Additive Diversity Partitioning and Hierarchical Null Model Testing

Description

In additive diversity partitioning, mean values of alpha diversity at lower levels of a sampling hierarchy are compared to the total diversity in the entire data set (gamma diversity). In hierarchical null model testing, a statistic returned by a function is evaluated according to a nested hierarchical sampling design (`hiersimu`).

Usage

```adipart(formula, data, index=c("richness", "shannon", "simpson"),
weights=c("unif", "prop"), relative = FALSE, nsimul=99, ...)
hiersimu(formula, data, FUN, location = c("mean", "median"),
relative = FALSE, drop.highest = FALSE, nsimul=99, ...)
## S3 method for class 'adipart':
print(x, ...)
## S3 method for class 'hiersimu':
print(x, ...)
```

Arguments

 `formula` A two sided model formula in the form `y ~ x`, where `y` is the community data matrix with samples as rows and species as column. Right hand side (`x`) must contain factors referring to levels of sampling hierarchy, terms from right to left will be treated as nested (first column is the lowest, last is the highest level). These variables must be factors in order to unambiguous handling. Interaction terms are not allowed. `data` A data frame where to look for variables defined in the right hand side of `formula`. If missing, variables are looked in the global environment. `index` Character, the diversity index to be calculated (see Details). `weights` Character, `"unif"` for uniform weights, `"prop"` for weighting proportional to sample abundances to use in weighted averaging of individual alpha values within strata of a given level of the sampling hierarchy. `relative` Logical, if `TRUE` then alpha and beta diversity values are given relative to the value of gamma for function `adipart`. `nsimul` Number of permutation to use if `matr` is not of class 'permat'. If `nsimul = 0`, only the `FUN` argument is evaluated. It is thus possible to reuse the statistic values without using a null model. `FUN` A function to be used by `hiersimu`. This must be fully specified, because currently other arguments cannot be passed to this function via `...`. `location` Character, identifies which function (mean or median) is to be used to calculate location of the samples. `drop.highest` Logical, to drop the highest level or not. When `FUN` evaluates only arrays with at least 2 dimensions, highest level should be dropped, or not selected at all. `x` An object to print. `...` Other arguments passed to functions, e.g. base of logarithm for Shannon diversity, or `method`, `thin` or `burnin` arguments for `oecosimu`.

Details

Additive diversity partitioning means that mean alpha and beta diversity adds up to gamma diversity, thus beta diversity is measured in the same dimensions as alpha and gamma (Lande 1996). This additive procedure is than extended across multiple scales in a hierarchical sampling design with i = 1, 2, 3, ..., m levels of sampling (Crist et al. 2003). Samples in lower hierarchical levels are nested within higher level units, thus from i=1 to i=m grain size is increasing under constant survey extent. At each level i, α_i denotes average diversity found within samples.

At the highest sampling level, the diversity components are calculated as

beta_m = gamma - alpha_m

For each lower sampling level as

beta_i = alpha_i+1 - alpha_i

Then, the additive partition of diversity is

gamma = alpha_1 + sum(beta_i)

Average alpha components can be weighted uniformly (`weight="unif"`) to calculate it as simple average, or proportionally to sample abundances (`weight="prop"`) to calculate it as weighted average as follows

alpha_i = sum(D_ij*w_ij)

where D_{ij} is the diversity index and w_{ij} is the weight calculated for the jth sample at the ith sampling level.

The implementation of additive diversity partitioning in `adipart` follows Crist et al. 2003. It is based on species richness (S, not S-1), Shannon's and Simpson's diversity indices stated as the `index` argument.

The expected diversity components are calculated `nsimul` times by individual based randomisation of the community data matrix. This is done by the `"r2dtable"` method in `oecosimu` by default.

`hiersimu` works almost the same as `adipart`, but without comparing the actual statistic values returned by `FUN` to the highest possible value (cf. gamma diversity). This is so, because in most of the cases, it is difficult to ensure additive properties of the mean statistic values along the hierarchy.

Value

An object of class 'adipart' or 'hiersimu' with same structure as 'oecosimu' objects.

Author(s)

P'eter S'olymos, solymos@ualberta.ca

References

Crist, T.O., Veech, J.A., Gering, J.C. and Summerville, K.S. (2003). Partitioning species diversity across landscapes and regions: a hierarchical analysis of α, β, and gamma-diversity. Am. Nat., 162, 734–743.

Lande, R. (1996). Statistics and partitioning of species diversity, and similarity among multiple communities. Oikos, 76, 5–13.

See `oecosimu` for permutation settings and calculating p-values.

Examples

```data(mite)
data(mite.xy)
data(mite.env)
## Function to get equal area partitions of the mite data
cutter <- function (x, cut = seq(0, 10, by = 2.5)) {
out <- rep(1, length(x))
for (i in 2:(length(cut) - 1))
out[which(x > cut[i] & x <= cut[(i + 1)])] <- i
return(as.factor(out))}
## The hierarchy of sample aggregation
levsm <- data.frame(
l1=as.factor(1:nrow(mite)),
l2=cutter(mite.xy\$y, cut = seq(0, 10, by = 2.5)),
l3=cutter(mite.xy\$y, cut = seq(0, 10, by = 5)),
l4=cutter(mite.xy\$y, cut = seq(0, 10, by = 10)))
## Let's see in a map
par(mfrow=c(1,3))
plot(mite.xy, main="l1", col=as.numeric(levsm\$l1)+1)
plot(mite.xy, main="l2", col=as.numeric(levsm\$l2)+1)
plot(mite.xy, main="l3", col=as.numeric(levsm\$l3)+1)
par(mfrow=c(1,1))