Correlation, Regression, and Causality

1. 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

2. Association does not mean causality Why?

3. SSRI & Depression Conceptualization: Dr. Smith believesthat 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.

4. Correct Conclusion: SSRI use is positively associated with depression. Results: Mean ± sd BDI scores were 50 ± 18 for those taking SSRI’s, and 15 ± 8 for those not taking SSRI’s. Incorrect Conclusion: SSRI use increases depression.

5. 0.5 Mean Daily CaloricIntake (unit=100 cal/day) Weight (lbs) Causal Modeling Notation for Discussing Study Design Independent variable 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.

6. Mean Daily CaloricIntake (unit=100 cal/day) 0.5 Weight (lbs) - 0.5 Mean Daily Activity (unit=100 cal/day) 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.

7. Treatment withSSRI (Coded yes=1, no=0) 35.0 BDI score ‘Causal’ Model Using a Categorical Independent Variable Independent variable 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.

8. Was diagnosed withsevere depression (yes=1, no=0) 50.0 BDI score What is actually going on? 0.80 Treatment withSSRI (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.

9. 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.

10. 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.” Question: Does mindfulness training (MT) increase working memory capacity (WMC) and positive emotions in subjects who are under stress?

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

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

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

14. b MindfulnessTraining (MT) Yes=1, No=0 MT Practice Time c Working Memory Capacity(WMC) Trait Mindfulness Spuriously Increases cobserved 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

15. MT Practice Time c x1 Working Memory Capacity(WMC) x2 Lots of variables may spuriously increase cobs x3 x4 y1 Trait Mindfulness y2 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

16. MT Practice Time c Working Memory Capacity(WMC) If you randomize subjects to Practice Time, this sets all x’s to 0 y1 Trait Mindfulness y2 y3 Pos Affect y4 IQ ?? c(obs) = c + x1y1 + x2y2 + x3y3 + x4y4 + …. + xnyn . This now becomes c(obs) = c + 0. Observed r = direct effect

17. 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.

18. Correct Conclusion: CAS treatment is positively associated with death at 9 months post. Results: 9-month death rates were 4% for CEA, 5% for CAS. Incorrect Conclusion: CEA produces better outcomes than CAS.

19. Tx: CAS=1, CEA=0 c x1 Death at 9 months x2 Lots of variables may spuriously increase cobs x3 x4 y1 Contralateralcarotid occlusion y2 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

20. 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

21. 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.

22. 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).

23. Problems in interpreting correlations

24. Correlation & Regression r = .933

25. Effect of Non-Linearity

26. Effect of Non-Linearity r = .18 Correlation is not a good statistic to use to measure non-linear relationships

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

28. Outlier Effect r = .093

29. Outlier Effect r = -.237

30. Effect of Subgroups Diagnosis A Diagnosis B

31. Effect of Subgroups Dx A Dx B