1. Regression To The Mean 林 建 甫 C.F. Jeff Lin, MD. PhD.
2019/2/25
Regression To The Mean
林 建 甫
C.F. Jeff Lin, MD. PhD.
台 北 大 學 統 計 研 究 所 助 理 教 授
台 北 榮 民 總 醫 院 生 物 統 計 顧 問
美 國 密 西 根 大 學 生 物 統 計 博 士
2019/2/25
Jeff Lin, MD. PhD.

2. Pre-test Post-test Comparison Regression to the Mean
2019/2/25
Pre-test Post-test Comparison Regression to the Mean
2019/2/25
Jeff Lin, MD. PhD.

3. Regression Toward the Mean (Definition)
A variable extreme on its first measurement will be closer to the center of the distribution for a later measurement (on the average).
Results from within person variability (temporal variability and measurement error), i.e., 2 e 
2019/2/25
Jeff Lin, MD. PhD.

4. Examples Studies Comparing Ambulatory and Office BP Measurements
Pickering et al. “How common is white coat hypertension.” JAMA, 1988.
Office
DBP
Awake
DBP
Difference
Normal volunteers
(<90 mmHg)
“Borderline Hypertensives”
( mmHg)
“Established Hypertensives”
(≥105 mmHg)
2019/2/25
Jeff Lin, MD. PhD.

5. The Simple Explanation...
when you select a group from the extreme end of a distribution...
selected
group’s
mean
overall
mean
...they will do better on a subsequent measure
their mean on the first measure appears to “regress toward the mean” of the second measure
overall
mean
regression to the mean
2019/2/25
Jeff Lin, MD. PhD.

6. Example I: pretest posttest
if the first measure is a pretest and you select the low scorers...
...and the second measure is a posttest
regression to the mean will make it appear as though they gained from pre to post
posttest
pseudo-effect
2019/2/25
Jeff Lin, MD. PhD.

7. Example II: pretest posttest
if the first measure is a pretest and you select the high scorers...
...and the second measure is a posttest
regression to the mean will make it appear as though they lost from pre to post
posttest
pseudo-effect
2019/2/25
Jeff Lin, MD. PhD.

8. Some facts this is purely a statistical phenomenon
2019/2/25
this is purely a statistical phenomenon
this is a group phenomenon
some individuals will move opposite to this group trend
2019/2/25
Jeff Lin, MD. PhD.

9. Why Does It Happen?
2019/2/25
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?
2019/2/25
Jeff Lin, MD. PhD.

10. Why Does It Happen?
2019/2/25
regression artifacts occur whenever we sample asymmetrically from a distribution
regression artifacts occur with any two variables (not just pre and posttest) and even backwards in time!
2019/2/25
Jeff Lin, MD. PhD.

11. the absolute amount of regression to the mean depends on two factors:
What Does It Depend On?
the absolute amount of regression to the mean depends on two factors:
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
2019/2/25
Jeff Lin, MD. PhD.

12. The percent of regression to the mean is:
A Simple Formula
The percent of regression to the mean is:
2019/2/25
Jeff Lin, MD. PhD.

13. The percent of regression to the mean is:
A Simple Formula
The percent of regression to the mean is:
Prm = 100(1 - r)
2019/2/25
Jeff Lin, MD. PhD.

14. A Simple Formula The percent of regression to the mean is:
Prm = 100(1 - r)
where r is the correlation between the two measures
2019/2/25
Jeff Lin, MD. PhD.

15. A Simple Formula The percent of regression to the mean is:
Prm = 100(1 - r)
where r is the correlation
between the two 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
2019/2/25
Jeff Lin, MD. PhD.

16. For Example: Prm = 100(1 - r)
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
2019/2/25
Jeff Lin, MD. PhD.

17. assume a standardized test with a mean of 50
Example
assume a standardized test with a mean of 50
pretest 50
2019/2/25
Jeff Lin, MD. PhD.

18. Example pretest assume a standardized test with a mean of 50
you give your program to the lowest scorers and their mean is 30 30 50
2019/2/25
Jeff Lin, MD. PhD.

19. Example pretest posttest assume a standardized test with a mean of 50
you give your program to the lowest scorers and their mean is 30
assume that the correlation of pre-post is .5 30 50
posttest
2019/2/25
Jeff Lin, MD. PhD.

20. Example pretest posttest assume a standardized test with a mean of 50
you give your program to the lowest scorers and their mean is 30
assume that the correlation of pre-post is .5 30 50
the formula is:
posttest
2019/2/25
Jeff Lin, MD. PhD.

21. Example pretest posttest assume a standardized test with a mean of 50
you give your program to the lowest scorers and their mean is 30
assume that the correlation of pre-post is .5 30 50
the formula is:
Prm = 100(1 - r)
= 100(1-.5)
= 50%
posttest
2019/2/25
Jeff Lin, MD. PhD.
50%

22. Example pretest posttest assume a standardized test with a mean of 50
you give your program to the lowest scorers and their mean is 30
assume that the correlation of pre-post is .5 30 50
the formula is:
Prm = 100(1 - r)
= 100(1-.5)
= 50%
Therefore the mean will regress up 50% (from 30 to 50), leaving a final mean of 40 and a 10 point pseudo-gain 40
posttest
pseudo-effect
2019/2/25
Jeff Lin, MD. PhD.

23. Regression Toward the Mean
Measurable characteristics of an individual do not have constant values
Vary above and below the average value
If two sets of measurements are made on individuals, the correlation between the first and the second series of measurements will not be perfect.
It is often the case that the more distant a measured characteristics from the population means of that characteristic, the more variable the measurement tends to be.
2019/2/25
Jeff Lin, MD. PhD.

24. Regression Toward the Mean
2019/2/25
The more extreme the initial selection criterion (that is, the further from the population mean), the greater will be the regression toward the mean at the time of the next measurement.
Whenever participants are selected from a population on the basis of some measured characteristic, the mean of a subsequent measurement will be closer to the population mean than is the first measurement mean.
2019/2/25
Jeff Lin, MD. PhD.

25. Regression Towards the Mean
1. Regression towards the mean complicates the interpretation of many uncontrolled studies; frequently it is not recognized
2. In randomized clinical trials regression towards the mean can influence:
Cost, amount and nature of screening
Choice of baseline for within group comparisons
Hypothesized treatment effect based on risk factor change
2019/2/25
Jeff Lin, MD. PhD.

26. Regression Toward the Mean
An investigator cannot simply compare pre-intervention and post-intervention values in the intervention group.
She must compare post-intervention values in the intervention group with values obtained at similar times in the control group.
2019/2/25
Jeff Lin, MD. PhD.

27. Regression Toward the Mean
Use a more extreme value than the entrance criterion when the investigators screened people before baseline.
Enroll participant had his or her pressure recorded three times.
Only those whose second and third measure averaged 95 mmHg or greater were invited to the clinic for further evaluation.
2019/2/25
Jeff Lin, MD. PhD.

28. Strategies for Reducing Regression Toward the Mean
Multiple measurements; multiple visits
Baseline free of selection
Standardization of methods; training
2019/2/25
Jeff Lin, MD. PhD.

29. Thanks !
2019/2/25
Jeff Lin, MD. PhD.