Regression to the mean is a statistical phenomenon where extreme values are likely to be closer to the average upon subsequent measurements. In addiction research, it can cause mistaken conclusions that changes in outcomes are due to treatment when they are actually due to random fluctuations. Researchers should minimize measurement error, use comparison groups, statistical corrections, and make reasonable conclusions about the effects of regression to the mean.

1. 1
Regression to the Mean
Addictions Seminar
9/17/08
Kevin Cummins
The MEAN


2. 2
Outline
• Objective
• Definition
• Description
• Implications
• Recent addictions literature
• What to do about it

3. 3
Objective
• Be able to identify regression to the
mean
• Know how to respond to its presence
• Recognize that the concept is used
loosely in addictions

4. 4
Definition
• Regression to the Mean (RTM):
The statistical phenomenon stating that the
greater the deviation of a random variable
from its mean, the greater the probability that
a subsequent observation will deviate
less far.
• In other words, an extreme event is likely to
be followed by a less extreme event.

5. 5
When It Happens
• RTM occurs whenever a
unrepresentative sample is selected
from a population and then repeated
measures are taken.

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Example

7. 7
Simulation
• http://www.ruf.rice.edu/~lane/stat_sim/reg_to_mean/index.html
True Value
RecordedValue

8. 8
Why it Happens
0
.1
.2
.3
.4
y
-4 -2 0 2 4
x
y
y
x1

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Why it Happens
All Variance is Random Measurement Error
0
.1
.2
.3
.4
y
-4 -2 0 2 4
x
y
y0
.1
.2
.3
.4
y
-4 -2 0 2 4
x
y
y
x2x1

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Why it Happens
No Measurement Error
0
.1
.2
.3
.4
y
-4 -2 0 2 4
x
y
y
 x2x1

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Why it Happens
Person-Person Variation and a
Little Measurement Error
0
.1
.2
.3
.4
y
-4 -2 0 2 4
x
y
y
x2
x1  x2x1

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Why it Happens
Observed distribution with error
has thicker tails
0
.1
.2
.3
.4
-4 -2 0 2 4
x
yy
y
W/Error
No Error

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Components View:
V(yij
)  Person
2
 " Error "
2
V(yij
)  Person
2
 measurement
2
 unexplained
2
yij
   personi
 (measurement _error  unexplained _error)j
V(yij
)  Person
2
 " Error "
2
V(yij
)  Person
2
 measurement
2
 unexplained
2
yij
   personi
 (measurement _error  unexplained _error)j
Components which vary across measurements

14. 14
0
.1
.2
.3
.4
y
-4 -2 0 2 4
x
y
y
y
y
More subjects with true means below the cutoff are included in the sample than
excluded subjects with true means above the cutoff.
A
Distribution of repeated measurements on subjects
with the same true mean: A) mean above the cutoff,
and B) mean below the cutoff.
B

15. 15
Formal Description

16. 16
Formal Description of RTM
P(x)dx  P(x)dx
 j
 j
i
i

for i > j > 0
This is a stochastic property; it applies to
random variates.
As you move away from the mean, the proportion of
the distribution that lies closer to the mean increases
continuously.

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Outline
• Objective
• Definition
• Description
• Implications
• Recent addictions literature
• What to do about it

18. 18
Clinical Implications
• Diagnostic Tests
• New Treatments
• Public Health
• Clinical Performance
Adjustments
• Placebo Effect
Morton & Torgenson 2008
“[RTM] can result in
wrongly concluding
that change is due to
treatment when it is
due to chance”
Mistaken spontaneous
reversion
Application to clinical
outliers increases
RTM influenceTreating clusters
Random components
Interpreting change
as a placebo effect

19. 19
Implications: Longitudinal
Research
0
2
4
6
8
10
Measure
1 1.2 1.4 1.6 1.8 2
time
Meas ure
Meas ure
Mean

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Implications: Longitudinal
Research
Mean
0
2
4
6
8
wceiling
1 1.2 1.4 1.6 1.8 2
time
wceiling
wceiling
Mean
MetricwithCeiling
Max

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Implications for Longitudinal
Research
yijt
   time_ effectt
 personi
 errorj
If your sample is not representative of the
population there will be a “time effect” due to
regression to the mean.
Side Note: Distributional assumptions can be
violated.

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Components View:
Two Time Periods & Two Tx
V (yijt
)  Person
2
 "error "
2
yijkt
   treatmentk
+ developmentt
 personi
 errorijt
yijt
   time_ effectt
 personi
 errorijt
Each estimated with comparisons
WARNING: Avoid concluding that an
observed change is due to treatment
or development without comparisons
or corrections.

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Would you assume that distance of these cows above
sea level is measurement error?

24. 24
Outline
• Objective
• Definition
• Description
• Implications
• Recent addictions literature
• What to do about it

25. 25
Recent Addictions Literature:
Out Bicycling
Babor 2008
“How often have we heard [treatment
researchers] casually invoke the RTM
concept as a possible explanation for
general improvements in post-treatment
drinking behavior?”

26. 26
Recent Addictions Literature:
Stout
“RTM is used in a number of contexts in
addiction research”
Contexts of RTM
Context 1 & 2 are actually the presence of
random components. Stout distinguishes
measurement error and unexplained variation.
Context 3 is “Measurement Bias”/True Change

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Recent Addictions Literature:
Stout
“...under many circumstances RTM effects may
dwarf intervention effects...”
KMC
This will happen whenever unexplained variance is
high relative to intervention effect size.

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Ripatti
Benefit
Reduces error variance
Assumptions
Errors are assumed ~Niid
– RTM may still impact
Stationarity
– No trending modeled
Limitations on Interpretations
Confounding not addressed
yikt
   time_effectk
 f (yt1
)  errorit

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Recent Addictions Literature:
Finney
Finney suggests assessment at multiple time points
prior to treatment application
Big Assumption
Stationarity
Limitation
Confounding remains an issue
RTM not eliminated

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Recent Addictions Literature:
Finney
Observational Studies
• Matching
– Does not include the whole sample
• Covariate Adjustment
– RTM is not eliminated

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Recent Addictions Literature:
RTM in Addiction Research
Gmel 2008
Aim: provide statistical methods to disaggregate
change and estimate its components:
1. “True change”
2. “Random fluctuations”
3. “Measurement error”
y    treatment + development  true_ variation  person  error

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Recent Addictions Literature:
Gmel et al. 2008
• Oldham Method
Need a random sample?
• Tu Method
y1  y2  [(y1  y2 ) / 2]
y1  y2  y1 Under a corrected null Ho
Restricted use and interpretation & fixed effects not parsed

33. 33
Recent Addictions Literature:
Gmel: Barnett Method
Calculate the expected magnitude of RTM and subtract that from the
observed change.
Benefits: Useful when there is no comparison group
Requirements:
1. Need to know population variance and within-subject
variance, which must be constant.
2. Need to know population mean.
3. The population and errors must be normally distributed.
Issue: No recognition for sampling error.

34. 34
Recent Addictions Literature:
Gmel: Growth Curve Method
• “RTM is often reported based on the correlation
between initial status and observed change”
If the models structure is correct, RTM will reduce
because the error variance will shrink. The
reduction will be proportional to the number of
within-subject observations.

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Assumptions in the Addictions
Literature
• Stationary
• Normality
• Leaps beyond the limits of observational
studies

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What to do About RTM
• Minimize measurement error
• Repeated/independent measures
– Multiple Measures
– HLM
• Quality comparison groups
– Randomization
• Statistical Corrections
– Ripatti models
• Make reasonable conclusions

37. 37
Recent Addictions Literature:
RTM in Addiction Research
Finney 2008
Reducing RTM
• Reducing RTM is not necessary to obtain
unbiased treatment effects in RCT
• Take repeated measurements
– (ie reduce sampling variation)
– Finney suggests assessment at multiple time points
prior to treatment application

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Implications: Longitudinal
Research
y    treatment + development  true_ variation  person  error

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Recent Addictions Literature:
Finney 2008
“the aim of this paper is to raise awareness of RTM”
Comparative Studies
“RCTs do not eliminate RTM”
“...treatment-seeking patients would tend to improve
in the absence of treatment as a result of RTM”
“[true changes] fluctuate around a mean level of
functioning for an individual over time”
Conclusions: Don’t blindly ascribe change to Tx.

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Components View:
V(yij
)  Person
2
 " Error "
2
V(yij
)  Person
2
 measurement
2
 unexplained
2
V(yij
)  Person
2
 " Error "
2
V(yij
)  Person
2
 measurement
2
 unexplained
2
yijt
   time_ effectt
 personi
 errorj
Components which vary across measurements
Estimate with
distributionsEstimate with
comparisons

41. 41

42. 42
Objective
• Be able to identify regression to the
mean
• Know how to respond to its presence
• Recognize that the concept is used
loosely in addictions