relrisk.lpp: Nonparametric Estimate of Spatially-Varying Relative Risk on...

Description Usage Arguments Details Value Author(s) References See Also Examples

Description

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.

Usage

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## S3 method for class 'lpp'
relrisk(X, sigma, ..., 
           at = c("pixels", "points"),
           relative=FALSE,
           adjust=1, 
           casecontrol=TRUE, control=1, case,
           finespacing=FALSE)

Arguments

X

A multitype point pattern (object of class "lpp" which has factor valued marks).

sigma

The numeric value of the smoothing bandwidth (the standard deviation of Gaussian smoothing kernel) passed to density.lpp. Alternatively sigma may be a function which can be used to select the bandwidth. See Details.

...

Arguments passed to density.lpp to control the pixel resolution.

at

Character string specifying whether to compute the probability values at a grid of pixel locations (at="pixels") or only at the points of X (at="points").

relative

Logical. If FALSE (the default) the algorithm computes the probabilities of each type of point. If TRUE, it computes the relative risk, the ratio of probabilities of each type relative to the probability of a control.

adjust

Optional. Adjustment factor for the bandwidth sigma.

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 control in a bivariate point pattern. Ignored if there are more than 2 types of points.

finespacing

Logical value specifying whether to use a finer spatial resolution (with longer computation time but higher accuracy).

Details

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.

Value

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:

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.

Author(s)

Greg McSwiggan and \adrian.

References

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.

See Also

relrisk

Examples

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   ## 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"))

spatstat.linnet documentation built on July 17, 2021, 9:07 a.m.