Biostatistics in Practice Session 5: Associations and confounding Youngju Pak, Ph.D. Biostatisticianhttp://research.LABioMed.org/Biostat 1

Revisiting the Food Additives Study Unadjusted Adjusted What does “adjusted” mean? How is it done? From Table 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. Qu et al(2005), JCEM 90: Δ ΔY/ΔX

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

5 Correlation  Standard English word correlate to establish a mutual or reciprocal relation between b: to show correlation or a causal relationship between to establish a mutual or reciprocal relation between b: to show correlation or a causal relationship betweencorrelation  In statistics, it has a more precise meaning

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 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 Example: Physical Activity level vs. Abdominal height (Visceral Fat) (Visceral Fat)

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

8 Scatter Plot: r= 1.0

9 Scatter Plot: r= -1.0

10 Scatter Plot: r= 0

Anemic women: Anemia.sav n=20 Hb(g/dl)PCV(%) …… r expresses how well the data fits in a straight line. Here, Pearson’s r =0.673

Correlations in real data Correlations in real data

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

Correlation Depends on Ranges of X & Y 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. BA

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 Ymean from X. Estimation of Δ in Y mean for a unit change in X = Rate of change of Y mean as a unit change in X (slope = regression coefficient  measure “effect” of X on Y).

SLR Example eiei Minimizes Σe i 2 Range for Individuals Range for mean Statistical software gives all this info. Range for Individuals Range for individuals

Hypothesis testing for the true slope=0 H0: true slope = 0 vs. Ha: true slope ≠0, with the rule: Claim association (slope≠0) if t c =|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: t c =|mean/ SE(mean)| Formula for SE(slope) is in statistics books.

Example Software Output The regression equation is: Y mean = X Predictor Coeff StdErr T P Constant < X < S = R-Sq = 79.0% Predicted Values: X: 100 Fit: SE(Fit): % CI: % PI: Predicted y = (100) Range of Ys with 95% assurance for: Mean of all subjects with x=100. Individual with x= =2.16/0.112 should be between ~ -2 and 2 if “true” slope=0. Refers to Intercept

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

Multiple Lienar Regression : Geometric View LHCY is the Y (homocysteine) to be predicted from the two X’s: LCLC (folate) and LB12 (B 12 ). LHCY = b 0 + b 1 LCLC + b 2 LB12 is the equation of the plane Suppose multiple predictors are continuous. Geometrically, this is fitting a slanted plane to a cloud of points:

Multiple Regression: Software

Output: Values of b 0, b 1, and b 2 for LHCY mean = b 0 + b 1 LCLC + b 2 LB12

How Are Coefficients Interpreted? LHCY mean = b 0 + b 1 LCLC + b 2 LB12 Outcome Predictors LHCY LCLC LB12 LB12 may have both an independent and an indirect (via LCLC) association with LHCY Correlation b 1 ? b 2 ?

Coefficients: Meaning of their Values LHCY = b 0 + b 1 LCLC + b 2 LB12 Outcome Predictors Mean LHCY increases by b 2 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.

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

Another Example: HDL Cholesterol Std Coefficient Error t Pr > |t| Intercept <.0001 AGE BMI <.0001 BLC PRSSY DIAST GLUM SKINF LCHOL 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 = (Age) +… (LCHOL) www. Statistical Practice.com Output:

HDL Example: Coefficients Interpretation of coefficients on previous slide: 1.Need to use entire equation for making predictions. 2.Each coefficient measures the difference in mean LHDL 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 lower in a subject whose BMI is 1 unit greater, but is the same as the other subject on other factors. Continued …

HDL Example: Coefficients Interpretation of coefficients two slides back: 3.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.

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

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.

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 group differences over two gender group, i.e., of : (Trt.– control) Female and (Trt. – control) Male

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.