Description of the Trichoptera data set

Formatting

Data are originally stored in a list of two data frames for abundances and covariates. We first prepare the data see the corresponding vignette for easy use in the multivariate framework of PLNmodels. Offsets are automatically computed:

trichoptera <- prepare_data(trichoptera$Abundance, trichoptera$Covariate)

Data are now stored in the trichoptera data frame which includes 49 rows (the observations - or trapping nights) and 9 columns. As can be seen, there are 2 multivariate columns (matrices of counts and offsets) and 7 univariate columns (vectors of covariates):

str(trichoptera)

## 'data.frame':    49 obs. of  9 variables:
##  $ Abundance    : num [1:49, 1:17] 0 0 0 0 0 0 0 0 0 0 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:49] "1" "2" "3" "4" ...
##   .. ..$ : chr [1:17] "Che" "Hyc" "Hym" "Hys" ...
##  $ Temperature  : num  18.7 19.8 22 23 22.5 23.9 15 17.2 15.4 14.1 ...
##  $ Wind         : num  -2.3 -2.7 -0.7 2.3 2.3 -2 -4.7 -1 -2.7 -3.7 ...
##  $ Pressure     : num  998 1000 997 991 990 ...
##  $ Humidity     : num  60 63 73 71 62 64 93 84 88 75 ...
##  $ Cloudiness   : num  19 0 6 81 50 50 100 19 69 6 ...
##  $ Precipitation: num  0 0 0 0 0 0 1.6 0 1.6 0 ...
##  $ Group        : Factor w/ 12 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ Offset       : num  29 13 38 192 79 18 8 34 12 4 ...

We thus rely on a not so common use of the data.frame structure, where a column may be a matrix and not necessarily an atomic vector. This formatting is very handy for model specification with R formulas, especially in a multivariate setting like in PLNmodels.

Table of counts (abundancies)

The Abundance column is a \(49 \times 17\) matrix of abundancies (or counts) for the 17 species found during the 49 trapping nights.

trichoptera$Abundance %>% head() %>% knitr::kable()

In all other vignettes and journal papers associated with PLNmodels, the count table is denoted by \(\mathbf{Y}\) in the mathematical model and Y in the R environment.

A basic representation of the matrix of counts (here transposed and log-transformed), shows the typical huge dispersion between low and highly abundant species.

corrplot::corrplot(
  t(log(1 + trichoptera$Abundance)),
  is.corr = FALSE,
  addgrid.col = NA
)

log-counts in the trichoptera data set

Covariates (external meteorological effect, groups)

Additional information was collected during the sampling, which corresponds to external covariates the effect of which may or may not be taken into account in the model (depending on the question at play). In the trichoptera data set, those covariates correspond to meteorological factors plus a categorical variable indicating the family of the caughts specimens.

dplyr::select(trichoptera, -Offset, -Abundance) %>% head() %>% knitr::kable()

The design matrix arising from the covariates is denoted by \(\mathbf{X}\) in our mathematical model and X within the R environment.

Offsets and the compositionality issue

A common issue with (microbiological) ecological data is the compositionality problem: counts can only be compared to each other within a sample but not across samples as they depend on a sample-specific size-factor, which may induce spurious negative correlations of its own. Besides, the sampling of some particular species may be biased, for instance when different technologies are used to sample different types of species. Those technical biases can be encoded in a table of offsets. In the case at hand, we have a natural offset for each sample that corresponds to the total counts per night, specified by an \(49\) numeric of offsets. Note that the offset term remains the same in a given sample albeit sometimes one might include an offset specific to both the sample and the species. The formula syntax accepts either no offset, a vector or a matrix or specify the offsets term.

Here, we have a vector whose corresponding column is named Offset in in the trichoptera data frame:

trichoptera$Offset

##  [1]   29   13   38  192   79   18    8   34   12    4    4    3   49   33  600
## [16]  172   58   51   56  127   35   13   17    3   27   40   44    8    9 1599
## [31] 2980   88  135  327   66   90   63   15   14   20   70   53   95   43   62
## [46]  149   16   31   86

See the importation vignette and the function prepare_data() to learn more about how Offset can be computed in PLNmodels.

Offsets are denoted by \(\mathbf{O}\) in the mathematical model across other vignettes, and by O in the R environment.