1. Collinearity

2. Symptoms of collinearity Collinearity between independent variables – High r 2 High vif of variables in model Variables significant in simple regression, but not in multiple regression Variables not significant in multiple regression, but multiple regression model (as whole) significant Large changes in coefficient estimates between full and reduced models Large standard errors in multiple regression models despite high power

3. Collinearity and confounding independent variables Two independent variables, correlated with each other, where both influence the response

4. Methods Truth: y = 10 + 3x 1 + 3x 2 + N(0,2) x 1 = U[0,10] x 2 = x 1 + N(0,z) where z = U[0.5,20] Run simple regression between y and x 1 Run multiple regression between y and x 1 + x 2 No interactions!

5. Simple regression: y~x 1

9. Multiple regression: y~x 1 +x 2

12. Collinearity and redundant independent variables Two independent variables, correlated with each other, where only one influences the response, although we don’t know which one

13. Methods Truth: y = 10 + 3x 1 + N(0,2) x 1 = U[0,10] x 2 = x 1 + N(0,z) where z = U[0.5,20] Run simple regression between y and x 1 Run multiple regression between y and x 1 + x 2 No interactions!

14. Simple regression: y~x 1

17. Simple regression: y~x 2

20. Multiple regression: y~x 1 +x 2

26. What to do? Be sure to calculate collinearity and vif among independent variables (before you start your analysis) Pay attention to how coefficient estimates and variable significance change as variables are removed or added Be careful to identify potentially confounding variables prior to data collection

27. Is a variable redundant or confounding? Think! Extreme collinearity – Redundant Large changes in coefficient estimates of both variables between full and reduced models – Confounding Large changes in coefficient estimates of one variable between full and reduced models – Redundant – full model estimate close to zero Uncertain – assume confounding – Multiple regression always produces unbiased estimates (on average) regardless of type of collinearity

28. What to do? Confounding variables Be sure to sample in a manner that eliminates collinearity – Collinearity may be due to real collinearity or sampling artifact Use multiple regression – May have large standard errors if strong collinearity Include confounding variables even if non- significant Get more data – Decreases standard errors (vif)

29. What to do? Redundant variables Determine which variable explains response best using P-values from regression and changes in coefficient estimates with variable addition and removal Do not include redundant variable in final model – Reduces vif Try a variable reduction technique like PCA