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.maskto 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
rat 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 datasetXis treated as an unmarked point pattern. By default (ifbaselineis 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 radiusrcentred at$u$, and another intensity$\beta_0$outside the circle. Ifbaselineis given, then it should be a pixel image or afunction(x,y). The null hypothesis is an inhomogeneous Poisson process with intensity proportional tobaseline. The alternative hypothesis is an inhomogeneous Poisson process with intensitybeta1 * baselineinside the circle, andbeta0 * baselineoutside the circle. - If
method="binomial"then the test statistic is based on binomial likelihood. The datasetXmust 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 radiusrcentred 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)