# How to treat Correlated covariates in linear regression analysis?

I have multiple covariates shown on the plot. AGE and DIABDUR are strongly correlated. To proceed with regression analysis do I include only one of those two covariates, AGE or DIABDUR or maybe do I create and how some interaction term between those, AGE_DIABDUR? I have 1395 observations. I am planning to do regression analysis in Plink using --glm.

How one can model this?

If I do:

oneway.model <- lm(PHENO ~ C1+C2+HBA1C+DIABDUR+AGE+SEX, data = a)

summary(oneway.model)

Call:
lm(formula = PHENO ~ C1 + C2 + HBA1C + DIABDUR + AGE + SEX, data = a)

Residuals:
Min       1Q   Median       3Q      Max
-1.05752 -0.35084 -0.01828  0.38080  0.95758

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.557103   0.099065   5.624 2.28e-08 ***
C1           1.153912   1.082741   1.066   0.2867
C2          -0.080059   1.568669  -0.051   0.9593
HBA1C        0.017581   0.007578   2.320   0.0205 *
DIABDUR      0.027316   0.001945  14.043  < 2e-16 ***
AGE          0.002478   0.001939   1.278   0.2015
SEX         -0.029370   0.024561  -1.196   0.2320
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4411 on 1307 degrees of freedom
(81 observations deleted due to missingness)
Multiple R-squared:  0.2255,    Adjusted R-squared:  0.222
F-statistic: 63.43 on 6 and 1307 DF,  p-value: < 2.2e-16


How one can interpret this? And what is the better model to try?

If I do it for all covariates:

mlr <- lm(PHENO~., data = a)
summary(mlr)

Call:
lm(formula = PHENO ~ ., data = a)

Residuals:
Min       1Q   Median       3Q      Max
-1.14737 -0.26029 -0.09105  0.27511  1.01946

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.0290078  0.0904134   0.321   0.7484
DIABDUR      0.0190938  0.0017525  10.895  < 2e-16 ***
HBA1C        0.0442987  0.0067641   6.549 8.32e-11 ***
ESRD         0.5117510  0.0251496  20.348  < 2e-16 ***
SEX         -0.0348553  0.0215128  -1.620   0.1054
AGE         -0.0001901  0.0017016  -0.112   0.9111
C1           1.7647042  0.9460201   1.865   0.0624 .
C2          -0.0275111  1.3697721  -0.020   0.9840
C3          -0.8154629  1.6310213  -0.500   0.6172
C4          -0.7681733  1.7158284  -0.448   0.6544
C5           0.6422739  1.8036192   0.356   0.7218
C6          -0.5205025  1.8344969  -0.284   0.7767
C7          -1.0526507  1.8609067  -0.566   0.5717
C8           1.2688756  1.9080009   0.665   0.5062
C9          -0.2403904  1.8996609  -0.127   0.8993
C10         -2.0195728  1.9165077  -1.054   0.2922
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3851 on 1298 degrees of freedom
(81 observations deleted due to missingness)
Multiple R-squared:  0.4136,    Adjusted R-squared:  0.4069
F-statistic: 61.04 on 15 and 1298 DF,  p-value: < 2.2e-16


• What is your concern about the correlation? Depending on your goal in doing this analysis, the correlation might not matter at all. – Dave Nov 11 '20 at 0:20
• I need to find out how those variables effect my trait. Pheno is designation for case or control. I suppose I would need to run some sort of linear regression and find out how those covariates effect the Beta – anamaria Nov 12 '20 at 0:14
• Correlation between covariates impacts the standard errors on the coefficients, but the estimates are unbiased (under the usual assumptions for linear regression that I suspect you’re making). What is your concern about the correlation? From where I am, it looks like you can say that DIABDUR has an estimated effect of $0.027316$. – Dave Nov 12 '20 at 0:37
• Thank you for being patient with me. So my concern is which covariates I should keep in analysis...I just updated the post listing the effect of all covariates. Looking at all of them which ones would you keep for the regression analysis? – anamaria Nov 12 '20 at 0:52
• Much of that will fall to your domain knowledge. – Dave Nov 12 '20 at 1:12