# Knox Test for Space-Time Interaction

`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).

## Usage

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

## Arguments

- 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.

## Note

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.

## Value

an object of class `"knox"`

(inheriting from `"htest"`

),
which is a list with the following components:

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

- data.name
a character string giving the supplied

`dt`

and`ds`

arguments.- statistic
the number of close pairs.

- parameter
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`

.- p.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"`

.- alternative
the character string

`"greater"`

(this is a one-sided test).- null.value
the expected number of close pairs in the absence of space-time interaction.

- table
the contingency table of

`dt <= eps.t`

and`ds <= eps.s`

.

The `plot`

-method invisibly returns `NULL`

.

A `toLatex`

-method exists, which generates LaTeX code for the
contingency table associated with the Knox test.

## See also

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.

## References

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.

## Examples

```
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$events)), eps.s = 50,
simulate.p.value = FALSE
)
knoxtest
## The Poisson approximation works well for these data since
## the proportion of close pairs is rather small (204/56280).
.opt <- options(xtable.comment = FALSE)
## contingency table in LaTeX
toLatex(knoxtest)
options(.opt)
## Obtain the p-value via a Monte Carlo permutation test,
## where the permutations can be computed in parallel
## (using forking on Unix-alikes and a cluster on Windows, see ?plapply)
knoxtestMC <- knox(
dt = dist(imdepiB$events$time), eps.t = 30,
ds = dist(coordinates(imdepiB$events)), eps.s = 50,
simulate.p.value = TRUE, B = 99, # limited here for speed
.parallel = 2, .seed = 1, .verbose = FALSE
)
knoxtestMC
plot(knoxtestMC)
```