Fast Backward Variable Selection

Performs a slightly inefficient but numerically stable version of fast backward elimination on factors, using a method based on Lawless and Singhal (1978). This method uses the fitted complete model and computes approximate Wald statistics by computing conditional (restricted) maximum likelihood estimates assuming multivariate normality of estimates. fastbw deletes factors, not columns of the design matrix. Factors requiring multiple d.f. will be retained or dropped as a group. The function prints the deletion statistics for each variable in turn, and prints approximate parameter estimates for the model after deleting variables. The approximation is better when the number of factors deleted is not large. For ols, the approximation is exact for regression coefficients, and standard errors are only off by a factor equal to the ratio of the mean squared error estimate for the reduced model to the original mean squared error estimate for the full model.

If the fit was from ols, fastbw will compute the usual \(R^2\) statistic for each model.

Usage

fastbw(fit, rule=c("aic", "p"),
       type=c("residual", "individual", "total"),
       sls=.05, aics=0, eps=.Machine$double.eps,
       k.aic=2, force=NULL)

# S3 method for class 'fastbw'
print(x, digits=4, estimates=TRUE, ...)

Arguments

fit: fit object with Varcov(fit) defined (e.g., from ols, lrm, cph, psm, glmD)
rule: Stopping rule. Defaults to "aic" for Akaike's information criterion. Use rule="p" to use \(P\)-values
type: Type of statistic on which to base the stopping rule. Default is "residual" for the pooled residual chi-square. Use type="individual" to use Wald chi-square of individual factors.
sls: Significance level for staying in a model if rule="p". Default is .05.
aics: For rule="aic", variables are deleted until the chi-square - k.aic times d.f. would rise above aics. Default aics is zero to use the ordinary AIC. Set aics to say 10000 to see all variables deleted in order of descending importance.
eps: Singularity criterion, default is 1E-14.
k.aic: multiplier to compute AIC, default is 2. To use BIC, set k.aic equal to \(\log(n)\), where \(n\) is the effective sample size (number of events for survival models).
force: a vector of integers specifying parameters forced to be in the model, not counting intercept(s)
x: result of fastbw
digits: number of significant digits to print
estimates: set to FALSE to suppress printing table of approximate coefficients, SEs, etc., after variable deletions
...: ignored

Value

a list with an attribute kept if bw=TRUE, and the following components:

result: matrix of statistics with rows in order of deletion.
names.kept: names of factors kept in final model.
factors.kept: the subscripts of factors kept in the final model
factors.deleted: opposite of factors.kept.
parms.kept: column numbers in design matrix corresponding to parameters kept in the final model.
parms.deleted: opposite of parms.kept.
coefficients: vector of approximate coefficients of reduced model.
var: approximate covariance matrix for reduced model.
Coefficients: matrix of coefficients of all models. Rows correspond to the successive models examined and columns correspond to the coefficients in the full model. For variables not in a particular sub-model (row), the coefficients are zero.

Author

Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com

References

Lawless, J. F. and Singhal, K. (1978): Efficient screening of nonnormal regression models. Biometrics 34:318–327.

Examples

if (FALSE) { # \dontrun{
fastbw(fit, optional.arguments)     # print results
z <- fastbw(fit, optional.args)     # typically used in simulations
lm.fit(X[,z$parms.kept], Y)         # least squares fit of reduced model
} # }