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 1955 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

eigh

t

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

Test

sco

re

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

Test

sco

re

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

ght

Height x Weight

R2 = 0.87

y = 6.7157x - 309.78100

120

140

160

180

200

220

65 70 75 80Height

Wei

ght

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

Test

Per

form

ance

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

SBP

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

SBP

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