# scanLRTS

##### Likelihood Ratio Test Statistic for Scan Test

Calculate the Likelihood Ratio Test Statistic for the Scan Test, at each spatial location.

##### Usage

```
scanLRTS(X, r, ...,
method = c("poisson", "binomial"),
baseline = NULL, case = 2,
alternative = c("greater", "less", "two.sided"),
saveopt = FALSE,
Xmask = NULL)
```

##### Arguments

- X
- A point pattern (object of class
`"ppp"`

). - r
- Radius of circle to use. A single number or a numeric vector.
- ...
- Optional. Arguments passed to
`as.mask`

to determine the spatial resolution of the computations. - method
- Either
`"poisson"`

or`"binomial"`

specifying the type of likelihood. - baseline
- Baseline for the Poisson intensity, if
`method="poisson"`

. A pixel image or a function. - case
- Which type of point should be interpreted as a case,
if
`method="binomial"`

. Integer or character string. - alternative
- Alternative hypothesis:
`"greater"`

if the alternative postulates that the mean number of points inside the circle will be greater than expected under the null. - saveopt
- Logical value indicating to save the optimal value of
`r`

at each location. - Xmask
- Internal use only.

##### Details

This command computes, for all spatial locations `u`

,
the Likelihood Ratio Test Statistic $\Lambda(u)$
for a test of homogeneity at the location $u$, as described
below. The result is a pixel image giving the values of
$\Lambda(u)$ at each pixel.

The **maximum** value of $\Lambda(u)$ over all locations
$u$ is the *scan statistic*, which is the basis of
the *scan test* performed by `scan.test`

.

- If
`method="poisson"`

then the test statistic is based on Poisson likelihood. The dataset`X`

is treated as an unmarked point pattern. By default (if`baseline`

is not specified) the null hypothesis is complete spatial randomness CSR (i.e. a uniform Poisson process). At the spatial location$u$, the alternative hypothesis is a Poisson process with one intensity$\beta_1$inside the circle of radius`r`

centred at$u$, and another intensity$\beta_0$outside the circle. If`baseline`

is given, then it should be a pixel image or a`function(x,y)`

. The null hypothesis is an inhomogeneous Poisson process with intensity proportional to`baseline`

. The alternative hypothesis is an inhomogeneous Poisson process with intensity`beta1 * baseline`

inside the circle, and`beta0 * baseline`

outside the circle. - If
`method="binomial"`

then the test statistic is based on binomial likelihood. The dataset`X`

must be a bivariate point pattern, i.e. a multitype point pattern with two types. The null hypothesis is that all permutations of the type labels are equally likely. The alternative hypothesis is that the circle of radius`r`

centred at$u$has a higher proportion of points of the second type, than expected under the null hypothesis.

If `r`

is a vector of more than one value for the radius,
then the calculations described above are performed for
every value of `r`

. Then the maximum over `r`

is taken
for each spatial location $u$.
The resulting pixel value of `scanLRTS`

at a location
$u$ is the profile maximum of the Likelihood Ratio Test Statistic,
that is, the maximum of the
Likelihood Ratio Test Statistic for circles of all radii,
centred at the same location $u$.

If you have already performed a scan test using
`scan.test`

, the Likelihood Ratio Test Statistic
can be extracted from the test result using the
function `as.im.scan.test`

.

##### Value

- A pixel image (object of class
`"im"`

) whose pixel values are the values of the (profile) Likelihood Ratio Test Statistic at each spatial location.

##### Warning: window size

Note that the result of `scanLRTS`

is a pixel image
on a larger window than the original window of `X`

.
The expanded window contains the centre of any circle
of radius `r`

that has nonempty intersection with the original window.

##### References

Kulldorff, M. (1997)
A spatial scan statistic.
*Communications in Statistics --- Theory and Methods*
**26**, 1481--1496.

##### See Also

##### Examples

```
plot(scanLRTS(redwood, 0.1, method="poisson"))
sc <- scanLRTS(chorley, 1, method="binomial", case="larynx")
plot(sc)
scanstatchorley <- max(sc)
```

*Documentation reproduced from package spatstat, version 1.36-0, License: GPL (>= 2)*