Regression to the Mean

I want to write about several genetic matters that inform breeding and give us confidence when we brag about our dogs.  Unfortunately, to address some of the more complex issues, I foresee the need to establish a common understanding of terms.  Politicians take note.  A very popular and useful concept for understanding polygenic inheritance is Regression to the Mean.  Too often I hear it used incorrectly, and so the goal of this post is to explain the concept and hopefully align our future discussions. If you already know this stuff, let me know where I made it too confusing.

To start, the “mean” is defined as the arithmetic average of a set of numbers.  Or as we like to say, “the average is mean.”  So, in the simplest terms, regression to the mean implies movement from some extreme value back toward the middle.  And if that’s your understanding, you’re most of the way there!  A more detailed examination of regression, however, suggests that this process, like your phone service, has terms and conditions.

Movement is an odd way to put it because nothing really moves.  What we are really talking about is the observed change from one measurement to the next.  Either measuring the same thing twice or measuring two related things.  The amount of change is always governed to some degree by regression and in certain cases, can be predicted.  To make this prediction, we need to know how correlated or dependent the measurements are on each other; and the more accurate the prediction, the more we can untangle regression from true change.

Let’s use coin flipping as our first example.  Take a bunch of coins, flip each 100 times and count heads.  Set aside the coins that got the most heads and flip them another 100 times.  Regression to the mean would predict that these coins will average only 50 heads in this second trial, not the larger number that got them set aside from the first trial.  That’s because coins don’t have memories.  The trial of 100 flips is completely independent of the next trial of 100 flips.  Perfect and complete regression to the mean.  Yes, a single coin might end up with more heads in the second trial, but on average, the effect is predictable.

Now let’s give a bunch of high school students the math SAT and retest the kids with the highest scores.  They are likely to score high again.  That’s because the test is designed to be a valid assessment of math skills and has a feature called “reliability,” an indication of consistency.  If the test was not reliable, it couldn’t say much about the student’s math achievement and the results would appear random.  So, the second score is likely to be very close to the first, but it too will regress to the mean.  Just a tiny bit, but it should regress nonetheless.  And not only that, the more extreme the first score is the more regression we expect.  Regression works on everything, but it’s more noticeable on extreme values.

To understand why, consider a student who scored 650 on the first testing.  A full 150 points above the mean.  This is a fairly accurate estimate of their true math score, but there is always a bit of error in this kind of measurement.  Let’s say 30 points of error either way.  This means that our student is either a true 620 having a good test day, or a true 680 having a bad test day.  Retesting them will give us a clue as to which they really are.  Since there are more students in the world who score 620 than there are those who score 680, it’s more likely that the student is a true 620 and the second test will yield a score lower than 650.  Regression to the mean.  Of course, the student could score a 700 on retest, but we are talking about a big picture phenomenon.  On average, regression.

So, what does this mean for Tibetan Mastiffs?  Well, the traits that parents pass on to their puppies work like retesting.  The sire’s or dam’s score is the first test, and the puppy’s score is the retest.  Different dogs, but since there is a correlation between parent and offspring traits, regression to the mean is in play.  Stay tuned.