GPAs and high SAT scores will likely get into a higher-tier school and not attend, even if they are accepted. The range of students that actually attend the school is given by the blue dots in the plot in the introduction. These dots show a downward trend even though the overall population (red and blue dots) show an upward trend. This trend reversal is the “paradox,” though there is nothing truly paradoxical about it. It is the result of a trade-off between GPA and SAT scores in the people reviewed.

Why is this so important?

Because it shows up everywhere! It’s tempting to see counterintuitive correlations and try to create a story about them but they are often not surprising once we realize that the narrow pool selects between 2 dominating attributes. Consider the NBA where many players are selected by skill and height. Height does not correlate with performance in the NBA because a short player in the NBA must have abnormally high skill to have gotten to the restricted range known as the NBA. Similarly, the chance of random 7 footer in the population playing in the NBA is an order of magnitude more likely than an average height player to make it to the NBA. It’s the same reason why you shouldn’t be surprised when a small NFL player like Wes Welker or Steve Tasker is a badass (I see you 90s Bills. I also hated you, but Tasker was a maniac).

So now I’ll go back to the original statements and boldface the “range restrictors”.

  1. Smart students are less athletic.”
  2. Good books make bad movies.”
  3. “Height does not correlate with performance in the NBA
  4. “People winning engineering contests are not that good at their job.”
  5. “In a good restaurant, the least appetizing sounding items likely taste better.”

The visual versions of all of these can be found in @page_eco Twitter thread that inspired this post.

More examples

  • Grit and violinists

    David Epstein spots a “restriction of range” problem in the book Grit which cites a study of 30 violinists. When you squash the range of a variable that is correlated with the dependent variable you risk understating the correlation with the restricted variable. In this case, the sample was violinists who had already been accepted to a famous academy. We have squashed their innate talent even though it likely has a wide range.

    He also articulates the NBA example: If you studied the correlation of height to points scored in basketball for NBA players you find a jarring negative correlation but that is because you are selecting from a sample of abnormally tall players, to begin with. You’ve squashed the height variable, which would lead people to think that height has no impact on points scored.

  • Surgeons

    Nassim Taleb has warned that you should be wary of surgeons that look like stars who play surgeons on TV.

  • Hedge fund managers

    Not due diligence advice but I’m guessing you probably would have wanted to invest with a black or female hedge fund manager in say the 1980s.

Finally, one last example from Byrne Hobart who has a “range restriction” detector in his brain:

Institutional Investor highlights a study showing that CEOs get more authority relative to boards when they benefit from lucky economic conditions they had nothing to do with. There’s always some range restriction at work when analyzing the performance of CEOs. In a simple model, a CEO gets the job through some combination of a) underlying skill, and b) the ability to persuade. That persuasive skill means that we should expect the worst CEOs to be overcompensated, and the best to be underpaid. And it makes sense that one channel through which charisma works is in taking credit for good news and dodging blame for bad news, irrespective of who was really responsible for either. 

It’s not surprising Byrne spots Berkson’s Paradox everywhere. He wrote Brilliant Jerks, Crazy Hotties, and Other Artifacts of Range Restriction. (Link)

The Money Angle

  • Understanding Options and Decision-Making (Thread)

    In an old Barron’s Roundtable, Jeff Yass, the founder of SIG had strong words about how fundamental option theory is to decision making.

    Of course this sounds self-serving, from a guy who understood options as a young teen. But it reminds me of a more famous investor. Warren Buffet. I’ll rely on readers to find it but I remember Munger saying that Buffet was already thinking of options at a precocious age. While Buffet calls derivatives “weapons of mass destruction” his own investing history shows an explicit use of options (his put-selling maneuvers are well-documented…and critically path-resistant since they are not marked-to-market). I’m not a Buffet expert, but his use of “insurance float” sure looks like something that came out of the mind of a derivatives trader.

    • The Moontower Volatility Wiki is growing every week due to submissions from the online vol community. It also includes every post I’ve written on options, many of which try to use options theory to understand markets and think about probabilities.
    • Decision-making is a practice.
      • A pillar of sound decision-making is thinking in probabilities or as Annie Duke’s book is titled, Thinking In Bets. Here’s the notes I took on an interview with her which captures the essence.
      • This weekend I came across a great post by in the same vein by Jonathan Bales The Time I Sold Furbies For Money. I especially liked the bits about Belichick’s non-punt, and poker pro Phil Laak about learning what “5% feels like”. [Phil is a good friend of some friends I made in the options game so it was especially cool to see his thinking turn up in that post].

        I’ve previously commented on the neat analysis Bales himself did on the question of when you should “work for free”. You should follow @BalesFootball if you want to sharpen your “thinking like a gambler” sword.

  • A Personal Take

    I added thoughts on my days at SIG in response to the Yass thread. Here’s the text:

    When I was a Susq I heard Jeff speak a few times. It was always engaging.

    They were savage in my days there but the doubling down on tech and brains thru the years probably makes Jeff the richest dude in the world you never heard of (unless you look at pol donations, then you know). One of the talks was on the primacy of markets (Yass is an extreme libertarian, free-marketer, no fool should be allowed to keep their money type. Appealing views to many traders, esp when they are young). This post was one of his market lessons: Dinosaur Markets.

    One of my interactions with Jeff was a mystery to me:

    I remember when I was a 1st year mm on the Amex and I reported a giant trade that got crossed in AIG on the internal chat. I got a dm. “Pls call”. It was from Jeff. I was never so scared. Was I supposed to break that cross up? I called Jeff from an Amex phone and he just asked me for the trade details. Implied vols, who the broker was, what bank crossed it. I told him and he abruptly hung up. That was it. Still don’t know why of all the trades I’ve ever on reported why that warranted a call.

    Other times I’ve heard Jeff speak was on why the dot com bubble was not an example of market inefficiency and it goes back to understanding option theory and the relationship of volatility to positive skew and what drives volatility. I’ll write about that one sometime.

    He also speaks to every trading class for an hour that goes thru the 3 months of theory and mock trading in Bala Cynwyd. In my class he talked about career risk with NFL coaches affecting decisions (he defended an oft- mocked Barry Switzer decision to not punt)

    I will always be thankful for having worked and learned at SIG. I really didn’t have any business being hired there (2000 was the largest cohort bc $ was raining from the sky. They needed warm bodies to pick it up) and I think I’m proof that traders can be shaped and aren’t born. [By the way, this is very much why I try to teach what I’ve learned. Hopefully people smarter than me can build on it and let me invest in them 🙂 ]

    Incidentally, the head of HR who hired me gave important advice I always remember. When I explained I had a few higher offers she said:

    “You’ll be rich whatever you choose. Decide who you want to work with.”

    She knew SIG held the nuts.

Last Call

This week’s letter touched on probability and SIG.

Let’s put a bow on it with the interview questions I got in 1999…

  1. You flip a single die and will paid $1 times the number that comes up. How much would you pay to play?
    • Suppose I let you take a mulligan on the roll. Now how much would you pay (you are pricing an option now btw)?
  2. My batting avg is higher than yours for the first half of the season. It’s also higher than your for the second half of the season.

    Is it possible your avg for the full season is higher than mine?

    (Hint: Simpsons paradox)

  3. You are mid game that you have a wager on. Opponent offers to double the stakes or you automatically lose. (Like the doubling cube in backgammon)

    What’s the min probability of winning you need to continue playing?

  4. You’re down by 2 with seconds left in regulation basketball game and have a 50/50 chance of winning a game if it goes to overtime. You have a 50% 2-pt shooter and a 33% 3-pt shooter.

    Who do you give the ball to?

    (simple EV question)

  5. You are given $1,000,000 for free but there’s a catch. You must put all of it into play on roulette.

    What do you do?

  6. There’s a 30% chance of raining Saturday. 30% chance of raining Sunday.

    What’s the probability it rains at least one day?

To encourage you to try before looking up the answers, I’ll make it annoying…the answers are somewhere in this thread.

I wrapped that thread with a short post on Trading And Aptitude (Link)

From my actual life 

First, Happy Easter!

Some pics from Spring Break last week…we were with both of Yinh’s sibling’s families.

We started at home smoking chicken…

Spent time in Muir Woods…

Spent 3 nights in Healdsburg (an amazing rental called the Treehouse…sleeps 20 for under $1800/night)

And ended up in Sacramento to see more family (this is at the American River).

We got to meet our in-laws’ newest addition, Boba.

Needless to say, 8 kids are in love.

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