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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 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 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. 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