Multicollinearity

Multicollinearity (or intercorrelation) exists when at least some of the predictor variables are correlated among themselves. In observational studies, multicollinearity happens more often than not. So, we need to understand the effects of multicollinearity on regression analyses.

Example #1 n = 20 hypertensive individuals p-1 = 6 predictor variables

Example #1 BP Age Weight BSA Duration Pulse Age Weight BSA Duration Pulse Stress Blood pressure (BP) is the response.

What is effect on regression analyses if predictors are perfectly uncorrelated? x1 x2 y Pearson correlation of x1 and x2 = 0.000

The regression equation is y = x1 Predictor Coef SE Coef T P Constant x Analysis of Variance Source DF SS MS F P Regression Error Total Regress Y on X 1

Regress Y on X 2 The regression equation is y = x2 Predictor Coef SE Coef T P Constant x Analysis of Variance Source DF SS MS F P Regression Error Total

Regress Y on X 1 and X 2 The regression equation is y = x x2 Predictor Coef SE Coef T P Constant x x Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS x x

Regress Y on X 2 and X 1 The regression equation is y = x x1 Predictor Coef SE Coef T P Constant x x Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS x x

If predictors are perfectly uncorrelated, then… You get the same slope estimates regardless of the first-order regression model used. That is, the effect on the response ascribed to a predictor doesn’t depend on the other predictors in the model.

If predictors are perfectly uncorrelated, then… The sum of squares SSR(X 1 ) is the same as the sequential sum of squares SSR(X 1 |X 2 ). The sum of squares SSR(X 2 ) is the same as the sequential sum of squares SSR(X 2 |X 1 ). That is, the marginal contribution of one predictor variable in reducing the error sum of squares doesn’t depend on the other predictors in the model.

Same effects for “real data” with nearly uncorrelated predictors? BP Age Weight BSA Duration Pulse Age Weight BSA Duration Pulse Stress

Regress BP on Stress The regression equation is BP = Stress Predictor Coef SE Coef T P Constant Stress S = R-Sq = 2.7% R-Sq(adj) = 0.0% Analysis of Variance Source DF SS MS F P Regression Error Total

Regress BP on BSA The regression equation is BP = BSA Predictor Coef SE Coef T P Constant BSA S = R-Sq = 75.0% R-Sq(adj) = 73.6% Analysis of Variance Source DF SS MS F P Regression Error Total

Regress BP on BSA and Stress The regression equation is BP = BSA Stress Predictor Coef SE Coef T P Constant BSA Stress Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS BSA Stress

Regress BP on Stress and BSA The regression equation is BP = Stress BSA Predictor Coef SE Coef T P Constant Stress BSA Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS Stress BSA

If predictors are nearly uncorrelated, then… You get similar slope estimates regardless of the first-order regression model used. The sum of squares SSR(X 1 ) is similar to the sequential sum of squares SSR(X 1 |X 2 ). The sum of squares SSR(X 2 ) is similar to the sequential sum of squares SSR(X 2 |X 1 ).

What happens if the predictor variables are highly correlated? BP Age Weight BSA Duration Pulse Age Weight BSA Duration Pulse Stress

Regress BP on Weight The regression equation is BP = Weight Predictor Coef SE Coef T P Constant Weight S = R-Sq = 90.3% R-Sq(adj) = 89.7% Analysis of Variance Source DF SS MS F P Regression Error Total

Regress BP on BSA The regression equation is BP = BSA Predictor Coef SE Coef T P Constant BSA S = R-Sq = 75.0% R-Sq(adj) = 73.6% Analysis of Variance Source DF SS MS F P Regression Error Total

Regress BP on BSA and Weight The regression equation is BP = BSA Weight Predictor Coef SE Coef T P Constant BSA Weight Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS BSA Weight

Regress BP on Weight and BSA The regression equation is BP = Weight BSA Predictor Coef SE Coef T P Constant Weight BSA Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS Weight BSA

Effect #1 of multicollinearity When predictor variables are correlated, the regression coefficient of any one variable depends on which other predictor variables are included in the model. Variables in model b1b1 b2b2 X1X X2X X 1, X

Even correlated predictors not in the model can have an impact! Regression of territory sales on territory population, per capita income, etc. Against expectation, coefficient of territory population was determined to be negative. Competitor’s market penetration, which was strongly positively correlated with territory population, was not included in model. But, competitor kept sales down in territories with large populations.

Effect #2 of multicollinearity When predictor variables are correlated, the marginal contribution of any one predictor variable in reducing the error sum of squares varies, depending on which other variables are already in model. SSR(X 1 ) = SSR(X 1 |X 2 ) = SSR(X 2 ) = SSR(X 2 |X 1 ) = 2.81

Effect #3 of multicollinearity When predictor variables are correlated, the precision of the estimated regression coefficients decreases as more predictor variables are added to the model. Variables in model se(b 1 )se(b 2 ) X1X X2X X 1, X

What is the effect on estimating mean or predicting new response?

Weight Fit SE Fit 95.0% CI 95.0% PI (111.85,113.54) (108.94,116.44) BSA Fit SE Fit 95.0% CI 95.0% PI (112.76,115.38) (108.06,120.08) BSA Weight Fit SE Fit 95.0% CI 95.0% PI (111.93,113.83) (109.08, ) Effect #4 of multicollinearity on estimating mean or predicting Y High multicollinearity among predictor variables does not prevent good, precise predictions of the response (within scope of model).

The regression equation is BP = BSA Predictor Coef SE Coef T P Constant BSA S = R-Sq = 75.0% R-Sq(adj) = 73.6% Analysis of Variance Source DF SS MS F P Regression Error Total What is effect on tests of individual slopes?

The regression equation is BP = Weight Predictor Coef SE Coef T P Constant Weight S = R-Sq = 90.3% R-Sq(adj) = 89.7% Analysis of Variance Source DF SS MS F P Regression Error Total What is effect on tests of individual slopes?

The regression equation is BP = Weight BSA Predictor Coef SE Coef T P Constant Weight BSA Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS Weight BSA

Effect #5 of multicollinearity on slope tests When predictor variables are correlated, hypothesis tests for β k = 0 may yield different conclusions depending on which predictor variables are in the model. Variables in model b2b2 se(b 2 )tP-value X2X X 1, X

Summary comments Tests for slopes should generally be used to answer a scientific question and not for model building purposes. Even then, caution should be used when interpreting results when multicollinearity exists. (Think marginal effects.)

Summary comments (cont’d) Multicollinearity has little to no effect on estimation of mean response or prediction of future response.

Diagnosing multicollinearity Realized effects (changes in coefficients, changes in sequential sums of squares, etc.) of multicollinearity. Scatter plot matrices. Pairwise correlation coefficients among predictor variables. Variance inflation factors (VIF).