Diggle-Cressie-Loosmore-Ford and Maximum Absolute Deviation Tests

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

htest, spatial
dclf.test(X, ..., alternative=c("two.sided", "less", "greater"),
                      rinterval = NULL, use.theo=FALSE)

mad.test(X, ..., alternative=c("two.sided", "less", "greater"), rinterval = NULL, use.theo=FALSE)

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 (ob
Arguments passed to envelope. Useful arguments include fun to determine the summary function, nsim to specify the number of Monte Carlo simulations, verbose=FALS
The alternative hypothesis. A character string. The default is a two-sided alternative. See Details.
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.
Logical value determining whether to compare the summary function for the data to its theoretical value for CSR (use.theo=TRUE) or to the sample mean of simulations from CSR (use.theo=FALSE).

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).

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.

  • IfXis 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.
  • IfXis 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.
  • IfXis an envelope object generated byenvelope, then it should have been generated withsavefuns=TRUEorsavepatterns=TRUEso that it contains simulation results. These simulations will be treated as realisations from the null hypothesis.
  • AlternativelyXcould be a previously-performed test of the same kind (i.e. the result of callingdclf.testormad.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 settingsavefuns=TRUEorsavepatterns=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).


  • 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.

Handling Ties

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.


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, to appear. 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.

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.

See Also

envelope, dclf.progress

  • dclf.test
  • mad.test
dclf.test(cells, Lest, nsim=39)
  m <- mad.test(cells, Lest, verbose=FALSE, rinterval=c(0, 0.1), nsim=19)
  # extract the 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")
Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)

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