An overview of AnthropMMD

In AnthropMMD: An R Package for the Mean Measure of Divergence (MMD)

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
options(digits = 3)

library(AnthropMMD)

Graphical user interface

AnthropMMD was primarily designed as an R-Shiny application [@Santos18], that can be launched with the following instruction:

start_mmd()

Starting with the version 3.0.0, it can also be used from the command line, to make it convenient in a context of reproducible research.

Using AnthropMMD from the command line

Data formatting

The MMD is a dissimilarity measure among groups of individuals described by binary (presence-absence) traits [@Sjovold73; @Sjovold77; @Harris04]. The MMD formula only requires to know the sample sizes and relative trait frequencies within each group: providing the raw individual data is not mandatory.

Usually, the user will have recorded the data in one of the the two following formats.

Raw binary data

Most of the time, the data will be formatted as a classical dataframe with $n$ rows (one per individual) and $p+1$ columns (the first column must be a group indicator; the $p$ other columns correspond to the $p$ binary traits observed). Each trait has two possible values: $1$ if present, $0$ if absent (missing values are allowed). Rownames are optional but colnames (i.e., headers) are mandatory.

An artificial dataset, toyMMD, is available in the package to give an example of such a format:

data(toyMMD)
head(toyMMD)

It is not necessary for the Traits to be manually converted as factors prior to the analysis. toyMMD includes nine traits with many missing values, and the individuals belong to five groups:

str(toyMMD)

If the data were recorded as a raw binary dataset, they must be converted into a table of relative frequencies prior to the MMD analysis. The R function binary_to_table() with the argument relative = TRUE can perform this operation:

tab <- binary_to_table(toyMMD, relative = TRUE)
tab

Tables of sample sizes and absolute frequencies

Sometimes, in particular if the data were extracted from research articles rather than true exprimentation, the user will only know the sample sizes and absolute trait frequencies in each group. As previously stated, those data are perfectly sufficient to compute MMD values. Here is an example of such a table:

data(absolute_freqs)
print(absolute_freqs)

If there are $k$ groups in the data, the first $k$ rows (whose names must start with an N_ prefix, followed by the group labels) indicates the sample sizes of each trait in each group, and the last $k$ rows indicates the absolute trait frequencies. Such a table can be directly imported by the user, but must be converted to relative trait frequencies prior to the analysis, for instance by using the function table_relfreq():

tab <- table_relfreq(absolute_freqs)
print(tab)

Important remark

Due to rounding issues, it is clearly not advised for the user to submit directly a table of relative frequencies. If the absolute frequencies do not perfectly sum up to 1 for each trait, an error will be triggered.

A table of relative frequencies is the starting point of all subsequent analyses, but it should be computed from the data loaded in R; it should not be loaded itself.

A quick summary:

• if you load a raw binary dataframe, convert it to a table of relative frequencies with the function binary_to_table(relative = TRUE) as shown above;
• if you load a table of absolute frequencies, convert it to a table of relative frequencies with the function table_relfreq().

Trait selection

AnthropMMD proposes a built-in feature for trait selection, in order to discard the traits that could be observed on too few individuals, or that do not show enough variability among groups. The function select_traits() allows such a selection according to several strategies [@Harris04; @Santos18].

For instance, with the following instruction, we can discard the traits that could be observed on at least 10 individuals per group, and that exhibit no significant variability in frequencies among groups according to Fisher's exact tests:

tab_selected <- select_traits(tab, k = 10, strategy = "keepFisher")
tab_selected$filtered

Trait1 (which could be observed on only 8 individuals in Group B), Trait5 and Trait8 (whose variation in frequencies was not significant among groups) were removed from the data.

Compute the mean measure of divergence

Once the trait selection has been performed, the MMD can be computed with the mmd() function. With the following instruction, the MMD is computed using Anscombe's angular transformation of trait frequencies:

mmd.result <- mmd(tab_selected$filtered, angular = "Anscombe")
mmd.result

• The first component, $MMDMatrix, follows the presentation adopted in most research articles [@Donlon00]: the true MMD values are indicated above the diagonal, and their standard deviations are indicated below the diagonal.
• A MMD value can be considered as significant if it is greater than twice its standard deviation. Significant MMD values are indicated by a * in the component $MMDSignif.
• The component $MMDSym is a symmetrical matrix of MMD values, with a null diagonal. This matrix of dissimilarities can be used to perform a multidimensional scaling or a hierarchical clustering to visualize the distances among the groups.
• Finally, p-values for non-nullity of each MMD value are given (in the lower triangular part of the matrix returned) by the component $MMDpval. Theoretical details for this test of significance can be found in @Souza77.

Graphical representations of MMD

Multidimensional scaling

Although mmd.result$MMDSym can perfectly be passed to your favourite function to produce a MDS plot, AnthropMMD also proposes a built-in generic function for such a graphical representation: plot().

The MDS can be computed using the classical stats::cmdscale() function (and the produces a metric MDS), or several variants of MDS algorithms implemented in the R package smacof.

For instance, we plot here the MDS coordinates computed with one variant of SMACOF algorithms:

par(cex = 0.8)
plot(x = mmd.result, method = "interval", 
     gof = TRUE, axes = TRUE, xlim = c(-1.2, 0.75))

The argument gof = TRUE displays goodness of fits statistics for the MDS configurations directly on the plot.

Hierarchical clustering

AnthropMMD does not propose a built-in function for hierarchical clustering, but such a plot can easily be obtained with the usual R functions.

library(cluster)
par(cex = 0.8)
plot(agnes(mmd.result$MMDSym), which.plots = 2, main = "Dendrogram of MMD dissimilarities")

Alternative: analysis using bootstrapped samples

Starting with AnthropMMD 4.0.0, a bootstrap method introduced by Daniel Fidalgo and co-workers [@Fidalgo2021; @Fidalgo2022] is now available. Please note that:

• this method is only available when the input data is a "raw binary dataset", as described earlier in this vignette (see Section 2.1.1);
• this method requires longer computations than the standard MMD, so that it is not available through the graphical user interface.

Let's start again from the dataset toyMMD. The first step is to perform the resampling and the MMD computations using the separate function mmd_boot(). @Fidalgo2022 suggest a value of B = 100 bootstrap experiments, it is thus the default value for the corresponding argument in this function. Below, for this simple example, we will use a value of B = 50. Be careful: the computation time (in the current implementation) will grow quite fast for high values of B.

Note that mmd_boot() takes care itself of the trait selection. All arguments usually submitted to select_traits() can also be passed to mmd_boot(); see example below.

## Load the example data once again:
data(toyMMD)
## Compute MMD among bootstrapped samples:
set.seed(2023) # set seed for reproducibility
resboot <- mmd_boot(
    data = toyMMD,
    B = 50, # number of bootstrap samples
    angular = "Anscombe",
    strategy = "keepFisher", # strategy for trait selection
    k = 10 # minimal number of observations required per trait
)

Now you can simply plot() the result of your computations, and set various parameters to customize your plot:

## MDS plot for bootstrapped samples:
plot(
    x = resboot,
    method = "interval", # algorithm used for MDS computation
    level = 0.95, # confidence level for the contour lines
    gof = TRUE # display goodness of fit statistic
)

This plot displays the contour lines of a 2D kernel density estimation, as in Figure 3 from @Fidalgo2022.

References

AnthropMMD documentation built on Aug. 8, 2023, 5:12 p.m.