Regression to the Mean. The Simple Explanation... When you select a group from the extreme end of a...

Regression to the MeanRegression to the MeanRegression to the MeanRegression to the Mean

The Simple Explanation...The Simple Explanation...The Simple Explanation...The Simple Explanation...

When you select a When you select a group from the group from the

extreme end of a extreme end of a distribution...distribution...




SelectedSelectedgroup’sgroup’smeanmean

OverallOverallmeanmean




the group will do the group will do betterbetter on a on a

subsequent measure.subsequent measure.










Where it Where it would would have have been been

with no with no regressionregression





subsequent measuresubsequent measure



Where itsWhere itsmean ismean is






The group mean on The group mean on the first measure the first measure

appears to “regress appears to “regress toward the mean” of toward the mean” of the second measure.the second measure.









The group mean on The group mean on the first measure the first measure

appears to “regress appears to “regress toward the mean” of toward the mean” of the second measure.the second measure.



Regression to the meanRegression to the mean


Example I:Example I:Example I:Example I:

If the first measure If the first measure is a pretest and is a pretest and

you select the you select the lowlow scorers...scorers...

PretestPretest




...and the second ...and the second measure is a posttestmeasure is a posttest

PretestPretest

PosttestPosttest


if the first measure if the first measure is a pretest and is a pretest and


...and the second ...and the second measure is a posttest,measure is a posttest,

PretestPretest

PosttestPosttest

regression to the regression to the mean will make it mean will make it

appear as though the appear as though the group group gainedgained from from

pre to post.pre to post.

Pseudo-effectPseudo-effect

Example II:Example II:Example II:Example II:


you select the you select the highhigh scorers...scorers...

PretestPretest

Example II:Example II:Example II:Example II:

if the first measure if the first measure is a pretest and is a pretest and



PretestPretest

PosttestPosttest



PretestPretest

PosttestPosttest

regression to the regression to the mean will make it mean will make it

appear as though the appear as though the group group lostlost from pre from pre

to post.to post.

Pseudo-effectPseudo-effect



Some FactsSome FactsSome FactsSome Facts

• This is purely a statistical phenomenon.• This is a group phenomenon.• Some individuals will move opposite to

this group trend.

Why Does It Happen?Why Does It Happen?Why Does It Happen?Why Does It Happen?

• For low scorers, you have taken the lowest x%. What are the chances they will be the lowest x% on the second measure?

• For high scorers, you have taken the highest x%. What are the chances they will be the highest x% on the second measure?

Why Does It Happen?Why Does It Happen?Why Does It Happen?Why Does It Happen?

• Regression artifacts occur whenever you sample asymmetrically from a distribution.

• Regression artifacts occur with any two variables (not just pre and posttest) and even backwards in time!

What Does It Depend On?What Does It Depend On?What Does It Depend On?What Does It Depend On?

• The degree of asymmetry (i.e., how far from the overall mean of the first measure the selected group's mean is)

• The correlation between the two measures

The absolute amount of regression to the mean depends on two factors:

A Simple FormulaA Simple FormulaA Simple FormulaA Simple Formula

The percent of regression to the mean isThe percent of regression to the mean isThe percent of regression to the mean isThe percent of regression to the mean is


The percent of regression to the mean is:The percent of regression to the mean is:The percent of regression to the mean is:The percent of regression to the mean is:

PPrmrm = 100(1 - r) = 100(1 - r)


The percent of regression to the mean isThe percent of regression to the mean isThe percent of regression to the mean isThe percent of regression to the mean is

PPrmrm = 100(1 - r) = 100(1 - r)

Where Where rr is the correlation between the two is the correlation between the two measures.measures.


The percent of regression to the mean is:The percent of regression to the mean is:The percent of regression to the mean is:The percent of regression to the mean is:

PPrmrm = 100(1 - r) = 100(1 - r)

Where Where rr is the correlation between the two is the correlation between the two measures.measures.

The formula tells the %, but the actual amount depends on how far the group mean is from the overall mean on

the selection variable.

For Example:For Example:For Example:For Example:

• If r = 1, there is no (i.e., 0%) regression to the mean.

• If r = 0, there is 100% regression to the mean.

• If r = .2, there is 80% regression to the mean.

• If r = .5, there is 50% regression to the mean.

PPrmrm = 100(1 - r) = 100(1 - r)

ExampleExampleExampleExampleAssume a standardized Assume a standardized test with a mean of 50.test with a mean of 50. PretestPretest

5050

ExampleExampleExampleExampleAssume a standardized Assume a standardized test with a mean of 50test with a mean of 50 PretestPretest

5050

You give your program You give your program to the lowest scorers to the lowest scorers and their mean is 30.and their mean is 30.

3030

ExampleExampleExampleExampleAssume a standardized Assume a standardized test with a mean of 50.test with a mean of 50. PretestPretest

PosttestPosttest

5050


3030Assume that the correlation Assume that the correlation of pre-post is .5.of pre-post is .5.

ExampleExampleExampleExampleAssume a standardized Assume a standardized test with a mean of 50.test with a mean of 50.

The formula is…The formula is…

PretestPretest

PosttestPosttest

5050




The formula isThe formula is

PretestPretest

PosttestPosttest

5050



PPrmrm = 100(1 - r)= 100(1 - r)

= 100(1-.5)= 100(1-.5)= 50%= 50%

50%50%


The formula isThe formula is

PretestPretest

PosttestPosttest

Therefore the mean Therefore the mean will regress up 50% will regress up 50%

(from 30 to 50), (from 30 to 50), leaving a final mean leaving a final mean of 40 and a 10 point of 40 and a 10 point

pseudo-gain.pseudo-gain. Pseudo-effectPseudo-effect

5050



PPrmrm = 100(1 - r)= 100(1 - r)

= 100(1-.5)= 100(1-.5)= 50%= 50%

4040

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