# relrisk.ppp

##### 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.

##### Usage

```
## S3 method for class '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 p - ...
- 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 prob - 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 i

##### 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$j$.

`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`

. - (if

##### References

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

##### See Also

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`

##### Examples

```
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 1.42-2, License: GPL (>= 2)*