# berman.test

##### Berman's Tests for Point Process Model

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

##### Usage

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

## S3 method for class 'ppm':
berman.test(model, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)

## S3 method for class 'lpp':
berman.test(X, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)

## S3 method for class 'lppm':
berman.test(model, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)

##### Arguments

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

or`"lpp"`

). - model
- A fitted point process model (object of class
`"ppm"`

or`"lppm"`

). - 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.

##### Details

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.

`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$Z_2$.

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.

##### Value

- 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.

##### Warning

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

##### References

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.

##### See Also

##### Examples

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

*Documentation reproduced from package spatstat, version 1.42-2, License: GPL (>= 2)*