title: Robust regression p values with lmrob
body: <p>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 <code>lmrob</code> in the package <code>robustbase</code>, 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:</p>
  <pre class="lang-r prettyprint-override"><code>library(robustbase)
  set.seed(4)
  ax&lt;-data.frame(a1=rnorm(20,3),
                 a2=rnorm(20,5),
                 a3=rnorm(20,4),
                 a4=rnorm(20,6),
                 a5=rnorm(20,2))
   
  axm&lt;-lmrob(a1~a2*a3*a4*a5,data=ax)
  summary(axm)
  </code></pre>
  <p>And the output:</p>
  <pre><code>        Call:
  lmrob(formula = a1 ~ a2 * a3 * a4 * a5, data = ax)
   \--&gt; method = &quot;MM&quot;
  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(&gt;|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 ( &lt; 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 
             &quot;bisquare&quot;         &quot;nonsingular&quot;         &quot;.vcov.avar1&quot;                  &quot;SM&quot; 
  seed : int(0) 
  </code></pre>
  <p>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.</p>
  <p>What is happening here? I am doing the regression wrong?</p>
  <p>Edit: I put a seed for which the p values are extremely low for replicability.</p>