Inferential Permutation Tests for Maximum Entropy Models

maxent.test performs the permutation tests proposed by Shipley (2010) for maximum entropy models. Two different null hypotheses can be tested: 1) the information encoded in the entire set of constraints $C$ is irrelevant for predicting the probabilities, and 2) the information encoded in subset $B$ of the entire set of constraints $C = {A U B}$ is irrelevant for predicting the probabilities. A plot can be returned to facilitate interpretation.

models, distribution, math
maxent.test(model, obs, sub.c, nperm = 99, quick = TRUE, alpha = 0.05, plot = TRUE)
list returned by maxent.
vector, matrix or data frame of observed probabilities of the states (columns).
character or numeric vector specifying the subset of constraints $B$ associated with null hypothesis 2. If missing, null hypothesis 1 is tested.
number of permutations for the test.
if TRUE, the algorithm stops when alpha is outside the confidence interval of the P-value. Can be useful to speed up the routine.
desired alpha-level for the test. Only relevant if quick is TRUE.
if TRUE, a plot is returned to facilitate interpretation.

maxent.test is a direct translation of the permutation tests described by Shipley (2010). Please refer to this article for details.

Using quick = FALSE will return the true null probability for a given nperm. However, if nperm is large (a rule-of-thumb is $>=999$ permutations for allowing inference at $alpha$ = 0.05), this can take a very long time. Using quick = TRUE is a much faster and highly recommended alternative if one is only interested in accepting/rejecting the null hypothesis at the specified $alpha$-level given by argument alpha.

If maxent was run with multiple data sets (i.e. if constr had more than one row), then maxent.test performs the test for all sets simultaneously, following the ‘omnibus’ procedure described by Shipley (2010).

The following measure of fit between observed and predicted probabilities is returned:

$$\mathrm{fit}=1-\frac{{\displaystyle \sum_{j=1}^{D} \sum_{i=1}^{S}\left(o_{ij}-p_{ij}\right)^{2}}}{\displaystyle \sum_{j=1}^{D} \sum_{i=1}^{S}\left(o_{ij}-q_{ij}\right)^{2}}$$

where $o_ij$, $o_ij$, and $q_ij$ are the observed, predicted and prior probabilities of state $i$ from data set $j$, respectively, $S$ is the number of states, and $D$ the number of data sets (i.e. rows in obs). A value of 1 indicates perfect predictive capacity, while a value near zero indicates that the constraints provide no additional information beyond what is already contained in the prior $q$ (Sonnier et al. 2009).


measure of fit giving the predictive ability of the entire set of constraints $C$, beyond that already provided by the prior distribution.
measure of fit giving the predictive ability of the subset of constraints $A$, beyond that already provided by the prior distribution; only returned if sub.c is specified
Pearson $r^2$ between observed and predicted probabilities, using the entire set of constraints $C$
Pearson $r^2$ between observed and predicted probabilities, using the subset of constraints $A$; only returned if sub.c is specified
Pearson $r^2$ between observed and prior probabilities; only returned when sub.c is missing
observed statistic used for the permutation test; see Shipley (2010)
number of permutations; can be smaller than the specified nperm when quick is TRUE
approximate confidence intervals of the P-value


maxent.test is a computationally intensive function. The tests can take a very long time when nperm is large and quick = FALSE. It is highly recommended to use quick = TRUE because of this, unless you are interested in obtaining the true null probability.


Sonnier, G., Shipley, B., and M. L. Navas. 2009. Plant traits, species pools and the prediction of relative abundance in plant communities: a maximum entropy approach. Journal of Vegetation Science in press.

Shipley, B. (2010) Inferential permutation tests for maximum entropy models in ecology. Ecology in press.

See Also

maxent to run the maximum entropy model that is required by maxent.test.

Another faster version of maxent.test for multicore processors called maxent.testMC is available from Etienne Laliberté ( It's exactly the same as maxent.test but makes use of the multicore, doMC, and foreach packages. Because of this, maxentMC only works on POSIX-compliant OS's (essentially anything but Windows).

  • maxent.test

# example with tussock dataset
traits <- tussock$trait[, c(2:7, 11)] # use only continuous traits
traits <- na.omit(traits) # remove 2 species with NA's
abun <- tussock$abun[, rownames(traits)] # abundance matrix
abun <- t(apply(abun, 1, function(x) x / sum(x) )) # relative abundances
agg <- functcomp(traits, abun) # community-aggregated traits
traits <- t(traits) # transpose matrix

# run maxent on site 1 (first row of abun), all species
pred.abun <- maxent(agg[1,], traits)

## Not run: 
# # do the constraints give predictive ability?
# maxent.test(pred.abun, obs = abun[1,], nperm = 49)
# # are height, LDMC, and leaf [N] important constraints?
# maxent.test(pred.abun, obs = abun[1,], sub.c = c("height",
# "LDMC", "leafN"), nperm = 49)
# ## End(Not run)
Documentation reproduced from package FD, version 1.0-12, License: GPL-2

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