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Confounding and collinearity

# Confounding and collinearity

## Introduction

In this blog article, I will discuss about the bias introduced in estimation of coefficient of a given explanatory variable due to the presence of confounding factors. After that, I will try to demonstrate about the effect of variable collinearity on estimation of coefficient.

## Underlying structure

Let us assume that we know about the underlying relationship between the explanatory variable (E), outcome variable (O), confounder ( C ), variable correlated with E (VE) and variable correlated with O (VO).

Let us assume that the real underlying relation between various variables are as given below:

O = 2.E + 2.C + 2.VO + 0.VE + error

E, C and VE are correlated with each other with correlation coef of 0.7 each.

All the random variables are normally distributed, with mean 0 and sd 5.  error is distributed as standard normal distribution.


The directed acyclic graph depicts the relationship between the variables. ## Samples for simulation

library(MASS)
library(dplyr)

##
## Attaching package: 'dplyr'
##
## The following objects are masked from 'package:Hmisc':
##
##     src, summarize
##
## The following objects are masked from 'package:lubridate':
##
##     intersect, setdiff, union
##
## The following object is masked from 'package:MASS':
##
##     select
##
## The following objects are masked from 'package:stats':
##
##     filter, lag
##
## The following objects are masked from 'package:base':
##
##     intersect, setdiff, setequal, union

set.seed(100)
sam <- mvrnorm(n = 10000, mu = c(0, 0, 0), Sigma = matrix(c(5, 4, 4, 4, 5, 4,
4, 4, 5), 3, 3, byrow = T))
df <- data.frame(E = sam[, 1], C = sam[, 2], VE = sam[, 3])
df <- mutate(df, VO = rnorm(10000, 0, 5), O = 2 * E + 2 * C + 2 * VO + rnorm(10000))


## Scenarios

### Scenario 1 (Only E as covariate, typical univariate analysis)

mod.E <- lm(O ~ E, data = df)
summary(mod.E)

##
## Call:
## lm(formula = O ~ E, data = df)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -40.47  -7.05  -0.05   7.01  45.04
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   0.0069     0.1034    0.07     0.95
## E             3.6420     0.0465   78.40   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.3 on 9998 degrees of freedom
## Multiple R-squared:  0.381,  Adjusted R-squared:  0.381
## F-statistic: 6.15e+03 on 1 and 9998 DF,  p-value: <2e-16

confint(mod.E)

##               2.5 % 97.5 %
## (Intercept) -0.1959 0.2097
## E            3.5509 3.7330


The model overestimates coefficient of E, its value is the sum of coef of E and C. This type of bias is known as bias due to confounding and occurs when confounders are not included as co-variates.

### Scenario 2 (E and C as covariates)

This model is incorrect as it does not include the other covariate, VO.

mod.EC <- lm(O ~ E + C, data = df)
summary(mod.EC)

##
## Call:
## lm(formula = O ~ E + C, data = df)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -35.99  -6.75  -0.11   6.71  43.32
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -0.0194     0.0998   -0.19     0.85
## E             2.0293     0.0741   27.40   <2e-16 ***
## C             2.0231     0.0740   27.35   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.98 on 9997 degrees of freedom
## Multiple R-squared:  0.424,  Adjusted R-squared:  0.424
## F-statistic: 3.68e+03 on 2 and 9997 DF,  p-value: <2e-16

confint(mod.EC)

##              2.5 % 97.5 %
## (Intercept) -0.215 0.1762
## E            1.884 2.1745
## C            1.878 2.1681


Due to addition of C, the estimate of E is unbiased. Exclusion of VO has not influenced the coefficient of E.

### Scenario 3 (E and VO as covariates)

E and VO are not correlated. C has been excluded from the model.

mod.EVO <- lm(O ~ E + VO, data = df)
summary(mod.EVO)

##
## Call:
## lm(formula = O ~ E + VO, data = df)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -9.638 -1.938  0.005  1.954 12.780
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.03893    0.02884    1.35     0.18
## E            3.59175    0.01295  277.27   <2e-16 ***
## VO           2.00167    0.00581  344.39   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.88 on 9997 degrees of freedom
## Multiple R-squared:  0.952,  Adjusted R-squared:  0.952
## F-statistic: 9.88e+04 on 2 and 9997 DF,  p-value: <2e-16

confint(mod.EVO)

##                2.5 %  97.5 %
## (Intercept) -0.01761 0.09546
## E            3.56636 3.61714
## VO           1.99028 2.01307


There is again overestimation of coefficient of E (as in scenario 1), due to exclusion of C. Estimate of VO is unbiased as it has no correlation with either E, VO or VE.

### Scenario 4 (E and VE as covariates)

E and VE are correlated, but VE is not associated with O.

mod.EVE <- lm(O ~ E + VE, data = df)
summary(mod.EVE)

##
## Call:
## lm(formula = O ~ E + VE, data = df)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -41.55  -7.00  -0.02   6.93  43.46
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   0.0106     0.1026     0.1     0.92
## E             2.8584     0.0761    37.6   <2e-16 ***
## VE            0.9857     0.0761    12.9   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.3 on 9997 degrees of freedom
## Multiple R-squared:  0.391,  Adjusted R-squared:  0.391
## F-statistic: 3.21e+03 on 2 and 9997 DF,  p-value: <2e-16

confint(mod.EVE)

##               2.5 % 97.5 %
## (Intercept) -0.1905 0.2117
## E            2.7093 3.0075
## VE           0.8364 1.1350


Estimates of both E and VE are biased. But coefficient of E is less biased than scenario 1. Reason for biasness of coefficient of VE is unknown to me.

### Scenario 5 (E, VE, VO, C as covariates) (Complete model)

mod.comp <- lm(O ~ E + C + VO + VE, data = df)
summary(mod.comp)

##
## Call:
## lm(formula = O ~ E + C + VO + VE, data = df)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -3.434 -0.677 -0.014  0.678  3.616
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.01297    0.01000    1.30     0.19
## E            1.98794    0.00826  240.62   <2e-16 ***
## C            1.99993    0.00830  240.81   <2e-16 ***
## VO           2.00033    0.00202  992.51   <2e-16 ***
## VE           0.01214    0.00831    1.46     0.14
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1 on 9995 degrees of freedom
## Multiple R-squared:  0.994,  Adjusted R-squared:  0.994
## F-statistic: 4.29e+05 on 4 and 9995 DF,  p-value: <2e-16

confint(mod.comp)

##                 2.5 %  97.5 %
## (Intercept) -0.006635 0.03258
## E            1.971745 2.00413
## C            1.983651 2.01621
## VO           1.996381 2.00428
## VE          -0.004159 0.02844


Unbiased coefficients of all the covariates.

### Scenario 6 (E, VE, C as covariates)

mod.EVEC <- lm(O ~ E + VE + C, data = df)
summary(mod.EVEC)

##
## Call:
## lm(formula = O ~ E + VE + C, data = df)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -36.18  -6.74  -0.08   6.71  43.21
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -0.0185     0.0998   -0.19     0.85
## E             1.9892     0.0824   24.13   <2e-16 ***
## VE            0.0920     0.0830    1.11     0.27
## C             1.9817     0.0829   23.92   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.98 on 9996 degrees of freedom
## Multiple R-squared:  0.424,  Adjusted R-squared:  0.424
## F-statistic: 2.45e+03 on 3 and 9996 DF,  p-value: <2e-16

confint(mod.EVEC)

##                2.5 % 97.5 %
## (Intercept) -0.21411 0.1771
## E            1.82758 2.1507
## VE          -0.07061 0.2546
## C            1.81930 2.1441


Addition of C has removed the biasedness from the coefficients of E and VE (see Scenario 4).

### Scenario 7 (E, VO and C as covariates)

mod.EVOC <- lm(O ~ E + VO + C, data = df)
summary(mod.EVOC)

##
## Call:
## lm(formula = O ~ E + VO + C, data = df)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -3.427 -0.676 -0.014  0.680  3.641
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.01285    0.01000    1.29      0.2
## E            1.99323    0.00742  268.50   <2e-16 ***
## VO           2.00036    0.00202  992.52   <2e-16 ***
## C            2.00539    0.00741  270.45   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1 on 9996 degrees of freedom
## Multiple R-squared:  0.994,  Adjusted R-squared:  0.994
## F-statistic: 5.72e+05 on 3 and 9996 DF,  p-value: <2e-16

confint(mod.EVOC)

##                 2.5 %  97.5 %
## (Intercept) -0.006751 0.03246
## E            1.978682 2.00779
## VO           1.996410 2.00431
## C            1.990857 2.01993


Performance of this model is almost the same as Scenario 5

## Performance of different scenarios

ModelsComplete (E, C, VO, VE)E, C, VEE, C, VOE, CE, VEE, VOE only
E (2)1.9879, 0.00831.9892, 0.08241.9932, 0.00742.0293, 0.07412.8584, 0.07613.5918, 0.0133.642, 0.0465
C (2)1.9999, 0.00831.9817, 0.08292.0054, 0.00742.0231, 0.074---
VE (0)0.0121, 0.00830.092, 0.083--0.9857, 0.0761--
VO (2)2.0003, 0.002-2.0004, 0.002--2.0017, 0.0058-

E (2): means covariate E and its actual value.

–,–: coefficient, standard error.

## Conclusions

Models with C, will have unbiased estimate of E. Models with correlated variables (E, C, VE) has larger SE due to near collinearity.