Nonparametric Estimate of Spatially-Varying Relative Risk on a Network
Given a multitype point pattern on a linear network, this function estimates the spatially-varying probability of each type of point, or the ratios of such probabilities, using kernel smoothing.
## S3 method for class 'lpp'
relrisk(X, sigma, ...,
at = c("pixels", "points"),
relative=FALSE,
adjust=1,
casecontrol=TRUE, control=1, case,
finespacing=FALSE)X |
A multitype point pattern (object of class |
sigma |
The numeric value of the smoothing bandwidth
(the standard deviation of Gaussian smoothing kernel)
passed to |
... |
Arguments passed to |
at |
Character string specifying whether to compute the probability values
at a grid of pixel locations ( |
relative |
Logical.
If |
adjust |
Optional. Adjustment factor for the bandwidth |
casecontrol |
Logical. Whether to treat a bivariate point pattern as consisting of cases and controls, and return only the probability or relative risk of a case. Ignored if there are more than 2 types of points. See Details. |
control |
Integer, or character string, identifying which mark value corresponds to a control. |
case |
Integer, or character string, identifying which mark value
corresponds to a case (rather than a control)
in a bivariate point pattern.
This is an alternative to the argument |
finespacing |
Logical value specifying whether to use a finer spatial resolution (with longer computation time but higher accuracy). |
The command relrisk is generic and can be used to
estimate relative risk in different ways.
This function relrisk.lpp is the method for point patterns
on a linear network (objects of class "lpp").
It computes nonparametric estimates of relative risk
by kernel smoothing.
If X is a bivariate point pattern
(a multitype point pattern consisting of two types of points)
then by default,
the points of the first type (the first level of marks(X))
are treated as controls or non-events, and points of the second type
are treated as cases or events. Then by default this command computes
the spatially-varying probability of a case,
i.e. the probability p(u)
that a point at location u on the network
will be a case. If relative=TRUE, it computes the
spatially-varying relative risk of a case relative to a
control, r(u) = p(u)/(1- p(u)).
If X is a multitype point pattern with m > 2 types,
or if X is a bivariate point pattern
and casecontrol=FALSE,
then by default this command computes, for each type j,
a nonparametric estimate of
the spatially-varying probability of an event of type j.
This is the probability p[j](u)
that a point at location u on the network
will belong to type j.
If relative=TRUE, the command computes the
relative risk of an event of type j
relative to a control,
r[j](u) = p[j](u)/p[k](u),
where events of type k are treated as controls.
The argument control determines which type k
is treated as a control.
If at = "pixels" the calculation is performed for
every location u on a fine pixel grid over the network, and the result
is a pixel image on the network representing the function p(u),
or a list of pixel images representing the functions
p[j](u) or r[j](u)
for j = 1,...,m.
An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as NA.
If at = "points" the calculation is performed
only at the data points x[i]. By default
the result is a vector of values
p(x[i]) giving the estimated probability of a case
at each data point, or a matrix of values
p[j](x[i]) giving the estimated probability of
each possible type j at each data point.
If relative=TRUE then the relative risks
r(x[i]) or r[j](x[i]) are
returned.
An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as Inf.
Estimation is performed by a Nadaraja-Watson type kernel smoother (McSwiggan et al., 2019).
The smoothing bandwidth sigma
should be a single numeric value, giving the standard
deviation of the isotropic Gaussian kernel.
If adjust is given, the smoothing bandwidth will be
adjust * sigma before the computation of relative risk.
Alternatively, sigma may be a function that can be applied
to the point pattern X to select a bandwidth; the function
must return a single numerical value; examples include the
functions bw.relrisklpp and bw.scott.iso.
Accuracy depends on the spatial resolution of the density
computations. If the arguments dx and dt are present,
they are passed to density.lpp to determine the
spatial resolution. Otherwise, the spatial resolution is determined
by a default rule that depends on finespacing and sigma.
If finespacing=FALSE (the default), the spatial resolution is
equal to the default resolution for pixel images.
If finespacing=TRUE, the spatial resolution is much finer
and is determined by a rule which guarantees higher accuracy,
but takes a longer time.
If X consists of only two types of points,
and if casecontrol=TRUE,
the result is a pixel image on the network (if at="pixels")
or a vector (if at="points").
The pixel values or vector values
are the probabilities of a case if relative=FALSE,
or the relative risk of a case (probability of a case divided by the
probability of a control) if relative=TRUE.
If X consists of more than two types of points,
or if casecontrol=FALSE, the result is:
(if at="pixels")
a list of pixel images on the network,
with one image for each possible type of point.
The result also belongs to the class "solist" so that it can
be printed and plotted.
(if at="points")
a matrix of probabilities, with rows corresponding to
data points x[i], and columns corresponding
to types j.
The pixel values or matrix entries
are the probabilities of each type of point if relative=FALSE,
or the relative risk of each type (probability of each type divided by the
probability of a control) if relative=TRUE.
If relative=FALSE, the resulting values always lie between 0
and 1. If relative=TRUE, the results are either non-negative
numbers, or the values Inf or NA.
Greg McSwiggan and Adrian Baddeley Adrian.Baddeley@curtin.edu.au.
McSwiggan, G., Baddeley, A. and Nair, G. (2019) Estimation of relative risk for events on a linear network. Statistics and Computing 30 (2) 469–484.
## case-control data: 2 types of points
set.seed(2020)
X <- superimpose(A=runiflpp(20, simplenet),
B=runifpointOnLines(20, as.psp(simplenet)[1]))
plot(X)
plot(relrisk(X, 0.2))
plot(relrisk(X, 0.2, case="B"))
head(relrisk(X, 0.2, at="points"))
## cross-validated bandwidth selection
plot(relrisk(X, bw.relrisklpp, hmax=0.3))
## more than 2 types
if(interactive()) {
U <- chicago
sig <- 170
} else {
U <- do.call(superimpose,
split(chicago)[c("theft", "cartheft", "burglary")])
sig <- 50
}
plot(relrisk(U, sig))
head(relrisk(U, sig, at="points"))
plot(relrisk(U, sig, relative=TRUE, control="theft"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.