summary.lmridge {lmridge}

Summarizing Linear Ridge Regression Fits

Description

The summary method for class "lmridge" for scalar or vector biasing parameter K (Cule and De lorio, 2012).

Usage

## S3 method for class 'lmridge'
summary(object, ...)
## S3 method for class 'summary.lmridge'
print(x, digits = max(3, getOption("digits") - 3),
           signif.stars = getOption("show.signif.stars"), ...)

Arguments

object

An "lmridge" object, typically generated by a call to lmridge.

x

An object of class summary.lmridge for the print.summary.lmridge.

signif.stars

logical: if TRUE, p-values are additionally encoded visually as significance starts in order to help scanning of long coefficient tables. It default to the show.signif.stars slot of options.

digits

The number of significant digits to use when printing.

...

Not presently used in this implementation.

Details

print.summary.lmridge tries to be smart about formatting the coefficients, standard errors etc. and additionally gives 'significance stars' if signif.stars is TRUE.

Value

The function summary computes and returns a list of summary statistics of the fitted linear ridge regression model for scalar or vector value biasing parameter K given as argument in lmridge function. All summary information can be called using list object summaries.

coefficients

A p \times 5 matrix with columns for the scaled estimated, descaled estimated coefficients, scaled standard error, scaled t-statistics, and corresponding p-value (two-tailed). The Intercept term is computed by the relation \hat{\beta}_{R_{0K}}=\overline{y}-\sum_{j=1}^{p}\overline{X}_j \hat{\beta}_{R_{0K}}. The standard error of intercept term is computed as, SE(\hat{\beta}_{R_{0K}})=\sqrt{Var(\overline{y}) +\overline{X}_j^2 diag[Cov(\hat{\beta}_{R_{0K}})]}.

stats

Ridge related statistics of R-squared, adjusted R-squared, F-statistics for testing of coefficients, AIC and BIC values for given biasing parameter K.

rmse1

Minimum MSE value for given biasing parameter K.

rmse2

Value of K at which MSE is minimum.

K

Value of given biasing parameter.

df1

Numerator degrees of freedom for p-value of F-statistics.

df2

Denominator degrees of freedom for p-value of F-statistics.

fpvalue

p-value for each F-statistics.

Author(s)

Muhammad Imdad Ullah, Muhammad Aslam

References

Cule, E. and De lorio, M. (2012). A semi-Automatic method to guide the choice of ridge parameter in ridge regression. arXiv:1205.0686v1 [stat.AP].

Hoerl, A. E., Kennard, R. W., and Baldwin, K. F. (1975). Ridge Regression: Some Simulation. Communication in Statistics, 4, 105-123. doi:10.1080/03610927508827232.

Hoerl, A. E. and Kennard, R. W., (1970). Ridge Regression: Biased Estimation of Nonorthogonal Problems. Technometrics, 12, 55-67. doi:10.1080/00401706.1970.10488634.

Imdad, M. U. Addressing Linear Regression Models with Correlated Regressors: Some Package Development in R (Doctoral Thesis, Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan), 2017.

See Also

The ridge model fitting lmridge

Examples

mod <- lmridge(y~., as.data.frame(Hald), K = c(0, 0.0132, 0.1))
summary(mod)

## coefficients for first biasing parameter
summary(mod)$summaries[[1]]$coefficients
summary(mod)$summaries[[1]][[1]]

## ridge related statistics from summary function
summary(mod)$summaries[[1]]$stats

## Ridge F-test's p-value
summary(mod)$summaries[[1]]$fpvalue