Regression to the Mean

From: Subject: Regression to the Mean Date: Wed, 5 Nov 2003 15:42:32 -0500 MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_000_0028_01C3A3B3.6D71E4C0"; type="text/html" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1165 This is a multi-part message in MIME format. ------=_NextPart_000_0028_01C3A3B3.6D71E4C0 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: http://trochim.human.cornell.edu/kb/regrmean.htm Regression to the Mean

3D"Regression
[ Home ]=20

A regression threat, also known as a = "regression=20 artifact" or "regression to the mean" is a statistical phenomenon = that=20 occurs whenever you have a nonrandom sample from a population and = two=20 measures that are imperfectly correlated. The figure shows the = regression=20 to the mean phenomenon. The top part of the figure shows the = pretest=20 distribution for a population. Pretest scores are "normally" = distributed,=20 the frequency distribution looks like a "bell-shaped" curve. = Assume that=20 the sample for your study was selected exclusively from the low = pretest=20 scorers. You can see on the top part of the figure where their = pretest=20 mean is -- clearly, it is considerably below the population = average. What=20 would we predict the posttest to look like? First, let's assume = that your=20 program or treatment doesn't work at all (the "null" case). Our = naive=20 assumption would be that our sample would score just as badly on = the=20 posttest as they did on the pretest. But they don't! The bottom of = the=20 figure shows where the sample's posttest mean would have been = without=20 regression and where it actually is. In actuality, the sample's = posttest=20 mean wound up closer to the posttest population mean than their = pretest=20 mean was to the pretest population mean. In other words, the = sample's mean=20 appears to regress toward the mean of the population from = pretest=20 to posttest.

Why Does It=20 Happen?

Let's start with a simple explanation and work from there. To = see why=20 regression to the mean happens, consider a concrete case. In your = study=20 you select the lowest 10% of the population based on their pretest = score.=20 What are the chances that on the posttest that exact group will = once again=20 constitute the lowest ten percent? Not likely. Most of them will = probably=20 be in the lowest ten percent on the posttest, but if even just a = few are=20 not, then their group's mean will have to be closer to the = population's=20 posttest than it was to the pretest. The same thing is true on the = other=20 end. If you select as your sample the highest ten percent pretest = scorers,=20 they aren't likely to be the highest ten percent on the posttest = (even=20 though most of them may be in the top ten percent). If even just a = few=20 score below the top ten percent on the posttest their group's = posttest=20 mean will have to be closer to the population posttest mean than = to their=20 pretest mean.

Here are a few things you need to know about the regression to = the mean=20 phenomenon:

It is a = statistical=20 = phenomenon.

Regression toward the mean occurs for two reasons. First, it = results=20 because you asymmetrically sampled from the population. If you = randomly=20 sample from the population, you would observe (subject to random = error)=20 that the population and your sample have the same pretest = average.=20 Because the sample is already at the population mean on the = pretest, it=20 is impossible for them to regress towards the mean of the = population any=20 more!

It is a group=20 = phenomenon.

You cannot tell which way an individual's score will move = based on=20 the regression to the mean phenomenon. Even though the group's = average=20 will move toward the population's, some individuals in the group = are=20 likely to move in the other=20 direction.

It happens between = any two=20 = variables.

Here's a common research mistake. You run a program and don't = find=20 any overall group effect. So, you decide to look at those who = did best=20 on the posttest (your "success" stories!?) and see how much they = gained=20 over the pretest. You are selecting a group that is extremely = high on=20 the posttest. They won't likely all be the best on the pretest = as well=20 (although many of them will be). So, their pretest mean has to = be closer=20 to the population mean than their posttest one. You describe = this nice=20 "gain" and are almost ready to write up your results when = someone=20 suggests you look at your "failure" cases, the people who score = worst on=20 your posttest. When you check on how they were doing on the = pretest you=20 find that they weren't the worst scorers there. If they had been = the=20 worst scorers both times, you would have simply said that your = program=20 didn't have any effect on them. But now it looks worse than that = -- it=20 looks like your program actually made them worse relative to the = population! What will you do? How will you ever get your grant = renewed?=20 Or your paper published? Or, heaven help you, how will you ever = get=20 tenured?

What you have to realize, is that the pattern of results I = just=20 described will happen anytime you measure two measures! It will = happen=20 forwards in time (i.e., from pretest to posttest). It will = happen=20 backwards in time (i.e., from posttest to pretest)! It will = happen=20 across measures collected at the same time (e.g., height and = weight)! It=20 will happen even if you don't give your program or=20 treatment.

It is a relative=20 = phenomenon.

It has nothing to do with overall maturational trends. Notice = in the=20 figure above that I didn't bother labeling the x-axis in either = the=20 pretest or posttest distribution. It could be that everyone in = the=20 population gains 20 points (on average) between the pretest and = the=20 posttest. But regression to the mean would still be operating, = even in=20 that case. That is, the low scorers would, on average, be = gaining more=20 than the population gain of 20 points (and thus their mean would = be=20 closer to the = population's).

You can have regression = up or=20 = down.

If your sample consists of below-population-mean scorers, the = regression to the mean will make it appear that they move = up=20 on the other measure. But if your sample consists of = high=20 scorers, their mean will appear to move down = relative to=20 the population. (Note that even if their mean increases, they = could be=20 losing ground to the population. So, if a high-pretest-scoring = sample=20 gains five points on the posttest while the overall sample gains = 15, we=20 would suspect regression to the mean as an alternative = explanation [to=20 our program] for that relatively low = change).

The more extreme the = sample group,=20 the greater the regression to the = mean.

If your sample differs from the population by only a little = bit on=20 the first measure, their won't be much regression to the mean = because=20 there isn't much room for them to regress -- they're already = near the=20 population mean. So, if you have a sample, even a nonrandom one, = that is=20 a pretty good subsample of the population, regression to the = mean will=20 be inconsequential (although it will be present). But if your = sample is=20 very extreme relative to the population (e.g., the lowest or = highest=20 x%), their mean is further from the population's and has more = room to=20 regress.

The less correlated the = two=20 variables, the greater the regression to the=20 = mean.

The other major factor that affects the amount of regression = to the=20 mean is the correlation between the two variables. If the two = variables=20 are perfectly correlated -- the highest scorer on one is = the=20 highest on the other, next highest on one is next highest on the = other,=20 and so on -- there will no be regression to the mean. But this = is=20 unlikely to ever occur in practice. We know from measurement = theory that=20 there is no such thing as "perfect" measurement -- all = measurement is=20 assumed (under the true = score=20 model) to have some random error in measurement. It is only = when the=20 measure has no random error -- is perfectly reliable -- that we = can=20 expect it will be able to correlate perfectly. Since that just = doesn't=20 happen in the real world, we have to assume that measures have = some=20 degree of unreliability, and that relationships between measures = will=20 not be perfect, and that there will appear to be regression to = the mean=20 between these two measures, given asymmetrically sampled=20 subgroups.

The Formula for the = Percent of=20 Regression to the Mean

You can estimate exactly the percent of regression to the mean = in any=20 given situation. The formula is:

P_rm =3D 100(1 - r)

where:

P_rm =3D the percent of regression to the mean
r = =3D the=20 correlation between the two measures

Consider the following four = cases:

	if r =3D 1, there is no = (i.e., 0%)=20 regression to the mean =
	if r =3D .5, there is 50% = regression to=20 the mean =
	if r =3D .2, there is 80% = regression to=20 the mean =
	if r =3D 0, there is 100% = regression to=20 the mean =

In the first case, the two variables are perfectly correlated = and there=20 is no regression to the mean. With a correlation of .5, the = sampled group=20 moves fifty percent of the distance from the = no-regression=20 point to the mean of the population. If the correlation is a small = .20,=20 the sample will regress 80% of the distance. And, if there is no=20 correlation between the measures, the sample will "regress" all = the way=20 back to the population mean! It's worth thinking about what this = last case=20 means. With zero correlation, knowing a score on one measure gives = you=20 absolutely no information about the likely score for that person = on the=20 other measure. In that case, your best guess for how any person = would=20 perform on the second measure will be the mean of that second = measure.

Estimating and Correcting = Regression=20 to the Mean

Given our percentage formula, for any given = situation we can=20 estimate the regression to the mean. All we need to know is the = mean of=20 the sample on the first measure the population mean on both = measures, and=20 the correlation between measures. Consider a simple example. Here, = we'll=20 assume that the pretest population mean is 50 and that we select a = low-pretest scoring sample that has a mean of 30. To begin with, = let's=20 assume that we do not give any program or treatment (i.e., the = null case)=20 and that the population is not changing over time on the = characteristic=20 being measured (i.e., steady-state). Given this, we would predict = that the=20 population mean would be 50 and that the sample would get a = posttest score=20 of 30 if there was no regression to the mean. Now, assume = that the=20 correlation is .50 between the pretest and posttest for the = population.=20 Given our formula, we would expect that the sampled group would = regress=20 50% of the distance from the no-regression point to the population = mean,=20 or 50% of the way from 30 to 50. In this case, we would observe a = score of=20 40 for the sampled group, which would constitute a 10-point = pseudo-effect=20 or regression artifact.

Now, let's relax some of the initial assumptions. = For=20 instance, let's assume that between the pretest and posttest the=20 population gained 15 points on average (and that this gain was = uniform=20 across the entire distribution, that is, the variance of the = population=20 stays the same across the two measurement occasions). In this = case, a=20 sample that had a pretest mean of 30 would be expected to get a = posttest=20 mean of 45 (i.e., 30+15) if there is no regression to the mean = (i.e.,=20 r=3D1). But here, the correlation between pretest and posttest is = .5 so we=20 expect to see regression to the mean that covers 50% of the = distance from=20 the mean of 45 to the population posttest mean of 65. That is, we = would=20 observe a posttest average of 55 for our sample, again a = pseudo-effect of=20 10 points.

Regression to the mean is one of the trickiest threats to = validity. It=20 is subtle in its effects, and even excellent researchers sometimes = fail to=20 catch a potential regression artifact. You might want to learn = more about=20 the regression to the mean phenomenon. One good way to do that = would be to=20 simulate the phenomenon. If you're not familiar with simulation, = you can=20 get a good introduction in the The Simulation Book. If you already understand the = basic=20 idea of simulation, you can do a manual (dice rolling) simulation of regression = artifacts or=20 a computerized simulation of regression = artifacts.=20 =

Copyright =A92002, William M.K. = Trochim, All=20 Rights Reserved
Purchase a = printed=20 copy of the Research Methods Knowledge Base
Last Revised: = 06/29/2000

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