Robust regression p values with lmrob - r

Question

I am trying to do robust multiple regression for a dataset where a few outliers don't allow me to see the underlying patterns through the usual linear models.
I am using the function lmrob in the package robustbase, and I was surprised by the number of significant relationships that I found. I decided to try the method with random data, this is the code:
library(robustbase)
set.seed(4)
ax<-data.frame(a1=rnorm(20,3),
a2=rnorm(20,5),
a3=rnorm(20,4),
a4=rnorm(20,6),
a5=rnorm(20,2))
axm<-lmrob(a1~a2*a3*a4*a5,data=ax)
summary(axm)
And the output:
Call:
lmrob(formula = a1 ~ a2 * a3 * a4 * a5, data = ax)
\--> method = "MM"
Residuals:
1 2 3 4 5 6 7 8 9 10 11 12 13
-34.740270 -0.049493 -0.044379 0.002770 0.219825 0.041285 0.156152 -0.072825 0.034824 -0.014757 -0.088263 -0.185045 -0.079679
14 15 16 17 18 19 20
-0.045121 -0.007576 0.008813 0.010451 0.015716 0.060781 0.040187
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1160.5907 94.0095 -12.35 0.000247 ***
a2 205.6910 15.8689 12.96 0.000204 ***
a3 327.9787 24.2161 13.54 0.000172 ***
a4 193.2384 15.7300 12.29 0.000252 ***
a5 734.2203 49.8960 14.71 0.000124 ***
a2:a3 -57.6229 4.0533 -14.22 0.000142 ***
a2:a4 -33.5644 2.6130 -12.85 0.000212 ***
a3:a4 -54.1622 4.0438 -13.39 0.000180 ***
a2:a5 -138.8395 9.2697 -14.98 0.000116 ***
a3:a5 -198.4961 12.3168 -16.12 8.67e-05 ***
a4:a5 -123.0895 8.2792 -14.87 0.000119 ***
a2:a3:a4 9.3344 0.6659 14.02 0.000150 ***
a2:a3:a5 37.1371 2.2502 16.50 7.89e-05 ***
a2:a4:a5 23.0014 1.5152 15.18 0.000110 ***
a3:a4:a5 32.9766 2.0388 16.18 8.55e-05 ***
a2:a3:a4:a5 -6.0817 0.3660 -16.62 7.68e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Robust residual standard error: 0.4039
Multiple R-squared: 0.9861, Adjusted R-squared: 0.934
Convergence in 5 IRWLS iterations
Robustness weights:
observation 1 is an outlier with |weight| = 0 ( < 0.005);
9 weights are ~= 1. The remaining 10 ones are
2 3 5 7 8 11 12 13 14 19
0.9986 0.9989 0.9732 0.9864 0.9970 0.9957 0.9810 0.9965 0.9989 0.9979
Algorithmic parameters:
tuning.chi bb tuning.psi refine.tol rel.tol scale.tol solve.tol eps.outlier
1.548e+00 5.000e-01 4.685e+00 1.000e-07 1.000e-07 1.000e-10 1.000e-07 5.000e-03
eps.x warn.limit.reject warn.limit.meanrw
1.150e-09 5.000e-01 5.000e-01
nResample max.it best.r.s k.fast.s k.max maxit.scale trace.lev mts compute.rd
500 50 2 1 200 200 0 1000 0
fast.s.large.n
2000
psi subsampling cov compute.outlier.stats
"bisquare" "nonsingular" ".vcov.avar1" "SM"
seed : int(0)
According to this, I understand that the other random variables are related to the first one, and have high predictive power over it, which makes no sense.
What is happening here? I am doing the regression wrong?
Edit: I put a seed for which the p values are extremely low for replicability.

Answer 1

I think I may have found the explanation to such high p-values: turns out that the MM estimate with small sample sizes (as in my example with 20) is prone to type 1 error. One of the authors of the robustbase package has published an article proposing an alternative estimate for this cases, but I'm afraid that for my data it doesn't work so much better.