Biostatistics in Practice Session 5: Associations and confounding

1. Biostatistics in PracticeSession 5: Associations and confounding Youngju Pak, Ph.D. Biostatistician http://research.LABioMed.org/Biostat

2. Revisiting the Food Additives Study From Table 3 Unadjusted What does “adjusted” mean? How is it done? Adjusted

3. Goal One of Session 5 Earlier: Compare means for a single measure among groups. Use t-test, ANOVA. Session 5: Relate two or more measures. Use correlation or regression. ΔY/ΔX Δ Qu et al(2005), JCEM 90:1563-1569.

4. Goal Two of Session 5 Try to isolate the effects of different characteristics on an outcome. Previous slide: Gender GH Peak BMI

5. Correlation Standard English word correlate to establish a mutual or reciprocal relation between <correlate activities in the lab and the field> b: to show correlation or a causal relationship between In statistics, it has a more precise meaning 5

6. Correlation in Statistics Correlation: measure of the strength of LINEAR association Positive correlation: two variables move to the same direction  As one variable increase, other variables also tends to increase LINEARLY or vice versa. Example: Weight vs Height Negative correlation: two variables move opposite of each other.  As one variable increases, the other variable tends to decrease LINEARLY or vice versa (inverse relationship). Example: Physical Activity level vs. Abdominal height (Visceral Fat) 6

7. Pearson r correlation coefficient r can be any value from -1 to +1 r = -1 indicates a perfect negative LINEAR relationship between the two variables r = 1 indicates a perfect positive LINEAR relationship between the two variables r = 0 indicates that there is no LINEAR relationship between the two variables 7

8. Scatter Plot: r= 1.0 8

9. Scatter Plot: r= -1.0 9

10. Scatter Plot: r= 0 10

11. Anemic women: Anemia.sav n=20 r expresses how well the data fits in a straight line. Here, Pearson’s r =0.673

12. Correlations in real data

13. Logic for Value of Correlation - + - + Σ(X-Xmean) (Y-Ymean) √Σ(X-Xmean)2Σ(Y-Ymean)2 Pearson’s r = Statistical software gives r.

14. Correlation Depends on Ranges of X & Y A B Graph B contains only the graph A points in the ellipse. Correlation is reduced in graph B. Thus: correlations for the same quantities X and Y may be quite different in different study populations.

15. Simple Linear Regression (SLR) • X and Y now assume unique roles: • Y is an outcome, response, output, dependent variable. • X is an input, predictor, explanatory, independent variable. • Regression analysis is used to: • Measure more than X-Y association, as with correlation. • Fit a straight line through the scatter plot, for: • Prediction of Ymeanfrom X. • Estimationof Δ in Ymeanfor a unit change in X • = Rate of change of Ymean as a unit change in X • (slope = regression coefficient •  measure “effect” of X on Y).

16. SLR Example Minimizes Σei2 ei Range for Individuals Range for individuals Range for Individuals Range for mean Statistical software gives all this info.

17. Hypothesis testing for the true slope=0 H0: true slope = 0 vs. Ha: true slope ≠0, with the rule: Claim association (slope≠0) if tc=|slope/SE(slope)| > t ≈ 2. There is a 5% chance of claiming an X-Y association that really does not exist. Note similarity to t-test for means: tc=|mean/ SE(mean)| Formula for SE(slope) is in statistics books.

18. Example Software Output The regression equation is: Ymean= 81.6 + 2.16 X Predictor CoeffStdErr T P Constant 81.64 11.47 7.12 <0.0001 X 2.1557 0.1122 19.21 <0.0001 S = 21.72 R-Sq = 79.0% Predicted Values: X: 100 Fit: 297.21 SE(Fit): 2.17 95% CI: 292.89 - 301.52 95% PI: 253.89 - 340.52 19.21=2.16/0.112 should be between ~ -2 and 2 if “true” slope=0. Refers to Intercept Predicted y = 81.6 + 2.16(100) Range of Ys with 95% assurance for: Mean of all subjects with x=100. Individual with x=100.

19. Multiple Regression We now generalize to prediction from multiple characteristics. The next slide gives a geometric view of prediction from two factors simultaneously.

20. Multiple Lienar Regression: Geometric View Suppose multiple predictors are continuous. Geometrically, this is fitting a slanted plane to a cloud of points: www.StatisticalPractice.com LHCY is the Y (homocysteine) to be predicted from the two X’s: LCLC (folate) and LB12 (B12). LHCY = b0 + b1LCLC + b2LB12 is the equation of the plane

21. Multiple Regression: Software

22. Multiple Regression: Software Output: Values of b0, b1, and b2 for LHCYmean= b0 + b1LCLC + b2LB12

23. How Are Coefficients Interpreted? LHCYmean= b0 + b1LCLC + b2LB12 Outcome Predictors LB12 may have both an independent and an indirect (via LCLC) association with LHCY LCLC b1 ? LHCY Correlation b2 ? LB12

24. Coefficients: Meaning of their Values LHCY = b0 + b1LCLC + b2LB12 Outcome Predictors Mean LHCY increases by b2 for a 1-unit increase in LB12 … if other factors (LCLC) remain constant, or … adjusting for other factors in the model (LCLC) May be physiologically impossible to maintain one predictor constant while changing the other by 1 unit.

26. Figure 2. Determine the relative and combined explanatory power of age, gender, BMI, ethnicity, and sport type on the markers. * * for age, gender, and BMI.

27. Another Example: HDL Cholesterol Output: Std Coefficient Error t Pr > |t| Intercept 1.16448 0.28804 4.04 <.0001 AGE -0.00092 0.00125 -0.74 0.4602 BMI -0.01205 0.00295 -4.08 <.0001 BLC 0.05055 0.02215 2.28 0.0239 PRSSY -0.00041 0.00044 -0.95 0.3436 DIAST 0.00255 0.00103 2.47 0.0147 GLUM -0.00046 0.00018 -2.50 0.0135 SKINF 0.00147 0.00183 0.81 0.4221 LCHOL 0.31109 0.10936 2.84 0.0051 The predictors of log(HDL) are age, body mass index, blood vitamin C, systolic and diastolic blood pressures, skinfold thickness, and the log of total cholesterol. The equation is: Log(HDL) mean = 1.16 - 0.00092(Age) +…+ 0.311(LCHOL) www. Statistical Practice .com

28. HDL Example: Coefficients • Interpretation of coefficients on previous slide: • Need to use entire equation for making predictions. • Each coefficient measures the difference in meanLHDL between 2 subjects if the factor differs by 1 unit between the two subjects, and if all other factors are the same. E.g., expected LHDL is 0.012 lower in a subject whose BMI is 1 unit greater, but is the same as the other subject on other factors. Continued …

29. HDL Example: Coefficients • Interpretation of coefficients two slides back: • P-values measure how strong the association of a factor with Log(HDL) is , if other factors do not change. • This is sometimes expressed as “after accounting for other factors” or “adjusting for other factors”, and is called independent association. • SKINF probably is associated. Its p=0.42 says that it has no additional info to predict LogHDL, after accounting for other factors such as BMI.

30. Special Cases of Multiple Regression So far, our predictors were all measured over a continuum, like age or concentration. This is simply called multiple regression. When some predictors are grouping factors like gender or ethnicity, regression has other special names: ANOVA Analysis of Covariance

31. Analysis of Variance • All predictors are grouping factors. • One-way ANOVA: Only 1 predictor that may have only 2 “levels”, such as gender, or more levels, such as ethnicity. • Two-way ANOVA: Two grouping predictors, such as decade of age and genotype.

32. Two way ANOVA • Interaction in 2-way ANOVA: Measures whether the effect of one factor depends on the other factor. Difference of a difference in outcome. E.g., (Trt.-– control)Female– (Trt.– control)Male • The effect of treatment, adjusted for gender, is a weighted average of groupdifferences overtwo gender group, i.e., of : (Trt.– control)Femaleand (Trt.– control)Male

33. Analysis of Covariance • At least one primary predictor is a grouping factor, such as treatment group , and at least one predictor is continuous, such as age, called a “covariate”. • Interest is often on comparing the groups. • The covariate is often a nuisance. • Confounder: A covariate that both co-varies with the outcome and is distributed differently in the groups.