Probably Overthinking It

Is Simpson’s Paradox just a mathematical curiosity, or does it happen in real life? And if it happens, what does it mean? To answer these questions, I’ve been searching for natural examples in data from the General Social Survey (GSS).

• A few weeks ago I posted this article, where I group GSS respondents by their decade of birth and plot changes in their opinions over time. Among questions related to faith in humanity, I found several instances of Simpson’s paradox; for example, in every generation, people have become more optimistic over time, but the overall average is going down over time. The reason for this apparent contradiction is generational replacement: as old optimists die, they are being replaced by young pessimists.
• In this followup article, I group people by level of education and plot their opinions over time, and again I found several instances of Simpson’s paradox. For example, at every level of education, support for legal abortion has gone down over time (at least under some conditions). But the overall level of support has increased, because over the same period, more people have achieved higher levels of education.
• In the most recent article, I group people by decade of birth again, and plot their opinions as a function of age rather than time. I found some of the clearest instances of Simpson’s paradox so far. For example, if we plot support for interracial marriage as a function of age, the trend is downward; older people are less likely to approve. But within every birth cohort, support for interracial marriage increases as a function of age.

With so many examples, we are starting to see a pattern:

• Examples of Simpson’s paradox are confusing at first because they violate our expectation that if a trend goes in the same direction in every group, it must go in the same direction when we put the groups together.
• But now we realize that this expectation is naive: mathematically, it does not have to be true, and in practice, there are several reasons it can happen, including generational replacement and period effects.
• Once explained, the examples we’ve seen so far have turned out not to be very informative. Rather than revealing useful information about the world, it seems like Simpson’s paradox is most often a sign that we are not looking at the data in the most effective way,

But before I give up, I want to give it one more try.

A more systematic search

Each example of Simpson’s paradox involves three variables:

• On the x-axis, I’ve put time, age, and a few other continuous variables.
• On the y-axis, I’ve put the fraction of people giving the most common response to questions about opinions, attitudes, and world view.
• And I have grouped respondents by decade of birth, age, sex, race, religion, and several other demographic variables.

At this point I have tried a few thousand combinations and found about ten clear-cut instances of Simpson’s paradox. So I’ve decided to make a more systematic search. From the GSS data I selected 119 opinion questions that were asked repeatedly over more than a decade, and 12 demographic questions I could sensibly use to group respondents.

With 119 possible variables on the x-axis, the same 119 possibilities on the y-axis, and 12 groupings, there are a 84,118 sensible combinations. When I tested them, 594 produced computational errors of some kind, in most cases because some variables have logical dependencies on others. Among the remaining combinations, I found 19 instances of Simpson’s paradox.

So one conclusion we can reach immediately is that Simpson’s paradox is rare in the wild, at least with data of this kind. But let’s look more closely at the 19 examples.

Many of them turn out to be statistical noise. For example, the following figure shows responses to a question about premarital sex on the y-axis, responses to a question about confidence in the press on the x-axis, with respondents grouped by political alignment.

As confidence in the press declines from left to right, the overall fraction of people who think premarital sex is “not wrong at all” declines slightly. But within each political group, there is a slight increase.

Although this example meets the requirements for Simpson’s paradox, it is unlikely to mean much. Most of these relationships are not statistically significant, which means that if the GSS had randomly sampled a different group of people, it is plausible that these trends might have gone the other way.

And this should not be surprising. If there is no relationship between two variables in reality, the actual trend is zero and the trend we see in a random sample is equally likely to be positive or negative. Under this assumption, we can estimate the probability of seeing a Simpson paradox by chance:

• If the overall trend is positive, the trend in all three groups has to be negative, which happens one time in eight.
• If the overall trend is negative, the trend in all three groups has to be positive, which also happens one time in eight.

When there are more groups, Simpson’s paradox is less likely to happen by chance. Even so, since we tried so many combinations, it is only surprising that we did find more.

A few good examples

Most of the examples I found are like the previous one. The relationships are so weak that the trends we see are mostly random, which means we don’t need a special explanation for Simpson’s paradox. But I found a few examples where the Simpsonian reversal is probably not random and, even better, it makes sense.

For example, the following figure shows the fraction of people who would support a gun law as a function of how often they pray, grouped by sex.

Within each group, the overall trend is downward: the more you pray, the less likely you are to favor gun control. But the overall trend goes the other way: people who pray more are more likely to support gun control. Before you proceed, see if you can figure out what’s going on.

At this point you might guess that there is a correlation of some kind between the variable on the x-axis and the groups. In this example, there is a substantial difference in how much men and women pray. The following figure shows how much:

And that’s why average support for gun control increases as a function of prayer:

• The low-prayer groups are mostly male, so average support for gun control is closer to the male response, which is lower.
• The high-prayer groups are mostly female, so the overall average is closer to the female response, which is higher.

On one hand, this result is satisfying because we were able to explain something surprising. But having made the effort, I’m not sure we have learned much. Let’s look at one more example.

The GSS includes the following question about a hypothetical open housing law:

Suppose there is a community-wide vote on the general housing issue. There are two possible laws to vote on. One law says that a homeowner can decide for himself whom to sell his house to, even if he prefers not to sell to [someone because of their race or color]. The second law says that a homeowner cannot refuse to sell to someone because of their race or color. Which law would you vote for?

The following figure shows the fraction of people who would vote for the second law, grouped by race and plotted as a function of income (on a log scale).

In every group, support for open housing increases as a function of income, but the overall trend goes the other way: people who make more money are less likely to support open housing.

At this point, you can probably figure out why:

• White respondents are less likely to support this law than Black respondents and people of other races, and
• People in the higher income groups are more likely to be white.

So the overall average in the lower income groups is closer to the non-white response; the overall average in the higher income groups is closer to the white response.

Summary

Is Simpson’s paradox a mathematical curiosity, or does it happen in real life?

Based on my exploration (and a similar search in a different dataset), if you go looking for Simpson’s paradox in real data, you will find it. But it is rare: I tried almost 100,000 combinations, and found only about 100 examples. And a large majority of the examples I found were just statistical noise.

What does Simpson’s paradox tell us about the data, and about the world?

In the examples I found, Simpson’s paradox doesn’t reveal anything about the world that is useful to know. Mostly it creates confusion, especially for people who have not encountered it before. Sometimes it is satisfying to figure out what’s going on, but if you create confusion and then resolve it, I am not sure you have made net progress. If Simpson’s paradox is useful, it is as a warning that the question you are asking and the way you are looking at the data don’t quite go together.

To paraphrase Henny Youngman,

The patient says, “Doctor, it’s confusing when I look at the data like this.”
The doctor says, “Then don’t do that!”

As people get older, do they become more racist, sexist, and homophobic? To find out, you could use data from the General Social Survey (GSS), which asks questions like:

• Do you think there should be laws against marriages between Blacks/African-Americans and whites?
• Should a man who admits[mfn]If you find the wording of this question problematic, remember that it was written in 1970 and reflects mainstream views at the time. It persists because, in order to support time series analysis, the GSS generally avoids changing the wording of questions.[/mfn] that he is a homosexual be allowed to teach in a college or university, or not?
• Tell me if you agree or disagree with this statement: Most men are better suited emotionally for politics than are most women.

If you plot the answers to these questions as a function of age, you find that older people are, in fact, more racist, sexist, and homophobic than younger people. But that’s not because they are old; it’s because they were born, raised, and educated during a time when large fractions of the population were racist, sexist homophobes.

In other words, it’s primarily a cohort effect, not an age effect. We can see that if we group respondents by birth cohort and plot their responses by age. Here are the results for the first question:

The circle markers show the proportion of respondents who got this question wrong (no other way to put it); the lines show local regressions through the markers.

The dashed gray line shows the overall trend, if we don’t group by cohort. Sure enough, when this question was asked between 1972 and 2002, older respondents were substantially more likely to support laws against marriage between people of difference races.

But when we group by decade of birth, we see:

• A cohort effect: people born later are less racist.
• A period effect: within every cohort, people get less racist over time.

The results are similar for the second question:

If you thought the racism was bad, get a load of the homophobia!

But again, all birth cohorts became more tolerant over time (even the people born in the 19-aughts, though it doesn’t look it). And again, there is no age effect; people do not become homophobic as they age.

They don’t get more sexist, either:

Simpson’s Paradox

These are all examples of Simpson’s paradox, where the trend in every group goes in one direction, and the overall trend goes in the other direction. It’s called a paradox because many people find it counterintuitive at first. But once you have seen a few examples, like the ones I wrote about this, this, and this previous article, it gets to be less of a surprise.

And if you pay attention, it can be a hint that there is something wrong with your model. In this case, it is a symptom that we are looking at the data the wrong way. If we suspect that the changes we see are due to cohort and period, rather than age, we can check by plotting over time, rather than age, like this:

Every cohort is less racist than its predecessor, every cohort gets less racist over time, and the overall trend goes in the same direction, so Simpson’s paradox is resolved.

Or maybe it persists in a weaker form: the overall trend is steeper than the trend in any of the cohorts, because in addition to the cohort effect and the period effect, we also see the effect of generational replacement.

This article is part of a series where I search the GSS for examples of Simpson’s paradox. More coming soon!

Is Simpson’s paradox a mathematical curiosity or something that matters in practice? To answer this question, I’m searching the General Social Survey (GSS) for examples. Last week I published the first batch, examples where we group people by decade of birth and plot their opinions over time. In this article I present the next batch, grouping by education and plotting over time.

The first example I found is in the responses to this question: “Please tell me whether or not you think it should be possible for a pregnant woman to obtain a legal abortion if she is married and does not want any more children?”

If we group respondents by the highest degree they have earned and compute the fraction who answer “yes” over time, the results meet the criteria for Simpson’s paradox: in every group, the trend over time is downward, but if we put the groups together, the overall trend is upward.

However, if we plot the data, we see that this example is not entirely satisfying.

The markers show the fraction of respondents in each group who answered “yes”; the lines show local regressions through the markers.

In all groups, support for legal abortion (under the specified condition) was decreasing until the 1990s, then started to increase. If we fit a straight line to these curves, the estimated slope is negative. And if we fit a straight line to the overall curve, the estimated slope is positive.

But in both cases, the result doesn’t mean very much because we’re fitting a line to a curve. This is one of many examples I have seen where Simpson’s paradox doesn’t happen because anything interesting is happening in the world; it is just an artifact of a bad model.

This example would have been more interesting in 2002. If we run the same analysis using data from 2002 or earlier, we see a substantial decrease in all groups, and almost no change overall. In that case, the paradox is explained by changes in educational level. Between 1972 and 2002, the fraction of people with a college degree increased substantially. Support for abortion was decreasing in all groups, but more and more people were in the high-support groups.

Free speech

We see a similar pattern in many of the questions related to free speech. For example, the GSS asks, “Suppose an admitted Communist wanted to make a speech in your community. Should he be allowed to speak, or not?” The following figure shows the fraction of respondents at each education level who say “allowed to speak”, plotted over time.

The differences between the groups are big: among people with a bachelor’s or advanced degree, almost 90% would allow an “admitted” Communist to speak; among people without a high school diploma it’s less than 50%. (If you are curious about the wording of questions like this, remember that many GSS questions were written in the 1970s and, for purposes of comparison over time, they avoid changing the text.)

The responses have changed only slightly since 1973: in most groups, support has increased a little; among people with a junior college degree, it has decreased a little.

But overall support has increased substantially, for the same reason as in the previous example: the number of people at higher levels of education increased during this interval.

Whether this is an example of Simpson’s paradox depends on the definition. But it is certainly an example where we see one story if we look at the overall trend and another story if we look at the subgroups.

Other questions related to free speech show similar trends. For example, the GSS asks: “There are always some people whose ideas are considered bad or dangerous by other people. For instance, somebody who is against all churches and religion. If some people in your community suggested that a book he wrote against churches and religion should be taken out of your public library, would you favor removing this book, or not?”

The following figure shows the fraction of respondents who say the book should not be removed:

Again, respondents with more education are more likely to support free speech (and probably less hostile to the non-religious, as well). But in this case support is increasing among people with less education. So the overall trend we see is really the sum of two trends: increases within some groups in addition to shifts between groups.

In this example, the overall slope is steeper than the estimated slope in any group. That would be surprising if you expected the overall slope to be like a weighted average of the group slopes. But as all of these examples show, it’s not.

This article presents examples of Simpson’s paradox, and related patterns, when we group people by education level and plot their responses over time. In the next article we’ll see what happens when we groups people by age.

Years ago I told one of my colleagues about my Data Science class and he asked if I taught Simpson’s paradox. I said I didn’t spend much time on it because, I opined, it is a mathematical curiosity unlikely to come up in practice. My colleague was shocked and dismayed because, he said, it comes up all the time in his field (psychology).

Ever since then, I have been undecided about Simpson’s paradox. A lot of people think it’s important, but when I look for compelling examples, I find the same three stories over and over: Berkeley admissions, kidney stones, and Derek Jeter’s batting average. Yawn.

But last week I found a more interesting example:

1. Between 2000 and 2018, real wages for full-time employees increased.
2. Over the same period, real wages decreased for people at every level of education.

If that seems impossible, go read my article about it.

I also found an article (“Can you Trust the Trend: Discovering Simpson’s Paradoxes in Social Data“) where the authors search a dataset for instances of Simpson’s paradox and describe the ones they find.

And that got me thinking about my old friend, the General Social Survey (GSS). So I’ve started searching the GSS for instances of Simpson’s paradox. I’ll report what I find, and maybe we can get a sense of (1) how often it happens, (2) whether it matters, and (3) what to do about it.

I’ll start with examples where the x-variable is time. For y-variables, I use about 120 questions from the GSS. And for subgroups, I use race, sex, political alignment (liberal-conservative), political party (Democrat-Republican), religion, age, birth cohort, social class, and education level. That’s about 1000 combinations.

Of these, about 10 meet the strict criteria for Simpson’s paradox, where the trend in all subgroups goes in the same direction and the overall trend goes in the other direction. On examination, most of them are not very interesting. In most cases, the actual trend is nonlinear, so the parameters of the linear model don’t mean very much.

But a few of them turn out to be interesting, at least to me. For example, the following figure shows the fraction of respondents who think “most of the time people try to be helpful”, plotted over time, grouped by decade of birth. The markers show the percentage in each group during each interval; the lines show local regressions.

Within each group, the trend is positive: apparently, people get more optimistic about human nature as they age. But overall the trend is negative. Why? Because of generational replacement. People born before 1940 are substantially more optimistic than people born later; as old optimists die, they are being replaced by young pessimists.

Based on this example, we can go looking for similar patterns in other variables. For example, here are the results from a related question about fairness.

Again, old optimists are being replaced by young pessimists.

For a similar question about trust, the results are a little more chaotic:

Some groups are going up and others down, so this example doesn’t meet the criteria for Simpson’s paradox. But it shows the same pattern of generational replacement.

Old conservatives, young liberals

Questions related to prohibition show similar patterns. For example, here are the responses to a question about whether pornography should be illegal.

In almost every group, support for banning pornography has increased over time. But recent generations are substantially more tolerant on this point, so overall support for prohibition is decreasing.

The results for legalizing marijuana are similar.

In most groups, support for legalization has gone down over time; nevertheless, through the power of generational replacement, overall support is increasing.

So far, I think this is more interesting than Derek Jeter’s batting average. More examples coming soon!