Title: | An R Package for the Mean Measure of Divergence (MMD) |
---|---|
Description: | Offers a graphical user interface for the calculation of the mean measure of divergence, with facilities for trait selection and graphical representations <doi:10.1002/ajpa.23336>. |
Authors: | Frédéric Santos [aut, cre] |
Maintainer: | Frédéric Santos <[email protected]> |
License: | CeCILL-2 | file LICENSE |
Version: | 4.0.3 |
Built: | 2024-11-24 03:41:21 UTC |
Source: | https://gitlab.com/f-santos/anthropmmd |
Offers a graphical user interface for the calculation of the mean measure of divergence, with facilities for trait selection and graphical representations.
Frédéric Santos, [email protected]
Harris, E. F. and Sjøvold, T. (2004) Calculation of Smith's mean measure of divergence for intergroup comparisons using nonmetric data. Dental Anthropology, 17(3), 83–93.
Irish, J. (2010) The mean measure of divergence: Its utility in model-free and model-bound analyses relative to the Mahalanobis D2 distance for nonmetric traits. American Journal of Human Biology, 22, 378–395. doi: 10.1002/ajhb.21010
Nikita, E. (2015) A critical review of the mean measure of divergence and Mahalanobis distances using artificial data and new approaches to the estimation of biodistances employing nonmetric traits. American Journal of Physical Anthropology, 157, 284–294. doi: 10.1002/ajpa.22708
Santos, F. (2018) AnthropMMD: an R package with a graphical user interface for the mean measure of divergence. American Journal of Physical Anthropology, 165(1), 200–205. doi: 10.1002/ajpa.23336
Fidalgo, D., Hubbe, M. and Wesolowski, V. (2021) Population History of Brazilian South and Southeast Shellmound Builders Inferred through Dental Morphology. American Journal of Physical Anthropology, 176(2), 192–207. doi: 10.1002/ajpa.24342
## Not run: start_mmd()
## Not run: start_mmd()
This artifical dataset includes 200 individuals described by 9 binary traits and splitted into 5 groups. To fit with commonly observed datasets in past sciences, a substantial amount of missing values have been added at random on this dataset.
data(absolute_freqs)
data(absolute_freqs)
A matrix with 10 rows and 9 columns:
Trait1
summary statistics for this trait
Trait2
summary statistics for this trait
Trait3
summary statistics for this trait
Trait4
summary statistics for this trait
Trait5
summary statistics for this trait
Trait6
summary statistics for this trait
Trait7
summary statistics for this trait
Trait8
summary statistics for this trait
Trait9
summary statistics for this trait
This function allows to get a summary of sample sizes and frequencies
for each trait in each group. It is also mandatory to apply this
function before using the mmd
function, since the latter
only accepts table of frequencies, and cannot work with raw binary data.
binary_to_table(data, relative = FALSE)
binary_to_table(data, relative = FALSE)
data |
A binary (0/1 for presence/absence of traits) data frame
with |
relative |
Boolean. Indicates if the last rows of the table must contain frequencies (i.e., number of individuals having a given trait) or relative frequencies (i.e., proportions). |
A matrix with rows (
being the number of groups in
the dataset) and
columns (one per trait).
The first
rows are the sample sizes, the last
rows are
trait frequencies.
Frédéric Santos, [email protected]
Santos, F. (2018) AnthropMMD: an R package with a graphical user interface for the mean measure of divergence. American Journal of Physical Anthropology, 165(1), 200–205. doi: 10.1002/ajpa.23336
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and relative frequencies: binary_to_table(toyMMD, relative = TRUE)
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and relative frequencies: binary_to_table(toyMMD, relative = TRUE)
Compute various MMD results, typically using a table returned by the
function binary_to_table
with the argument
relative = TRUE
.
mmd(data, angular = c("Anscombe", "Freeman"), correct = TRUE, all.results = TRUE)
mmd(data, angular = c("Anscombe", "Freeman"), correct = TRUE, all.results = TRUE)
data |
A table of sample sizes and frequencies |
angular |
Choice of a formula for angular transformation: either Anscombe or Freeman-Tukey transformation. |
correct |
Boolean; whether to apply the correction for small
sample sizes (should be |
all.results |
Boolean; whether to compute all four matrices
described below as results. If FALSE, only the matrix |
A list with four components:
MMDMatrix |
Following the presentation adopted in many research articles, a matrix filled with MMD values above the diagonal, and standard deviations of MMD below the diagonal. |
MMDSym |
A symmetrical matrix of MMD values, where negative values are replaced by zeroes. |
MMDSignif |
A matrix where any pair of traits having a significant MMD value is indicated by a star, ‘*’. |
MMDpval |
A matrix filled with MMD values above the diagonal, and p-values below the diagonal. |
Frédéric Santos, [email protected]
de Souza, P. and Houghton, P. (1977). The mean measure of divergence and the use of non-metric data in the estimation of biological distances. Journal of Archaeological Science, 4(2), 163–169. doi: 10.1016/0305-4403(77)90063-2
Harris, E. F. and Sjøvold, T. (2004) Calculation of Smith's mean measure of divergence for intergroup comparisons using nonmetric data. Dental Anthropology, 17(3), 83–93.
Nikita, E. (2015) A critical review of the mean measure of divergence and Mahalanobis distances using artificial data and new approaches to the estimation of biodistances employing nonmetric traits. American Journal of Physical Anthropology, 157, 284–294. doi: 10.1002/ajpa.22708
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and relative ## frequencies: tab <- binary_to_table(toyMMD, relative = TRUE) tab ## Compute and display a symmetrical matrix of MMD values: mmd_out <- mmd(tab, angular = "Anscombe") mmd_out$MMDSym ## Significant MMD values are indicated by a star: mmd_out$MMDSignif
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and relative ## frequencies: tab <- binary_to_table(toyMMD, relative = TRUE) tab ## Compute and display a symmetrical matrix of MMD values: mmd_out <- mmd(tab, angular = "Anscombe") mmd_out$MMDSym ## Significant MMD values are indicated by a star: mmd_out$MMDSignif
Compute a matrix of MMD dissimilarities among bootstrapped samples of the original groups. The input data must be a “raw binary dataset”.
mmd_boot(data, angular = c("Anscombe", "Freeman"), B = 100, ...)
mmd_boot(data, angular = c("Anscombe", "Freeman"), B = 100, ...)
data |
A “raw binary dataset”, as defined in the man page
of |
angular |
Choice of a formula for angular transformation: either Anscombe or Freeman-Tukey transformation. |
B |
Numeric value: number of bootstrap samples. |
... |
Arguments for traits selection, passed to
|
This function sticks very close to Fidalgo et al's (2022) implementation. In particular, no correction for small sample sizes is applied in the MMD formula; see Fidalgo et al's (2021) for the rationale.
Note that only a “raw binary dataset” is allowed as input, since the resampling cannot be performed properly from a table of counts and frequencies.
To get a MDS plot of the dissimilarity matrix obtained with this
function, see plot.anthropmmd_boot
.
A symmetrical dissimilarity matrix of MMD values among original groups
and bootstrapped samples. This matrix is an R object of class
anthropmmd_boot
.
Frédéric Santos, [email protected]
D. Fidalgo, M. Hubbe and V. Vesolowski (2021). Population history of Brazilian south and southeast shellmound builders inferred through dental morphology. American Journal of Physical Anthropology 176(2), 192-207.
D. Fidalgo, V. Vesolowski and M. Hubbe (2022). Biological affinities of Brazilian pre-colonial coastal communities explored through boostrapped biodistances of dental non-metric traits. Journal of Archaeological Science 138, 105545.
## Not run: ## Load and visualize a raw binary dataset: data(toyMMD) head(toyMMD) ## Compute MMD among bootstrapped samples: resboot <- mmd_boot( data = toyMMD, B = 50, # number of bootstrap samples angular = "Anscombe", strategy = "excludeQNPT", # strategy for trait selection k = 10 # minimal number of observations required per trait ) ## View part of MMD matrix among bootstrapped samples: dim(resboot) print(resboot[1:15, 1:15]) ## End(Not run)
## Not run: ## Load and visualize a raw binary dataset: data(toyMMD) head(toyMMD) ## Compute MMD among bootstrapped samples: resboot <- mmd_boot( data = toyMMD, B = 50, # number of bootstrap samples angular = "Anscombe", strategy = "excludeQNPT", # strategy for trait selection k = 10 # minimal number of observations required per trait ) ## View part of MMD matrix among bootstrapped samples: dim(resboot) print(resboot[1:15, 1:15]) ## End(Not run)
This function plots a 2D MDS to represent the MMD dissimilarities
among the groups compared, after a bootstrap resampling performed with
mmd_boot
.
## S3 method for class 'anthropmmd_boot' plot(x, method = c("classical", "interval", "ratio", "ordinal"), level = 0.95, pch = 16, gof = FALSE, xlab = NA, ylab = NA, main = "MDS plot of original and bootstrapped samples", ...)
## S3 method for class 'anthropmmd_boot' plot(x, method = c("classical", "interval", "ratio", "ordinal"), level = 0.95, pch = 16, gof = FALSE, xlab = NA, ylab = NA, main = "MDS plot of original and bootstrapped samples", ...)
x |
An object of class |
.
method |
Algorithm used for MDS computation; see |
level |
Numeric value between 0 and 1, confidence level for the contour lines displayed after the kernel density estimate. |
pch |
Passed to |
gof |
Boolean; whether to display goodness of fit statistic on the plot. |
xlab |
Passed to |
ylab |
Passed to |
main |
Passed to |
... |
Other arguments possibly passed to |
In the current implementation, to stick to Fidalgo et al.'s (2022)
protocol, this function does not provide as much freedom as
plot.anthropmmd_result
as concenrs MDS parameters and
other analysis options.
This function returns no value by itself, and only plots a MDS in a new device.
Frédéric Santos, [email protected]
D. Fidalgo, V. Vesolowski and M. Hubbe (2022). Biological affinities of Brazilian pre-colonial coastal communities explored through boostrapped biodistances of dental non-metric traits. Journal of Archaeological Science 138, 105545.
start_mmd
, stats::cmdscale
## Not run: ## Load and visualize a raw binary dataset: data(toyMMD) head(toyMMD) ## Compute MMD among bootstrapped samples: resboot <- mmd_boot( data = toyMMD, B = 50, # number of bootstrap samples angular = "Anscombe", strategy = "excludeQNPT", # strategy for trait selection k = 10 # minimal number of observations required per trait ) ## MDS plot for bootstrapped samples: plot( x = resboot, method = "interval", # algorithm used for MDS computation level = 0.95 # confidence level for the contour lines ) ## End(Not run)
## Not run: ## Load and visualize a raw binary dataset: data(toyMMD) head(toyMMD) ## Compute MMD among bootstrapped samples: resboot <- mmd_boot( data = toyMMD, B = 50, # number of bootstrap samples angular = "Anscombe", strategy = "excludeQNPT", # strategy for trait selection k = 10 # minimal number of observations required per trait ) ## MDS plot for bootstrapped samples: plot( x = resboot, method = "interval", # algorithm used for MDS computation level = 0.95 # confidence level for the contour lines ) ## End(Not run)
This function plots a 2D or 3D MDS to represent the MMD
dissimilarities among the groups compared. Various MDS methods are
proposed, and most of them are based on the R package smacof
.
## S3 method for class 'anthropmmd_result' plot(x, method = c("classical", "interval", "ratio", "ordinal"), axes = FALSE, gof = FALSE, dim = 2, asp = TRUE, xlim = NULL, ...)
## S3 method for class 'anthropmmd_result' plot(x, method = c("classical", "interval", "ratio", "ordinal"), axes = FALSE, gof = FALSE, dim = 2, asp = TRUE, xlim = NULL, ...)
x |
An object of class |
.
method |
Specification of MDS type. |
axes |
Boolean: should the axes be displayed on the plot? |
gof |
Boolean: should goodness of fit statistics be displayed on the topleft corner of the plot? More details below. |
dim |
Numeric value, 2 or 3. Indicates the maximal dimension
desired for the MDS plot. It should be noted that, even with
|
asp |
Boolean. If |
xlim |
Parameter passed to |
... |
Other arguments possibly passed to |
Axes and scale. Making all axes use the same scale is strongly recommended in all cases (Borg et al., 2013). For a 3D-plot, since the third axis carries generally only a very small percentage of the total variability, you might want to uncheck this option to better visualize the distances along the third axis. In this case, the axes scales must be displayed on the plot, otherwise the plot would be misleading.
Goodness of fit values. (i) For classical metric MDS, a
common statistic is given: the sum of the eigenvalues of the first
two axes, divided by the sum of all eigenvalues. It indicates the
fraction of the total variance of the data represented in the MDS
plot. This statistic comes from the $GOF
value returned by
the function stats::cmdscale
. (ii) For SMACOF methods, the
statistic given is the $stress
value returned by the function
smacof::smacofSym
It indicates the final stress-1 value. A
value very close to 0 corresponds to a perfect fit. (iii) For both
approaches, a 'rho' value is also given, which is the Spearman's
correlation coefficient between real dissimilarities (i.e., MMD
values) and distances observed on the MDS plot (Dzemyda et
al.,2013). A value very close to 1 indicates a perfect fit.
This function returns no value by itself, and only plots a MDS in a new device.
Frédéric Santos, [email protected]
G. Dzemyda, O. Kurasova and J. Zilinskas (2013) Multidimensional Data Visualization, Springer, chap. 2, p. 39–40.
I. Borg, P. Groenen and P. Mair (2013) Applied Multidimensional Scaling, Springer, chap. 7, p. 79.
start_mmd
, stats::cmdscale
, smacof::smacofSym
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and relative ## frequencies: tab <- binary_to_table(toyMMD, relative = TRUE) tab ## Compute and display a symmetrical matrix of MMD values: mmd_out <- mmd(tab, angular = "Freeman") ## Plot a classical metric MDS in two dimensions: plot(x = mmd_out, method = "classical", axes = TRUE, gof = TRUE, dim = 2)
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and relative ## frequencies: tab <- binary_to_table(toyMMD, relative = TRUE) tab ## Compute and display a symmetrical matrix of MMD values: mmd_out <- mmd(tab, angular = "Freeman") ## Plot a classical metric MDS in two dimensions: plot(x = mmd_out, method = "classical", axes = TRUE, gof = TRUE, dim = 2)
This function provides several strategies to discard some useless traits (non-polymorphic, non-discriminatory, etc.) upstream the MMD analysis.
select_traits(tab, k = 10, strategy = c("none", "excludeNPT", "excludeQNPT", "excludeNOMD", "keepFisher"), OMDvalue = NULL, groups, angular = c("Anscombe", "Freeman"))
select_traits(tab, k = 10, strategy = c("none", "excludeNPT", "excludeQNPT", "excludeNOMD", "keepFisher"), OMDvalue = NULL, groups, angular = c("Anscombe", "Freeman"))
tab |
A table of sample sizes and frequencies, typically returned
by the function |
k |
Numeric value: the required minimal number of individuals per group. Any trait that could be taken on fewer individuals in at least one group will be removed from the dataset. This allows to select only the traits with a sufficient amount of information in each group. |
strategy |
Strategy for trait selection, i.e. for the removal of
non-polymorphic traits. The four options are fully described in
Santos (2018) and in the help page of |
OMDvalue |
To be specified if and only if |
groups |
A factor or character vector, indicating the group to be
considered in the analysis. Since some groups can have a very low
sample size, this will allow to discard those groups in order to
facilitate the trait selection via the argument
|
angular |
Formula for angular transformation, see Harris and Sjøvold (2004). Useful only for the calculation of overall measure of divergence. |
A list with two components:
filtered |
The dataset filtered according to the user-defined criteria. |
OMD |
The “overall measure of divergence” for each trait. |
Frédéric Santos, [email protected]
Harris, E. F. and Sjøvold, T. (2004) Calculation of Smith's mean measure of divergence for intergroup comparisons using nonmetric data. Dental Anthropology, 17(3), 83–93.
Santos, F. (2018) AnthropMMD: an R package with a graphical user interface for the mean measure of divergence. American Journal of Physical Anthropology, 165(1), 200–205. doi: 10.1002/ajpa.23336
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and ## relative frequencies: tab <- binary_to_table(toyMMD, relative = TRUE) tab ## Filter this dataset to keep only those traits that have at ## least k=10 individuals in each group: select_traits(tab, k = 10) ## Only Trait1 is excluded. ## Filter this dataset to keep only those traits that have at ## least k=11 individuals in each group, and show significant ## differences at Fisher's exact test: select_traits(tab, k = 11, strategy = "keepFisher") ## Traits 1, 5 and 8 are excluded.
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and ## relative frequencies: tab <- binary_to_table(toyMMD, relative = TRUE) tab ## Filter this dataset to keep only those traits that have at ## least k=10 individuals in each group: select_traits(tab, k = 10) ## Only Trait1 is excluded. ## Filter this dataset to keep only those traits that have at ## least k=11 individuals in each group, and show significant ## differences at Fisher's exact test: select_traits(tab, k = 11, strategy = "keepFisher") ## Traits 1, 5 and 8 are excluded.
Launches a graphical user interface (GUI) for the calculation of the mean measure of divergence.
start_mmd() StartMMD()
start_mmd() StartMMD()
The GUI of AnthropMMD is completely autonomous: reading the data file and specifying the parameters of the analysis are done through the interface. Once the dataset is loaded, the output reacts dynamically to any change in the analysis settings.
AnthropMMD accepts .CSV or .TXT data files, but does not support .ODS or .XLS(X) files. Two types of data input formats can be used:
A ‘Raw binary dataset’ (one row for each individual, one column for each variable). The first column must be the group indicator, and the other columns are binary data for the traits studied, where 1 indicates the presence of a trait, and 0 its absence. Row names are optional for this type of file. An example of valid data file can be found as Supporting Information online in Santos (2018).
A ‘Table of n's and absolute frequencies for each
group’, i.e. a dataset of sample sizes and absolute
frequencies. This type of dataset has rows
(
being the number of groups compared) and
columns (
being the number of traits studied). The first
lines must be the group n's for each trait, and the last
lines are absolute frequencies for each trait (i.e. the
number of times the trait is present). Row names are mandatory for
this type of file. The first
rows must be labelled with
names beginning with ‘N_’, such as: N_GroupA, N_GroupB, ...,
N_GroupK. The last
rows should be labelled with names
beginning with ‘Freq_’, such as: Freq_GroupA, ...,
Freq_GroupK. An example of valid data file can be found as
Supporting Information online in Santos (2018).
For both data types, column names are strongly recommended for better interpretability of the results.
One can choose between Anscombe or Freeman-Tukey formula for angular transformation (cf. Harris and Sjøvold 2004; Irish 2010).
‘Only retain the traits with this minimal number of individuals per group’: the traits with fewer individuals in at least one active group will not be considered in the analysis.
‘Exclusion strategy’: a careful selection of traits is crucial when using MMD (cf. Harris and Sjøvold 2004 for a complete explanation), and the user should probably “exclude the traits that are nondiscriminatory across groups” (Irish 2010).
‘Exclude nonpolymorphic traits’ removes all the traits showing no variability at all, i.e. with the same value (‘0’ or ‘1’) for all individuals.
‘Exclude quasi-nonpolymorphic traits’ also removes the traits whose variability is only due to a single individual: for example, a trait with only one positive observation in the whole dataset.
‘Use Fisher's exact test’ implements the advice given by Harris and Sjøvold (2004) to select contributory traits, defined as those “showing a statistically significant difference between at least one pair of the groups being evaluated”. Fisher's exact tests are performed for each pair of groups, and the traits showing no intergroup difference at all are excluded. Note that if you have a large number of groups (say, 10 groups), a trait with strictly equal frequencies for the last 8 groups may be considered as useful according to this criterion if there is a significant difference for the first two groups. This criterion will select all traits that can be useful for a given pair of groups, even if they are nondiscriminatory for all the other ones.
‘Exclude traits with overall MD’ lower than a given threshold: it is a simple way of removing the traits with quite similar frequencies across groups (the ‘overall MD’ is defined as the sum of the variable's measures of divergence over all pairs of groups). This criterion aims to select the traits whose frequency differs substantially across most or all groups.
These four options are designed to avoid negative MMD values.
Some groups/populations can be manually excluded from the analysis. This may be useful if very few individuals belonging to a given population could be recorded for the variables retained by the criteria described above.
A MDS plot and a hierarchical clustering, done using MMD
dissimilarities as inputs, are displayed in the last two tabs. As MMD
can sometimes be negative, those negatives values are replaced by
zeros, so that the MMD matrix can be seen as a symmetrical distance
matrix. Please note that the classical two-dimensional metric MDS plot
cannot be displayed if there is only one positive eigenvalue. Several
MDS options are proposed, cf. the help page of the smacofSym
function from the R package smacof
for detailed technical
information.
The function returns no value by itself, but all results can be individually downloaded through the graphical interface.
The ‘true’ MMD values (i.e., which can be negative in the case of small samples with similar traits frequencies, cf. Irish 2010) and their standard deviations are presented in the matrix labelled ‘MMD values (upper triangular part) and associated SD values (lower triangular part)’.
A MMD value can be considered as significant if it is greater than twice its standard deviation. Significance is assessed in another ad-hoc table of results.
The negative MMD values, if any, are replaced by zeros in the ‘Symmetrical matrix of MMD values’.
The R console is not available when the GUI is active. To exit the GUI, type Echap (on MS Windows systems) or Ctrl+C (on Linux systems) in the R console.
On 14-inch (or smaller) screens, for convenience, it may be necessary to decrease the zoom level of your web browser and/or to turn on fullscreen mode.
Frédéric Santos, [email protected]
Harris, E. F. and Sjøvold, T. (2004) Calculation of Smith's mean measure of divergence for intergroup comparisons using nonmetric data. Dental Anthropology, 17(3), 83–93.
Irish, J. (2010) The mean measure of divergence: Its utility in model-free and model-bound analyses relative to the Mahalanobis D2 distance for nonmetric traits. American Journal of Human Biology, 22, 378–395. doi: 10.1002/ajhb.21010
Nikita, E. (2015) A critical review of the mean measure of divergence and Mahalanobis distances using artificial data and new approaches to the estimation of biodistances employing nonmetric traits. American Journal of Physical Anthropology, 157, 284–294. doi: 10.1002/ajpa.22708
Santos, F. (2018) AnthropMMD: an R package with a graphical user interface for the mean measure of divergence. American Journal of Physical Anthropology, 165(1), 200–205. doi: 10.1002/ajpa.23336
## An example of valid binary dataset: data(toyMMD) head(toyMMD) ## An example of valid table: data(absolute_freqs) absolute_freqs ## Launch the GUI: ## Not run: start_mmd()
## An example of valid binary dataset: data(toyMMD) head(toyMMD) ## An example of valid table: data(absolute_freqs) absolute_freqs ## Launch the GUI: ## Not run: start_mmd()
Mostly used as an internal function, but could also be convenient to transform frequencies (i.e., number of individuals having a given trait) into relative frequencies (i.e., proportions).
table_relfreq(tab)
table_relfreq(tab)
tab |
A table of sample sizes and frequencies, such as the tables
returned by the function |
The last rows (
being the number of groups) of
tab
are simply transformed to relative frequencies.
Frédéric Santos, [email protected]
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and frequencies: tab <- binary_to_table(toyMMD, relative = FALSE) tab ## Convert this table into relative frequencies: table_relfreq(tab)
## Load and visualize a binary dataset: data(toyMMD) head(toyMMD) ## Convert this dataframe into a table of sample sizes and frequencies: tab <- binary_to_table(toyMMD, relative = FALSE) tab ## Convert this table into relative frequencies: table_relfreq(tab)
This artifical dataset includes 200 individuals described by 9 binary traits and splitted into 5 groups. To fit with commonly observed datasets in past sciences, a substantial amount of missing values have been added at random on this dataset.
data(toyMMD)
data(toyMMD)
A data frame with 200 observations on the following 10 variables:
Group
a factor with 5 levels (group indicator)
Trait1
a numeric vector of zeroes and ones
Trait2
a numeric vector of zeroes and ones
Trait3
a numeric vector of zeroes and ones
Trait4
a numeric vector of zeroes and ones
Trait5
a numeric vector of zeroes and ones
Trait6
a numeric vector of zeroes and ones
Trait7
a numeric vector of zeroes and ones
Trait8
a numeric vector of zeroes and ones
Trait9
a numeric vector of zeroes and ones