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)

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.

• Ifmethod="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.
• Ifmethod="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.

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

scan.test, as.im.scan.test

Aliases

• scanLRTS

Examples

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