Multicollinearity in Blood Pressure Data

Sameer Mathur

What is the effect on regression analyses if the predictors are nearly uncorrelated versus highly correlated?

Multicollinearity

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Reading Data

Let's focus on the relationships among the response BP and the predictors Weight; BSA and Stress.

# reading data
bp.df <- read.delim("BloodPressureData.txt")
# attaching data columns of the dataframe
attach(bp.df)
# dimension of the dataframe
dim(bp.df)

[1] 20  8

Regression Analysis

Model 1: BP ~ Weight

summary(lm(BP ~ Weight, data = bp.df))


Call:
lm(formula = BP ~ Weight, data = bp.df)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.6933 -0.9318 -0.4935  0.7703  4.8656 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.20531    8.66333   0.255    0.802    
Weight       1.20093    0.09297  12.917 1.53e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.74 on 18 degrees of freedom
Multiple R-squared:  0.9026,    Adjusted R-squared:  0.8972 
F-statistic: 166.9 on 1 and 18 DF,  p-value: 1.528e-10

Model 2a: BP ~ Stress

summary(lm(BP ~ Stress, data = bp.df))


Call:
lm(formula = BP ~ Stress, data = bp.df)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6394 -3.3014  0.0722  2.2181  9.9287 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 112.71997    2.19345  51.389   <2e-16 ***
Stress        0.02399    0.03404   0.705     0.49    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.502 on 18 degrees of freedom
Multiple R-squared:  0.02686,   Adjusted R-squared:  -0.0272 
F-statistic: 0.4969 on 1 and 18 DF,  p-value: 0.4899

Model 3a: BP ~ Weight + Stress

summary(lm(BP ~ Weight + Stress, data = bp.df))


Call:
lm(formula = BP ~ Weight + Stress, data = bp.df)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.4865 -0.9395  0.1950  0.5080  4.0023 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.71023    8.09054   0.211   0.8351    
Weight       1.19522    0.08683  13.765  1.2e-10 ***
Stress       0.01924    0.01006   1.913   0.0727 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.625 on 17 degrees of freedom
Multiple R-squared:  0.9199,    Adjusted R-squared:  0.9105 
F-statistic: 97.59 on 2 and 17 DF,  p-value: 4.807e-10

Model 2b: BP ~ BSA

summary(lm(BP ~ BSA, data = bp.df))


Call:
lm(formula = BP ~ BSA, data = bp.df)

Residuals:
   Min     1Q Median     3Q    Max 
-5.314 -1.963 -0.197  1.934  4.831 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   45.183      9.392   4.811  0.00014 ***
BSA           34.443      4.690   7.343 8.11e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.79 on 18 degrees of freedom
Multiple R-squared:  0.7497,    Adjusted R-squared:  0.7358 
F-statistic: 53.93 on 1 and 18 DF,  p-value: 8.114e-07

Model 3b: BP ~ Weight + BSA

summary(lm(BP ~ Weight + BSA, data = bp.df))


Call:
lm(formula = BP ~ Weight + BSA, data = bp.df)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.8932 -1.1961 -0.4061  1.0764  4.7524 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   5.6534     9.3925   0.602    0.555    
Weight        1.0387     0.1927   5.392 4.87e-05 ***
BSA           5.8313     6.0627   0.962    0.350    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.744 on 17 degrees of freedom
Multiple R-squared:  0.9077,    Adjusted R-squared:  0.8968 
F-statistic: 83.54 on 2 and 17 DF,  p-value: 1.607e-09

Beta Coefficient Table