scbM: Calculate simultaneous confidence bands for penalized splines

In AdaptFitOS: Adaptive Semiparametric Additive Regression with Simultaneous Confidence Bands and Specification Tests

Calculate simultaneous confidence bands for penalized splines

Description

Calculates simultaneous (uniform) confidence bands for the mixed model representation of penalized splines based on volume-of-tube formula. Simultaneous confidence bands cover the entire curve with a prescribed level of confidence and allow us to assess the estimation uncertainty for the whole curve. In contrast to pointwise confidence bands, they permit statements about the statistical significance of certain features in the underlying curve.

Usage

scbM(object,select=NULL,drv=0,level=0.95,div=1000,
			calc.stdev=TRUE,bayes=FALSE)

Arguments

object	an asp object.
select	vector specifying which curves in an additive model should be considered. If NULL, all curves are considered.
drv	the derivative order. Defaults to 0, i.e. the estimated function itself is plotted. First and second derivatives are supported.
level	level of confidence.
div	precision for the integral used for calculation of the length of the curve, default is 1000.

calc.stdev	TRUE to compute standard deviation and confidence bands for each value of the covariates. Computationally intensive for large data sets. Use plot.scbm() or plot.asp() to compute standard deviation and bounds only for a grid. If FALSE only critical values are computed.
bayes	FALSE for confidence bands with (approximate) frequentist coverage probability, TRUE for (approximate) Bayesian coverage probability. See Krivobokova et al. (2010) for details.

Details

Returns a scbm object and prints critical values. The resulting confidence bands have (approximate) frequentist coverage probabilities with automatic bias correction (see references). Makes use of the libtube library by Catherine Loader (see package locfit).

Value

A list object of class scbm containing

aspobject	an asp object.
drv	the derivative order.
crit	a list of critical values.
sigma2	the variance of the residuals.
cov.coef	a list of covariance matrices of spline coefficients in the mixed model framework.
Stdev	the standard deviations of estimates. Only given if calc.stdev=TRUE.
fitted	a list of fitted values. Only given if calc.stdev=TRUE.
lcb	a list of lower bounds of confidence bands. Only given if calc.stdev=TRUE.
ucb	a list of upper bounds of confidence bands. Only given if calc.stdev=TRUE.
...	further

References

Krivobokova, T., Kneib, T., and Claeskens, G. (2010)
Simultaneous confidence bands for penalized spline estimators. Journal of the American Statistical Association, 105(490):852-863.

Wiesenfarth, M., Krivobokova, T., Klasen, S., Sperlich, S. (2012).
Direct Simultaneous Inference in Additive Models and its Application to Model Undernutrition. Journal of the American Statistical Association, 107(500): 1286-1296.

Examples

## Not run: 
beta=function(l,m,x) 
		return(gamma(l+m)*(gamma(l)*gamma(m))^(-1)*x^(l-1)*(1-x)^(m-1))
f1 = function(x) return((0.6*beta(30,17,x)+0.4*beta(3,11,x))*1/0.958)
f2 = function(z) return((sin(2*pi*(z-0.5))^2)*1/.3535)
f3 = function(z) 
		return((exp(-400*(z-0.6)^2)+
				5/3*exp(-500*(z-0.75)^2)+2*exp(-500*(z-0.9)^2))*1/0.549)
center=function(x) return(x-mean(x))

set.seed(1)
N <- 500
x1 = runif(N,0,1)
x2 = runif(N,0,1)
x3 = runif(N,0,1)


kn1 <- default.knots(x1,40)
kn2 <- default.knots(x2,40)
kn3 <- default.knots(x3,40)
kn.var3 <- default.knots(kn3,5)

y <- f1(x1)+f2(x2)+f3(x3)+0.3*rnorm(N)

# fit model with last smoothing parameter adaptive
  fit2b=  asp2(y~f(x1, basis="os", degree=3, knots=kn1, adap=FALSE)
                +f(x2, basis="os", degree=3, knots=kn2, adap=FALSE)
                +f(x3, basis="os", degree=3, knots=kn3, adap=TRUE,
                       var.knots=kn.var3, var.basis="os", var.degree=3),
                niter = 20, niter.var = 200)

  # compute 95
  # You could skip this and use "fit2b" indstead of "scb2b" later on, 
  # however, if N is large, computing the SCBs various times can take 
  # some time if you don't need fitted values and bounds for all covariate points
  # (can be computationally intensive due to large matrix dimensions),
  # set calc.stdev=F such that these are not computed.
    scb2b<- scbM(fit2b,calc.stdev=FALSE)
    plot(scb2b,pages=1)

  # plot first derivative of f(x1)
    scb2bdrv<- scbM(fit2b,drv=1,calc.stdev=FALSE)
    plot(scb2bdrv,select=1)
    #the following would give the same result
    #plot(fit2b,select=1,drv=1)
    #different style
    plot(scb2bdrv,select=1,scb.lty="blank", shade=TRUE,
    									shade.col="steelblue")

## End(Not run)

AdaptFitOS documentation built on July 21, 2022, 5:10 p.m.