Exploring Pascal’s Wager

Blaise Pascal was a 17th-century mathematician/philosopher most famous for introducing a certain way to organize numbers, now known as Pascal’s Triangle:

Uh…seems…useful?

Pascal is also known for writing short thought experiments in philosophy and mathematics. One of these is an argument called Pascal’s Wager, which goes something like this:

“Look: if god exists, and you believe in him, you get eternal life…if you believe in god and it turns out that he doesn’t exist, what have you really lost? Not that much, you’re going to die anyway.

If you don’t believe in god, because of your human pride, and you shake your fists at him, but it turns out he does exist, INFINITE TORTURE. And if you don’t believe in god and he doesn’t exist, then cool.

So really, why not believe in god? Because, at worst, you will die forever, just like you would if you didn’t believe in him. At best, you’ll have eternal life.

But if you choose not to believe in god, you’re playing with infinite odds that you will actually be tortured and mutilated by demons that tear your flesh apart and pour lemon juice on it.”

– David Pizarro, Episode 113 of the Very Bad Wizards podcast

 

Before I explain why I’m writing about Pascal’s Wager, here’s a simpler version of the problem. Suppose you’re shown a large room of brightly colored balls:

Go on, but if you don’t get to the point soon I’m jumping in.

A person approaches you and tells you that one of the balls has the word “GOD” written on it, and all the others read “Just Earth, Stars, Universes, and Solar-systems” (J.E.S.U.S., for short). They then offer you the following game:

If you give me one dollar, that large crane arm over there will dive in and pick one ball. If it reads “GOD”, you get ONE MILLION DOLLARS.

 

THE CLAWWWW EXISTSSSS

The question: is it a good idea to play? There’s actually an equation for the amount of money you’d win on average, if you played the game a near-infinite amount of times. This equation, called the “expected value“, looks like this:

$$E_{\text{PLAY}} = \frac{1}{\text{total }\#\text{ of balls}}\times \$999,999 + \frac{\# \text{ J.E.S.U.S. balls}}{\text{total }\#\text{ of balls}}\times (-\;\$1) $$

(Achievement unlocked: use of the phrase “jesus balls” in a math expression.)

The expression for $E_{\text{PLAY}}$ is simply net winnings weighted by probability of winning, plus net loss weighted by probability of loss. In gambling, if $E_{\text{PLAY}}$ is greater than zero, reason suggests you should go in. If the expected value is less than or equal to zero, however, you should pass.

So what happens in our ball game? Let’s say there are 10,000 total balls in the room. Even with those small odds of drawing the single GOD ball, the expected value calculation actually favors buying in:

$$E_{\text{PLAY}} = \frac{1}{10,000}\times \$999,999 + \frac{9,999}{10,000}\times (-\;\$1) \approx \$100$$

Intuitively, the reason this works is that the amount of money you get for winning greatly outweighs the odds of losing. However, you can imagine versions of the game where this is not the case: if there are significantly more balls, for instance, or if you are being offered significantly less money.

Pascal’s argument is, essentially, that the decision whether or not to believe in god is a version of the ball game in which you’re being offered INFINITE MONIES. Looking at the expected value equation, we can see that increasing the winnings indefinitely makes playing (believing) a good idea, no matter how much the game costs.

Wait, what about hell? Right, good question. The ball game isn’t a perfect analogy to the original Wager, which also has the threat of eternal damnation if you mistakenly disbelieve. So, we have to add this rule to the game:

Even if you don’t play, I’ll still draw a ball. If it reads “GOD”, you owe me ONE MILLION DOLLARS.

 

Now there’s a potential (and likely) cost for not playing! This just makes the decision easier. Here’s the overall expected value of not playing (“passing”):

$$E_{\text{PASS}} = \frac{1}{10,000}\times\left(-\;\$999,999\right) + \frac{9,999}{10,000}\times \$1 \approx -\;\$100$$

So, the larger the winnings and losings, the more obvious the choice becomes: believe in god (play for \$1) and expect basically infinite reward; disbelieve (pass and keep \$1) and expect eternal punishment.

I’ve never found the analogy of roulette to religious belief particularly compelling. First, there are too many concessions you have to make before it’s realistic: that lifetime belief in god costs as little as a lottery ticket; that there aren’t an infinite number of other potential gods who won’t make this bargain; that you can be hand-wavy with complex concepts like infinity and human value.

Even if you grant those concessions, there’s still the problem that you can no more decide to believe in god than you can decide who you fall in love with. This basic fact severs any link between religious belief and gambling, and is the main reason I was never overly impressed with Pascal’s argument.

So, the face-palming was real when Pascal’s Wager popped up as the primary subject of the latest episode from my favorite podcast Very Bad Wizards. At first, I couldn’t decide if going through the episode was worth the risk that they’d mangle probability rules, or run through lots of trivial objections.

Eventually, however, my hunger for the infrequent VBW epistemology adventures got the better of me, and I wasn’t disappointed. The majority of the first half of the episode (the latter half introduced a variant on the Wager) focused on precisely the aforementioned problem of belief and choice, and many of the philosophical/psychological intricacies involved.

In the episode, the hosts eventually arrived at the same conclusion that has had me continually unimpressed with the Wager: the type of gods for which the Wager is relevant require belief to be a conscious, free act. Essentially, they said (heavily paraphrased): “Well, the argument sort of works if you’re Jewish, because then being religious is in fact mostly participation in activities. But Pascal wasn’t Jewish, so…yeah.”

This post picks up where the VBW discussion left off. The fact is, the majority of Christians actually do think that a believer is something you get up every day and decide to be. As David pointed out in the episode, this is arguably one of most important structural supports of the Christian mindset. Like David, I was raised and confirmed Catholic, so I have some perspective here. From first-hand experience, I know that Pascal’s Wager remains pretty convincing to a lot of folks.

So, I don’t think David & Tamler satisfactorily closed the issue for, say, Jews who think keeping kosher their whole life is a good bargain for the chance at eternal life, or Christians who really think belief is a choice. The natural next question for me is: can we concede that belief is reducible to an opt-in checkbox, and still find an intuitive counter to Pascal?

I think yes, and it has to do with how we value time when life is limited.

And here I was just about to dive in and spend eternity not reading this blog post.


A key assumption of Pascal’s Wager is that a year of life has a constant value, regardless of whether or not heaven and hell exist. This is like saying “I could eat a hamburger every day, forever, and each day it will taste just as good as the very first one!”

For most people, this doesn’t work for hamburgers, and it doesn’t work for time, either. Suppose you knew you were going to die at age 80. How much would you pay for an extra year? What if you knew you were going to die at age 40? Surely, a year is worth more in the latter case. At the other extreme, if you know you’ll live for 1,000 years, a year should be worth far less.

Now, what if you knew you would never die? How much would you pay for an extra year? A quick answer is, of course, zero: you don’t need extra years if you’re immortal.

The point of these questions is to illustrate that, in some sense, an 80-year life might be at least as valuable as an endless life, all things considered. Compared with eternity, any finite amount of time is fleeting and therefore precious. This is felt deeply by many atheists:

“Death is a deadline…knowing life is temporary brings focus to our lives, inspires us to treasure the people and experiences we encounter, and motivates us to do something valuable with the short time we have.”

Greta Christina, from Comforting Thoughts About Death That Have Nothing To Do With God.

 

If years of a finite life are more valuable, that should affect the outcome of Pascal’s Wager. To see this work out, let’s define some terms:

  • Let $P_G$ be the probability that god exists. Referencing the ball game, $P_G$ is like 1 over the total number of balls.
  • Let $Y_E$ by the number of years living on earth, and $Y_H$ the number of years living in either heaven or hell. In what follows, I’ll assume these numbers are finite. The only way to assess what happens when heaven is “infinite” is to consider the resulting expected values when $Y_H$ is arbitrarily large.
  • Finally, we need some notion of a “flourishing value” as a believer, atheist, or resident of heaven. Let’s write these as $F_B$, $F_A$, and $F_H$ (respectively). These numbers can be positive or negative, and represent how much fun you’re having (on average, over long periods of time) in each state.

…for instance, $F_B < F_A$ means that on average, there’s a net cost to belief. The assumption that $F_B < F_A$ is part of the original Wager, which claims that even though belief may be a cost, the expected values work out in the end. However, I won’t assume this in any of the following calculations.

We can assume that $F_H$ is a positive number (heaven is fun guyz), and for simplicity, the level of flourishing in hell is just its negative version, $-F_H$. This reflects the symmetry inherent to standard conceptions of heaven and hell.

Let’s re-derive the conclusion of the original Wager. As a correct believer, I would get $Y_E$ years valued at $F_B$, and $Y_H$ years valued at $F_H$, resulting in a full life value of $Y_EF_B + Y_HF_H$. However, since I don’t know whether or not god exists, I have to factor this in to the overall expected value:

$$ E_B = P_G \cdot \left(Y_EF_B + Y_HF_H\right) + (1\,- P_G)\cdot Y_EF_B $$

The above equation is directly analogous to the one used in the ball game. To decide if belief is rational, we have to calculate the expected value of non-belief. As a correct non-believer, I would get just $Y_E$ years valued at $F_A$, giving a full life value of $Y_EF_A$. As an incorrect non-believer, I get a huge value loss from years in hell*. Factoring this in:

$$ E_A = P_G \cdot \left(Y_EF_A\,- Y_HF_H\right) + (1\,- P_G)\cdot Y_EF_A $$

Notice that in the first part of this equation, $Y_HF_H$ is subtracted, since if god exists, an atheist spends $Y_H$ years in hell, each valued at $-F_H$. Here’s the expression for the difference between $E_B$ and $E_A$:

$$ E_B\, – E_A = Y_E\cdot (F_B\, – F_A) + 2P_GY_HF_H$$

This equation is illustrative. If $F_B < F_A$, the first term adds a finite, negative value that does not depend on $P_G$. The second term is always positive, and gets larger as $Y_H$ grows. So, the trade-off in the original Wager is made clear: for any non-zero probability of god $P_G$, there is a length of time spent in heaven $Y_H$ that would offset any cost incurred by belief.

Phew. Ok. Now let’s calculate $E_B – E_A$ if we value years realistically, that is, if years are valued inversely proportional to total years lived. For instance, as a correct believer, I would get $Y_E$ years valued at $F_B / (Y_E + Y_H)$, and $Y_H$ years valued at $F_H / (Y_E + Y_H)$. Factoring this in:

$$ E_B = P_G \cdot \left(Y_E\cdot\frac{F_B}{Y_E + Y_H} + Y_H\cdot\frac{F_H}{Y_E + Y_H}\right) + (1 \,- P_G)\cdot Y_E\cdot\frac{F_B}{Y_E} $$

For non-belief:

$$ E_A = P_G \cdot \left(Y_E\cdot\frac{F_A}{Y_E + Y_H}\,- Y_H\cdot\frac{F_H}{Y_E + Y_H}\right) + (1\,- P_G)\cdot Y_E\cdot\frac{F_A}{Y_E} $$

Deriving the new expected value difference takes some algebra, but here’s a simplified form:

$$ E_B\,- E_A = \left(P_G \cdot \frac{Y_E}{Y_E + Y_H} + 1 \,- P_G\right)\cdot (F_B\, – F_A) + 2P_G\cdot\frac{Y_H}{Y_E + Y_H}\cdot F_H$$

This equation is much less illustrative, so bear with me. The mathematics of infinity are needed to fully understand what’s going on here. Consider that as $Y_H$ becomes very large, the fraction $\frac{Y_E}{Y_E+Y_H}$, seen in the first term, approaches 0. And $\frac{Y_H}{Y_E + Y_H}$, seen in the second term, becomes close to 1.

So how do we rule on this case when $Y_H$ “equals infinity”? We just have to consider $E_B\, – E_A$ when $Y_H$ is arbitrarily large. This means that we replace those fractions in $E_B\, – E_A$ that depend on $Y_H$ with their “limiting” values (0 and 1). When we do this, we get a much simpler expression:

$$ E_B\,- E_A =(1 \,- P_G)\cdot(F_B\, – F_A) + 2P_GF_H$$

This is even simpler than $E_B-E_A$ from the original Wager. Moreover, it has a strikingly different conclusion: a rational decision about belief now depends on all of $P_G$, $F_A$, $F_B$, and $F_H$. If $P_G$ is very small, then $F_H$ must be large enough to outweigh the (very likely) cost of belief.

Of course, the believer could simply respond to this new equation with some optimism: “$F_H$ is, of course, very large! How could it not be? And there are so many benefits to belief, maybe $F_B > F_A$! Then $E_B-E_A$ will always be positive, qed”. Such a response is valid, certainly. But the point of this analysis is that, with a simple, realistic assumption about how we value years in a finite life, we force Pascal adherents to argue for a priori assumptions on $P_G$, $F_A$, $F_B$, and $F_H$. In the original Wager, these values didn’t matter.

Let’s think for a second about what that means. For the record, I’m an atheist. It’ll be hard to convince me that $P_G$ shouldn’t be super, duper small, at least not without some seriously strong assumptions. Also, I don’t see $F_H$ as being too large. The god of the bible was generally a dick, Jesus wasn’t much better, and most religious conceptions of heaven sound pretty boring (or awful). Finally, $F_B$ is clearly lower than $F_A$, at least for me. I mean, I get 2 full 24-hour days back per year just by not going to church. Thus, it’s easy for me to plug in realistic numbers for these constants that justify non-belief.

There’s one more realistic assumption that makes things even worse for Pascal. What if, as the years go on, each year is just a little less fun than the last? This is certainly reasonable to imagine**, and approximately encapsulates what I meant by “heaven sounds pretty boring.”

Mathematically, let’s say that each subsequent year is $Q$ times as fun, where $Q$ is a number between 0 and 1 (non-inclusive). Relegating the math to the end, this assumption results in a simple rule. For any $Q$, $P_G$, $Y_E$, and $F_H$, there exists $Y_H$ large enough such that

$$ \text{$E_B\; – E_A < 0$ when $F_B-F_A<0$, and $E_B\; – E_A \geq 0$ when $F_B \,- F_A\geq0$.}$$

Essentially, this says that if heaven is infinitely long, the rationality of belief depends only on the per-year cost of belief; importantly, it doesn’t depend on the probability of god! We’ve assumed only that (I) years are valued inversely proportional to how long you’re alive, and (II) each year is a little less fun. Rewording again: no matter how likely ($P_G$) or fun ($F_H$) god & heaven are, the fact that eternity is a long, long time makes my enjoyment of life on earth pretty dang valuable.

These analyses have better-illustrated parts of the Pascal debate that I’ve read and heard throughout the years. I think many atheists pass off Pascal’s Wager as ludicrous because they’re already intuitively doing the sorts of calculations I’ve put forth. They instantly realize “well, I don’t think heaven’s that exciting”, or “finite life is pretty valuable”, and how those feelings should play into Pascal’s Wager.

Relatedly, rigorously formulating the Pascal scenarios above can help to illuminate the differences between deeper assumptions that believers and atheists bring to the table. For instance, I think assumptions (I) and (II) well-approximate ways people really value time. Believers may not concede that ground so easily.

Ideas similar to those in this post are, I suspect, covered in the references mentioned in objection #2 to premise 1 of the Wager, though I haven’t read them. I was surprised not to find anything like them in the rational wiki article on the Wager, as I find much of what’s there less intuitive. Regardless, I wanted to make an accessible, fully explanatory post for this particular counter. I’d be interested in more recent related work: please send it along.

*Yes, I’m fully aware the current Pope said atheists aren’t necessarily going to hell. That actually makes things worse for Pascal adherents; it can be incorporated in the equations by setting the flourishing level of hell equal to zero.

**The result still holds as long as years eventually start becoming less fun. We can allow for usual up/down-ticks that happen over the course of a standard earthly, mortal life, but the math notation is far more complicated.


Math for the strong counter

If each year is a little “less fun” at rate $Q$, we have to break down the values of the years and sum them. This analysis requires some high-school level pre-calc, which I won’t explain fully (aside from a link or two). First, the values received by a correct believer (CB) and incorrect believer (IB) in this case are

$$ \text{CB} = \sum_{y = 1}^{Y_E} \frac{F_B}{Y_E + Y_H}Q^{y – 1} + \sum_{y = Y_E + 1}^{Y_H + Y_E}\frac{F_H}{Y_E + Y_H}Q^{y – 1},\;\;\;\;\text{IB} = \sum_{y = 1}^{Y_E} \frac{F_B}{Y_E}Q^{y – 1}$$

We can simplify these expressions using the formula for a geometric series:

$$ \text{CB} = \frac{(1 \,- Q^{Y_E})F_B + (Q^{Y_E} \,- Q^{Y_E + Y_H})F_H}{(1 – Q)(Y_E + Y_H)},\;\;\;\;\text{IB} = \frac{(1 \,- Q^{Y_E})F_B}{(1 \,- Q)Y_E}$$

With these shorthands, the expected value of belief is just $E_B = P_G\cdot \text{CB} + (1 – P_G)\cdot \text{IB}$. Similar algebra gives the expressions for an incorrect atheist (IA) and correct atheist (CA) as

$$ \text{IA} = \frac{(1 \,- Q^{Y_E})F_A \,- (Q^{Y_E} \,- Q^{Y_E + Y_H})F_H}{(1 – Q)(Y_E + Y_H)},\;\;\;\;\text{CA} = \frac{(1 \,- Q^{Y_E})F_A}{(1 \,- Q)Y_E}$$

…and similarly, $E_A = P_G\cdot\text{IA} + (1 – P_G)\cdot \text{CA}$. With some more algebra, we can obtain an expression for the expected value difference:

$$E_B – E_A = \frac{P_G}{1 – Q}\left[\frac{(F_B – F_A)(1 – Q^{Y_E}) + 2F_H(Q^{Y_E} – Q^{Y_E + Y_H})}{Y_E + Y_H}\right] + \frac{1 – P_G}{1 – Q}\left[\frac{(F_B – F_A)(1 -Q^{Y_E})}{Y_E}\right]$$

Now, when $Y_H$ increases without bound, $Q^{Y_E + Y_H}$ approaches 0, and $Y_E + Y_H$ grows toward infinity. This means that the limiting value of the big, messy first term is zero. Therefore, for fixed values of $P_G$, $Q$, $F_B$, $F_A$, $F_H$, and $Y_E$, we have

$$E_B \,- E_A \approx \frac{1 \,- P_G}{1 \,- Q}\left[\frac{(F_B \,- F_A)(1 \,-Q^{Y_E})}{Y_E}\right]$$

for large enough $Y_H$. The multiplicands other than $F_B – F_A$ on the right-hand-side are all positive. This implies that the sign of the expected value difference is determined by $F_B – F_A$.

Cry wolf

This post is in response to a SlateStarCodex article published shortly after the 2016 U.S. presidential election. The author Scott Alexander argues that President-Elect Trump is not “openly racist”, and therefore much of the response to the Trump campaign/administration was/is overblown.

Despite the organized and detailed write-up, the article ignores a lot of relevant data and literature. This is strange given the amount of positive attention it got from well-known scholars:

Those are just from my personal feed, and I follow 53 people. I’m sure there are more based on Pinker‘s vast popularity. It was, for instance, mentioned positively by Paul Bloom on this episode of Sam Harris’s podcast. A search for the article’s title on Twitter will show the level of traction and positive energy it received overall.

Among my Facebook friends, the article was shared by those with backgrounds in statistics and economics. A week or two after I began writing this post, a friend of mine (data scientist doing fraud detection for a major credit card company) discovered the article independently and emailed it to me with the subject line “This is why I’m not worried”. I’m still seeing it going around and getting accolades for its supposedly rigorous, data-cognizant approach:ss1_2

So, I decided that a detailed response would be well-worth the effort.


Alexander opens with the observation that Trump lost votes (percentage-wise, compared with Romney) from white voters:

From an NYT exit poll report

From an NYT exit poll report

This result may seem counter-intuitive. Consider, though, that many white people are not racist, or at least are turned off by obvious appeals to racial prejudice. Moderates and conservatives who fit that description were probably still able to vote for Mitt Romney in good conscience. So, if the theory is that racist sentiments were uniquely important to Trump’s campaign, these data may actually corroborate that. Trump kept all the white republicans who loved or could tolerate his rhetoric, and lost the rest.

This seems to be the simple version of what actually happened, according to some research from UMass (I was linked to it first on Twitter, but here’s one press release). Schaffner et. al. gave convincing evidence that (1) lack of college education predicts racist and sexist sentiments, and (2) racist and sexist sentiments are the strongest predictors of support for Trump. Amazingly, the data that motivated their hypotheses in the first place was taken from the same NYT exit poll report Alexander cited:

Schaffner, MacWilliams, Nteta (2017)

Schaffner, MacWilliams, Nteta (2017)

The Schaffner study is well-worth a full read. For now, you’ll find a few of their main results down the page, when I address other parts of the SSC article.


In part II of his post, Alexander focuses on a number of Trump’s statements that, when taken literally, seem to contradict the claim that he is openly racist: maybe even that he values some vague conception of social diversity.

But Alexander’s “representative sample” don’t distinguish Trump from someone who is out to be popular, or pay lip-service and appeal to queasy Republicans. Given Trump’s history and character, these are better explanations of his statements.

And that’s being charitable. Trump’s phrasing and verbal tics, like prefacing the name of a minority group with “the”, indicate xenophobic attitudes. Amid the bullshitting and half-formed sentences, his only cogent thoughts about black communities apparently involve racist generalizations. From the quotes in Alexander’s piece:

 

‘I employ thousands and thousands of Hispanics. I love the people. They’re great workers. They’re fantastic people and they want legal immigration… The Hispanics are going to get those jobs, and they’re going to love Trump.’

‘And at the end of four years I guarantee that I will get over 95% of the African-American vote. I promise you. Because I will produce for the inner-cities and I will produce for the African-Americans.’

 

There’s a reason we never saw President Obama tweeting pictures of himself eating American-ized Mexican food, or improvising half-assed shows of support for the LGBQT community by grabbing audience props (both were cited in Alexander’s piece as examples of Trump’s benignity). These are not actions taken by someone with honest concerns for social justice. Someone with such concerns would make them explicit parts of a continually stated and reinforced political platform.

To all this, Alexander says:

 

“And if you believe he’s lying, fine. Yet I notice that people accusing Trump of racism use the word ‘openly’ like a tic. He’s never just ‘racist’ or ‘white supremacist’. He’s always ‘openly racist’ and ‘openly white supremacist.”

 

True: if Trump is racist, he’s technically not being “open” about it around cameras, in a way that a longtime KKK member might be around friends. But who cares? Here’s one reason why SA thinks we should:

 

“This, I think, is the first level of crying wolf. What if, one day, there is a candidate who hates black people so much that he doesn’t go on a campaign stop to a traditionally black church in Detroit, talk about all of the contributions black people have made to America, promise to fight for black people, and say that his campaign is about opposing racism in all its forms? What if there’s a candidate who does something more like, say, go to a KKK meeting and say that black people are inferior and only whites are real Americans?”

 

I think that, nowadays, such an openly racist candidate becoming a serious contender is next to impossible. This is a good thing. It means you can’t write Mein Kampf and take control of a world power eight years later. It means most people are decent enough to expect at least a serious apology for public racism, and nobody wants to think of themselves as racist. But it also means that people with deep-seated bigotries who run for office are likely to behave exactly like Trump has for the past two years: putting out a handful of genuinely racist statements surrounded by a bunch of disingenuous appeals, excuses, and distractions.

If I’m wrong, we’ll have to say something like “hey everyone, I know we went nuts over Trump, but this is even worse. It’s actually the worst.” And, if a future front-runner is as openly racist as Alexander seemingly needs him/her to be, most people won’t have any problems accepting that. Anyone who would say “haha nope, sorry, you used up your chance with Trump, I’m not falling for that again” is someone whose reason is not worth appealing to in the first place.


The majority of the data Alexander gives are to be found in part III, within his counters to some common talking points. I respond to these below.

 

1. Is Trump getting a lot of his support from white supremacist organizations?

 

It’s noteworthy that Alexander didn’t provide any links to articles claiming that Trump is getting, specifically, “a lot of his support” from white supremacist organizations. That’s because, as far as I can tell, there aren’t any that use that exact phrase. If you Google the question above, you’ll actually get a lot of great articles detailing public support Trump has received from white supremacist organizations. But none of them say Trump received “a lot of” his support from them, percentage-wise, the claim to which Alexander seemed to be responding in remarking how small the KKK is relative to the media attention they received. Mathematically, Trump could not have gotten “a lot of” his support from white supremacist organizations, because there aren’t “a lot of” them to begin with.

So, this is a straw-man. Also, since when did Internet-Articles-Per-Person become the standard metric for determining when enough is enough? Let’s apply Alexander’s exact logic here in another setting. The search string “Saddam Hussein” returns about 20 million results from Google. There is only one Saddam Hussein. So, assuming at least one of every 10,000 results is a relevant article, that means there are at least 2,000 articles for every one person in the I am Saddam Hussein club! Wow, what an overblown reaction we had to that organization!

 

2. Is Trump getting a lot of his support from online white nationalists and the alt-right?”

 

Same problem. Figures like “% of Trump supporters who are X” are uninformative without

  1. knowing the overall % of people who are X, and
  2. a comparison to, say, “% of Clinton supporters who are X”.

Noticing this can work for both sides. A headline like “80 Percent of GOP Voters Say Trump’s Racist Comments Are ‘Totally Fine’” should immediately raise the question “well, what percentage of Democrat voters said the same?”. If the answer is close to 80, the first result doesn’t mean much. (It wasn’t close to 80.)

Regarding white nationalists and the alt-right, noticing that (a) the active alt-right is a small minority, and (b) almost all of them were ardently anti-Clinton, makes it completely uninteresting that the alt-right and KKK comprised only a small percentage of Trump supporters. What matters is that they comprised any percentage at all. For all the awfulness they bring to the world, groups like the alt-right and the KKK are, perhaps, effective red-flags. When they show support for something or someone, and when we see behavior like this and this from adherents to a front-runner candidate, we should be freaked: whether or not the candidate accepts the support or condones the behavior.

 

3. Is Trump getting a lot of his support from people who wouldn’t join white nationalist groups, aren’t in the online alt-right, but still privately hold some kind of white supremacist position?

 

This question suggests the crucial idea that Trump tapped into widespread but mostly dormant racist sentiment, and that this directly contributed to his success. In his response, however, Alexander answers an entirely different question. His logic is hard to follow: for some reason, he starts pointing to surveys showing that racist sentiments have gone down, overall, during the past century. These polls should surprise no one (especially since a focus on Southern states can change the picture), and they don’t tell us anything about Trump’s campaign.

At the time Alexander published his post, there was a ton of easily-accessible, relevant data on this issue. In mid-September, this author used the 2016 ANES Pilot Survey (conducted in January) to show that, other than party affiliation, racial resentment is the best predictor of support for Trump:

clintonVtrump1

According to that same analysis, racist resentment is also strongly correlated with dissatisfaction with President Obama:

obama

Schaffner et. al. found the same kinds of results in their study, which relied on a roughly 2,000-sample YouGov survey conducted in October. In their regression model predicting the 2-party vote, the coefficients for sexism and racism were double that for economic dissatisfaction:

Schaffner, MacWilliams, Nteta (2017)

Schaffner, MacWilliams, Nteta (2017)

Here’s how the magnitude of those variables affect the predicted probability of a Trump vote, with the others variables held fixed at their averages:

Schaffner, MacWilliams, Nteta (2017)

Schaffner, MacWilliams, Nteta (2017)

Finally, they found this large effect of sexism and racism to be unique to the 2016 election:

Schaffner, MacWilliams, Nteta (2017)

Schaffner, MacWilliams, Nteta (2017)

 

Here’s a compendium of similar results from yet others surveys conducted before November. Notably, a Reuters poll (March 2016) found that Trump supporters have far-and-away the worst racist attitudes toward black people:

reuters1

 

4. Aren’t there a lot of voters who, although not willing to vote for David Duke or even willing to express negative feelings about black people on a poll, still have implicit racist feelings, the kind where they’re nervous when they see a black guy on a deserted street at night?

Probably. And this is why I am talking about crying wolf.”

 

Again, we shouldn’t need Trump to come right out with “I think white people are the superior race and here’s why”. This strange obsession with precision of language doesn’t seem as important to me as un-normalizing Trump and not giving ground to authoritarian sentiments. It’s clear from the data that racism was a key part of the Trump campaign, well ahead of economic anxiety.

 

5. But even if Donald Trump isn’t openly white supremacist, didn’t he get an endorsement from KKK leader David Duke? Didn’t he refuse to reject that endorsement? Doesn’t that mean that he secretly wants to court the white supremacist vote?

The answer is no on all counts.”

 

If there’s a consequential difference between an official political endorsement and something like the following Tweet, I’m not aware of it.

 

6. What about Trump’s ‘drugs and crime’ speech about Mexicans?

 

Alexander contrasted Trump’s words on illegal immigration from Mexico with McCain’s and those from Bill Clinton’s platform, and argued that Trump’s approach was basically identical. But there are two noteworthy differences. The first is that Trump others the immigrants by saying “They’re not sending you” (emphasis mine). This is in line with the theory that Trump’s rise to power is in large part due to activation of authoritarian sentiments via xenophobic language. The second is his use of the word “rapists”. To me, this indicates that Trump intended to stir feelings of physical and moral disgust against Mexicans, whereas the words of the other candidates point more to concerns about societal well-being.

 

7. What about the border wall? Doesn’t that mean Trump must hate Mexicans?

 

I never really thought so. It seemed more like one of his “decision-maker” or “businessman” signals.

 

8. Isn’t Trump anti-immigrant?

 

His supporters are. And his cabinet picks continue to dance around Trump’s more controversial campaign talking points on immigration, like a national Muslim registry. The fact that they’re still dancing should be horrifying enough.

 

9. Don’t Trump voters oppose the Emancipation Proclamation that freed the slaves?

This was in New York Times, Vox, Huffington Post, Time, et cetera. It’s very misleading. See Snopes for full explanation.

 

The survey statistic was that 20% of Trump voters said they oppose the “executive order which freed all slaves in the states that were in rebellion against the federal government.” The Snopes article basically said this was because they care more about not allowing executive orders than literally freeing slaves. I’m not sure how this makes anything better.

 

10. Isn’t Trump anti-Semitic?

 

I’m not sure, and I’ve personally never heard this issue brought up.

 

11. Don’t we know that Trump voters are motivated by racism because somebody checked and likelihood of being a Trump voter doesn’t correlate with some statistic or other supposedly measuring economic anxiety?

 

Economic anxiety is a predictor of Trump support. Just not nearly as much as racism or sexism sentiments. See the previously referenced analyses.

 

12. Don’t we know that Trump voters are motivated by racism because despite all the stuff about economic anxiety, rich people were more likely to vote Trump than poor people?

 

No, we know that Trump voters are motivated by racism (better said, racist sentiments predict a Trump vote) because we have data to support it.

The rest of the talking points (13.-17.) are interesting but not as important to the ideas in this post. I’ll leave them for another time, if necessary.


In part IV, Alexander writes:

 

Why am I harping on this?

I work in mental health. So far I have had two patients express Trump-related suicidal ideation. One of them ended up in the emergency room, although luckily both of them are now safe and well. I have heard secondhand of several more.

Like Snopes, I am not sure if the reports of eight transgender people committing suicide due to the election results are true or false. But if they’re true, it seems really relevant that Trump denounced North Carolina’s anti-transgender bathroom law, and proudly proclaimed he would let Caitlyn Jenner use whatever bathroom she wanted in Trump Tower, making him by far the most pro-transgender Republican president in history.

 

Ok, but Mike Pence is straightfowardly anti-gay and anti-transgender, Chief Strategist Steve Bannon is Steve Bannon, and Congress is now controlled by the Republicans. Are we supposed to depend on Trump’s convictions, moral scruples, and work ethic to keep things in check?

 

Listen. Trump is going to be approximately as racist as every other American president.

 

With the exception of President Obama, I might agree here, if we’re talking about Trump the person. For the most part, it seems to me like Trump will say whatever he thinks necessary to get people to like him and remain powerful. Ten years ago, he was a Democrat, and he said all the right things. For the last few years, he was Republican, and he rode a wave of racist and authoritarian sentiments into the White House by saying all the right things. When he wants to appear pro-diversity, he says all the right things, or at least right-enough things for people like Alexander to back him up.

Trump may not hold rigorous, calculated beliefs about white supremacy in his heart-of-hearts. He may “just” be an extraordinarily crass, emotionally immature, sociopathic bully who will put these traits to work against women, minorities, and people with disabilities to feel better about himself and gain support. He may be at least as much of a Michael Scott as he is a David Duke. How any of this is supposed to make us less worried is beyond me.


Let’s look at Alexander’s “confidences” in various events not happening during a Trump presidency. These are listed at the tail-end of his piece:

 

  • Total hate crimes incidents as measured here will be not more than 125% of their 2015 value at any year during a Trump presidency, conditional on similar reporting methodology [confidence: 80%]
  • Gay marriage will remain legal throughout a Trump presidency [confidence: 95%]
  • Neither Trump nor any of his officials (Cabinet, etc) will endorse the KKK, Stormfront, or explicit neo-Nazis publicly, refuse to back down, etc, and keep their job [confidence: 99%].
  • No large demographic group (> 1 million people) get forced to sign up for a “registry” [confidence: 95%]
  • No large demographic group gets sent to internment camps [confidence: 99%]
  • Number of deportations during Trump’s four years will not be greater than Obama’s 8 [confidence: 90%]

 

Taking the confidences as true probabilities, then, at least one of

  1. total hate crimes will be more than 125% of their 2015 value during at least one of the next four years,
  2. gay marriage will become illegal,
  3. at least one of Trump’s officials will endorse the KKK, Stormfront, or explicit neo-Nazis publicly and refuse to back down,
  4. a large demographic group will get forced to sign up for a registry,
  5. a large demographic group will get sent to internment camps, or
  6. the number of deportations during Trump’s 4 years will be greater than Obama’s 8

will occur with probability*

1 – (.80)(.95)(.99)(.95)(.99)(.90) = .3631

So, according to Alexander, the chance of at least one shocking injustice occurring during the Trump presidency is 36.3%: eight points above Trump’s chances of winning the presidency, as reported by FiveThirtyEight on November 8th.

And he’s telling us that we’re crying wolf.

At the end of his post, Alexander says

 

“If you disagree with me, come up with a bet and see if I’ll take it.”

 

Ok. Above, I didn’t even account for the ever-increasing chances of some global catastrophe brought on by an unstable sociopath becoming the most powerful person in the world. So, let’s add one more item to the list:

7. Trump, Putin, Jong-un, or any high-ranking military official from those leaders’ countries threatens a nuclear strike

I’ll give Alexander 1:1 odds on at least one of 1-7 happening during Trump’s presidency. By his accounting, these odds are generous. I’ll put up $500.00, and if something happens and I win, it will go toward helping people through whatever it is. Here’s my contact info.

 


*Under the assumption that the events are independent, which is not quite fair. However, any reasonable model with dependence would not affect my number’s comparability to Trump’s election-day chances. For instance, the probability in question is at least .20, again using Alexander’s confidences.

 

How to know to switch: Monty Hall without the math

You may have encountered, in a book, article, or possibly a classroom, the now well-known game show puzzle sometimes called the Monty Hall problem. It goes like this: you are a contestant on a game show, presented with three closed doors. Behind one of the doors is a new car; the other two, goats.

Here’s how the game proceeds:

  1. First, you choose any door you’d like.

  2. Second, the host opens a door you didn’t choose, which (purposefully) reveals a goat.Step 2

  3. Third, you have the option to switch your initial guess.Step 3

Then the door you finally settled on is opened: and you get whatever it reveals! The puzzle is this: to get the car, is it to your benefit to switch after the host opens a door? That is, will your odds of winning the car increase if you change your initial guess?

The mathematical solution to this problem is complicated to understand completely without a college-level understanding of probability. In fact, when the puzzle was first widely publicized by Marilyn vos Savant, it fooled hundreds of respected mathematicians and physicists. To this day, a basic internet search for the problem will turn up countless forum posts asking for help in understanding, and as many diverse attempts to explain the answer, often to no avail.

But the answer should be obvious, right? After the host reveals one of the goats, you must have a 50-50 shot at the car: it shouldn’t matter whether or not you switch doors. That turns out not to be correct: you’re actually twice as likely to take home the car if you switch your guess!

The issue people usually have with this puzzle is that the answer is non-intuitive: even if you understand the math behind the right answer, when you put yourself in the story, it doesn’t feel like switching should make a difference. The goal of this short post is to conjure the right gut reaction for this game – to call into being the intuition needed to “feel-out” the answer.

Consider a modification of the game in which there are not three but one hundred doors. There is still only 1 car, but now there are 99 goats. The rules can be generalized, like this:

  1. First, you choose any door you’d like.Step 1, Part 2

  2. Second, the host purposefully reveals all but one goat (and that goat is behind either your door or the final door).Step 2, Part 2

  3. Third, you have the option to switch your initial guess.Step 3, Part 2

As in the three-door game, by step three you are down to two options: either your initial guess was correct, or the final door has the car. Now, ask yourself: does this choice still feel like a 50-50 shot? In any 100-door game, it is quite likely (99% chance) that your first guess will be wrong, and therefore that the car is among the other doors. And, since the host will always end up opening all but one non-car door, the best strategy is to switch your guess.

The same principle applies in the 3-door game. More likely than not, your first guess was wrong: you only had a 1/3 chance of being right. After your initial guess, the host shows you all but one goat*. So, under a day-to-day interpretation of probability, there’s a 2/3rds chance the car is behind the final door.

*Notice that the 3-door game is the smallest game in which the host can take this action. In the 2-door game, it really is a 50-50 shot at the end: but that’s because your first guess is a 50-50 shot, and the host can do nothing.

 

Understanding Mixed-Effects Models with Cersei Lannister



This post is mostly for people who know a bit about doing statistics and want a primer on mixed-effects models. I’ve used the plainest language possible, so even if you don’t fit that bill, you might still find it readable. At the very least, you’ll get to laugh at my nerdy attempts to make statistics more entertaining to non-experts. My main goals are:

  1. Give accessible examples (the last being an online services simulation) of fitting mixed-effects models in R
  2. Discuss the interpretation of and motivation for mixed-effects models
  3. Provide short, educational GOT fan-fic, with pictures

Some of this is in response to a great post from folks at Google. They show a cool result about the predictive efficacy of a mixed-effects model, so I decided to try a related experiment. However, their introduction of the model did not thoroughly enough (for me) distinguish it from a Bayesian approach. So I’ll be discussing this as well.

I’ll provide some code I used, but not all of it. Most code I provide in-line is meant to display the basic functionality of the R library I chose to fit the mixed-effects models, called lme4. I put some other code at the end, and the rest is available upon request.

Hat tip to this cool website for providing me endless Westeros-style names to use! In the following example, set in the Game of Thrones universe, I go over what it means to have different experimental units in a regression model. This is an important concept to understanding why we might want to use a mixed-effects model in the first place. Enter the Lannister family:

Jaime’s archers

Jaime was preparing for a long journey, on official Lannister business. Knowing that things would likely come to swords, he assembled a small force to accompany him. He had some time to take special care when outfitting his troops, so he decided to run an experiment with two different kinds of bows.

Normally, Lannister forces use bows made from Northwood, as common knowledge says they are stronger than Southwood bows. However, since Jaime is a Southerner, he prefers Southwood bows out of pride and convenience. He wanted to see if Southwood bows cause a difference in archer range.

So, he chose three archers randomly from the kingdom’s forces, who each would shoot a Northwood bow and a Southwood bow 10 times. He bullied some local city-folk into recording the distance of the shots, in meters. Some of the resulting data is shown below:

##   Distance_      Bow_          Name_
## 1       163 Northwood Deonte Skinner
## 2       163 Northwood Deonte Skinner
## 3       159 Northwood Deonte Skinner
## 4       149 Northwood Deonte Skinner
## 5       152 Northwood Deonte Skinner
## 6       155 Northwood Deonte Skinner

The other two archers were named Fabiar Darklyn and Margan Drumm. Jaime chose to fit the following linear regression model to his experiment’s data: \[
Y_{ik} = \mu + \tau\mathbb{1}(i = 2) + \epsilon_{ik}
\]
Above, \(Y_{ik}\) is the \(k\)-th shot from bow \(i = 1,2\), and the Southwood bow is considered to be the second bow. The variables \(\mu\) and \(\tau\) represent (respectively) the mean shot distance and mean effect of the Southwood bow. The final term \(\epsilon_{ik}\) is the random error (the Winds of Westeros, perhaps) contributing to the \(k\)-th shot from the \(i\)-th bow. There are \(3\times 10 = 30\) shots per bow, since each markman shoots each bow 10 times. The errors are independent shot-to-shot and Normally distributed with constant variance \(\sigma^2_e\). Here’s the summary of Jaime’s analysis:

lm0 <- lm(Distance_ ~ Bow_, data = JaimeExp1$Data)
summary(lm0)
## 
## Call:
## lm(formula = Distance_ ~ Bow_, data = JaimeExp1$Data)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -25.84 -15.59 -10.21  23.75  34.70 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    167.885      3.678  45.643   <2e-16 ***
## Bow_Southwood   -9.767      5.202  -1.878   0.0655 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 20.15 on 58 degrees of freedom
## Multiple R-squared:  0.0573, Adjusted R-squared:  0.04105 
## F-statistic: 3.526 on 1 and 58 DF,  p-value: 0.06546

We see that when the archers switched to a Southwood bow, their shot distances decreased by about 10 meters, on average. However, Jaime realized his p-value was not below that magical 5% threshold! He strutted merrily out of the experiment grounds, confident that his results must be due to unfair winds. (Clearly, Jaime was absent for some important Statistical Lectures of the Crown.)

A most excellent result

“A most excellent result”

Adding archer terms

He was, of course, stopped by his brother Tyrion, who observed the whole experiment. “Now Jaime, can’t you see that you chose archers of wildly different statures? Only a fool would not account for this in his analysis.” Grumpily, Jaime re-fit his model with additional terms for the archers:

\[
Y_{ijk} = \mu + \tau\mathbb{1}(i = 2) + \alpha_j + \epsilon_{ik}
\]

Above, \(Y_{ijk}\) is the \(k\)-th shot (now \(k\) goes from 1 to 10) from the \(j\)-th archer when he is using the \(i\)-th bow. The \(\alpha_j\) parameters represent archer \(j\)’s baseline skill. Here were Jaime’s results, using the new model:

lm1 <- lm(Distance_ ~ Bow_ + Name_, data = JaimeExp1$Data)
summary(lm1)
## 
## Call:
## lm(formula = Distance_ ~ Bow_ + Name_, data = JaimeExp1$Data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.1652  -3.4554  -0.5064   2.5754  13.1611 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          156.427      1.238 126.386  < 2e-16 ***
## Bow_Southwood         -9.767      1.238  -7.891 1.17e-10 ***
## Name_Fabiar Darklyn   38.585      1.516  25.454  < 2e-16 ***
## Name_Margan Drumm     -4.212      1.516  -2.779  0.00741 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.794 on 56 degrees of freedom
## Multiple R-squared:  0.9485, Adjusted R-squared:  0.9457 
## F-statistic: 343.6 on 3 and 56 DF,  p-value: < 2.2e-16

(A small detail you’ll notice above is that there is no estimate for the 1st archer effect. The estimate of that effect is couched in the Intercept term; to predict shot distance from the 1st archer, we simply leave out all archer terms).

After accounting for the different archer, the estimated effect of the Southwood bow became extremely statistically significant. These results weren’t as pleasing to Jaime, who stormed past a smirking Tyrion. “Well? What read your results?” he heard as he slammed the training field gate behind him.

So what happened? In truth, there are two separable sources of error around the bow effect: per-shot random error, and variance among archer shooting strengths. Jaime (well, actually Tyrion) controlled for this in the second model by including archer effects.

I have some experience in these matters

“I have some experience in these matters”

If you’ve never encountered this statistical situation before, a visualization can help illustrate why it’s important to include archer effects, or in general, “experimental unit” effects. Here’s what Jaime’s data look like, without knowledge of the particular archer:

The red shots look somewhat above the blue shots, but something’s fishy. The shot points seem to be layered, somehow. Now let’s shape the points differently for different archers:

Now we can see clearly that the red shots are usually higher than the green shots, and the shots cluster around distinct locations that depend on the archer. By including archer terms in our model, we’re able to estimate the noise variance around the archer centers, rather than the overall center. This is beneficial from (at least) two standpoints:

  1. Interpretation: the variance due to archer skill is clearly not random error (Winds of Westeros), so a model that does not account for the marksman will give a highly innacurate estimate of the per-shot variance.
  2. Estimation of \(\tau\): Loosely speaking, we are better able to “see” the effect of the Southwood bow when we separate the shots by archer.

After cooling down a little, Jaime realized that (despite not achieving the result he desired) his findings could be of use to the Lannister army. He decided to run a larger experiment with 50 archers to solidify the results. Here was the outcome:

lm2 <- lm(Distance_ ~ Bow_ + Name_, data = JaimeExp2$Data)
summary(lm2)
## 
## Call:
## lm(formula = Distance_ ~ Bow_ + Name_, data = JaimeExp2$Data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.4424  -3.3524  -0.1689   3.2859  14.8538 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            168.8198     1.1263 149.883  < 2e-16 ***
## Bow_Southwood          -10.4792     0.3154 -33.221  < 2e-16 ***
## Name_Aldo Apperford    -12.2736     1.5772  -7.782 1.86e-14 ***
## Name_Antorn Feller      34.5155     1.5772  21.884  < 2e-16 ***
## Name_Arren Durrandon    29.3949     1.5772  18.637  < 2e-16 ***
## Name_Ashtin Orme        11.5839     1.5772   7.345 4.44e-13 ***
## Name_Barret Foral      -16.6079     1.5772 -10.530  < 2e-16 ***
## Name_Barrock Ryswell     6.4531     1.5772   4.091 4.65e-05 ***
## Name_Brack Mallister   -53.4039     1.5772 -33.860  < 2e-16 ***
## Name_Brandeth Erenford   6.1860     1.5772   3.922 9.41e-05 ***
## Name_Brayan Trante     -35.2405     1.5772 -22.344  < 2e-16 ***
## Name_Charres Apperford   0.3317     1.5772   0.210 0.833494    
## Name_Clarrik Parge       8.2105     1.5772   5.206 2.37e-07 ***
## Name_Codin Sadelyn      37.8102     1.5772  23.973  < 2e-16 ***
## Name_Culler Vyrwel       6.8461     1.5772   4.341 1.57e-05 ***
## Name_Curtass Shiphard  -33.6791     1.5772 -21.354  < 2e-16 ***
## Name_Denzin Garner     -21.6985     1.5772 -13.758  < 2e-16 ***
## Name_Deran Swygert       5.3668     1.5772   3.403 0.000695 ***
## Name_Donnal Brackwell  -18.5525     1.5772 -11.763  < 2e-16 ***
## Name_Dontar Brask        3.0880     1.5772   1.958 0.050534 .  
## Name_Dran Kenning       -4.3976     1.5772  -2.788 0.005406 ** 
## Name_Dwan Plumm        -14.3560     1.5772  -9.102  < 2e-16 ***
## Name_Eddard Meadows    -24.9352     1.5772 -15.810  < 2e-16 ***
## Name_Erock Conklyn      11.5451     1.5772   7.320 5.28e-13 ***
## Name_Evin Rane         -33.1439     1.5772 -21.014  < 2e-16 ***
## Name_Fabiar Darklyn    -33.2426     1.5772 -21.077  < 2e-16 ***
## Name_Harden Blest      -30.0566     1.5772 -19.057  < 2e-16 ***
## Name_Howar Clifton      -2.6749     1.5772  -1.696 0.090221 .  
## Name_Jaesse Cantell    -15.7270     1.5772  -9.971  < 2e-16 ***
## Name_Jares Grell         3.6472     1.5772   2.312 0.020966 *  
## Name_Jarvas Bywater      8.6005     1.5772   5.453 6.32e-08 ***
## Name_Jeffary Mudd       -4.4122     1.5772  -2.797 0.005254 ** 
## Name_Jeran Parsin       -8.9081     1.5772  -5.648 2.14e-08 ***
## Name_Kober Maeson       -5.4609     1.5772  -3.462 0.000559 ***
## Name_Koryn Staunton     32.5297     1.5772  20.625  < 2e-16 ***
## Name_Marak Herston     -51.0331     1.5772 -32.357  < 2e-16 ***
## Name_Mikal Netley      -11.2779     1.5772  -7.151 1.72e-12 ***
## Name_Mykal Ryser       -21.1308     1.5772 -13.398  < 2e-16 ***
## Name_Nelsor Baxter     -19.2805     1.5772 -12.225  < 2e-16 ***
## Name_Portar Parge       36.7979     1.5772  23.331  < 2e-16 ***
## Name_Raman Brakker      36.6719     1.5772  23.251  < 2e-16 ***
## Name_Randar Condon      -4.3076     1.5772  -2.731 0.006428 ** 
## Name_Roberd Spyre        3.6156     1.5772   2.292 0.022098 *  
## Name_Rolan Morrass      25.9665     1.5772  16.464  < 2e-16 ***
## Name_Samn Whitehill    -18.9911     1.5772 -12.041  < 2e-16 ***
## Name_Sarrac Ridman      -9.2904     1.5772  -5.890 5.34e-09 ***
## Name_Seamas Knigh        3.9833     1.5772   2.526 0.011713 *  
## Name_Seldan Harker     -21.1080     1.5772 -13.383  < 2e-16 ***
## Name_Sharun Mudd       -27.9034     1.5772 -17.692  < 2e-16 ***
## Name_Zandren Kneight   -53.3039     1.5772 -33.797  < 2e-16 ***
## Name_Zarin Goodbrother -21.9110     1.5772 -13.892  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.988 on 949 degrees of freedom
## Multiple R-squared:  0.9585, Adjusted R-squared:  0.9564 
## F-statistic: 438.8 on 50 and 949 DF,  p-value: < 2.2e-16

Having disregarded much of his scientific education, Jaime felt quite overwhelmed with this massive list of parameter estimates. In particular, he was confused by the few archers without a statistically significant estimate. Should he re-run the experiment without them?

(If you’re wondering why all the estimated standard errors for the archer effect estimates are the same, it’s because they all fired the same number of shots. Intuitively, this means we should have the same amount of certainty in each estimate.)

This time, Cersei happened to be hanging out around the training grounds. Overhearing his concerns, she decided to step in. “Jaime, don’t trouble yourself with those worrisome parameter estimates for the archer. I looked at your charts from the first experiment. It is certainly proper to account for archers in the model, but the only quantity we should care about right now is the variance of their skills.” Later that afternoon she developed Westeros’s first linear mixed-effects model, in which archer effects are considered to be random draws from a larger population.