correlation, regression, and causality richard l. amdur, ph.d. chief, biostatistics & data...
TRANSCRIPT
Correlation, Regression, and Causality
Richard L. Amdur, Ph.D.Chief, Biostatistics & Data Management Core
DC VAMCAssistant Professor, Depts. of Psychiatry & Surgery
Georgetown University Medical Center
Association does not mean causality
Why?
SSRI & Depression
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.
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.
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)
0.5
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.
Weight (lbs)
Independent variable Dependent variableEffect size
Mean Daily Caloric Intake(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.
Weight (lbs)
Mean Daily Activity(unit=100 cal/day)
- 0.5
‘Causal’ Model Using a Categorical Independent Variable
Treatment withSSRI(Coded yes=1, no=0)
35.0
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.
BDI score
Independent variable Dependent variableEffect size
What is actually going on?
Treatment withSSRI(Coded yes=1, no=0)
0.80
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 SSRIBDI 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.
BDI score
Was diagnosed withsevere depression(yes=1, no=0)
50.0
-5.0
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 high-stress contexts.”
Author’s Model of Mindfulness EffectsMT increases WMC, WMC increases PA , both WMC & PA increase Job Performance
MindfulnessTraining (MT)
Working MemoryCapacity(WMC)
PositiveAffect (PA)
JobPerformance
a
Mindfulness Effects are Mediated by Practice Time
MindfulnessTraining (MT)
Working MemoryCapacity(WMC)
PositiveAffect (PA)
JobPerformance
MindfulnessPractice Time
b
c(obs)
a = bc(obs)
Mindfulness Effects: The observed effect of Practice Time on WMC may be spurious
MindfulnessPractice Time
Pre-MTWorking Memory
Pre-MTPositiveAffect
Post-MTWorking Memory
Post-MTPositiveAffect
x
y
Pre-MT
TraitMindfulness
During-MT Post-MT
JobPerformance
c
Trait Mindfulness Spuriously Increases cobserved
MindfulnessTraining (MT)Yes=1, No=0 Working
MemoryCapacity(WMC)
c
Trait Mindfulness
MT Practice Time
y
x
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
b
Lots of variables may spuriously increase cobs
Working MemoryCapacity(WMC)
c
Trait Mindfulness
MT Practice Time
y1
x1
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
Pos Affect
IQ
??
y2
x2
y3
x3
y4
x4
If you randomize subjects to Practice Time, this sets all x’s to 0
Working MemoryCapacity(WMC)
c
Trait Mindfulness
MT Practice Time
y1
c(obs) = c + x1y1 + x2y2 + x3y3 + x4y4 + …. + xnyn . This now becomes c(obs) = c + 0.
Observed r = direct effect
Pos Affect
IQ
??
y2
y3
y4
Carotid Arterial Stent vs. Surgical Repair (endarterectomy) for
carotid stenosis
Study Design:Examine a large database to determine outcomes following treatment.
Conceptualization:Dr. Smith believes that if CAS works better than CEA, then patients who received CAS should live longer than those who received CEA.
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
Death at 9 months
c
Contralateralcarotid occlusion
Tx: CAS=1, CEA=0
y1
x1
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
CHF
Recent MI
Unstableangina
y2
x2
y3
x3
y4
x4
Severe COPD Age > 80
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 indicesElixhauser, 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 & RegressionSubject Height Weight
1 66 125
2 68 150
3 70 160
4 72 195
5 73 180
6 74 175
7 76 200
8 77 205
Mean 72 173.75
SD 3.82 27.48
r = .933
64 66 68 70 72 74 76 7890
110
130
150
170
190
210
230
f(x) = 6.7156862745098 x − 309.779411764706R² = 0.870022713577157
Height x Weight
HeightW
eig
ht
Effect of Non-Linearity
4
5
6
7
8
9
10
11
12
13
14
0 2 4 6 8 10
Arousal level
Mem
ory
Tes
t sc
ore
Effect of Non-Linearity
R2 = 0.0323
4
5
6
7
8
9
10
11
12
13
14
0 2 4 6 8 10
Arousal level
Mem
ory
Tes
t sc
ore
Correlation is not a good statistic to use to measure non-linear relationships
r = .18
Effect of Extreme Score
Height x Weight with Outlier
y = 4.9329x - 176.87
R2 = 0.5474
100
120
140
160
180
200
220
65 70 75 80
Height
Wei
gh
t
Height x Weight
R2 = 0.87
y = 6.7157x - 309.78
100
120
140
160
180
200
220
65 70 75 80
Height
Weig
ht
r = .933r = .740
Outlier EffectR2 = 0.0086
0
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10 12
Arousal
Tes
t P
erfo
rman
ce
r = .093
Outlier Effect
R2 = 0.05630
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10 12
r = -.237
Effect of Subgroups
70
80
90
100
110
120
130
0 10 20 30 40 50 60 70
med dose
SB
P
Diagnosis A
Diagnosis B
Effect of Subgroups
R2 = 0.0003
70
80
90
100
110
120
130
0 10 20 30 40 50 60 70
med dose
SB
P
R2 = 0.96684
86
88
90
92
94
96
0 10 20 30 40 50
R2 = 0.9708
108
110
112
114
116
118
120
122
124
126
128
0 10 20 30 40 50 60 70
Dx A
Dx B