Convenience Sampling and Mensa

Substack’s algorithm for figuring out what I like to read is still serving up some odd stuff, but this time it found a good article. In The Mensa Fallacy Emil Kirkegaard takes on Karpinski et al (2018) which claimed that high intelligence is a risk factor for both psychological and physiological disease. The claim in the paper is to a novel finding, but Kirkegaard cites a number of studies that would indicate problems with Karpinski’s arguments.

While the language in the post is a bit on the irreverent side, this seems to be a good example of the general difficulty with convenience studies. The latter are all over science, and many times used without much thought. You have a theory about a population $P$ but it’s hard to actually sample $p\in P$? Luckily for you there is a population $Q$ that is easy to sample such that $q\in Q\Rightarrow q\in P$. That’s where I’ve often seen it end. Maybe there’s an acknowledgment that $p\in P\nRightarrow p\in Q$, but that’s obviously not surprising or else $P=Q$ and there wasn’t a problem to begin with.

So what’s the problem? We’ve done random sampling on $Q$ and we have a nice unbiased estimator ${\hat{\theta }}_Q$ for our population. Is my estimator also unbiased for $P$? A priori we have no grounds to think so. A popular illustration: assume I want to know the average height at my school or university (population $P$). I have no idea how tall random students are and it’s awkward to ask, but luckily the basketball team (population $Q$) publishes player stats that include the height of the players. So I go online, put all the stats in an Excel sheet, calculate the mean, and call it a day. That result would clearly not be particularly valuable for estimating the average height of all students at the school/uni. The problem is that a random sample of $Q$ does not necessarily translate into a random sample of $P$.

Subtle versions of this problem show up in real life research all the time. If we know something about (relevant) ways in which $Q$ differs from $P$, then we can make some adjustments to correct for the bias introduced by our sampling technique. Aside from us sampling from a specific group, the issue can also crop up when we try to “hijack” an existing RCT for secondary analysis, as for example discussed in this post on the Kindergarten study that I’ve been meaning to comment on for a bit. A related issue that requires care is the use of surrogate endpoints in clinical trials, where for example an easy and cheap to acquire lab value functions as a stand-in for the actual issue of interest that may be difficult, expensive or unpleasant to collect. Without a good understanding of the relationship between a population defined by certain lab value cutoffs compared to the endpoint of interest, this may be not interpretable or provide much weaker evidence than we’d like.