Additive Diversity Partitioning and Hierarchical Null Model Testing

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(...)
# Default S3 method
adipart(y, x, index=c("richness", "shannon", "simpson"),
    weights=c("unif", "prop"), relative = FALSE, nsimul=99,
    method = "r2dtable", ...)
# S3 method for class 'formula'
adipart(formula, data, index=c("richness", "shannon", "simpson"),
    weights=c("unif", "prop"), relative = FALSE, nsimul=99,
    method = "r2dtable", ...)

hiersimu(...)
# Default S3 method
hiersimu(y, x, FUN, location = c("mean", "median"),
    relative = FALSE, drop.highest = FALSE, nsimul=99,
    method = "r2dtable", ...)
# S3 method for class 'formula'
hiersimu(formula, data, FUN, location = c("mean", "median"),
    relative = FALSE, drop.highest = FALSE, nsimul=99,
    method = "r2dtable", ...)

Arguments

y: A community matrix.
x: A matrix with same number of rows as in y, columns coding the levels of sampling hierarchy. The number of groups within the hierarchy must decrease from left to right. If x is missing, function performs an overall decomposition into alpha, beta and gamma diversities.
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 be grouping variables 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). The formula will add a unique indentifier to rows and constant for the rows to always produce estimates of row-level alpha and overall gamma diversities. You must use non-formula interface to avoid this behaviour. 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 permutations to use. If nsimul = 0, only the FUN argument is evaluated. It is thus possible to reuse the statistic values without a null model.
method: Null model method: either a name (character string) of a method defined in make.commsim or a commsim function. The default "r2dtable" keeps row sums and column sums fixed. See oecosimu for Details and Examples.
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.
...: 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 diversities add up to gamma diversity, thus beta diversity is measured in the same dimensions as alpha and gamma (Lande 1996). This additive procedure is then extended across multiple scales in a hierarchical sampling design with $i = 1, 2, 3, \ldots, m$ 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 $i$, $\alpha_i$ 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 $$\gamma = \alpha_1 + \sum_{i=1}^m \beta_i$$

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 $$\alpha_i = \sum_{j=1}^{n_i} D_{ij} w_{ij}$$ where $D_{ij}$ is the diversity index and $w_{ij}$ is the weight calculated for the $j$th sample at the $i$th 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 in the same way 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.

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 $\alpha$, $\beta$, 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.

Author

Péter Sólymos, solymos@ualberta.ca

Examples

## NOTE: 'nsimul' argument usually needs to be >= 99
## here much lower value is used for demonstration

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(out)}
## The hierarchy of sample aggregation
levsm <- with(mite.xy, data.frame(
    l1=1:nrow(mite),
    l2=cutter(y, cut = seq(0, 10, by = 2.5)),
    l3=cutter(y, cut = seq(0, 10, by = 5)),
    l4=rep(1, nrow(mite))))
## Let's see in a map
par(mfrow=c(1,3))
plot(mite.xy, main="l1", col=as.numeric(levsm$l1)+1, asp = 1)
plot(mite.xy, main="l2", col=as.numeric(levsm$l2)+1, asp = 1)
plot(mite.xy, main="l3", col=as.numeric(levsm$l3)+1, asp = 1)

par(mfrow=c(1,1))
## Additive diversity partitioning
adipart(mite, index="richness", nsimul=19)
#> adipart object
#> 
#> Call: adipart(y = mite, index = "richness", nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> options:  index richness, weights unif
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic     SES   mean   2.5%    50%  97.5% Pr(sim.)  
#> alpha.1    15.114 -28.979 22.393 21.978 22.429 22.765     0.05 *
#> gamma      35.000   0.000 35.000 35.000 35.000 35.000     1.00  
#> beta.1     19.886  28.979 12.607 12.235 12.571 13.022     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## the next two define identical models
adipart(mite, levsm, index="richness", nsimul=19)
#> adipart object
#> 
#> Call: adipart(y = mite, x = levsm, index = "richness", nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> options:  index richness, weights unif
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic      SES     mean     2.5%      50%  97.5% Pr(sim.)  
#> alpha.1    15.114 -54.1889 22.35038 22.10429 22.37143 22.584     0.05 *
#> alpha.2    29.750 -25.1503 34.81579 34.50000 34.75000 35.000     0.05 *
#> alpha.3    33.000   0.0000 35.00000 35.00000 35.00000 35.000     0.05 *
#> gamma      35.000   0.0000 35.00000 35.00000 35.00000 35.000     1.00  
#> beta.1     14.636   9.9124 12.46541 12.10000 12.45000 12.779     0.05 *
#> beta.2      3.250  15.2208  0.18421  0.00000  0.25000  0.500     0.05 *
#> beta.3      2.000   0.0000  0.00000  0.00000  0.00000  0.000     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
adipart(mite ~ l2 + l3, levsm, index="richness", nsimul=19)
#> adipart object
#> 
#> Call: adipart(formula = mite ~ l2 + l3, data = levsm, index =
#> "richness", nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> options:  index richness, weights unif
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic     SES      mean      2.5%       50%  97.5% Pr(sim.)  
#> alpha.1    15.114 -51.771 22.378947 22.100714 22.400000 22.587     0.05 *
#> alpha.2    29.750 -25.578 34.736842 34.500000 34.750000 35.000     0.05 *
#> alpha.3    33.000 -17.206 34.973684 34.725000 35.000000 35.000     0.05 *
#> gamma      35.000   0.000 35.000000 35.000000 35.000000 35.000     1.00  
#> beta.1     14.636  11.165 12.357895 12.025714 12.428571 12.686     0.05 *
#> beta.2      3.250  15.455  0.236842  0.000000  0.250000  0.500     0.05 *
#> beta.3      2.000  17.206  0.026316  0.000000  0.000000  0.275     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## Hierarchical null model testing
## diversity analysis (similar to adipart)
hiersimu(mite, FUN=diversity, relative=TRUE, nsimul=19)
#> hiersimu object
#> 
#> Call: hiersimu(y = mite, FUN = diversity, relative = TRUE, nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> 
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic     SES    mean    2.5%     50%  97.5% Pr(sim.)  
#> level_1   0.76064 -52.201 0.93892 0.93378 0.93866 0.9444     0.05 *
#> leve_2    1.00000   0.000 1.00000 1.00000 1.00000 1.0000     1.00  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
hiersimu(mite ~ l2 + l3, levsm, FUN=diversity, relative=TRUE, nsimul=19)
#> hiersimu object
#> 
#> Call: hiersimu(formula = mite ~ l2 + l3, data = levsm, FUN = diversity,
#> relative = TRUE, nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> 
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>      statistic      SES    mean    2.5%     50%  97.5% Pr(sim.)  
#> unit   0.76064  -60.783 0.93927 0.93425 0.93883 0.9435     0.05 *
#> l2     0.89736 -116.072 0.99795 0.99617 0.99808 0.9991     0.05 *
#> l3     0.92791 -421.205 0.99932 0.99902 0.99935 0.9995     0.05 *
#> all    1.00000    0.000 1.00000 1.00000 1.00000 1.0000     1.00  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## Hierarchical testing with the Morisita index
morfun <- function(x) dispindmorisita(x)$imst
hiersimu(mite ~., levsm, morfun, drop.highest=TRUE, nsimul=19)
#> hiersimu object
#> 
#> Call: hiersimu(formula = mite ~ ., data = levsm, FUN = morfun,
#> drop.highest = TRUE, nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> 
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>    statistic     SES     mean     2.5%      50%   97.5% Pr(sim.)  
#> l1   0.52070  6.4968  0.35648  0.31140  0.35449  0.3978     0.05 *
#> l2   0.60234 12.5142  0.15359  0.08900  0.14525  0.2039     0.05 *
#> l3   0.67509 17.6125 -0.19189 -0.26300 -0.20619 -0.1149     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1