Author: Mark V. Murphy
Date: 2021-10-13

# dissCqN

#### Mark V. Murphy

#### 2021-10-13

Let’s use `dissCqN` to calculate compositional dissimilarity for a dataset of tree species counts from tropical forests of Barro Colorado Island, Panama (Condit et al. 2002), provided with the `vegan` package. There are 50 sampled 1-hectare forest plots (rows) and 225 total tree species (columns). Let’s take a look at a sample:

    # install.packages(c("dissCqN", "vegan"))
    library(dissCqN)
    data(BCI, package = "vegan")
    rownames(BCI) <- paste0("plot_", 1:nrow(BCI))
    knitr::kable(
      BCI[sample(1:nrow(BCI), 5),
          sample(1:ncol(BCI), 5)]
    )

| Astronium.graveolens | Casearia.arborea | Amaioua.corymbosa | Zuelania.guidonia | Zanthoxylum.ekmanii
--- | --- | --- | --- | --- | ---
plot_47 | 0 | 1 | 0 | 1 | 2
plot_8 | 2 | 3 | 0 | 0 | 4
plot_24 | 0 | 1 | 0 | 0 | 0
plot_9 | 0 | 9 | 0 | 0 | 13
plot_5 | 0 | 4 | 0 | 0 | 3

We can calculate multiple-assemblage dissimilarity for the forest plots for, say, orders q = 0-6:

    q <- 0:6
    (BCI.diss.mult <- dissCqN(BCI, q))
    #>        C0N        C1N        C2N        C3N        C4N        C5N        C6N
    #> 0.03017387 0.11635124 0.36695937 0.68341668 0.89674072 0.97536758 0.99488515

And plot a dissimilarity profile for q (Jost, Chao, and Chazdon 2011):

    par(bg = "#303030", fg = "#cccccc")  # dark plot scheme
    plot(q, BCI.diss.mult, xlab = expression(italic("q")), ylab = "Dissimilarity",
         col.axis = "#cccccc", col.lab = "#cccccc")
    lines(q, BCI.diss.mult)

We can see that when only presence-absence of trees is considered (q = 0, or Sørensen dissimilarity), dissimilarity is very low (0.03) – implying no major difference in species composition between the sampled assemblages. When tree abundance is taken into account however, dissimilarity steadily increases through the values of q as changes in relative abundance of common species have increasing influence on community difference (reaching >0.99 for q = 6). Plotting such profiles can help to illustrate the dominance of common species in the community, with steeper increases associated with higher dominance. If under-sampling of the community is suspected, dissimilarity for q > 0 or q > 1 is likely to be more reliable for assessing difference, as the common species are probably still sampled relatively accurately using the abundance-based measures.

What about pairwise dissimilarity? Simply set `pairwise = TRUE`:

    system.time(
      BCI.diss.pair <- dissCqN(BCI, pairwise = TRUE)
    )
    #>    user  system elapsed
    #>    8.21    0.05    8.27

This generates a symmetrical pairwise dissimilarity matrix for each value of q (0–2 only), returned in a list:

    lapply(BCI.diss.pair, function(i) i[1:5, 1:5])
    #> $C0N
    #>           plot_1    plot_2    plot_3    plot_4    plot_5
    #> plot_1 0.0000000 0.2768362 0.3005464 0.2834225 0.2989691
    #> plot_2 0.2768362 0.0000000 0.2873563 0.2808989 0.3081081
    #> plot_3 0.3005464 0.2873563 0.0000000 0.2717391 0.2984293
    #> plot_4 0.2834225 0.2808989 0.2717391 0.0000000 0.2615385
    #> plot_5 0.2989691 0.3081081 0.2984293 0.2615385 0.0000000
    #>
    #> $C1N
    #>           plot_1    plot_2    plot_3    plot_4    plot_5
    #> plot_1 0.0000000 0.1306597 0.1760514 0.1919161 0.1942899
    #> plot_2 0.1306597 0.0000000 0.1356817 0.1658189 0.1919815
    #> plot_3 0.1760514 0.1356817 0.0000000 0.1637515 0.1732861
    #> plot_4 0.1919161 0.1658189 0.1637515 0.0000000 0.1783943
    #> plot_5 0.1942899 0.1919815 0.1732861 0.1783943 0.0000000
    #>
    #> $C2N
    #>           plot_1    plot_2    plot_3    plot_4    plot_5
    #> plot_1 0.0000000 0.1243699 0.2089619 0.2074992 0.2415559
    #> plot_2 0.1243699 0.0000000 0.1089096 0.1332090 0.2308209
    #> plot_3 0.2089619 0.1089096 0.0000000 0.1556811 0.2009035
    #> plot_4 0.2074992 0.1332090 0.1556811 0.0000000 0.2877408
    #> plot_5 0.2415559 0.2308209 0.2009035 0.2877408 0.0000000

Note that another way to calculate multiple-assemblage dissimilarity might be to take the means of these pairwise matrices (omitting self-comparisons):

    sapply(BCI.diss.pair, function(i) {
      sum(i) / (prod(dim(i)) - dim(i)[1])
    })
    #>       C0N       C1N       C2N
    #> 0.3399075 0.2826509 0.3273713

These values are biased however, as they never take into account species shared by more than two assemblages, and there is also the problem of covariance between the pairwise comparisons (Diserud and Ødegaard 2007). Multiple-assemblage measures should generally be preferred when assessing overall community difference for randomly-sampled assemblages. However, when assessing change across ordered assemblages, for example along geographic or environmental gradients, pairwise comparisons can be more appropriate.

Subsets of assemblages can also be compared pairwise (using names or indices), if only some comparisons are of interest:

    dissCqN(BCI, pairwise = TRUE, compare.sub = list(1:2, 3:5))
    #> $C0N
    #>           plot_3    plot_4    plot_5
    #> plot_1 0.3005464 0.2834225 0.2989691
    #> plot_2 0.2873563 0.2808989 0.3081081
    #>
    #> $C1N
    #>           plot_3    plot_4    plot_5
    #> plot_1 0.1760514 0.1919161 0.1942899
    #> plot_2 0.1356817 0.1658189 0.1919815
    #>
    #> $C2N
    #>           plot_3    plot_4    plot_5
    #> plot_1 0.2089619 0.2074992 0.2415559
    #> plot_2 0.1089096 0.1332090 0.2308209

This could potentially save time if the total number of assemblages in the dataset is very large.

#### References

Condit, Richard, Nigel Pitman, Egbert G. Leigh, Jérôme Chave, John Terborgh, Robin B. Foster, Percy Núñez, et al. 2002. “Beta-Diversity in Tropical Forest Trees.” Science 295 (5555): 666–69. https://doi.org/c8cjfz.
Diserud, Ola H, and Frode Ødegaard. 2007. “A Multiple-Site Similarity Measure.” Biology Letters 3 (1): 20–22. https://doi.org/bwhfx6.
Jost, Lou, Anne Chao, and Robin L. Chazdon. 2011. “Compositional Similarity and Beta Diversity.” In Biological Diversity: Frontiers in Measurement and Assessment, edited by Anne E. Magurran and Brian J. McGill, 66–84. Oxford, UK: Oxford University Press.