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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

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Association does not mean causality Why?

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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.

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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.

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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 variableDependent variableEffect size

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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

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‘Causal’ Model Using a Categorical Independent Variable Treatment with SSRI (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 variableDependent variableEffect size

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What is actually going on? Treatment with SSRI (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 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. BDI score Was diagnosed with severe depression (yes=1, no=0)

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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.

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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.”

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Author’s Model of Mindfulness Effects MT increases WMC, WMC increases PA, both WMC & PA increase Job Performance Mindfulness Training (MT) Working Memory Capacity (WMC) Positive Affect (PA) Job Performance a

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Mindfulness Effects are Mediated by Practice Time Mindfulness Training (MT) Working Memory Capacity (WMC) Positive Affect (PA) Job Performance Mindfulness Practice Time b c (obs) a = bc (obs)

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Mindfulness Effects: The observed effect of Practice Time on WMC may be spurious Mindfulness Practice Time Pre-MT Working Memory Pre-MT Positive Affect Post-MT Working Memory Post-MT Positive Affect x y Pre-MT Trait Mindfulness During-MTPost-MT Job Performance c

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Trait Mindfulness Spuriously Increases c observed Mindfulness Training (MT) Yes=1, No=0 Working Memory Capacity (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

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Lots of variables may spuriously increase c obs Working Memory Capacity (WMC) c Trait Mindfulness MT Practice Time y1y1 x1x1 c (obs) = c + x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 + …. + x n y n There may be many unmeasured variables creating spurious effects, so c (obs) >>> c Observed r = direct effect + spurious effect Pos Affect IQ ?? y2y2 x2x2 y3y3 x3x3 y4y4 x4x4

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If you randomize subjects to Practice Time, this sets all x’s to 0 Working Memory Capacity (WMC) c Trait Mindfulness MT Practice Time y1y1 c (obs) = c + x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 + …. + x n y n. This now becomes c (obs) = c + 0. Observed r = direct effect Pos Affect IQ ?? y2y2 y3y3 y4y4

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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.

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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.

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Lots of variables may spuriously increase c obs Death at 9 months c Contralateral carotid occlusion Tx: CAS=1, CEA=0 y1y1 x1x1 c (obs) = c + x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 + …. + x n y n There may be many unmeasured variables creating spurious effects, so c (obs) >>> c Observed r = direct effect + spurious effect CHF Recent MI Unstable angina y2y2 x2x2 y3y3 x3x3 y4y4 x4x4 Severe COPD Age > 80

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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

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Comorbidity indices Elixhauser, A., Steiner, C., Harris, D. R., & Coffey, R. M. (1998). Comorbidity measures for use with administrative data. Med Care, 36, Goldstein, L. B., Samsa, G. P., Matchar, D. B., & Horner, R. D. (2004). Charlson Index comorbidity adjustment for ischemic stroke outcome studies. Stroke, 35, 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, 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.

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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).

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Problems in interpreting correlations

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Correlation & Regression SubjectHeightWeight Mean SD r =.933

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Effect of Non-Linearity

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Correlation is not a good statistic to use to measure non-linear relationships r =.18

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Effect of Extreme Score r =.933 r =.740

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Outlier Effect r =.093

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Outlier Effect r = -.237

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Effect of Subgroups Diagnosis A Diagnosis B

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Effect of Subgroups Dx A Dx B

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