# relrisk.ppp

0th

Percentile

##### Nonparametric Estimate of Spatially-Varying Relative Risk

Given a multitype point pattern, this function estimates the spatially-varying probability of each type of point, or the ratios of such probabilities, using kernel smoothing. The default smoothing bandwidth is selected by cross-validation.

Keywords
methods, smooth, spatial
##### Usage
# S3 method for ppp
relrisk(X, sigma = NULL, ..., varcov = NULL, at = "pixels",
relative=FALSE,
se=FALSE,
casecontrol=TRUE, control=1, case)
##### Arguments
X

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

sigma

Optional. The numeric value of the smoothing bandwidth (the standard deviation of isotropic Gaussian smoothing kernel). Alternatively sigma may be a function which can be used to select a different bandwidth for each type of point. See Details.

Arguments passed to bw.relrisk to select the bandwidth, or passed to density.ppp to control the pixel resolution.

varcov

Optional. Variance-covariance matrix of anisotopic Gaussian smoothing kernel. Incompatible with sigma.

at

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.

se

Logical value indicating whether to compute standard errors as well.

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.

##### Details

The command relrisk is generic and can be used to estimate relative risk in different ways.

This function relrisk.ppp is the method for point pattern datasets. 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 spatial location $u$ 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 spatial location $u$ 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 spatial location $u$ on a fine pixel grid, and the result is a pixel image representing the function $p(u)$ or a list of pixel images representing the functions $p_j(u)$ or $r_j(u)$ for $j = 1,\ldots,m$. 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 simple Nadaraja-Watson type kernel smoother (Diggle, 2003). The smoothing bandwidth can be specified in any of the following ways:

• sigma is a single numeric value, giving the standard deviation of the isotropic Gaussian kernel.

• sigma is a numeric vector of length 2, giving the standard deviations in the $x$ and $y$ directions of a Gaussian kernel.

• varcov is a 2 by 2 matrix giving the variance-covariance matrix of the Gaussian kernel.

• sigma is a function which selects the bandwidth. Bandwidth selection will be applied separately to each type of point. An example of such a function is bw.diggle.

• sigma and varcov are both missing or null. Then a common smoothing bandwidth sigma will be selected by cross-validation using bw.relrisk.

If se=TRUE then standard errors will also be computed, based on asymptotic theory, assuming a Poisson process.

##### Value

If se=FALSE (the default), the format is described below. If se=TRUE, the result is a list of two entries, estimate and SE, each having the format described below.

If X consists of only two types of points, and if casecontrol=TRUE, the result is a pixel image (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, 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.

##### References

Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.

There is another method relrisk.ppm for point process models which computes parametric estimates of relative risk, using the fitted model.

See also bw.relrisk, density.ppp, Smooth.ppp, eval.im

• relrisk.ppp
##### Examples
# NOT RUN {
p.oak <- relrisk(urkiola, 20)
if(interactive()) {
plot(p.oak, main="proportion of oak")
plot(eval.im(p.oak > 0.3), main="More than 30 percent oak")
plot(split(lansing), main="Lansing Woods")
p.lan <- relrisk(lansing, 0.05, se=TRUE)
plot(p.lan$estimate, main="Lansing Woods species probability") plot(p.lan$SE, main="Lansing Woods standard error")
wh <- im.apply(p.lan\$estimate, which.max)
types <- levels(marks(lansing))
wh <- eval.im(types[wh])
plot(wh, main="Most common species")
}
# }

Documentation reproduced from package spatstat, version 1.49-0, License: GPL (>= 2)

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