Tests the goodness-of-fit of a Poisson point process model using methods of Berman (1986).

`berman.test(...)`# S3 method for ppp
berman.test(X, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)

An object of class `"htest"`

(hypothesis test)
and also of class `"bermantest"`

,
containing the results of the test. The return value can be
plotted (by `plot.bermantest`

) or printed
to give an informative summary of the test.

- X
A point pattern (object of class

`"ppp"`

or`"lpp"`

).- covariate
The spatial covariate on which the test will be based. An image (object of class

`"im"`

) or a function.- which
Character string specifying the choice of test.

- alternative
Character string specifying the alternative hypothesis.

- ...
Additional arguments controlling the pixel resolution (arguments

`dimyx`

and`eps`

passed to`as.mask`

) or other undocumented features.

The meaning of a one-sided test must be carefully scrutinised: see the printed output.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

These functions perform a goodness-of-fit test of a Poisson point process model fitted to point pattern data. The observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same values under the model, are compared using either of two test statistics \(Z_1\) and \(Z_2\) proposed by Berman (1986). The \(Z_1\) test is also known as the Lawson-Waller test.

The function `berman.test`

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

or `"lpp"`

)
and point process models (`"ppm"`

or `"lppm"`

).

If

`X`

is a point pattern dataset (object of class`"ppp"`

or`"lpp"`

), then`berman.test(X, ...)`

performs a goodness-of-fit test of the uniform Poisson point process (Complete Spatial Randomness, CSR) for this dataset.If

`model`

is a fitted point process model (object of class`"ppm"`

or`"lppm"`

) then`berman.test(model, ...)`

performs a test of goodness-of-fit for this fitted model. In this case,`model`

should be a Poisson point process.

The test is performed by comparing the observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same covariate under the model. Thus, you must nominate a spatial covariate for this test.

The argument `covariate`

should be either a `function(x,y)`

or a pixel image (object of class `"im"`

containing the values
of a spatial function.
If `covariate`

is an image, it should have numeric values,
and its domain should cover the observation window of the
`model`

. If `covariate`

is a function, it should expect
two arguments `x`

and `y`

which are vectors of coordinates,
and it should return a numeric vector of the same length
as `x`

and `y`

.

First the original data point pattern is extracted from `model`

.
The values of the `covariate`

at these data points are
collected.

Next the values of the `covariate`

at all locations in the
observation window are evaluated. The point process intensity
of the fitted model is also evaluated at all locations in the window.

If

`which="Z1"`

, the test statistic \(Z_1\) is computed as follows. The sum \(S\) of the covariate values at all data points is evaluated. The predicted mean \(\mu\) and variance \(\sigma^2\) of \(S\) are computed from the values of the covariate at all locations in the window. Then we compute \(Z_1 = (S-\mu)/\sigma\). Closely-related tests were proposed independently by Waller et al (1993) and Lawson (1993) so this test is often termed the Lawson-Waller test in epidemiological literature.If

`which="Z2"`

, the test statistic \(Z_2\) is computed as follows. The values of the`covariate`

at all locations in the observation window, weighted by the point process intensity, are compiled into a cumulative distribution function \(F\). The probability integral transformation is then applied: the values of the`covariate`

at the original data points are transformed by the predicted cumulative distribution function \(F\) into numbers between 0 and 1. If the model is correct, these numbers are i.i.d. uniform random numbers. The standardised sample mean of these numbers is the statistic \(Z_2\).

In both cases the null distribution of the test statistic is the standard normal distribution, approximately.

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

containing the
results of the hypothesis test. The print method for this class
gives an informative summary of the test outcome.

Berman, M. (1986)
Testing for spatial association between a point process
and another stochastic process.
*Applied Statistics* **35**, 54--62.

Lawson, A.B. (1993)
On the analysis of mortality events around a
prespecified fixed point.
*Journal of the Royal Statistical Society, Series A*
**156** (3) 363--377.

Waller, L., Turnbull, B., Clark, L.C. and Nasca, P. (1992)
Chronic Disease Surveillance and testing of
clustering of disease and exposure: Application to
leukaemia incidence and TCE-contaminated dumpsites
in upstate New York.
*Environmetrics* **3**, 281--300.

`cdf.test`

,
`quadrat.test`

,
`ppm`

```
# Berman's data
X <- copper$SouthPoints
L <- copper$SouthLines
D <- distmap(L, eps=1)
# test of CSR
berman.test(X, D)
berman.test(X, D, "Z2")
```

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