Perform the Diggle (1986) / Cressie (1991) / Loosmore and Ford (2006) test or the Maximum Absolute Deviation test for a spatial point pattern.

```
dclf.test(X, …, alternative=c("two.sided", "less", "greater"),
rinterval = NULL, leaveout=1,
scale=NULL, clamp=FALSE, interpolate=FALSE)
```mad.test(X, …, alternative=c("two.sided", "less", "greater"),
rinterval = NULL, leaveout=1,
scale=NULL, clamp=FALSE, interpolate=FALSE)

X

Data for the test.
Either a point pattern (object of class `"ppp"`

, `"lpp"`

or other class), a fitted point process model (object of class `"ppm"`

,
`"kppm"`

or other class), a simulation envelope (object of class
`"envelope"`

) or a previous result of `dclf.test`

or
`mad.test`

.

…

Arguments passed to `envelope`

.
Useful arguments include `fun`

to determine the summary
function, `nsim`

to specify the number of Monte Carlo
simulations, `verbose=FALSE`

to turn off the messages,
`savefuns`

or `savepatterns`

to save the simulation
results, and `use.theory`

described under Details.

alternative

The alternative hypothesis. A character string. The default is a two-sided alternative. See Details.

rinterval

Interval of values of the summary function argument `r`

over which the maximum absolute deviation, or the integral,
will be computed for the test. A numeric vector of length 2.

leaveout

Optional integer 0, 1 or 2 indicating how to calculate the deviation between the observed summary function and the nominal reference value, when the reference value must be estimated by simulation. See Details.

scale

Optional. A function in the R language which determines the
relative scale of deviations, as a function of
distance \(r\). Summary function values for distance `r`

will be *divided* by `scale(r)`

before the
test statistic is computed.

clamp

Logical value indicating how to compute deviations
in a one-sided test. Deviations of the observed
summary function from the theoretical summary function are initially
evaluated as signed real numbers, with large positive values indicating
consistency with the alternative hypothesis.
If `clamp=FALSE`

(the default), these values are not changed.
If `clamp=TRUE`

, any negative values are replaced by zero.

interpolate

Logical value specifying whether to calculate the \(p\)-value
by interpolation.
If `interpolate=FALSE`

(the default), a standard Monte Carlo test
is performed, yielding a \(p\)-value of the form \((k+1)/(n+1)\)
where \(n\) is the number of simulations and \(k\) is the number
of simulated values which are more extreme than the observed value.
If `interpolate=TRUE`

, the \(p\)-value is calculated by
applying kernel density estimation to the simulated values, and
computing the tail probability for this estimated distribution.

An object of class `"htest"`

.
Printing this object gives a report on the result of the test.
The \(p\)-value is contained in the component `p.value`

.

If the observed value of the test statistic is equal to one or more of the
simulated values (called a *tied value*), then the tied values
will be assigned a random ordering, and a message will be printed.

These functions perform hypothesis tests for goodness-of-fit of a point pattern dataset to a point process model, based on Monte Carlo simulation from the model.

`dclf.test`

performs the test advocated by Loosmore and Ford (2006)
which is also described in Diggle (1986), Cressie (1991, page 667, equation
(8.5.42)) and Diggle (2003, page 14). See Baddeley et al (2014) for
detailed discussion.

`mad.test`

performs the ‘global’ or
‘Maximum Absolute Deviation’ test described by Ripley (1977, 1981).
See Baddeley et al (2014).

The type of test depends on the type of argument `X`

.

If

`X`

is some kind of point pattern, then a test of Complete Spatial Randomness (CSR) will be performed. That is, the null hypothesis is that the point pattern is completely random.If

`X`

is a fitted point process model, then a test of goodness-of-fit for the fitted model will be performed. The model object contains the data point pattern to which it was originally fitted. The null hypothesis is that the data point pattern is a realisation of the model.If

`X`

is an envelope object generated by`envelope`

, then it should have been generated with`savefuns=TRUE`

or`savepatterns=TRUE`

so that it contains simulation results. These simulations will be treated as realisations from the null hypothesis.Alternatively

`X`

could be a previously-performed test of the same kind (i.e. the result of calling`dclf.test`

or`mad.test`

). The simulations used to perform the original test will be re-used to perform the new test (provided these simulations were saved in the original test, by setting`savefuns=TRUE`

or`savepatterns=TRUE`

).

The argument `alternative`

specifies the alternative hypothesis,
that is, the direction of deviation that will be considered
statistically significant. If `alternative="two.sided"`

(the
default), both positive and negative deviations (between
the observed summary function and the theoretical function)
are significant. If `alternative="less"`

, then only negative
deviations (where the observed summary function is lower than the
theoretical function) are considered. If `alternative="greater"`

,
then only positive deviations (where the observed summary function is
higher than the theoretical function) are considered.

In all cases, the algorithm will first call `envelope`

to
generate or extract the simulated summary functions.
The number of simulations that will be generated or extracted,
is determined by the argument `nsim`

, and defaults to 99.
The summary function that will be computed is determined by the
argument `fun`

(or the first unnamed argument in the list
`…`

) and defaults to `Kest`

(except when
`X`

is an envelope object generated with `savefuns=TRUE`

,
when these functions will be taken).

The choice of summary function `fun`

affects the power of the
test. It is normally recommended to apply a variance-stabilising
transformation (Ripley, 1981). If you are using the \(K\) function,
the normal practice is to replace this by the \(L\) function
(Besag, 1977) computed by `Lest`

. If you are using
the \(F\) or \(G\) functions, the recommended practice is to apply
Fisher's variance-stabilising transformation
\(\sin^{-1}\sqrt x\) using the argument
`transform`

. See the Examples.

The argument `rinterval`

specifies the interval of
distance values \(r\) which will contribute to the
test statistic (either maximising over this range of values
for `mad.test`

, or integrating over this range of values
for `dclf.test`

). This affects the power of the test.
General advice and experiments in Baddeley et al (2014) suggest
that the maximum \(r\) value should be slightly larger than
the maximum possible range of interaction between points. The
`dclf.test`

is quite sensitive to this choice, while the
`mad.test`

is relatively insensitive.

It is also possible to specify a pointwise test (i.e. taking
a single, fixed value of distance \(r\)) by specifing
`rinterval = c(r,r)`

.

The argument `use.theory`

passed to `envelope`

determines whether to compare the summary function for the data
to its theoretical value for CSR (`use.theory=TRUE`

)
or to the sample mean of simulations from CSR
(`use.theory=FALSE`

).

The argument `leaveout`

specifies how to calculate the
discrepancy between the summary function for the data and the
nominal reference value, when the reference value must be estimated
by simulation. The values `leaveout=0`

and
`leaveout=1`

are both algebraically equivalent (Baddeley et al, 2014,
Appendix) to computing the difference `observed - reference`

where the `reference`

is the mean of simulated values.
The value `leaveout=2`

gives the leave-two-out discrepancy
proposed by Dao and Genton (2014).

Baddeley, A., Diggle, P.J., Hardegen, A., Lawrence, T., Milne,
R.K. and Nair, G. (2014) On tests of spatial pattern based on
simulation envelopes.
*Ecological Monographs* **84**(3) 477--489.

Baddeley, A., Hardegen, A., Lawrence, T., Milne, R.K. and Nair,
G. (2015) *Pushing the envelope*. In preparation.

Besag, J. (1977)
Discussion of Dr Ripley's paper.
*Journal of the Royal Statistical Society, Series B*,
**39**, 193--195.

Cressie, N.A.C. (1991)
*Statistics for spatial data*.
John Wiley and Sons, 1991.

Dao, N.A. and Genton, M. (2014)
A Monte Carlo adjusted goodness-of-fit test for
parametric models describing spatial point patterns.
*Journal of Graphical and Computational Statistics*
**23**, 497--517.

Diggle, P. J. (1986).
Displaced amacrine cells in the retina of a
rabbit : analysis of a bivariate spatial point pattern.
*J. Neuroscience Methods* **18**, 115--125.

Diggle, P.J. (2003)
*Statistical analysis of spatial point patterns*,
Second edition. Arnold.

Loosmore, N.B. and Ford, E.D. (2006)
Statistical inference using the *G* or *K* point
pattern spatial statistics. *Ecology* **87**, 1925--1931.

Ripley, B.D. (1977)
Modelling spatial patterns (with discussion).
*Journal of the Royal Statistical Society, Series B*,
**39**, 172 -- 212.

Ripley, B.D. (1981)
*Spatial statistics*.
John Wiley and Sons.

# NOT RUN { dclf.test(cells, Lest, nsim=39) m <- mad.test(cells, Lest, verbose=FALSE, rinterval=c(0, 0.1), nsim=19) m # extract the p-value m$p.value # variance stabilised G function dclf.test(cells, Gest, transform=expression(asin(sqrt(.))), verbose=FALSE, nsim=19) ## one-sided test ml <- mad.test(cells, Lest, verbose=FALSE, nsim=19, alternative="less") ## scaled mad.test(cells, Kest, verbose=FALSE, nsim=19, rinterval=c(0.05, 0.2), scale=function(r) { r }) # }