`knox.Rd`

Given temporal and spatial distances as well as corresponding critical
thresholds defining what “close” means, the function
`knox`

performs Knox (1963, 1964) test for space-time interaction.
The corresponding p-value can be calculated either by the Poisson
approximation or by a Monte Carlo permutation approach (Mantel, 1967)
with support for parallel computation via `plapply`

.
There is a simple `plot`

-method showing a `truehist`

of
the simulated null distribution together with the expected and observed
values.
This implementation of the Knox test is due to Meyer et al. (2016).

```
knox(dt, ds, eps.t, eps.s, simulate.p.value = TRUE, B = 999, ...)
# S3 method for knox
plot(x, ...)
```

- dt,ds
numeric vectors containing temporal and spatial distances, respectively. Logical vectors indicating temporal/spatial closeness may also be supplied, in which case

`eps.t`

/`eps.s`

is ignored. To test for space-time interaction in a single point pattern of \(n\) events, these vectors should be of length \(n*(n-1)/2\) and contain the pairwise event distances (e.g., the lower triangle of the distance matrix, such as in`"dist"`

objects). Note that there is no special handling of matrix input, i.e., if`dt`

or`ds`

are matrices, all elements are used (but a warning is given if a symmetric matrix is detected).- eps.t,eps.s
Critical distances defining closeness in time and space, respectively. Distances lower than or equal to the critical distance are considered “"close"”.

- simulate.p.value
logical indicating if a Monte Carlo permutation test should be performed (as per default). Do not forget to set the

`.Random.seed`

via an extra`.seed`

argument if reproducibility is required (see the ... arguments below). If`simulate.p.value = FALSE`

, the Poisson approximation is used (but see the note below).- B
number of permutations for the Monte Carlo approach.

- ...
arguments configuring

`plapply`

:`.parallel`

,`.seed`

, and`.verbose`

. By default, no parallelization is performed (`.parallel = 1`

), and a progress bar is shown (`.verbose = TRUE`

).

For the`plot`

-method, further arguments passed to`truehist`

.- x
an object of class

`"knox"`

as returned by the`knox`

test.

The Poisson approximation works well if the proportions of close pairs in both time and space are small (Kulldorff and Hjalmars, 1999), otherwise the Monte Carlo permutation approach is recommended.

an object of class `"knox"`

(inheriting from `"htest"`

),
which is a list with the following components:

a character string indicating the type of test performed, and whether the Poisson approximation or Monte Carlo simulation was used.

a character string giving the supplied `dt`

and
`ds`

arguments.

the number of close pairs.

if `simulate.p.value = TRUE`

, the number
`B`

of permutations, otherwise the `lambda`

parameter of
the Poisson distribution, i.e., the same as `null.value`

.

the p-value for the test. In case
`simulate.p.value = TRUE`

, the p-value from the Poisson
approximation is still attached as an attribute `"Poisson"`

.

the character string `"greater"`

(this is a
one-sided test).

the expected number of close pairs in the absence of space-time interaction.

the contingency table of `dt <= eps.t`

and
`ds <= eps.s`

.

Sebastian Meyer

The function `mantel.randtest`

in package ade4
implements Mantel's (1967) space-time interaction test, i.e., using
the Pearson correlation between the spatial and temporal distances of
all event pairs as the test statistic, and assessing statistical
significance using a Monte Carlo permutation approach as with
`simulate.p.value`

here in the `knox`

function.
To combine information from different scales `eps.t`

and
`eps.s`

while also handling edge effects, the space-time
K-function test available via `stKtest`

can be used.
Function `epitest`

tests epidemicity in a
`"twinstim"`

point process model.

Knox, G. (1963):
Detection of low intensity epidemicity: application to cleft lip and palate.
*British Journal of Preventive & Social Medicine*, **17**, 121-127.

Knox, E. G. (1964):
The detection of space-time interactions.
*Journal of the Royal Statistical Society. Series C (Applied
Statistics)*, **13**, 25-30.

Kulldorff, M. and Hjalmars, U. (1999):
The Knox method and other tests for space-time interaction.
*Biometrics*, **55**, 544-552.

Mantel, N. (1967):
The detection of disease clustering and a generalized regression approach.
*Cancer Research*, **27**, 209-220.

Meyer, S., Warnke, I., Rössler, W. and Held, L. (2016):
Model-based testing for space-time interaction using point processes:
An application to psychiatric hospital admissions in an urban area.
*Spatial and Spatio-temporal Epidemiology*, **17**, 15-25.
doi: 10.1016/j.sste.2016.03.002
.
Eprint: https://arxiv.org/abs/1512.09052.

```
data("imdepi")
imdepiB <- subset(imdepi, type == "B")
## Perfom the Knox test using the Poisson approximation
knoxtest <- knox(
dt = dist(imdepiB$events$time), eps.t = 30,
ds = dist(coordinates(imdepiB$events)), eps.s = 50,
simulate.p.value = FALSE
)
knoxtest
## The Poisson approximation works well for these data since
## the proportion of close pairs is rather small (204/56280).
## contingency table in LaTeX
toLatex(knoxtest)
if (surveillance.options("allExamples")) {
## Obtain the p-value via a Monte Carlo permutation test,
## where the permutations can be computed in parallel
## (using forking on Unix-alikes and a cluster on Windows, see ?plapply)
knoxtestMC <- knox(
dt = dist(imdepiB$events$time), eps.t = 30,
ds = dist(coordinates(imdepiB$events)), eps.s = 50,
simulate.p.value = TRUE, B = 999,
.parallel = 2, .seed = 1, .verbose = FALSE
)
knoxtestMC
plot(knoxtestMC)
}
```