---
title: "Environmental and geographic standardisation case study"
author: "G. T. Antell"
date: "24 October 2023"
output: rmarkdown::html_vignette
bibliography: habitat-case-study-refs.bib
vignette: >
  %\VignetteIndexEntry{Environmental and geographic standardisation case study}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---




## Goals and limitations of the vignette

This vignette is written to model a realistic project flow for ecological occurrence data, providing example uses of `divvy` functions in `R` code chunks to analyse geographic range size disparities between comparison categories. The data and hypothesis are representative of many real palaeobiology studies, with the hope of serving as a template for comparing ecology between environments, regions, or clades---for instance, investigations about whether diversity was greater in Gondwanaland or Laurasia, or whether biogeographic connectedness was greater within reefal or non-reefal assemblages.

Results herein should not be taken as a conclusive answer to the example research question. For a publication-quality study, ideally one would further vet both taxonomic and environmental assignments in the original public database download, for instance. Similarly, while this vignette illustrates two lines of analysis, additional follow-up inquiries would be prudent to pursue before arguing a definitive palaeoecological interpretation of the results to the research community.

The vignette consists of the following parts:

1. Background on example research question
1. Formatting point occurrence data
1. Developing environmental categories (raster data)
1. Categorising taxon preferences for environments
1. Spatial subsampling to regions of equal area and dispersion
1. Hypothesis testing of assemblage-based range-size metric
1. Hypothesis testing of per-taxon range-size metrics
1. References

## Background on geographic ranges between marine environments

Palaeobiologists often track ecological parameters through time, where, as the main `divvy` package vignette illustrates, it is crucial to standardise differences in spatial coverage between time steps. However, it is equally important to apply spatial standardisation for comparisons between environment or habitat types, even if data occur within one time step. For instance, consider two habitat types that cover different geographic extents. To determine fairly which habitat hosts more diverse communities, one must account for the difference in spatial coverage. By first subsampling each habitat type into equivalent spatial units, one can then directly and fairly compare richness or other ecological metrics between environments.

In the marine realm, one of the most important environmental axes of ecological differentiation is water depth. Diversity, geographic range size, and other aspects of community structure vary along a gradient of near-shore to deep-water environments. Here, we divide this environmental spectrum coarsely into shallow (proximal) vs. deep (distal) marine habitats. 'Shallow' habitats encompass deltaic, (sub)tidal, and shelf margin environments. 'Deep' habitats include continental slope and offshore settings, generally beginning at 140 m [@Duran2018].

The ecological attribute this case study focusses on is geographic range size, a trait with widespread importance. For instance, range size determines extinction risk to a large degree [@Harnik2012; @Saupe2015]. Range size might influence origination propensity as well, although researchers dispute whether this relationship is positive [@Rosenzweig1995; @Maurer2001], negative [@Hansen1980; @Jablonski2003], or unimodal [@Gaston2003; @Allmon2014].

The environmental niche breadth of a taxon strongly controls its geographic distribution; the maximum attainable extent of a taxon's range size is the extent of occurrence of that taxon's suitable habitat. Two consequences of this relationship are that (1) taxa with broad niches (generalists or eurytopic taxa) are expected to occupy larger ranges than those with narrow niches (specialists or stenotopic taxa), and (2) for environments within a given specialisation, a larger geographic distribution of those habitats permits taxa to occupy larger ranges. Taxa may be more restricted than suitable environmental conditions would predict (realised niche breadth), due to biotic or historic factors, but taxa cannot exceed the bounds set by environmental suitability. Since environmental characteristics are more homogeneous in the cold, dark waters of the deep sea than in the topologic complexity of shallow settings, ecologists have long theorised geographic ranges to be larger for deep-dwelling than shallow-water species [@Coe1946; @McClain2010; @Costello2017]. Most empirical tests of this hypothesis have relied upon modern data, however, with relatively little evidence from periods of Earth history when ocean circulation and continental configuration were entirely unlike those of today.

<a id="atlas_map"></a>
The following sections walk through an analysis to test the expectations geographic ranges are wider for (1) eurytopic than stenotopic taxa and (2) deep- than shallow-water taxa, using brachiopod occurrences from the Silurian (444-419 Ma) recorded in the [Paleobiology Database](https://paleobiodb.org/) (PBDB). Brachiopoda was a diverse clade that made up a large component of Paleozoic benthic communities [@Close2020; @Rojas2021]. In the Paleozoic, much of the world's landmass had coalesced at southermost latitudes into Gondwanaland, with the remaining continents scattered at the equator and separated by the Iapetus, Paleo-Tethys, Rheic, and Panthalassic oceans (map below). Hence, the geometry of a cosmopolitan marine species' range from the Silurian would differ distinctly from a cosmopolitan range today. Spatial subsampling is crucial to addressing the research questions because the preserved area and distribution of shallow and deep-water habitats also differ from each other.

![*Silurian palaeogeography, as reconstructed by the PALEOMAP PaleoAtlas project [@Scotese2016].*](PALEOMAP_atlas_map75a_Llandovery_430.png)

## Prepare taxonomic occurrence data

PBDB downloads of (1) all marine collections and (2) all Brachiopoda occurrences with accepted names at genus-ranking or below are available as attached datasets in the `divvy` package. Both datasets are subset to relevant columns and are cleaned to remove records missing coordinates. For more information, load the package and query the data objects from the console with `?collSilur` and `?occSilur`. The script to download and vet the raw data is available as `format-data.R` within the GitHub repository [data-raw folder (link)](https://github.com/GawainAntell/divvy/tree/main/data-raw).


```r
# terra and sf have all the necessary helper functions for spatial operations
library(terra)
library(sf)
library(divvy)
data(collSilur)
data(occSilur)
```

The taxonomic hierarchy and stratigraphic ranges of the occurrence data are standardised against the Sepkoski Compendium already, following the `fossilbrush` package vignette [@fossilbrush]. A further element of cleaning is removal of taxa that occur only once ('space-time singletons' *sensu* @Antell2020). These records contain insufficient information to contribute to geographic range-size calculations.

```r
nmFreq <- table(occSilur$genus)
sss <- nmFreq[nmFreq == 1] |> names()
sssRows <- occSilur$genus %in% sss
occSilur <- occSilur[ !sssRows, ]
length(sss)
```

```
#> [1] 52
```
This step removes 52 taxa with singular occurrence records.

To disregard trivial differences in coordinate values between points from the same or adjacent localities, the next data treatment is conversion of vector (point) data to raster grid cells. For each PBDB record, the ID and centroid coordinates of its encompassing raster cell are saved in new columns. These cell values will be used in spatial calculations later to improve computation time substantially compared to operations on vector point data.

```r
# initialise Equal Earth projected coordinates
rWorld <- rast()
prj <- 'EPSG:8857'
rPrj <- project(rWorld, prj, res = 200000) # 200,000m is approximately 2 degrees
values(rPrj) <- 1:ncell(rPrj)

# coordinate column names for the current and target coordinate reference system
xyCartes <- c('paleolng','paleolat')
xyCell   <- c('cellX','cellY')

# collection data
llColl <- vect(collSilur, geom = xyCartes, crs = 'epsg:4326')
prjColl <- project(llColl, prj)
collSilur$cell <- cells(rPrj, prjColl)[,'cell']
collSilur[, xyCell] <- xyFromCell(rPrj, collSilur$cell)

# taxon occurrences
llOccs <- vect(occSilur, geom = xyCartes, crs = 'epsg:4326')
prjOccs <- project(llOccs, prj)
occSilur$cell <- cells(rPrj, prjOccs)[,'cell']
occSilur[, xyCell] <- xyFromCell(rPrj, occSilur$cell)
```

## Categorise environment associated with occurrences

To compare ecological parameters between preferred environments, the 23 categories in the PBDB `environment` field must first be reclassified as a binary variable for proximal and deep-water settings. The following division of marine environments is copied from scripts of @Antell2020, who adapted the classification of @Nurnberg2013.

```r
proxTerms <- c("coastal indet.", "delta front", "delta plain",
               "deltaic indet.", "estuary/bay", "foreshore", "interdistributary bay",
               "lagoonal", "lagoonal/restricted shallow subtidal",
               "marginal marine indet.", "open shallow subtidal",
               "fluvial-deltaic indet.", "paralic indet.",
               "peritidal", "prodelta", "sand shoal",
               "shallow subtidal indet.", "shoreface",
               "transition zone/lower shoreface",
               "intrashelf/intraplatform reef", "reef, buildup or bioherm",
               "perireef or subreef", "platform/shelf-margin reef")
deepTerms <- c("basinal (carbonate)", "basinal (siliceous)",
               "basinal (siliciclastic)", "deep-water indet.",
               "deep subtidal indet.", "deep subtidal ramp",
               "deep subtidal shelf", "offshore", "offshore indet.",
               "offshore shelf", "slope", "submarine fan", "offshore ramp",
               "basin reef", "slope/ramp reef")
```

The most accurate environmental information will be that associated directly with occurrence records. Where available, that is the information this script will prioritise. However, there are many instances of a blank or indeterminate `environment` field. In those cases, a way to infer probable environment is to determine the predominant environment in the local grid cell based on the environment recorded from any PBDB collections in that cell. A collection in the PBDB can contain any number of occurrences of any number of clades but is tied to a point coordinate---therefore, collections are a rich source of environmental data that is largely independent of occurrences for an individual clade.

`divvy` provides the `classRast()` function to convert environmental occurrence points to a raster grid with a dominant environmental class in each cell. In the use below, the function classifies grid cells as proximal or deep-water whenever more than 80% of collections are affiliated with a single environment. The template raster passed to `classRast()` here has the same resolution, extent, and coordinate reference system as applied above to the occurrence data. <a id="raster_map"></a>

```r
collSilur$bath <- NA
collSilur$bath[collSilur$environment %in% proxTerms] <- 'prox'
collSilur$bath[collSilur$environment %in% deepTerms] <- 'deep'
bathRast <- classRast(rPrj, dat = collSilur, cutoff = 0.8,
                      xy = xyCell, env = 'bath')
plot(bathRast, col = c("#1B9E77", "#D95F02", "black"))
```

<div class="figure" style="text-align: center">
<img src="classify-collections-enviro-1.png" alt="World map where a mix of grid cells labelled deep, shallow, and indeterminate are scattered around the outline of Gondwanaland and Laurasia. Most of the points fall close to the coastlines and are concentrated around Laurasia." width="650" height="325" />
<p class="caption">*Predominant environment recorded in PBDB collections*</p>
</div>
A plot of the environmental distribution inferred from collection data traces the mid-latitude continents [(figured above)](#atlas_map), albeit with gaps. Very few occurrences appear along the coasts of Gondwanaland, perhaps largely due to the under-sampling and under-reporting of the corresponding modern continents relative to North America and Europe. Also note that sampling of northern extents of Laurasia occurs almost exclusively in deep-water settings.

It is now possible to infer binary water depth for brachiopod occurrence records. Where available, the classification will reflect the environment directly associated with occurrences, and otherwise will be inferences from the collections-based raster.

```r
# create a new 'bathCell' column to note the collections-inferred environment
occSilur$bathCell <- occSilur$bath <- extract(bathRast, prjOccs)[['mainClass']]

# refine the enviro where possible based on observations directly linked to occs
occSilur$bath[occSilur$environment %in% proxTerms] <- 'prox'
occSilur$bath[occSilur$environment %in% deepTerms] <- 'deep'
```

A quick inspection reveals roughly equal numbers of occurrences in each environment. However, there are half again as many shallow as deep-water grid cells as indicated by all marine collections---a result that points to the necessity of subsampling. Without accounting for the greater observation of proximal habitats, if shallow-water specialists turn out to have equal or larger geographic occupancies than deep-water specialists it might be impossible to tell if this result is due purely to sampling or also to ecology.

```r
table(occSilur$bath) # environment inferred directly from records
```

```
#> 
#>   deep   prox indet. 
#>   4966   4124   2035
```

```r
freq(bathRast, bylayer = FALSE)
```

```
#>    value count
#> 1   deep   141
#> 2   prox   226
#> 3 indet.    70
```

Inspection also reveals the majority of environments inferred from collection data nearby do match the environments associated with occurrence records directly. Most of the discrepancies occur when collection-derived environment was indeterminate and occurrence data contained specific environment terms.

```r
# what proportion of environments associated with individual records
# match environmental inferences from nearby collections' data?
which(occSilur$bath == occSilur$bathCell) |>
  length()/nrow(occSilur)
```

```
#> [1] 0.6516729
```

```r
# compare number of indeterminate enviro from collections vs above from records
table(occSilur$bathCell)
```

```
#> 
#>   deep   prox indet. 
#>   3616   3330   4158
```

## Assign per-taxon environmental preferences

Now that the data are standardised to the same raster gridding and environmental classification, it is possible to infer the environmental preferences and niche breadth of all taxa. The custom function below labels a focal taxon as a specialist for an environment if more than a threshold proportion of its occurrences (here, 70%) lie within a single habitat type.

```r
# categorise preferences as binary or nonspecific/eurytopic
pref <- function(env, cutoff){
  envTbl <- table(env)
  if (sum(envTbl[1:2])==0) { 'unk' } else {
    envTbl <- envTbl[1:2] / sum(envTbl[1:2]) # proportional enviro abundance
    if (any(envTbl[1:2] > cutoff)) { # check if either envrio is predominant
      ifelse(envTbl[[1]] > cutoff, names(envTbl)[1], names(envTbl)[2])
    } else { 'both' }
  }
}
# The bathymetry column of occurrences is an ordered factor, so levels
# 1 and 2 subset by the function correspond to the two environment types
levels(occSilur$bath)
```

```
#> [1] "deep"   "prox"   "indet."
```

```r
# tag each row as from a taxon with deep, proximal, or eurytopic preference
taxa <- unique(occSilur$genus)
occSilur$pref <- NA
cutoff <- 0.7
for (tax in taxa){
 taxBool <- occSilur$genus == tax
 envOccs <- occSilur$bath[taxBool]
 occSilur$pref[taxBool] <- pref(envOccs, cutoff)
}
```

A simple way to split the data into comparison groups used in analysis below is to divide the occurrence dataset into three `data.frame`s: one for deep-water specialists, one for proximal-environment specialists, and one for eurytopic taxa with respect to depth. This data structure is straightforward to illustrate subsampling in the section below but is not the only way to proceed with analysis; for instance, one could organise the data subsets in a `list` or `array` instead, and then perform subsampling across that structure's indices.

```r
deep <- occSilur[ occSilur$pref == 'deep', ] |>
  uniqify(taxVar = 'genus', xy = xyCell)
prox <- occSilur[ occSilur$pref == 'prox', ] |>
  uniqify(taxVar = 'genus', xy = xyCell)
eury <- occSilur[ occSilur$pref == 'both', ] |>
  uniqify(taxVar = 'genus', xy = xyCell)
# the divvy uniqify() function removes duplicate taxon-cell combinations
# for smaller memory use and faster calculations below

# how many taxa specialise in each?
taxaD <- unique(deep$genus)
taxaP <- unique(prox$genus)
taxaE <- unique(eury$genus)
paste(length(taxaD), 'deep;', length(taxaP), 'shallow;', length(taxaE), 'both')
```

```
#> [1] "94 deep; 61 shallow; 100 both"
```
The 70% threshold used to define taxon affinities leads to approximately equal numbers of deep and eurytopic taxa and a smaller sample of proximal specialists.

## Spatially subsample each environment

It is now an appropriate point in the workflow to standardise the spatial coverage of occurrences in each comparison group. `divvy` provides several options of spatial subsampling functions; in the code chunk below, we apply the `cookies()` subsampling routine to the datasets of deep, proximal, and eurytopic taxa. The arguments `reps`, `siteQuota`, and `radius` define the number of subsampling iterations run, the number of localities to include in each subsample, and the maximum radius of each circular subsampling region, respectively.

The value of `reps` is set at a number high enough to ensure all possible points are captured in at least one subsample, while keeping the returned object (list of subsampled `data.frame`s) at a manageable size. Increasing the number of `reps` only slightly prolongs the computational time of subsampling but will linearly slow the runtime of range-size calculations on subsamples in the following sections. In Phanerozoic analyses, @Antell2020 performed 500 replicates of circular subsampling per time step, and @Close2017 did 100 (20 replicates from each of 5 regions defined by minimum spanning trees).

The values of `siteQuota` and `radius` are a trade-off between increasing the sample size of localities within each subsample to precisely characterise geographic ranges while limiting the total extent of the subsampling region not to exceed the approximate size of an ocean basin or biogeographic zone.


```r
reps <- 500 # number of subsamples to draw
siteQuota <- 15 # number of sites (equal-area grid cells) in each subsample
r <- 1000 # radial distance in km

set.seed(8)
sampD <- cookies(dat = deep,
                 xy = xyCell, iter = reps,
                 nSite = siteQuota,
                 r = r, weight = TRUE,
                 crs = prj, output = 'full')

set.seed(9)
sampP <- cookies(dat = prox,
                 xy = xyCell, iter = reps,
                 nSite = siteQuota,
                 r = r, weight = TRUE,
                 crs = prj, output = 'full')

set.seed(10)
sampE <- cookies(dat = eury,
                 xy = xyCell, iter = reps,
                 nSite = siteQuota,
                 r = r, weight = TRUE,
                 crs = prj, output = 'full')
```

For further details of subsampling using `divvy`, including the `clustr()` function to define subsamples based on minimum spanning tree diameter instead of circular regional constraints, consult the other vignettes and @Antell2023.

## Summed Common Occurrence Rate between environments

We need to parameterise geographic range sizes in some way to make quantitative comparisons between ecological groups. One way to do so is with a measure of cosmopolitanism calculated on assemblage-level data (rather than individual taxa), such as the Summed Common species/taxon Occurrence Rate (SCOR). SCOR reflects the degree to which taxa are common/widespread and is decoupled from richness or abundance [@Hannisdal2012]. `divvy` provides the `sdSumry()` function to compute summary statistics about the geographic coverage and diversity of a dataset, including SCOR; conveniently, `sdSumry()` can take a `list` of subsampled occurrence data subsets as its data argument and return a single `data.frame` with columns for the summary variables and a row for each subsample.

```r
metaD <- sdSumry(sampD, xyCell, 'genus', prj)
metaP <- sdSumry(sampP, xyCell, 'genus', prj)
metaE <- sdSumry(sampE, xyCell, 'genus', prj)
```

SCOR is calculated as the sum across taxa of the log probability of incidence: very widespread taxa make a large contribution to an assemblage SCOR, while rare taxa have relatively little influence. When a taxon is present in *all* sampled locations, its log probability of incidence is infinite. Infinity is nonsensical in an empirical comparison, so let's screen out subsamples with those values (12 in total, of 1500 subsamples).

```r
# check for infinity values of SCOR and remove them
is.infinite(metaD$SCOR) |> any() # none in deep habitat dataset
```

```
#> [1] FALSE
```

```r
is.infinite(metaP$SCOR) |> any() # but some in the prox habitat dataset
```

```
#> [1] TRUE
```

```r
is.infinite(metaE$SCOR) |> any() # and some in the all-habitats dataset
```

```
#> [1] TRUE
```

```r
infP <- is.infinite(metaP$SCOR)
metaP <- metaP[ !infP, ]
infE <- is.infinite(metaE$SCOR)
metaE <- metaE[ !infE, ]
sum(infP)
```

```
#> [1] 10
```

```r
sum(infE)
```

```
#> [1] 2
```

For the permutation hypothesis tests below, it would be ideal for comparison groups to match in sample size. The previous step removed 10 samples from one group, 2 from another, and none from the third---a difference that could be remedied by omitting another 8 and 10 samples from the latter two groups, respectively. Samples could be selected at random. However, one could argue it would unfairly increase the odds of supporting the hypothesis of larger ranges in deep than shallow habitats to randomly omit 10 deep samples but selectively omit the 10 shallow samples where at least one taxon was universally distributed. For a conservative comparison, one could instead omit deep and eurytopic samples with largest values of SCOR.

```r
# make sample size (of subsamples) equal, by removing ones with largest SCOR
lowestN <- min(nrow(metaD), nrow(metaP), nrow(metaE))
sortD <- order(metaD$SCOR, decreasing = FALSE)
metaD <- metaD[sortD, ]
metaD <- metaD[1:lowestN, ]
sortE <- order(metaE$SCOR, decreasing = FALSE)
metaE <- metaE[sortE, ]
metaE <- metaE[1:lowestN, ]
```
Ultimately, the choice of how to perform this selection step makes little difference to results, because it affects only 2% of subsamples.

At last we can compare the degree to which taxa of each ecological group are widespread, within normalised amounts of habitat. There are various test statistics one could calculate. Here are custom functions for a bootstrapping approach that capitalises on the existing division of subsamples to generate nonparametric estimates of how probable SCOR values from one group are to exceed those of another.

```r
# how often does the value in group 1 exceed that in group 2 paired samples?
# randomly shuffle subsample ordering before comparing
testDiff <- function(dat1, dat2, vTest){
  n <- nrow(dat1)
  dat1 <- dat1[ sample(1:n), ]
  dat2 <- dat2[ sample(1:n), ]
  h1obs <- dat1[, vTest] > dat2[, vTest]
  sum(h1obs) / n
}

# repeat the comparison many times to generate a 95% confidence interval
diffDist <- function(dat1, dat2, vTest, nReps){
  testStatDist <- testDiff(dat1, dat2, vTest) |> replicate(n = nReps)
  avg <- mean(testStatDist)
  ci95 <- c(0.025, 0.975)
  null95 <- quantile(testStatDist, ci95)
  names(null95) <- c('lwrCI', 'uprCI')
  c(null95, 'mean' = avg)
}
```

When we apply the boostrapping test, the results indicate a random region of deep-water habitat contains more taxa with widespread ranges than a proximal habitat two-thirds of the time. Assemblages of eurytopic taxa from any random region almost always contained representatives with more widespread ranges than assemblages of either habitat specialist.

```r
# do deep-water environments harbor more cosmopolitan taxa?
diffDist(metaD, metaP, vTest = 'SCOR', nReps = 1000)
```

```
#>     lwrCI     uprCI      mean 
#> 0.6122449 0.6592347 0.6349163
```

```r
# what about eurytopic taxa vs. deep-water specialists?
diffDist(metaE, metaD, vTest = 'SCOR', nReps = 1000)
```

```
#>     lwrCI     uprCI      mean 
#> 0.9755102 0.9938776 0.9844980
```

```r
# and if eurytopic species are more widespread than deep,
# they'll also be more widespread than proximal-dwellers
# (since more deep-water species are widespread than proximal are)
diffDist(metaE, metaP, vTest = 'SCOR', nReps = 1000)
```

```
#>     lwrCI     uprCI      mean 
#> 0.9816327 0.9979592 0.9894939
```

Another way one could compare SCOR between groups is inspecting the distribution of values in each comparison group. Overall, SCOR values from assemblages of deep-water taxa were slightly larger than those of proximal taxa, while eurytopic taxa were substantially more common than either of those groups (SCOR values 4--5 times as large). Again, note that because these results are derived from spatially standardised data, these numbers account for the fact that eurytopic taxa as defined here had many more cells of suitable habitat available to occupy.

```r
summary(metaD$SCOR)
```

```
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   2.183   3.174   3.689   3.700   4.153   5.878
```

```r
summary(metaP$SCOR)
```

```
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   1.090   2.028   3.297   3.135   3.942   5.550
```

```r
summary(metaE$SCOR)
```

```
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   3.383   9.796  15.042  14.959  19.822  28.715
```

### Contrast results from non-subsampled data

What would the results and conclusions have been without going to the effort to spatially subsample the original occurrence data? For comparison with the average SCOR values listed above, the following code uses the `sdSumry()` function again, this time on the non-standardised full datasets.

```r
sdSumry(deep, xyCell, 'genus', prj) |> round(digits = 2)
```

```
#>      nOcc nLoc centroidX centroidY latRange greatCircDist meanPairDist minSpanTree SCOR nTax
#> [1,]  713  250  -3192759  -1735760   166.32      33567.84       8071.8    130946.3    3   94
```

```r
sdSumry(prox, xyCell, 'genus', prj) |> round(digits = 2)
```

```
#>      nOcc nLoc centroidX centroidY latRange greatCircDist meanPairDist minSpanTree SCOR nTax
#> [1,]  432  204  -3766508   -957619   166.32      33567.84      7845.98      127435 2.27   61
```

```r
sdSumry(eury, xyCell, 'genus', prj) |> round(digits = 2)
```

```
#>      nOcc nLoc centroidX centroidY latRange greatCircDist meanPairDist minSpanTree SCOR nTax
#> [1,] 3462  404  -3134553  -1227392   166.32      33567.84      8321.72    168862.1 9.46  100
```

In this case the overall direction of results is the same (deep-water taxa tend to be slightly more widespread than shallow-water taxa, and eurytopic taxa are most widespread of all). However, subsampling makes the difference between eurytopic taxa and specialists more pronounced---SCOR values from raw data are estimated about three times larger (rather than 4--5 times larger) compared to shallow- and deep-water taxa.

## Average taxon range size between environments

The preceding section illustrates a way one might quantitatively compare an ecological variable calculated from assemblage or community data. There are many cases where one would be interested in a taxon-specific variable instead, e.g. body size or other traits. The present section demonstrates a way to extract and summarise focal taxon data from subsampled assemblage data. The trait of interest in this instance is geographic range size, which can be parameterised in a variety of ways. The `rangeSize()` function in `divvy` returns five measures of geographic distribution, two of which the following code compares between ecological study groups.


```r
# custom function to find all occurrences of focal taxon in a subsample
# and return summary measures of geographic distribution from them
subsampRange <- function(taxa, df, taxCol, coordCols, crs){
  taxRange <- function(taxon){
    taxBool <- df[, taxCol] == taxon
    if (sum(taxBool) == 0){ # case where taxon unsampled
      rep(NA, 7)
    } else {
      taxCoords <- df[taxBool, coordCols]
      rangeSize(taxCoords, crs)
    }
  }
  taxa <- sort(taxa)
  rngMat <- sapply(taxa, taxRange)
  rownames(rngMat) <- c('occurrences', 'centroidX', 'centroidY', 'latRange',
                        'greatCircDist', 'meanPairDist', 'minSpanTree')
  rngDf <- t(rngMat) |> data.frame()
  return(rngDf)
}

# run the function on all subsamples of each comparison set. warning:
# these are intensive computations so might be slow (minutes of runtime)
rngD <- lapply(sampD, FUN = subsampRange, taxa = taxa, taxCol = 'genus',
               coordCols = xyCell, crs = prj)
rngP <- lapply(sampP, FUN = subsampRange, taxa = taxa, taxCol = 'genus',
               coordCols = xyCell, crs = prj)
rngE <- lapply(sampE, FUN = subsampRange, taxa = taxa, taxCol = 'genus',
               coordCols = xyCell, crs = prj)
```

The above code chunk produces a `list` of range-size data for all genera in all subsamples. The lines below average all values for each focal taxon to consolidate these intermediate calculations into per-taxon range data for final analysis.

```r
# summarize a focal species' range size data across all subsamples
taxRange <- function(a, rngL, singletons = FALSE){
  tmp <- lapply(rngL, function(x){
    aBool <- row.names(x) == a
    x[ aBool, ]
  })
  aDf <- do.call(rbind, tmp)
  # option: only consider subsamples where species is sampled multiple times
  if (singletons == FALSE){
    singl <- aDf$occ == 1
    aDf <- aDf[ !singl, ]
  }
  colMeans(aDf, na.rm = TRUE)
}

# apply summary function over all taxa in subsamples of preferred enviro
rngAvgD <- sapply(taxaD, taxRange, rngL = rngD) |>
  t() |> data.frame()
rngAvgP <- sapply(taxaP, taxRange, rngL = rngP) |>
  t() |> data.frame()
rngAvgE <- sapply(taxaE, taxRange, rngL = rngE) |>
  t() |> data.frame()
```
There are cases where a taxon is observed only once in a given subsample, which is uninformative for any geographic range measure except raw occurrence counts. The custom argument `singletons = FALSE` omits these cases from consideration for the focal taxon range-size average.

Some taxa are never observed multiple times even with many subsamples. After omitting these taxa, the sample size of genera in each comparison group shrinks to 43 in deep water, 24 in proximal habitats, and 95 occupying a breadth of depths.

```r
rngAvgD <- rngAvgD[ complete.cases(rngAvgD), ]
rngAvgP <- rngAvgP[ complete.cases(rngAvgP), ]
rngAvgE <- rngAvgE[ complete.cases(rngAvgE), ]
sapply(list('D' = rngAvgD, 'P' = rngAvgP, 'E' = rngAvgE), nrow)
```

```
#>  D  P  E 
#> 43 24 95
```

The previous section, assessing SCOR as an example of comparing assemblage-based metrics, demonstrated a bootstrapping approach to generate results. As a complementary example, the following chunk uses Frequentist two-sample tests to address each hypothesis. For simplicity, only two measure of geographic range are compared here (grid cell occupancy and summed length of minimum spanning tree); the reader could easily substitute any of the other three in the datasets (latitudinal range, great circle distance, and mean pairwise distance).

The occurrence or occupancy measure of range size is count data, so a Wilcoxon (Mann-Whitney) test is appropriate as a nonparametric test. The second range-size metric, summed length of minimum spanning tree, follows a normal distribution after square-root transformation; a student's t-test makes parametric comparisons on it.

```r
wilcox.test(rngAvgD$occurrences, rngAvgP$occurrences, alternative='g') # deep vs shallow
```

```
#> Warning in wilcox.test.default(rngAvgD$occurrences, rngAvgP$occurrences, : cannot compute exact p-value with ties
```

```
#> 
#> 	Wilcoxon rank sum test with continuity correction
#> 
#> data:  rngAvgD$occurrences and rngAvgP$occurrences
#> W = 538, p-value = 0.3876
#> alternative hypothesis: true location shift is greater than 0
```

```r
wilcox.test(rngAvgE$occurrences, rngAvgD$occurrences, alternative='g') # eurytopic vs deep
```

```
#> 
#> 	Wilcoxon rank sum test with continuity correction
#> 
#> data:  rngAvgE$occurrences and rngAvgD$occurrences
#> W = 2794, p-value = 0.0002696
#> alternative hypothesis: true location shift is greater than 0
```

```r
wilcox.test(rngAvgE$occurrences, rngAvgP$occurrences, alternative='g') # eurytopic vs shallow
```

```
#> 
#> 	Wilcoxon rank sum test with continuity correction
#> 
#> data:  rngAvgE$occurrences and rngAvgP$occurrences
#> W = 1603, p-value = 0.001076
#> alternative hypothesis: true location shift is greater than 0
```

```r
rngAvgD$minSpanTree <- sqrt(rngAvgD$minSpanTree)
rngAvgP$minSpanTree <- sqrt(rngAvgP$minSpanTree)
rngAvgE$minSpanTree <- sqrt(rngAvgE$minSpanTree)
t.test(rngAvgD$minSpanTree, rngAvgP$minSpanTree, alternative='g') # deep vs shallow
```

```
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  rngAvgD$minSpanTree and rngAvgP$minSpanTree
#> t = 0.10779, df = 38.352, p-value = 0.4574
#> alternative hypothesis: true difference in means is greater than 0
#> 95 percent confidence interval:
#>  -3.728928       Inf
#> sample estimates:
#> mean of x mean of y 
#>  26.92918  26.67442
```

```r
t.test(rngAvgE$minSpanTree, rngAvgD$minSpanTree, alternative='g') # eurytopic vs deep
```

```
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  rngAvgE$minSpanTree and rngAvgD$minSpanTree
#> t = 3.153, df = 82.129, p-value = 0.001129
#> alternative hypothesis: true difference in means is greater than 0
#> 95 percent confidence interval:
#>  2.119235      Inf
#> sample estimates:
#> mean of x mean of y 
#>  31.41559  26.92918
```

```r
t.test(rngAvgE$minSpanTree, rngAvgD$minSpanTree, alternative='g') # eurytopic vs shallow
```

```
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  rngAvgE$minSpanTree and rngAvgD$minSpanTree
#> t = 3.153, df = 82.129, p-value = 0.001129
#> alternative hypothesis: true difference in means is greater than 0
#> 95 percent confidence interval:
#>  2.119235      Inf
#> sample estimates:
#> mean of x mean of y 
#>  31.41559  26.92918
```

The results of per-taxon range size comparisons are less clear than those of assemblage-based SCOR comparisons above. There is no discernible difference in mean range size between deep- and shallow-water species whether measured as occurrence counts or minimum spanning tree distances. However, eurytopic taxa still measure larger than either specialist group, even by conservative thresholds for significance: eurytopic taxa occur in nearly one more grid cell of the 15 in every subsample, and span about 270 km farther in tree length connecting occupied cells. (Recall the tree lengths printed here are square-root transformed.)

```r
vars <- c('occurrences', 'minSpanTree')
sapply(list('D' = rngAvgD, 'P' = rngAvgP, 'E' = rngAvgE),
       function(x) apply(x[,vars], 2, mean)
       )
```

```
#>                     D         P        E
#> occurrences  2.863038  2.898537  3.66210
#> minSpanTree 26.929182 26.674421 31.41559
```

### Contrast results from non-subsampled data

As with the assemblage-based analysis above, we can investigate the inferences that would have resulted from analysing the original occurrence data, without subsampling.

```r
# calculate geographic range size on alloccurrences (not subsampled)
unsamp <- subsampRange(taxa, df = occSilur, taxCol = 'genus', coordCols = xyCell, crs = prj)
unsampD <- unsamp[rownames(unsamp) %in% taxaD,]
unsampP <- unsamp[rownames(unsamp) %in% taxaP,]
unsampE <- unsamp[rownames(unsamp) %in% taxaE,]

wilcox.test(unsampD$occurrences, unsampP$occurrences, alternative='g') # deep vs shallow
```

```
#> 
#> 	Wilcoxon rank sum test with continuity correction
#> 
#> data:  unsampD$occurrences and unsampP$occurrences
#> W = 2778, p-value = 0.6299
#> alternative hypothesis: true location shift is greater than 0
```

```r
wilcox.test(unsampE$occurrences, unsampD$occurrences, alternative='g') # eurytopic vs deep
```

```
#> 
#> 	Wilcoxon rank sum test with continuity correction
#> 
#> data:  unsampE$occurrences and unsampD$occurrences
#> W = 7878, p-value < 2.2e-16
#> alternative hypothesis: true location shift is greater than 0
```

```r
wilcox.test(unsampE$occurrences, unsampP$occurrences, alternative='g') # eurytopic vs shallow
```

```
#> 
#> 	Wilcoxon rank sum test with continuity correction
#> 
#> data:  unsampE$occurrences and unsampP$occurrences
#> W = 5177.5, p-value = 5.463e-14
#> alternative hypothesis: true location shift is greater than 0
```

```r
unsampE$minSpanTree <- sqrt(unsampE$minSpanTree)
unsampD$minSpanTree <- sqrt(unsampD$minSpanTree)
unsampP$minSpanTree <- sqrt(unsampP$minSpanTree)
t.test(unsampD$minSpanTree, unsampP$minSpanTree, alternative='g') # deep vs shallow
```

```
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  unsampD$minSpanTree and unsampP$minSpanTree
#> t = 1.4578, df = 128.32, p-value = 0.07368
#> alternative hypothesis: true difference in means is greater than 0
#> 95 percent confidence interval:
#>  -2.172339       Inf
#> sample estimates:
#> mean of x mean of y 
#> 105.78370  89.87555
```

```r
t.test(unsampE$minSpanTree, unsampD$minSpanTree, alternative='g') # eurytopic vs deep
```

```
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  unsampE$minSpanTree and unsampD$minSpanTree
#> t = 5.5061, df = 169.6, p-value = 6.687e-08
#> alternative hypothesis: true difference in means is greater than 0
#> 95 percent confidence interval:
#>  41.22068      Inf
#> sample estimates:
#> mean of x mean of y 
#>  164.7019  105.7837
```

```r
t.test(unsampE$minSpanTree, unsampD$minSpanTree, alternative='g') # eurytopic vs shallow
```

```
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  unsampE$minSpanTree and unsampD$minSpanTree
#> t = 5.5061, df = 169.6, p-value = 6.687e-08
#> alternative hypothesis: true difference in means is greater than 0
#> 95 percent confidence interval:
#>  41.22068      Inf
#> sample estimates:
#> mean of x mean of y 
#>  164.7019  105.7837
```

In the bootstrapped analysis of SCOR values, subsampling happened to amplify the differences between comparison groups. In the Frequentist analysis of per-taxon range size in this section, subsampling reduced the quantitative magnitude of differences between comparison groups. To some extent, this muting of absolute differences makes sense given the inherent truncation of geographic ranges due to subsampling. However, looking back at the [map of PBDB data distribution](#raster_map), it is also possible to hypothesise that biased sampling artificially inflated group differences. For instance, as noted when plotting the map, sampling coverage of deep-water habitats extends much farther around Laurasia compared to shallow-water habitats. Without accounting for this influence, it is impossible to attribute the greater minimum spanning tree length of deep-water species to ecology alone. After subsampling, although there may be fewer results that are clear, the analyst can trust the results that do emerge as meaningful for biological interpretation.

## References
