smooth.influence.measures: Nonparametric Regression Deletion Diagnostics

Description

These functions compute several regression (leave-one-out deletion) diagnostics for a fit smoothing spline (fit by ss), smooth model (fit by sm), or generalized smooth model (fit by gsm).

Usage

smooth.influence.measures(model, infl = smooth.influence(model))
# S3 method for ss
rstandard(model, infl = NULL, sd = model$sigma, 
          type = c("sd.1", "predictive"), ...)
# S3 method for sm
rstandard(model, infl = NULL, sd = model$sigma, 
          type = c("sd.1", "predictive"), ...)
# S3 method for gsm
rstandard(model, infl = NULL, 
          type = c("deviance", "pearson"), ...)
# S3 method for ss
rstudent(model, infl = influence(model, do.coef = FALSE), 
         res = infl$wt.res, ...)
# S3 method for sm
rstudent(model, infl = influence(model, do.coef = FALSE), 
         res = infl$wt.res, ...)
# S3 method for gsm
rstudent(model, infl = influence(model, do.coef = FALSE), ...)
# S3 method for ss
dfbeta(model, infl = NULL, ...)
# S3 method for sm
dfbeta(model, infl = NULL, ...)
# S3 method for gsm
dfbeta(model, infl = NULL, ...)
# S3 method for ss
dfbetas(model, infl = smooth.influence(model, do.coef = TRUE), ...)
# S3 method for sm
dfbetas(model, infl = smooth.influence(model, do.coef = TRUE), ...)
# S3 method for gsm
dfbetas(model, infl = smooth.influence(model, do.coef = TRUE), ...)
cov.ratio(model, infl = smooth.influence(model, do.coef = FALSE),
          res = weighted.residuals(model))
# S3 method for ss
cooks.distance(model, infl = NULL, res = weighted.residuals(model), 
               sd = model$sigma, hat = hatvalues(model), ...)
# S3 method for sm
cooks.distance(model, infl = NULL, res = weighted.residuals(model), 
               sd = model$sigma, hat = hatvalues(model), ...)
# S3 method for gsm
cooks.distance(model, infl = NULL, res = residuals(model, type = "pearson"), 
               dispersion = model$dispersion, hat = hatvalues(model), ...)
# S3 method for ss
hatvalues(model, ...)
# S3 method for sm
hatvalues(model, ...)
# S3 method for gsm
hatvalues(model, ...)

Arguments

model: an object of class "gsm" output by the gsm function, "sm" output by the sm function, or "ss" output by the ss function
infl: influence structure as returned by smooth.influence
res: (possibly weighted) residuals with proper defaults
sd: standard deviation to use, see defaults
dispersion: dispersion (for gsm objects) to use, see defaults
hat: hat values \(S_{ii}\), see defaults
type: type of residuals for rstandard
...: additional arguments (currently ignored)

Author

Nathaniel E. Helwig <helwig@umn.edu>

Details

Inspired by influence.measures and related functions in R's stats package.

The function smooth.influence.measures produces a class "infl" object, which displays the DFBETAS for each coefficient, DFFITS, covariance ratios, Cook's distance, and the diagonals of the smoothing matrix. Cases which are influential with respect to any of these measures are marked with an asterisk.

The S3 methods dfbetas, dffits, covratio, and cooks.distance provide direct access to the corresponding diagnostic quantities. The S3 methods rstandard and rstudent give the standardized and Studentized residuals, respectively. (These re-normalize the residuals to have unit variance, using an overall and leave-one-out measure of the error variance, respectively.)

Values for generalized smoothing models are approximations, as described in Williams (1987) (except that Cook's distances are scaled as \(F\) rather than chi-square values). THe approximations can be poor when some cases have large influence.

The optional infl, res, and sd arguments are there to encourage the use of these direct access functions in situations where the underlying basic influence measures, e.g., from smooth.influence, are already available.

For ss and sm objects, the code rstandard(*, type = "predictive") returns the leave-one-out (ordinary) cross-validation residuals, and the PRESS (PREdictive Sum of Squares) statistic is defined as

PRESS <- sum(rstandard(model, type = "predictive")^2)

Note that OCV = PRESS / n, where OCV = ordinary cross-validation criterion

References

See references listed in influence.measures

Williams, D. A. (1987). Generalized linear model diagnostics using the deviance and single case deletions. Applied Statistics, 36, 181-191. tools:::Rd_expr_doi("10.2307/2347550")

Examples

Run this code

# generate data
set.seed(1)
n <- 100
x <- seq(0, 1, length.out = n)
fx <- 2 + 3 * x + sin(2 * pi * x)
y <- fx + rnorm(n, sd = 0.5)

# fit models
mod.ss <- ss(x, y, nknots = 10)
mod.sm <- sm(y ~ x, knots = 10)
mod.gsm <- gsm(y ~ x, knots = 10)

# calculate influence
infl.ss <- smooth.influence.measures(mod.ss)
infl.sm <- smooth.influence.measures(mod.sm)
infl.gsm <- smooth.influence.measures(mod.gsm)

# standardized residuals
rstan.ss <- rstandard(mod.ss)
rstan.sm <- rstandard(mod.sm)
rstan.gsm <- rstandard(mod.gsm)

# studentized residuals
rstud.ss <- rstudent(mod.ss)
rstud.sm <- rstudent(mod.sm)
rstud.gsm <- rstudent(mod.gsm)

Run the code above in your browser using DataLab