Multicollinearity

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

120

110

53.25

47.75

97.325

89.375

2.125

1.875

8.275

4.425

72.5

65.5

120110

76.25

30.75

53.25

47.75

97.325

89.375 2.1

251.8

758.2

754.4

25 72.565.5

76.25

30.75

BP

Age

Weight

BSA

Duration

Pulse

Stress

n = 20 hypertensive individuals

p-1 = 6 predictor variables

Example #1

BP Age Weight BSA Duration PulseAge 0.659Weight 0.950 0.407BSA 0.866 0.378 0.875Duration 0.293 0.344 0.201 0.131Pulse 0.721 0.619 0.659 0.465 0.402Stress 0.164 0.368 0.034 0.018 0.312 0.506

Blood pressure (BP) is the response.

What is effect on regression analyses if predictors are perfectly

uncorrelated?

x1 x2 y 2 5 52 2 5 43 2 7 49 2 7 46 4 5 50 4 5 48 4 7 44 4 7 43

Pearson correlation of x1 and x2 = 0.000

The regression equation is y = 48.8 - 0.63 x1

Predictor Coef SE Coef T PConstant 48.750 4.025 12.11 0.000x1 -0.625 1.273 -0.49 0.641

Analysis of VarianceSource DF SS MS F PRegression 1 3.13 3.13 0.24 0.641Error 6 77.75 12.96Total 7 80.88

Regress Y on X1

Regress Y on X2

The regression equation is y = 55.1 - 1.38 x2

Predictor Coef SE Coef T PConstant 55.125 7.119 7.74 0.000x2 -1.375 1.170 -1.17 0.285


Regress Y on X1 and X2

The regression equation is y = 57.0 - 0.63 x1 - 1.38 x2

Predictor Coef SE Coef T PConstant 57.000 8.486 6.72 0.001x1 -0.625 1.251 -0.50 0.639x2 -1.375 1.251 -1.10 0.322


Source DF Seq SSx1 1 3.13x2 1 15.13

Regress Y on X2 and X1The regression equation is y = 57.0 - 1.38 x2 - 0.63 x1

Predictor Coef SE Coef T PConstant 57.000 8.486 6.72 0.001x2 -1.375 1.251 -1.10 0.322x1 -0.625 1.251 -0.50 0.639


Source DF Seq SSx2 1 15.13x1 1 3.13

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(X1) is the same as the sequential sum of squares SSR(X1|X2).

• The sum of squares SSR(X2) is the same as the sequential sum of squares SSR(X2|X1).

• 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?


Regress BP on Stress

The regression equation isBP = 113 + 0.0240 Stress

Predictor Coef SE Coef T PConstant 112.720 2.193 51.39 0.000Stress 0.02399 0.03404 0.70 0.490

S = 5.502 R-Sq = 2.7% R-Sq(adj) = 0.0%


Regress BP on BSA

The regression equation isBP = 45.2 + 34.4 BSA

Predictor Coef SE Coef T PConstant 45.183 9.392 4.81 0.000BSA 34.443 4.690 7.34 0.000

S = 2.790 R-Sq = 75.0% R-Sq(adj) = 73.6%


Regress BP on BSA and StressThe regression equation is BP = 44.2 + 34.3 BSA + 0.0217 Stress

Predictor Coef SE Coef T PConstant 44.245 9.261 4.78 0.000BSA 34.334 4.611 7.45 0.000Stress 0.02166 0.01697 1.28 0.219


Source DF Seq SSBSA 1 419.86Stress 1 12.26

Regress BP on Stress and BSAThe regression equation isBP = 44.2 + 0.0217 Stress + 34.3 BSA

Predictor Coef SE Coef T PConstant 44.245 9.261 4.78 0.000Stress 0.02166 0.01697 1.28 0.219BSA 34.334 4.611 7.45 0.000


Source DF Seq SSStress 1 15.04BSA 1 417.07

If predictors are nearlyuncorrelated, then…

• You get similar slope estimates regardless of the first-order regression model used.

• The sum of squares SSR(X1) is similar to the sequential sum of squares SSR(X1|X2).

• The sum of squares SSR(X2) is similar to the sequential sum of squares SSR(X2|X1).

What happens if the predictor variables are highly correlated?


Regress BP on Weight

The regression equation isBP = 2.21 + 1.20 Weight

Predictor Coef SE Coef T PConstant 2.205 8.663 0.25 0.802Weight 1.20093 0.09297 12.92 0.000

S = 1.740 R-Sq = 90.3% R-Sq(adj) = 89.7%


Regress BP on BSA



S = 2.790 R-Sq = 75.0% R-Sq(adj) = 73.6%


Regress BP on BSA and WeightThe regression equation isBP = 5.65 + 5.83 BSA + 1.04 Weight

Predictor Coef SE Coef T PConstant 5.653 9.392 0.60 0.555BSA 5.831 6.063 0.96 0.350Weight 1.0387 0.1927 5.39 0.000


Source DF Seq SSBSA 1 419.86Weight 1 88.43

Regress BP on Weight and BSAThe regression equation isBP = 5.65 + 1.04 Weight + 5.83 BSA

Predictor Coef SE Coef T PConstant 5.653 9.392 0.60 0.555Weight 1.0387 0.1927 5.39 0.000BSA 5.831 6.063 0.96 0.350


Source DF Seq SSWeight 1 505.47BSA 1 2.81

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

b1 b2

X1 1.20 ----

X2 ---- 34.4

X1, X2 1.04 5.83

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.


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(X1) = 505.47

SSR(X1|X2) = 88.43

SSR(X2) = 419.86

SSR(X2|X1) = 2.81


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(b1) se(b2)

X1 0.093 ----

X2 ---- 4.69

X1, X2 0.193 6.06

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

2.252.152.051.951.851.75

100

95

90

85

BSA

Wei

ght

(2,92)

Weight Fit SE Fit 95.0% CI 95.0% PI

92 112.7 0.402 (111.85,113.54) (108.94,116.44)

BSA Fit SE Fit 95.0% CI 95.0% PI 2 114.1 0.624 (112.76,115.38) (108.06,120.08)

BSA Weight Fit SE Fit 95.0% CI 95.0% PI

2 92 112.8 0.448 (111.93,113.83) (109.08, 116.68)

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).



S = 2.790 R-Sq = 75.0% R-Sq(adj) = 73.6%


What is effect on tests of individual slopes?

The regression equation isBP = 2.21 + 1.20 Weight

Predictor Coef SE Coef T PConstant 2.205 8.663 0.25 0.802Weight 1.20093 0.09297 12.92 0.000

S = 1.740 R-Sq = 90.3% R-Sq(adj) = 89.7%




The regression equation isBP = 5.65 + 1.04 Weight + 5.83 BSA

Predictor Coef SE Coef T PConstant 5.653 9.392 0.60 0.555Weight 1.0387 0.1927 5.39 0.000BSA 5.831 6.063 0.96 0.350


Source DF Seq SSWeight 1 505.47BSA 1 2.81

Effect #5 of multicollinearity on slope tests

When predictor variables are correlated, hypothesis tests for βk = 0 may yield different conclusions depending on which predictor variables are in the model.

Variables in model

b2 se(b2) t P-value

X2 34.4 4.7 7.34 0.000

X1, X2 5.83 6.1 0.96 0.350

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).

Multicollinearity

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