FRBmultiregGS: GS-Estimates for multivariate regression with bootstrap confidence intervals

Description

Computes GS-estimates for multivariate regression together with standard errors, confidence intervals and p-values based on the Fast and Robust Bootstrap.

Usage

# S3 method for formula
FRBmultiregGS(formula, data=NULL, ...)
# S3 method for default
FRBmultiregGS(X, Y, int = TRUE, R = 999, bdp = 0.5, conf = 0.95,
control=GScontrol(...), na.action=na.omit, ...)

Value

An object of class FRBmultireg which extends class mlm and contains at least the following components:

coefficients: GS-estimates of the regression coefficients

residuals: the residuals, that is response minus fitted values
fitted.values: the fitted values.
Sigma: GS-estimate of the error covariance matrix
scale: GS-estimate of the size of the multivariate errors
weights: implicit weights corresponding to the GS-estimates (i.e. final weights in the RWLS procedure for the intercept estimate)
outFlag: outlier flags: 1 if the robust distance of the residual exceeds the .975 quantile of (the square root of) the chi-square distribution with degrees of freedom equal to the dimension of the responses; 0 otherwise
SE: bootstrap standard errors corresponding to the regression coefficients
cov: bootstrap covariance matrix corresponding to the regression coefficients (in vectorized form)
CI.bca.lower: a matrix containing the lower bound of the bias corrected and accelerated confidence intervals for the regression coefficients
CI.bca.upper: a matrix containing the upper bound of the bias corrected and accelerated confidence intervals for the regression coefficients
CI.basic.lower: a matrix containing the lower bound of basic bootstrap intervals for the regression coefficients
CI.basic.upper: a matrix containing the upper bound of basic bootstrap intervals for the regression coefficients
p.bca: a matrix containing the p-values based on the BCa confidence intervals for the regression coefficients
p.basic: a matrix containing the p-values based on the basic bootstrap intervals for the regression coefficients
est: GS-estimates as returned by the call to GSest_multireg()
bootest: bootstrap results for the GS-estimates as returned by the call to GSboot_multireg()
conf: a copy of the conf argument
method: a list with following components: est = character string indicating that GS-estimates were used, and bdp = a copy of the bdp argument
control: a copy of the control argument
X, Y: either copies of the respective arguments or the corresponding matrices produced from formula
ROK: number of bootstrap samples actually used (i.e. not discarded due to too few distinct observations with positive weight)

Arguments

formula: an object of class formula; a symbolic description of the model to be fit.
data: data frame from which variables specified in formula are to be taken.
X: a matrix or data frame containing the explanatory variables.
Y: a matrix or data frame containing the response variables.
int: logical: if TRUE an intercept term is added to the model (unless it is already present in X)
R: number of bootstrap samples. Default is R=999.
bdp: required breakdown point. Should have \(0 < \) bdp \(\le 0.5\), the default is 0.5.
conf: confidence level of the bootstrap confidence intervals. Default is conf=0.95.
control: a list with control parameters for tuning the computing algorithm, see GScontrol().
na.action: a function which indicates what should happen when the data contain NAs. Defaults to na.omit.
...: allows for specifying control parameters directly instead of via control.

Author

Ella Roelant, Stefan Van Aelst and Gert Willems

Details

Generalized S-estimators are defined by minimizing the determinant of a robust estimator of the scatter matrix of the differences of the residuals (Roelant et al. 2009). Hence, this procedure is intercept free and only gives an estimate for the slope matrix. To estimate the intercept, we use the M-type estimator of location of Lopuhaa (1992) on the residuals with the residual scatter matrix estimate of the residuals as a preliminary estimate. This computation is carried out by a call to GSest_multireg(), which uses a fast-S-type algorithm (its tuning parameters can be changed via the control argument). The result of this call is also returned as the value est.

The Fast and Robust Bootstrap (Salibian-Barrera and Zamar 2002) is used to calculate so-called basic bootstrap confidence intervals and bias corrected and accelerated (BCa) confidence intervals (Davison and Hinkley 1997, p.194 and p.204 respectively). Apart from the intervals with the requested confidence level, the function also returns p-values for each coefficient corresponding to the hypothesis that the actual coefficient is zero. The p-values are computed as 1 minus the smallest level for which the confidence intervals would include zero. Both BCa and basic bootstrap p-values in this sense are given. The bootstrap calculation is carried out by a call to GSboot_multireg(), the result of which is returned as the value bootest. Bootstrap standard errors are returned as well.

Note: Bootstrap samples which contain too few distinct observations with positive weights are discarded (a warning is given if this happens). The number of samples actually used is returned via ROK.

In the formula-interface, a multivariate response is produced via cbind. For example cbind(x4,x5) ~ x1+x2+x3. All arguments from the default method can also be passed to the formula method.

The returned object inherits from class mlm such that the standard coef, residuals, fitted and predict functions can be used.

References

A.C. Davison and D.V. Hinkley (1997) Bootstrap Methods and their Application. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press.
H.P. Lopuhaa (1992) Highly efficient estimators of multivariate location with high breakdown point. The Annals of Statistics, 20, 398-413.
E. Roelant, S. Van Aelst and C. Croux (2009) Multivariate Generalized S-estimators. Journal of Multivariate Analysis, 100, 876--887.
M. Salibian-Barrera, S. Van Aelst and G. Willems (2008) Fast and robust bootstrap. Statistical Methods and Applications, 17, 41-71.
S. Van Aelst and G. Willems (2013), Fast and robust bootstrap for multivariate inference: The R package FRB. Journal of Statistical Software, 53(3), 1--32. tools:::Rd_expr_doi("10.18637/jss.v053.i03").

Examples

Run this code

data(schooldata)
school.x <- data.matrix(schooldata[,1:5])
school.y <- data.matrix(schooldata[,6:8])

#computes 25% breakdown point GS-estimate and 80% confidence intervals 
#based on 99 bootstrap samples:
GSres <- FRBmultiregGS(school.x, school.y, R=99, bdp = 0.25, conf = 0.8,nsamp=50)
#or using the formula interface
# \donttest{
GSres <- FRBmultiregGS(cbind(reading,mathematics,selfesteem)~., data=schooldata, 
          bdp = 0.25, conf = 0.8,R=99)
# }

#the print method just displays the coefficient estimates
GSres

#the summary function additionally displays the bootstrap standard errors and p-values 
#("BCA" method by default)
summary(GSres)

summary(GSres, confmethod="basic")

#ask explicitely for the coefficient matrix:
GSres$coefficients
# or equivalently,
coef(GSres)
#For the error covariance matrix:
GSres$Sigma
                                                              
#plot some bootstrap histograms for the coefficient estimates 
#(with "BCA" intervals by default) 
plot(GSres, expl=c("education", "occupation"), resp=c("selfesteem","reading"))

#plot bootstrap histograms for all coefficient estimates
plot(GSres)
#possibly the plot-function has made a selection of coefficients to plot here, 
#since 'all' may have been too many to fit on one page, see help(plot.FRBmultireg); 
#this is platform-dependent

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