nsxD: Generate a Basis Matrix for the first derivative of Natural Cubic Splines (with eXtensions)

Description

Generate the B-spline basis matrix for the first derivative of a natural cubic spline (with eXtensions).

Usage

nsxD(x, df = NULL, knots = NULL, intercept = FALSE,
    Boundary.knots = range(x), derivs = if (cure) c(2, 1) else c(2, 2),
    log = FALSE, centre = FALSE, 
    cure = FALSE, stata.stpm2.compatible = FALSE)

Value

A matrix of dimension length(x) * df where either df was supplied or if knots were supplied,

df = length(knots) + 1 + intercept. Attributes are returned that correspond to the arguments to ns, and explicitly give the knots, Boundary.knots etc for use by predict.nsxD().

nsxD() is based on the functions ns and spline.des. It generates a basis matrix for representing the family of piecewise-cubic splines with the specified sequence of interior knots, and the natural boundary conditions. These enforce the constraint that the function is linear beyond the boundary knots, which can either be supplied, else default to the extremes of the data. A primary use is in modeling formula to directly specify a natural spline term in a model.

The extensions from ns are: specification of the derivative constraints at the boundary knots; whether to centre the knots; incorporation of cure using derivatives; compatible knots with Stata's stpm2; and an indicator for a log-transformation of

x for calculating derivatives.

Arguments

x: the predictor variable. Missing values are allowed.
df: degrees of freedom. One can supply df rather than knots; ns() then chooses df - 1 - intercept + 4 - sum(derivs) knots at suitably chosen quantiles of x (which will ignore missing values).
knots: breakpoints that define the spline. The default is no knots; together with the natural boundary conditions this results in a basis for linear regression on x. Typical values are the mean or median for one knot, quantiles for more knots. See also Boundary.knots.
intercept: if TRUE, an intercept is included in the basis; default is FALSE.
Boundary.knots: boundary points at which to impose the natural boundary conditions and anchor the B-spline basis (default the range of the data). If both knots and Boundary.knots are supplied, the basis parameters do not depend on x. Data can extend beyond Boundary.knots
derivs: an integer vector of length 2 with values between 0 and 2 giving the derivative constraint order at the left and right boundary knots; an order of 2 constrains the second derivative to zero (f''(x)=0); an order of 1 constrains the first and second derivatives to zero (f'(x)=f''(x)=0); an order of 0 constrains the zero, first and second derivatives to zero (f(x)=f'(x)=f''(x)=0)
log: a Boolean indicating whether the underlying values have been log transformed; (deprecated: only used to calculate derivatives in rstpm2:::stpm2Old
centre: if specified, then centre the splines at this value (i.e. f(centre)=0) (default=FALSE)
cure: a Boolean indicated whether to estimate cure; changes the default derivs argument, such that the right boundary has the first and second derivatives constrained to zero; defaults to FALSE
stata.stpm2.compatible: a Boolean to determine whether to use Stata stpm's default knot placement; defaults to FALSE

References

Hastie, T. J. (1992) Generalized additive models. Chapter 7 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

Examples

Run this code

require(stats); require(graphics); require(splines)
nsx(women$height, df = 5)
summary(fm1 <- lm(weight ~ ns(height, df = 5), data = women))

## example of safe prediction
plot(women, xlab = "Height (in)", ylab = "Weight (lb)")
ht <- seq(57, 73, length.out = 200)
lines(ht, predict(fm1, data.frame(height=ht)))