Why are the residuals computed manually different from those computed by R?

Question

I have done a simple test using R, take a look at the code below:

First of all I have created three samples:

> a = rnorm(10)
> b = rnorm(10)
> c = rnorm(10)
> a
 [1] -0.2485833 -1.3077108  0.4019243  0.4453618  0.3024991 -0.4228684
 [7] -0.3817301  0.2195161  1.3693408 -1.1030199
> b
 [1]  0.1795048 -0.8764778  0.6097460 -1.0405654  0.8688749 -1.5191619
 [7]  0.8955941  0.7544877 -1.0576488  1.7317130
> c
 [1] -0.29437630 -0.96914835  0.16349019  0.08477029 -0.52696295  0.28241544
 [7] -1.49860175 -0.26208560 -0.60898891 -1.67154576

Then, I've done a linear regression on those three samples:

> mod = lm(a ~ b + c + 0)
> mod

Call:
lm(formula = a ~ b + c + 0)

Coefficients:
       b         c  
-0.02748   0.37456  

Perfect, now I have the residuals of the linear regression.

(note: I set the intercept to zero)

Now, I print the residuals:

> mod$residuals
         1          2          3          4          5          6          7 
-0.1333896 -0.9687917  0.3574424  0.3850173  0.5237530 -0.5703937  0.2041941 
         8          9         10 
 0.3384146  1.5683807 -0.4293429 

If I'm not wrong in this case(without intercept) the residuals are calculated doing:

a - (b*coef_b + c*coef_c)

If I do it manually I have:

> a - ((b*-0.02748) + (c*0.37456))
 [1] -0.1333890 -0.9687922  0.3574432  0.3850155  0.5237550 -0.5703965
 [7]  0.2041971  0.3384162  1.5683795 -0.4293383

if you look at the numbers closely you will notice a slight difference.

Could someone explain me the reason?

Thank you!

Answer 1

It looks like you're grabbing coef_b and coef_c from what you see from the model print out. R stores more digits than that so you should use coef(mod) to get the actual stored values.

So you should be able to do this:

a - coef(mod)[1]*b - coef(mod)[2]*c

Edit: To clarify what is going on. To get the prediction you could do something like

apred = coef(mod)[1]*b + coef(mod)[2]*c

If that seems wrong to you then you probably need to take a step back and review a few things about regression. The residual is just defined as the actual data point minus the prediction.

a - apred

replacing apred with how we defined it we get

a - (coef(mod)[1]*b + coef(mod)[2]*c)

distributing the negative into both terms in the parenthesis we get

a - coef(mod)[1]*b - coef(mod)[2]*c