## The selection-distortion effect: How selection changes correlations in surprising ways

Note: After I wrote this post, some people helpfully pointed out in the comments (as I suspected might happen) that this phenomenon has been documented before, first as Berkson’s Paradox and later as “conditioning on a collider” in Pearl’s causal framework. Hopefully the post is still an interesting discussion, but please call it one of those names and not “the selection-distortion effect.”

A little while back I ran across an idea buried in an old paper of Robyn Dawes that really opened my eyes. It was one of those things that seemed really simple and straightforward once I saw it. But I’d never run across it before.[1] The idea is this: when a sample is selected on a combination of 2 (or more) variables, the relationship between those 2 variables is different after selection than it was before, and not just because of restriction of range. The correlation changes in ways that, if you don’t realize it’s happening, can be surprising and potentially misleading. It can flip the sign of a correlation, or turn a zero correlation into a substantial one. Let’s call it the selection-distortion effect.

First, some background: Dawes was the head of the psychology department at the University of Oregon back in the 1970s. Merging his administrative role with his interests in decision-making, he collected data about graduate admissions decisions and how well they predict future outcomes. He eventually wrote a couple of papers based on that work for Science and American Psychologist. The Science paper, titled “Graduate admission variables and future success,” was about why the variables used to select applicants to grad school do not correlate very highly with the admitted students’ later achievements. Dawes’s main point was to demonstrate why, when predictor variables are negatively correlated with each other, they can be perfectly reasonable predictors as a set even though each one taken on its own has a low predictive validity among selected students.

However, in order to get to his main point Dawes had to explain why the correlations would be negative in the first place. He offered the explanation rather briefly and described it in the context of graduate admissions. But it actually refers to (I believe) a very general phenomenon. The key fact to grasp is this: Dawes found, consistently across multiple cohorts, that the correlation between GRE and GPA was negative among admitted students but positive among applicants.

This isn’t restriction of range. Restriction of range attenuates correlations – it pushes them toward zero. As I’ll show below, this phenomenon can easily flip signs and even make the absolute value of a correlation go from zero to substantial.

Instead, it is a result of a multivariate selection process. Grad school admissions committees select for both GRE and GPA. So the selection process eliminates people who are low on both, or really low on just one. Some people are very high on both, and they get admitted. But a lot of people who pass through the selection process are a bit higher on one than on the other (relative to each variable’s respective distributions). Being really excellent on one can compensate for only being pretty good on the other and get you across the selection threshold. It is this kind of implicitly compensatory relationship that makes the correlation more negative in the post-selection group than in the pre-selection group.

To illustrate, here is a figure from a simulation I ran. On the left X and Y are sampled from a standard normal distribution with a population correlation of rho =.30. The observed correlation among 500 cases is r = .26. On the right I have simulated a hard-threshold selection process designed to select cases in the top 50% of the population. Specifically, cases are selected if X + Y > 0. Among the 239 cases that passed the selection filter, the observed correlation is now r = -.25. The correlation hasn’t been attenuated — it has been flipped!

Eyeballing the plot on the right, it’s pretty obvious that a selection process has taken place — you can practically draw a diagonal line along the selection threshold. That’s because I created a hard threshold for illustrative purposes. But that isn’t necessary for the distortion effect to occur. If X and Y are just 2 of several things that predict selection, and/or if they are used in the selection process inconsistently (e.g., with random error as you might expect with human judges), you’ll still get the effect. So you can get it in samples where, if you only had the post-selection dataset to look at, it would not be at all obvious that it had been selected on those variables.

To illustrate, I ran another simulation. This time I set the population correlation to rho = .00 and added another uncorrelated variable, Z, to the selection process (which simulates a committee using things other than GRE and GPA to make its decisions). The observed pre-selection correlation between X and Y is r = .01; in the 253 cases that passed through the selection filter (X + Y + Z > 0), X and Y are correlated r = -.21. The correlation goes from nil to negative, increasing in absolute magnitude; and the scatterplot on the right looks a lot less chopped-off.

As I mentioned it above, once I wrapped my head around this phenomenon I started seeing it in a lot of places. Although Dawes found it among GPA and GRE, it is a statistical issue that is not particular to any one subject-matter domain. You will see it any time there is selection on a combination of 2 variables that allows them to compensate for each other to any degree. Thus both variables have to be part of one selection process: if you run a sample through 2 independent selection filters, one on X while ignoring Y and one on Y while ignoring X (so they cannot compensate for each other), the correlation will be attenuated by restriction of range but you will not observe the selection-distortion effect.[2]

Here are a few examples where I have started to wonder if something like this might be happening. These are all speculative but they fit the pattern.

1. Studies of intellectual ability and academic motivation among college students. You have to have some combination of intelligence and motivation in order to succeed academically and get into college. So the correlation between those two things is probably different among college students than in the pre-selection pool of applicants (and the general population), especially when looking at selective colleges. For example, in a sample of Penn students, Duckworth et al. (2007) reported that grit was negatively correlated with SAT scores. The authors described the finding as “surprising” and offered some possible explanations for it. I’d add the selection-distortion effect to the list of possible explanations.

To be clear, I am not saying that the negative correlation is “wrong.” That may well be a good unbiased estimate of the correlation at Penn. This is about what populations it would and wouldn’t generalize to. You might find something similar at  selective colleges and universities, but perhaps not in the general population. That’s something that anybody who studies ability and motivation in university subject pools should be aware of.

2. The correlation between research productivity and teaching effectiveness. In a recent op-ed, Adam Grant proposed that universities should create new research-only and teaching-only tenure tracks. Grant drew on sound thinking from organizational psychology that says that jobs should be organized around common skill sets. If you are going to create one job that requires multiple skills, they should be skills that are positively correlated so you can hire people who are good at all parts of their job. Grant combined that argument with evidence from Hattie & Marsh (1996) that among university professors, research productivity and teaching effectiveness have a correlation close to zero. On that basis he argued that we should split research and teaching into different positions.

However, it is plausible that the zero correlation among people who have been hired for R1 tenure-track jobs could reflect a selection-distortion effect. On the surface it may seem to people familiar with that selection process that research and teaching aren’t compensatory. But the studies in the Hattie & Marsh meta-analysis typically measured research productivity with some kind of quantitative metric like number of publications or citations, and overwhelmingly measured teaching effectiveness with student evaluations. Those 2 things are pretty close to 2 of the criteria that weigh heavily in hiring decisions: an established record of scholarly output (the CV) and oral presentation skills (the job talk). The latter is almost certainly related to student evaluations of teaching; indeed, I have heard many people argue that job talks are useful for that reason. Certainly it is plausible that in the hiring process there is some tradeoff between an outstanding written record and a killer job talk. There may be something similar on the self-selection side: Ph.D. grads who aren’t interested and good at some combination of research and teaching pursue other kinds of jobs. So it seems plausible to me that research and teaching ability (as these are typically indexed in the data Grant cites) could be positively correlated among Ph.D. graduates, and then the selection process is pushing that correlation in a negative direction.

1. The burger-fry tradeoff. Okay, admittedly kinda silly, but hear me out. Back when I was in grad school I noticed that my favorite places for burgers usually weren’t my favorite places for fries, and vice versa. I’m a enough of That Guy that I actually thought about it in correlational terms (“Gee, I wonder why there is a negative correlation between burger quality and fry quality”). Well years later I think I finally found the answer. The set of burger joints I frequented in town was already selected — I avoided the places with both terrible burgers and terrible fries. So yeah, among the selected sample of places I usually went to, there was a negative correlation. But I bet if you randomly sampled all the burger joints in town, you’d find a positive burger-fries correlation.

(Like I said, once I wrapped my head around the selection-distortion effect I started seeing it everywhere.)

What does this all mean? We as psychologists tend to be good at recognizing when we shouldn’t try to generalize about univariate statistics from unrepresentative samples. Like, you would not think that Obama’s approval rating in your subject pool is representative of his national approval. But we often try to draw generalizable conclusions about relationships between variables from unrepresentative samples. The selection-distortion effect is one way (of many) that that can go wrong. Correlations are sample statistics: at best they say something about the population and context they come from. Whether they generalize beyond that is an empirical question. When you have a selected sample, the selection-distortion effect can even give you surprising and even counterintuitive results if you are not on the lookout for it.

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1. Honestly, I’m more than a little afraid that somebody is going to drop into the comments and say, “Oh that? That’s the blahblahblah effect, everybody knows about that, here’s a link.”

2. Also, this may be obvious to the quantitatively-minded but “selection” is defined mechanistically, not psychologically — it does not matter if a human agent deliberately selected on X and Y, or even if it is just an artifact or side effect of some other selection process.

## 15 thoughts on “The selection-distortion effect: How selection changes correlations in surprising ways”

1. Milton Strauss says:

mind if I post this on facebook?…i think it is of some general interest to many of my friends… milton

1. Sanjay Srivastava says:

By all means!

1. Sanjay Srivastava says:

Rogier Kievet just pointed out the same thing on Twitter, giving me a nice forehead-slapping moment: https://twitter.com/rogierK/status/496367614269325313

I think the reason Simpson’s/Robinson’s Paradox didn’t jump out at me is that in the instances of that I tend to think about, the relationship between X and Y is conditional on aggregation/disaggregation on some 3rd variable. Like in the kidney stone example from the Wikipedia link, the relationship between treatment modality and outcome is different when you aggregate or disaggregate by stone size. And in the Berkeley gender-bias case, the relationship between gender and admissions rates is different when you aggregate or disaggregate over graduate programs.

But in the selection-distortion phenomenon I wrote about, the XY relationship changes when you aggregate or disaggregate on a linear combination of X and Y, not some 3rd variable. I’m not sure whether that is technically Simpson’s/Robinson’s Paradox or just something closely related, but either way the connection did not jump out at me before. (In my defense I totally anticipated this with Footnote 1.)

2. Berkson’s Paradox fits even better, I think!

3. Genobollocks says:

Was reminded of this immediately:
http://www.sciencedirect.com/science/article/pii/S0191886914003432

> Previously, negative associations between intelligence and conscientiousness have been reported and explained in terms of an ‘intelligence compensation hypothesis’ (ICH) whereby higher conscientiousness develops in order to compensate for lower cognitive ability. We argue that conscientious traits, especially those related to achievement, are just as likely to be reinforced by cognitive ability. We evidence this by showing that previous negative associations may be attributable to a compensatory sample selection effect arising because of the use of research samples comprised of participants with achievement above certain thresholds. The associations between conscientiousness and ability in the samples of adolescents and their parents from the Sibling Interaction and Behaviour Study (SIBS) and Minnesota Twin Family Study (MTFS) – which were not selected in this way – were either zero or positive. Further, artificially introducing selection on achievement into these samples biased the associations in the negative direction. Together, these results are consistent with the hypothesis that the true association between these constructs may be zero or positive at the population level but that the use of selected research samples has sometimes resulted in the appearance of a negative association.

They don’t mention Berkson either though :-)

4. On reflection (aka 20/20 hindsight), this effect is logical. The people who are being eliminated all have GPA and GRE scores that are positively correlated (albeit with the scores themselves having relatively small magnitude, since these are the poor performers on both measures). So the remainder are going to have correlations between the two that are, on average, more negative, simply because the bottom-left corner of the distribution “cloud” has been removed (cf. the “After” plot, above).

In the blog post cited by Gwern, this effect is even more pronounced, because another chunk of the people with a positive correlation between GPA and GRE — in this case, with scores of large magnitude — got taken by Harvard. :-) This corresponds to the top-right corner of the cloud in the “After” plot being removed as well. At that point the negative regression line draws itself.

And the takeaway lesson, if it needed to be said, is to be very careful in your assumptions about how people came to be in the group they’re in, unless you (or, even better, a blinded confederate) randomly assigned them there.

1. Sanjay Srivastava says:

Nick – the Slate post you linked above is a great visual explainer. And it makes it clear that what I was talking about is indeed Berkson’s paradox. The phenomenon does seem pretty straightforward and simple once you know it’s there, and especially when you see it from beginning to end as in the Slate piece. I think one of the reasons it’s hard to notice in practice is that often the only data you have is from the post-selection group, and the footprints of the effect are not necessarily obvious from visual inspection (as in the second simulation I ran). So you have to stop and try to imaginatively reverse-engineer whatever selection processes might have produced the sample at hand.

I didn’t mention it in the post but Dawes also talked about the effects that competition would have on selecting out cases that are very high on both. It’s not necessary, but it exacerbates the effect.

5. Oh that? ;) Looks like your Footnote 1 fears were realized… I still thought this was interesting and like the graphical illustrations very much.

6. lov says:

“Oh that? That’s … ” conditioning on a common effect. It makes the causes anticorrelated, even if they’re independent. See Causality by Judea Pearl.

7. Jan Zilinsky says:

Can we just add this whole post (or at least example 3) to the Berkson’s Paradox Wikipedia entry? :)

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