# mrpp: Multi Response Permutation Procedure and Mean Dissimilarity... In vegan: Community Ecology Package

## Description

Multiple Response Permutation Procedure (MRPP) provides a test of whether there is a significant difference between two or more groups of sampling units. Function `meandist` finds the mean within and between block dissimilarities.

## Usage

 ```1 2 3 4 5 6 7 8``` ```mrpp(dat, grouping, permutations = 999, distance = "euclidean", weight.type = 1, strata = NULL, parallel = getOption("mc.cores")) 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, ...) ```

## Arguments

 `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` a list of control values for the permutations as returned by the function `how`, or the number of permutations required, or a permutation matrix where each row gives the permuted indices. These are used 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. `parallel` Number of parallel processes or a predefined socket cluster. With `parallel = 1` uses ordinary, non-parallel processing. The parallel processing is done with parallel package. `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.

## Details

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; cf. Warton et al. 2012). 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 δ is the overall weighted mean of within-group means of the pairwise dissimilarities among sampling units. The 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 δ. It then permutes the sampling units and their associated pairwise distances, and recalculates δ based on the permuted data. It repeats the permutation step `permutations` times. The significance test is 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 -δ/E(δ), where E(δ) is the expected δ assessed as the average of dissimilarities.

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, CS (Van Sickle and Hughes, 2000). CS is defined for dissimilarities as Bbar-Wbar, where Bbar is the mean between cluster dissimilarity and Wbar is the mean within cluster dissimilarity with `weight.type = 1`. The function does not perform significance tests for these statistics, but you must use `mrpp` with appropriate `weight.type`. There is currently no significance test for CS, but `mrpp` with `weight.type = 1` gives the correct test for Wbar and a good approximation for CS. Function `plot` draws a dendrogram or a histogram of the result matrix based on the within-group and between group dissimilarities. The dendrogram is found with the method given in the `cluster` argument using function `hclust`. The terminal segments hang to within-cluster dissimilarity. If some of the clusters are more heterogeneous than the combined class, the leaf segment are reversed. The histograms are based on dissimilarities, but ore otherwise similar to those of Van Sickle and Hughes (2000): horizontal line is drawn at the level of mean between-cluster dissimilarity and vertical lines connect within-cluster dissimilarities to this line.

## Value

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

 `call ` Function call. `delta ` The overall weighted mean of group mean distances. `E.delta` expected delta, under the null hypothesis of no group structure. This is the mean of original dissimilarities. `CS` Classification strength (Van Sickle and Hughes, 2000). Currently not implemented and always `NA`. `n` Number of observations in each class. `classdelta` Mean dissimilarities within classes. The overall δ 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. The distribution of this item can be inspected with `permustats` function. `permutations` The number of permutations used. `control` A list of control values for the permutations as returned by the function `how`.

## Note

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.

## Author(s)

M. Henry H. Stevens HStevens@muohio.edu and Jari Oksanen.

## References

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 and R. M. Hughes 2000. Classification strengths of ecoregions, catchments, and geographic clusters of aquatic vertebrates in Oregon. J. N. Am. Benthol. Soc. 19:370–384.

Warton, D.I., Wright, T.W., Wang, Y. 2012. Distance-based multivariate analyses confound location and dispersion effects. Methods in Ecology and Evolution, 3, 89–101

`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.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```data(dune) data(dune.env) dune.mrpp <- with(dune.env, mrpp(dune, Management)) dune.mrpp # Save and change plotting parameters def.par <- par(no.readonly = TRUE) layout(matrix(1:2,nr=1)) plot(dune.ord <- metaMDS(dune), type="text", display="sites" ) with(dune.env, ordihull(dune.ord, Management)) with(dune.mrpp, { fig.dist <- hist(boot.deltas, xlim=range(c(delta,boot.deltas)), main="Test of Differences Among Groups") abline(v=delta); text(delta, 2*mean(fig.dist\$counts), adj = -0.5, expression(bold(delta)), cex=1.5 ) } ) par(def.par) ## meandist dune.md <- with(dune.env, meandist(vegdist(dune), Management)) dune.md summary(dune.md) plot(dune.md) plot(dune.md, kind="histogram") ```

### Example output   ```Loading required package: permute
This is vegan 2.4-3

Call:
mrpp(dat = dune, grouping = Management)

Dissimilarity index: euclidean
Weights for groups:  n

Class means and counts:

BF    HF    NM    SF
delta 10.03 11.08 10.66 12.27
n     3     5     6     6

Chance corrected within-group agreement A: 0.1246
Based on observed delta 11.15 and expected delta 12.74

Significance of delta: 0.001
Permutation: free
Number of permutations: 999

Run 0 stress 0.1192678
Run 1 stress 0.219292
Run 2 stress 0.1192679
... Procrustes: rmse 8.820187e-05  max resid 0.0002625276
... Similar to previous best
Run 3 stress 0.1183186
... New best solution
... Procrustes: rmse 0.0202672  max resid 0.06494471
Run 4 stress 0.1183186
... Procrustes: rmse 8.415528e-05  max resid 0.0002703275
... Similar to previous best
Run 5 stress 0.1809579
Run 6 stress 0.1808913
Run 7 stress 0.2075713
Run 8 stress 0.2465148
Run 9 stress 0.1183186
... New best solution
... Procrustes: rmse 2.533113e-05  max resid 7.662459e-05
... Similar to previous best
Run 10 stress 0.1183186
... Procrustes: rmse 1.948304e-05  max resid 6.912645e-05
... Similar to previous best
Run 11 stress 0.1183186
... Procrustes: rmse 0.000123975  max resid 0.0004025931
... Similar to previous best
Run 12 stress 0.1183186
... Procrustes: rmse 9.693875e-05  max resid 0.0003189633
... Similar to previous best
Run 13 stress 0.1192678
Run 14 stress 0.1183186
... Procrustes: rmse 9.748507e-06  max resid 3.252465e-05
... Similar to previous best
Run 15 stress 0.1192685
Run 16 stress 0.1889658
Run 17 stress 0.1809582
Run 18 stress 0.1192684
Run 19 stress 0.1192679
Run 20 stress 0.1192679
*** Solution reached
BF        HF        NM        SF
BF 0.4159972 0.4736637 0.7296979 0.6247169
HF 0.4736637 0.4418115 0.7217933 0.5673664
NM 0.7296979 0.7217933 0.6882438 0.7723367
SF 0.6247169 0.5673664 0.7723367 0.5813015
attr(,"class")
 "meandist" "matrix"
attr(,"n")
grouping
BF HF NM SF
3  5  6  6

Mean distances:
Average
within groups  0.5746346
between groups 0.6664172
overall        0.6456454

Summary statistics:
Statistic
MRPP A weights n        0.1423836
MRPP A weights n-1      0.1339124
MRPP A weights n(n-1)   0.1099842
Classification strength 0.1127012
```

vegan documentation built on May 2, 2019, 5:51 p.m.