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

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

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

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

- 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.

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.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

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`

.

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

`scan.test`

,
`as.im.scan.test`

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

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