# quadrat.test

##### Chi-Squared Dispersion Test for Spatial Point Pattern Based on Quadrat Counts

Performs a chi-squared test of Complete Spatial Randomness for a given point pattern, based on quadrat counts. Alternatively performs a chi-squared goodness-of-fit test of a fitted inhomogeneous Poisson model.

##### Usage

```
quadrat.test(X, ...)
## S3 method for class 'ppp':
quadrat.test(X, nx=5, ny=nx, ..., xbreaks=NULL, ybreaks=NULL, tess=NULL)
## S3 method for class 'ppm':
quadrat.test(X, nx=5, ny=nx, ..., xbreaks=NULL, ybreaks=NULL, tess=NULL)
## S3 method for class 'quadratcount':
quadrat.test(X, ...)
```

##### Arguments

- X
- A point pattern (object of class
`"ppp"`

) to be subjected to the goodness-of-fit test. Alternatively a fitted point process model (object of class`"ppm"`

) to be tested. Alternatively`X`

can be the result - nx,ny
- Numbers of quadrats in the $x$ and $y$ directions.
Incompatible with
`xbreaks`

and`ybreaks`

. - ...
- Ignored.
- xbreaks
- Optional. Numeric vector giving the $x$ coordinates of the
boundaries of the quadrats. Incompatible with
`nx`

. - ybreaks
- Optional. Numeric vector giving the $y$ coordinates of the
boundaries of the quadrats. Incompatible with
`ny`

. - tess
- Tessellation (object of class
`"tess"`

) determining the quadrats. Incompatible with`nx, ny, xbreaks, ybreaks`

.

##### Details

These functions perform $\chi^2$ tests of goodness-of-fit for a point process model, based on quadrat counts.

The function `quadrat.test`

is generic, with methods for
point patterns (class `"ppp"`

), split point patterns
(class `"splitppp"`

), point process models
(class `"ppm"`

) and quadrat count tables (class `"quadratcount"`

).

- if
`X`

is a point pattern, we test the null hypothesis that the data pattern is a realisation of Complete Spatial Randomness (the uniform Poisson point process). Marks in the point pattern are ignored. - if
`X`

is a split point pattern, then for each of the component point patterns (taken separately) we test the null hypotheses of Complete Spatial Randomness. See`quadrat.test.splitppp`

for documentation. - If
`X`

is a fitted point process model, then it should be a Poisson point process model. The data to which this model was fitted are extracted from the model object, and are treated as the data point pattern for the test. We test the null hypothesis that the data pattern is a realisation of the (inhomogeneous) Poisson point process specified by`X`

.

In all cases, the window of observation is divided
into tiles, and the number of data points in each tile is
counted, as described in `quadratcount`

.
The quadrats are rectangular by default, or may be regions of arbitrary shape
specified by the argument `tess`

.
The expected number of points in each quadrat is also calculated,
as determined by CSR (in the first case) or by the fitted model
(in the second case). Then we perform the
$\chi^2$ test of goodness-of-fit to the quadrat counts.

The return value is an object of class `"htest"`

.
Printing the object gives comprehensible output
about the outcome of the test.

The return value also belongs to
the special class `"quadrat.test"`

. Plotting the object
will display the quadrats, annotated by their observed and expected
counts and the Pearson residuals. See the examples.

##### Value

- An object of class
`"htest"`

. See`chisq.test`

for explanation.The return value is also an object of the special class

`"quadrat.test"`

, and there is a plot method for this class. See the examples.

##### See Also

`quadrat.test.splitppp`

,
`quadratcount`

,
`quadrats`

,
`quadratresample`

,
`chisq.test`

,
`kstest`

.

To test a Poisson point process model against a specific alternative,
use `anova.ppm`

.

##### Examples

```
data(simdat)
quadrat.test(simdat)
quadrat.test(simdat, 4, 3)
# quadrat counts
qS <- quadratcount(simdat, 4, 3)
quadrat.test(qS)
# fitted model: inhomogeneous Poisson
fitx <- ppm(simdat, ~x, Poisson())
quadrat.test(fitx)
te <- quadrat.test(simdat, 4)
residuals(te) # Pearson residuals
plot(te)
plot(simdat, pch="+", cols="green", lwd=2)
plot(te, add=TRUE, col="red", cex=1.4, lty=2, lwd=3)
sublab <- eval(substitute(expression(p[chi^2]==z),
list(z=signif(te$p.value,3))))
title(sub=sublab, cex.sub=3)
# quadrats of irregular shape
B <- dirichlet(runifpoint(6, simdat$window))
qB <- quadrat.test(simdat, tess=B)
plot(simdat, main="quadrat.test(simdat, tess=B)", pch="+")
plot(qB, add=TRUE, col="red", lwd=2, cex=1.2)
```

*Documentation reproduced from package spatstat, version 1.23-3, License: GPL (>= 2)*