mrpp {vegan}R Documentation

Multi Response Permutation Procedure of Within- versus Among-Group Dissimilarities


Multiple Response Permutation Procedure (MRPP) provides a test of whether there is a significant difference between two or more groups of sampling units.


mrpp(dat, grouping, permutations = 999, distance = "euclidean",
     weight.type = 1, strata)
meandist(dist, grouping, ...)
## S3 method for class 'meandist':
summary(object, ...)
## S3 method for class 'meandist':
plot(x, kind = c("dendrogram", "histogram"),  cluster = "average", 
     ylim, axes = TRUE, ...)


dat data matrix or data frame in which rows are samples and columns are response variable(s), or a dissimilarity object or a symmetric square matrix of dissimilarities.
grouping Factor or numeric index for grouping observations.
permutations Number of permutations to assess the significance of the MRPP statistic, delta.
distance Choice of distance metric that measures the dissimilarity between two observations . See vegdist for options. This will be used if dat was not a dissimilarity structure of a symmetric square matrix.
weight.type choice of group weights. See Details below for options.
strata An integer vector or factor specifying the strata for permutation. If supplied, observations are permuted only within the specified strata.
dist A dist object of dissimilarities, such as produced by functions dist, vegdist or designdist.
object, x A meandist result object.
kind Draw a dendrogram or a histogram; see Details.
cluster A clustering method for the hclust function for kind = "dendrogram". Any hclust method can be used, but perhaps only "average" and "single" make sense.
ylim Limits for vertical axes (optional).
axes Draw scale for the vertical axis.
... Further arguments passed to functions.


Multiple Response Permutation Procedure (MRPP) provides a test of whether there is a significant difference between two or more groups of sampling units. This difference may be one of location (differences in mean) or one of spread (differences in within-group distance). Function mrpp operates on a data.frame matrix where rows are observations and responses data matrix. The response(s) may be uni- or multivariate. The method is philosophically and mathematically allied with analysis of variance, in that it compares dissimilarities within and among groups. If two groups of sampling units are really different (e.g. in their species composition), then average of the within-group compositional dissimilarities ought to be less than the average of the dissimilarities between two random collection of sampling units drawn from the entire population.

The mrpp statistic delta is simply the overall weighted mean of within-group means of the pairwise dissimilarities among sampling units. The correct choice of group weights is currently not clear. The mrpp function offers three choices: (1) group size (n), (2) a degrees-of-freedom analogue (n-1), and (3) a weight that is the number of unique distances calculated among n sampling units (n(n-1)/2).

The mrpp algorithm first calculates all pairwise distances in the entire dataset, then calculates delta. It then permutes the sampling units and their associated pairwise distances, and recalculates a delta based on the permuted data. It repeats the permutation step permutations times. The significance test is simply the fraction of permuted deltas that are less than the observed delta, with a small sample correction. The function also calculates the change-corrected within-group agreement A = 1 -delta/E(delta), where E(delta) is the expected delta assessed as the average of permutations.

With weight.type = 3, the function also calculates classification strength (Van Sickle 1997) which is defined as the difference between average between group dissimilarities and within group dissimilarities. With weight.type = 3 the classification strength is a simple transformation of delta, and has the same permutation significance.

If the first argument dat can be interpreted as dissimilarities, they will be used directly. In other cases the function treats dat as observations, and uses vegdist to find the dissimilarities. The default distance is Euclidean as in the traditional use of the method, but other dissimilarities in vegdist also are available.

Function meandist calculates a matrix of mean within-cluster dissimilarities (diagonal) and between-cluster dissimilarities (off-diagonal elements), and an attribute n of grouping counts. Function summary finds the within-class, between-class and overall means of these dissimilarities, and the MRPP statistics with all weight.type options and the classification strength. The function does not allow significance tests for these statistics, but you must use mrpp with appropriate weight.type. Function plot draws a dendrogram or a histogram of the result matrix based on the within-group and between group dissimilarities. The dendrogram is given with the cluster argument which is passed to hclust. The terminal segments hang to within-cluster dissimilarity. If some of the clusters is more heterogeneous than the combined class, the leaf segment is reversed. The histogram is similar as the graphics used by Van Sickle (1997): horizontal line is drawn at the level of mean between-cluster dissimilarity and vertical lines connect within-cluster dissimilarities to this line.


The function returns a list of class mrpp with following items:

call Function call.
delta The overall weighted mean of group mean distances. expected delta, under the null hypothesis of no group structure. This is the mean of original dissimilarities.
CS Classification strength (Van Sickle 1997) with weight.type = 3 and NA with other weights.
n Number of observations in each class.
classdelta Mean dissimilarities within classes. The overall delta is the weighted average of these values with given weight.type
Pvalue Significance of the test.
A A chance-corrected estimate of the proportion of the distances explained by group identity; a value analogous to a coefficient of determination in a linear model.
distance Choice of distance metric used; the "method" entry of the dist object.
weight.type The choice of group weights used.
boot.deltas The vector of "permuted deltas," the deltas calculated from each of the permuted datasets.
permutations The number of permutations used.


This difference may be one of location (differences in mean) or one of spread (differences in within-group distance). That is, it may find a significant difference between two groups simply because one of those groups has a greater dissimilarities among its sampling units. Most mrpp models can be analysed with adonis which seems not suffer from the same problems as mrpp and is a more robust alternative.


M. Henry H. Stevens and Jari Oksanen.


B. McCune and J. B. Grace. 2002. Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon, USA.

P. W. Mielke and K. J. Berry. 2001. Permutation Methods: A Distance Function Approach. Springer Series in Statistics. Springer.

J. Van Sickle 1997. Using mean similarity dendrograms to evaluate classifications. Journal of Agricultural, Biological, and Environmental Statistics 2:370-388.

See Also

anosim for a similar test based on ranks, and mantel for comparing dissimilarities against continuous variables, and vegdist for obtaining dissimilarities, adonis is a more robust alternative in most cases.


dune.mrpp <- mrpp(dune, dune.env$Management)

# Save and change plotting parameters
def.par <- par(no.readonly = TRUE)

plot(dune.ord <- metaMDS(dune), type="text", display="sites" )
ordihull(dune.ord, dune.env$Management)

with(dune.mrpp, {
  fig.dist <- hist(boot.deltas, xlim=range(c(delta,boot.deltas)), 
                 main="Test of Differences Among Groups")
  text(delta, 2*mean(fig.dist$counts), adj = -0.5,
     expression(bold(delta)), cex=1.5 )  }
## meandist <- meandist(vegdist(dune), dune.env$Management)

[Package vegan version 1.16-32 Index]