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.
A point pattern (object of class "ppp").
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.
Either "poisson" or "binomial"
specifying the type of likelihood.
Baseline for the Poisson intensity, if method="poisson".
A pixel image or a function.
Which type of point should be interpreted as a case,
if method="binomial".
Integer or character string.
Alternative hypothesis: "greater" if the alternative
postulates that the mean number of points inside the circle
will be greater than expected under the null.
Logical value indicating to save the optimal value of r
at each location.
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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