[R] obtaining p-values for lm.ridge() coefficients (package 'MASS')

[Liviu Andronic]

Dear all
I'm familiarising myself with Ridge Regressions in R and the following
is bugging me: How does one get p-values for the coefficients obtained
from MASS::lm.ridge() output (for a given lambda)? Consider the
example below (adapted from PRA [1]):

> require(MASS)
> data(longley)
> gr <- lm.ridge(Employed ~ .,longley,lambda = seq(0,0.1,0.001))
> plot(gr)
> select(gr)
modified HKB estimator is 0.004275
modified L-W estimator is 0.0323
smallest value of GCV  at 0.003
> ##let's choose 0.03 for lambda
> coef(gr)[gr$lam == 0.03,]
             GNP.deflator          GNP   Unemployed Armed.Forces
Population         Year
-1620.429355     0.021060     0.007994    -0.013146    -0.007752
-0.101880     0.869116


But how does one obtain the customary 'lm' summary information for the
model above? I tried supplying the chosen lambda to Design::ols()
using its 'penalty' argument, but strangely the results differ. See
below.

> require(Design)
> Design::ols(Employed ~ GNP.deflator + GNP + Unemployed + Armed.Forces + Population + Year,data=longley,penalty = 0.03,tol=1e-12)

Linear Regression Model

Design::ols(formula = Employed ~ GNP.deflator + GNP + Unemployed +
    Armed.Forces + Population + Year, data = longley, penalty = 0.03,
    tol = 1e-12)

         n Model L.R.       d.f.         R2      Sigma
        16      78.22       7.05     0.9929     0.3438

Residuals:
     Min       1Q   Median       3Q      Max
-0.42480 -0.17774 -0.02169  0.16834  0.77203

Coefficients:
                    Value Std. Error       t Pr(>|t|)
Intercept    -1580.022688 518.501881 -3.0473 0.016006
GNP.deflator     0.023161   0.068862  0.3363 0.745322
GNP              0.008351   0.015160  0.5509 0.596839
Unemployed      -0.013057   0.002651 -4.9255 0.001178
Armed.Forces    -0.007691   0.002097 -3.6671 0.006406
Population      -0.096757   0.144240 -0.6708 0.521355
Year             0.847932   0.269614  3.1450 0.013812

Adjusted R-Squared: 0.9866

> ##the above seems more similar to a lambda of 0.032 than the one required (0.03)
> coef(gr)[gr$lam %in% c(0.03, 0.032),]  ##lm.ridge output
            GNP.deflator      GNP Unemployed Armed.Forces Population   Year
0.030 -1620      0.02106 0.007994   -0.01315    -0.007752   -0.10188 0.8691
0.032 -1580      0.02316 0.008351   -0.01306    -0.007691   -0.09676 0.8479


The difference between the coefficients of MASS::lm.ridge and
Design::ols is small, but they're different nonetheless and I'm
wondering if I'm doing something wrong. Perhaps Design::ols uses a
penalty different from the one specified? Alternatively, is there some
other way to perform diagnostics on Ridge Regression models?

Thank you
Liviu

[1] http://cran.r-project.org/doc/contrib/Faraway-PRA.pdf