Correlation, Regression, and
Causality
Richard L. Amdur, Ph.D.
Chief, Biostatistics & Data Management Core
DC VAMC
Assistant Professor, Depts. of Psychiatry & Surgery
Georgetown University Medical Center
Association does not mean
causality
Why?
SSRI & Depression
Conceptualization:
Dr. Smith believes that if SSRI’s reduce depression then people who take
SSRI’s should have less depression than those who do not take SSRI’s.
Study Design:
Do a survey of everyone who is currently present at the DCVA, to
determine if taking SSRI’s reduces depression. Find out whether or not
each person is currently taking an SSRI, and measure their level of
depression with the Beck Depression Inventory.
Results:
Mean ± sd BDI scores were 50 ± 18 for those taking SSRI’s, and 15 ± 8 for
those not taking SSRI’s.
Correct Conclusion:
SSRI use is positively associated with depression.
Incorrect Conclusion:
SSRI use increases depression.
Causal Modeling Notation for Discussing Study Design
Mean Daily
Caloric Intake
(unit=100 cal/day)
Independent variable
0.5
Weight (lbs)
Effect size
Dependent variable
Interpretation of path coefficient:
For every 1-unit increase in Daily Caloric Intake, there is an increase in weight of 0.5 units.
In this case, for every additional 100 calories taken in, subjects will gain ½ pound.
Mean Daily
Caloric Intake
(unit=100 cal/day)
0.5
Weight (lbs)
Mean Daily Activity
(unit=100 cal/day)
- 0.5
Interpretation of path coefficients:
For every 100cal/day increase in Daily Caloric Intake, there is an increase in weight of 0.5
pounds. For every 100 cal/day increase in activity, there is a decrease in weight of 0.5
pounds.
‘Causal’ Model Using a Categorical Independent Variable
Treatment with
SSRI
(Coded yes=1,
no=0)
Independent variable
35.0
BDI score
Effect size
Dependent variable
Interpretation:
For every 1-unit increase in Treatment, there is an increase in BDI score of 35 units.
In this case, subjects in treatment with an SSRI will have an average BDI score 35 points
higher than subjects not taking SSRIs.
What is actually going on?
Was diagnosed with
severe depression
(yes=1, no=0)
50.0
BDI score
0.80
Treatment with
SSRI
(Coded yes=1,
no=0)
-5.0
Interpretation:
80% of those diagnosed with depression are taking an SSRI.
Those diagnosed with depression have 50 points higher BDI scores.
Taking an SSRI reduces the BDI score by 5 points.
Observed SSRIBDI effect (35) = 50 x 0.80 – 5.0
Correct Conclusion:
After accounting for the effect of Pre-Treatment Depression, SSRI treatment has a direct
negative effect on depression score.
Case Study: the effect of mindfulness training
(MT) on working memory capacity (WMC) and
positive and negative emotions in subjects who
are under stress
Study Design:
One Marine unit was given MT, another was not.
Both units underwent stressful preparations for
deployment.
Question: Does mindfulness training (MT) increase
working memory capacity (WMC) and positive emotions
in subjects who are under stress?
Results:
“In the MT group, WMC decreased over time in those with
low MT practice time, but increased in those with high
practice time. Higher MT practice time also corresponded to
lower levels of negative affect and higher levels of positive
affect ….”
Conclusion:
“these findings suggest that sufficient MT practice may
protect against functional impairments associated with highstress contexts.”
Author’s Model of Mindfulness Effects
MT increases WMC, WMC increases PA , both WMC & PA increase Job Performance
a
Working
Memory
Capacity
(WMC)
Positive
Affect (PA)
Mindfulness
Training (MT)
Job
Performance
Mindfulness Effects are Mediated by Practice Time
Mindfulness
Training (MT)
Working
Memory
Capacity
(WMC)
b
Positive
Affect (PA)
c(obs)
Mindfulness
Practice Time
a = bc(obs)
Job
Performance
Mindfulness Effects: The observed effect of Practice Time on WMC
may be spurious
Post-MT
Working
Memory
Pre-MT
Working
Memory
y
Pre-MT
Positive
Affect
Post-MT
Positive
Affect
c
Trait
Mindfulness
Job
Performance
x
Mindfulness
Practice Time
Pre-MT
During-MT
Post-MT
Trait Mindfulness Spuriously Increases cobserved
Mindfulness
Training (MT)
Yes=1, No=0
b
MT Practice Time
x
y
Trait Mindfulness
Observed MT-Practice-time—WMC correlation [c(obs)] = c + xy
We know that since x and y are both positive, c(obs) > c
Observed r = direct effect + spurious effect
c
Working
Memory
Capacity
(WMC)
Lots of variables may spuriously increase cobs
MT Practice Time
c
x1
x2
x3
x4
y1
Trait Mindfulness
y2
Working
Memory
Capacity
(WMC)
y3
Pos Affect
y4
IQ
??
c(obs) = c + x1y1 + x2y2 + x3y3 + x4y4 + …. + xnyn
There may be many unmeasured variables creating spurious effects, so c(obs) >>> c
Observed r = direct effect + spurious effect
If you randomize subjects to Practice Time,
this sets all x’s to 0
MT Practice Time
y1
Trait Mindfulness
y2
c
Working
Memory
Capacity
(WMC)
y3
Pos Affect
y4
IQ
??
c(obs) = c + x1y1 + x2y2 + x3y3 + x4y4 + …. + xnyn . This now becomes c(obs) = c + 0.
Observed r = direct effect
Carotid Arterial Stent vs. Surgical
Repair (endarterectomy) for
carotid stenosis
Conceptualization:
Dr. Smith believes that if CAS works better than CEA, then patients who
received CAS should live longer than those who received CEA.
Study Design:
Examine a large database to determine outcomes following treatment.
Results:
9-month death rates were 4% for CEA, 5% for CAS.
Correct Conclusion:
CAS treatment is positively associated with death at 9 months post.
Incorrect Conclusion:
CEA produces better outcomes than CAS.
Lots of variables may spuriously increase cobs
Tx: CAS=1, CEA=0
c
x1
x2
x3
x4
y1
Contralateral
carotid occlusion
y2
Death
at
9 months
y3
CHF
y4
Recent MI
Unstable
angina
Severe COPD
Age > 80
c(obs) = c + x1y1 + x2y2 + x3y3 + x4y4 + …. + xnyn
There may be many unmeasured variables creating spurious effects, so c(obs) >>> c
Observed r = direct effect + spurious effect
Does regression modeling solve
this problem?
To some extent: only if you identify all the possible covariates that have x
& y effects, and you have reliable measures for each of these variables. In practice,
this is usually difficult to do. And you will not know if you’ve done it.
How about using a general comorbidity index as a
covariate:
For example, use Elixhauser score instead of individual variables
Comorbidity indices
Elixhauser, A., Steiner, C., Harris, D. R., & Coffey, R. M. (1998). Comorbidity measures
for use with administrative data. Med Care, 36, 8-27.
Goldstein, L. B., Samsa, G. P., Matchar, D. B., & Horner, R. D. (2004). Charlson Index
comorbidity adjustment for ischemic stroke outcome studies. Stroke, 35, 1941-1945.
Dominick, K. L., Dudley, T. K., Coffman, C. J., & Bosworth, H. B. (2005). Comparison of
three comorbidity measures for predicting health service use in patients with
osteoarthritis. Arthritis Rheum, 53, 666-672.
These indices create a single score which is a sum of all the
possible medical problems a patient could have:
TB, infection, HIV, cancers, thyroid disorder, DM, MS, epilepsy,
Headache, hyperlipidemia, gout, anemia, psychiatric disorders,
cataracts, dizziness, HTN, cardiac disorders, varicose veins,
bronchitis, asthma, abdominal hernia, etc.
• Useful to
correct for case mix in administrative
studies examining treatment outcomes across
hospitals or regions.
• The long list of disorders creates noise that
swamps the actual covariates of interest when
patients are the unit of analysis.
• Use of Propensity Scores is a better option
(but you still may have problems with unmeasured covariates, measures with poor
reliability, lack of group overlap).
Problems in interpreting
correlations
Correlation & Regression
Subject
1
Mean
SD
Height
Weight
66
125
Height x Weight
230
210
2
68
150
3
70
160
4
72
195
5
73
180
6
74
175
130
7
76
200
110
8
77
205
90
72
173.75
3.82
27.48
Weight
190
170
150
64
66
y = 6.7157x - 309.78
R² = 0.87
r = .933
68
70
72
Height
74
76
78
Effect of Non-Linearity
14
13
Memory Test score
12
11
10
9
8
7
6
5
4
0
2
4
6
Arousal level
8
10
Effect of Non-Linearity
14
13
Memory Test score
12
11
10
9
8
7
r = .18
6
5
4
0
2
4
6
Arousal level
8
R2 = 0.0323
Correlation is not a good statistic to use to measure
non-linear relationships
10
Effect of Extreme Score
Height x Weight
220
Height x Weight with Outlier
R2 = 0.87
200
220
180
Weight
R2 = 0.5474
200
160
140
Weight
180
120
y = 6.7157x - 309.78
160
100
65
70
75
Height
140
120
y = 4.9329x - 176.87
100
65
70
75
80
Height
r = .740
r = .933
80
Outlier Effect
R2 = 0.0086
10
9
Test Performance
8
7
6
5
r = .093
4
3
2
1
0
0
2
4
6
Arousal
8
10
12
Outlier Effect
10
9
8
7
6
r = -.237
5
4
3
2
1
R2 = 0.0563
0
0
2
4
6
8
10
12
Effect of Subgroups
130
120
Diagnosis A
SBP
110
100
90
Diagnosis B
80
70
0
10
20
30
40
m ed dose
50
60
70
Effect of Subgroups
96
130
94
92
Dx A
120
90
SBP
110
88
86
100
R2 = 0.966
84
0
90
10
20
30
40
50
80
128
126
70
124
0
10
20
30
40
m ed dose
50
60
70
R2 = 0.0003
Dx B
122
120
118
116
114
112
R2 = 0.9708
110
108
0
10
20
30
40
50
60
70