# quadrat.test

##### Dispersion Test for Spatial Point Pattern Based on Quadrat Counts

Performs a test of Complete Spatial Randomness for a given point pattern, based on quadrat counts. Alternatively performs a goodness-of-fit test of a fitted inhomogeneous Poisson model. By default performs chi-squared tests; can also perform Monte Carlo based tests.

##### Usage

`quadrat.test(X, ...)`# S3 method for ppp
quadrat.test(X, nx=5, ny=nx,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE, CR=1,
lambda=NULL, df.est=NULL,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)

# S3 method for ppm
quadrat.test(X, nx=5, ny=nx,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE, CR=1, df.est=NULL,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)

# S3 method for quadratcount
quadrat.test(X,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE, CR=1,
lambda=NULL, df.est=NULL,
...,
nsim=1999)

##### 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 of applying`quadratcount`

to a point pattern.- nx,ny
Numbers of quadrats in the \(x\) and \(y\) directions. Incompatible with

`xbreaks`

and`ybreaks`

.- alternative
Character string (partially matched) specifying the alternative hypothesis.

- method
Character string (partially matched) specifying the test to use: either

`method="Chisq"`

for the chi-squared test (the default), or`method="MonteCarlo"`

for a Monte Carlo test.- conditional
Logical. Should the Monte Carlo test be conducted conditionally upon the observed number of points of the pattern? Ignored if

`method="Chisq"`

.- CR
Optional. Numerical value. The exponent for the Cressie-Read test statistic. See Details.

- lambda
Optional. Pixel image (object of class

`"im"`

) or function (class`"funxy"`

) giving the predicted intensity of the point process.- df.est
Optional. Advanced use only. The number of fitted parameters, or the number of degrees of freedom lost by estimation of parameters.

- …
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"`

or something acceptable to`as.tess`

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

.- nsim
The number of simulated samples to generate when

`method="MonteCarlo"`

.

##### Details

These functions perform \(\chi^2\) tests or Monte Carlo 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`lambda`

is given then the null hypothesis is the Poisson process with intensity`lambda`

.)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 the Pearson \(X^2\) statistic
$$
X^2 = sum((observed - expected)^2/expected)
$$
is computed.

If `method="Chisq"`

then a \(\chi^2\) test of
goodness-of-fit is performed by comparing the test statistic
to the \(\chi^2\) distribution
with \(m-k\) degrees of freedom, where `m`

is the number of
quadrats and \(k\) is the number of fitted parameters
(equal to 1 for `quadrat.test.ppp`

). The default is to
compute the *two-sided* \(p\)-value, so that the test will
be declared significant if \(X^2\) is either very large or very
small. One-sided \(p\)-values can be obtained by specifying the
`alternative`

. An important requirement of the
\(\chi^2\) test is that the expected counts in each quadrat
be greater than 5.

If `method="MonteCarlo"`

then a Monte Carlo test is performed,
obviating the need for all expected counts to be at least 5. In the
Monte Carlo test, `nsim`

random point patterns are generated
from the null hypothesis (either CSR or the fitted point process
model). The Pearson \(X^2\) statistic is computed as above.
The \(p\)-value is determined by comparing the \(X^2\)
statistic for the observed point pattern, with the values obtained
from the simulations. Again the default is to
compute the *two-sided* \(p\)-value.

If `conditional`

is `TRUE`

then the simulated samples are
generated from the multinomial distribution with the number of “trials”
equal to the number of observed points and the vector of probabilities
equal to the expected counts divided by the sum of the expected counts.
Otherwise the simulated samples are independent Poisson counts, with
means equal to the expected counts.

If the argument `CR`

is given, then instead of the
Pearson \(X^2\) statistic, the Cressie-Read (1984) power divergence
test statistic
$$
2nI = \frac{2}{CR(CR+1)}
\sum_i \left[ \left( \frac{X_i}{E_i} \right)^CR - 1 \right]
$$
is computed, where \(X_i\) is the \(i\)th observed count
and \(E_i\) is the corresponding expected count.
The value `CR=1`

gives the Pearson \(X^2\) statistic;
`CR=0`

gives the likelihood ratio test statistic \(G^2\);
`CR=-1/2`

gives the Freeman-Tukey statistic \(T^2\);
`CR=-1`

gives the modified likelihood ratio test statistic \(GM^2\);
and `CR=-2`

gives Neyman's modified statistic \(NM^2\).
In all cases the asymptotic distribution of this test statistic is
the same \(\chi^2\) distribution as above.

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
`"quadrattest"`

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

##### References

Cressie, N. and Read, T.R.C. (1984)
Multinomial goodness-of-fit tests.
*Journal of the Royal Statistical Society, Series B*
**46**, 440--464.

##### See Also

`quadrat.test.splitppp`

,
`quadratcount`

,
`quadrats`

,
`quadratresample`

,
`chisq.test`

,
`cdf.test`

.

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

.

##### Examples

```
# NOT RUN {
quadrat.test(simdat)
quadrat.test(simdat, 4, 3)
quadrat.test(simdat, alternative="regular")
quadrat.test(simdat, alternative="clustered")
## Likelihood ratio test
quadrat.test(simdat, CR=0)
## Power divergence tests
quadrat.test(simdat, CR=-1)$p.value
quadrat.test(simdat, CR=-2)$p.value
# Using Monte Carlo p-values
quadrat.test(swedishpines) # Get warning, small expected values.
# }
# NOT RUN {
quadrat.test(swedishpines, method="M", nsim=4999)
quadrat.test(swedishpines, method="M", nsim=4999, conditional=FALSE)
# }
# NOT RUN {
# }
# NOT RUN {
# quadrat counts
qS <- quadratcount(simdat, 4, 3)
quadrat.test(qS)
# fitted model: inhomogeneous Poisson
fitx <- ppm(simdat ~ x)
quadrat.test(fitx)
# an equivalent test (results differ due to discretisation effects):
quadrat.test(simdat, lambda=predict(fitx), df.est=length(coef(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, Window(simdat)))
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 1.63-3, License: GPL (>= 2)*