Multiplicative Diversity Partitioning

In multiplicative 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 or the pooled samples (gamma diversity).

multipart(...)
# S3 method for default
multipart(y, x, index=c("renyi", "tsallis"), scales = 1,
    global = FALSE, relative = FALSE, nsimul=99, method = "r2dtable", ...)
# S3 method for formula
multipart(formula, data, index=c("renyi", "tsallis"), scales = 1,
    global = FALSE, relative = 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, two levels are assumed: each row is a group in the first level, and all rows are in the same group in the second level.
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, at least two levels specified). 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 entropy index to be calculated (see Details).
relative	Logical, if `TRUE` then beta diversity is standardized by its maximum (see Details).
scales	Numeric, of length 1, the order of the generalized diversity index to be used.
global	Logical, indicates the calculation of beta diversity values, see Details.
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.
...	Other arguments passed to `oecosimu`, i.e. `method`, `thin` or `burnin`.

Details

Multiplicative diversity partitioning is based on Whittaker's (1972) ideas, that has recently been generalised to one parametric diversity families (i.e. Rényi and Tsallis) by Jost (2006, 2007). Jost recommends to use the numbers equivalents (Hill numbers), instead of pure diversities, and proofs, that this satisfies the multiplicative partitioning requirements.

The current implementation of multipart calculates Hill numbers based on the functions renyi and tsallis (provided as index argument). If values for more than one scales are desired, it should be done in separate runs, because it adds extra dimensionality to the implementation, which has not been resolved efficiently.

Alpha diversities are then the averages of these Hill numbers for each hierarchy levels, the global gamma diversity is the alpha value calculated for the highest hierarchy level. When global = TRUE, beta is calculated relative to the global gamma value: $$\beta_i = \gamma / \alpha_{i}$$ when global = FALSE, beta is calculated relative to local gamma values (local gamma means the diversity calculated for a particular cluster based on the pooled abundance vector): $$\beta_ij = \alpha_{(i+1)j} / mean(\alpha_{ij})$$ where $j$ is a particular cluster at hierarchy level $i$. Then beta diversity value for level $i$ is the mean of the beta values of the clusters at that level, $\beta_{i} = mean(\beta_{ij})$.

If relative = TRUE, the respective beta diversity values are standardized by their maximum possible values ($mean(\beta_{ij}) / \beta_{max,ij}$) given as $\beta_{max,ij} = n_{j}$ (the number of lower level units in a given cluster $j$).

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

Value

An object of class "multipart" with same structure as "oecosimu" objects.

References

Jost, L. (2006). Entropy and diversity. Oikos, 113, 363--375.

Jost, L. (2007). Partitioning diversity into independent alpha and beta components. Ecology, 88, 2427--2439.

Whittaker, R. (1972). Evolution and measurement of species diversity. Taxon, 21, 213--251.

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=cutter(y, cut = seq(0, 10, by = 10))))
## Multiplicative diversity partitioning
multipart(mite, levsm, index="renyi", scales=1, nsimul=19)
#> multipart object
#> 
#> Call: multipart(y = mite, x = levsm, index = "renyi", scales = 1,
#> nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> options:  index renyi, scales 1, global FALSE
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic      SES    mean    2.5%     50%   97.5% Pr(sim.)  
#> alpha.1    8.0555  -64.394 12.2036 12.0813 12.2105 12.2904     0.05 *
#> alpha.2   11.2353  -76.305 14.0792 14.0375 14.0726 14.1638     0.05 *
#> alpha.3   12.0064 -385.486 14.1363 14.1273 14.1359 14.1441     0.05 *
#> gamma     14.1603    0.000 14.1603 14.1603 14.1603 14.1603     1.00  
#> beta.1     1.3568   34.501  1.1560  1.1474  1.1549  1.1664     0.05 *
#> beta.2     1.0710   27.947  1.0041  0.9985  1.0046  1.0065     0.05 *
#> beta.3     1.1794  453.847  1.0017  1.0011  1.0017  1.0023     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
multipart(mite ~ ., levsm, index="renyi", scales=1, nsimul=19)
#> multipart object
#> 
#> Call: multipart(formula = mite ~ ., data = levsm, index = "renyi",
#> scales = 1, nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> options:  index renyi, scales 1, global FALSE
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic      SES     mean     2.5%      50%   97.5% Pr(sim.)  
#> alpha.1    8.0555  -64.100 12.19792 12.10148 12.18494 12.3025     0.05 *
#> alpha.2   11.2353  -95.995 14.08305 14.03774 14.07722 14.1401     0.05 *
#> alpha.3   12.0064 -258.881 14.13405 14.11471 14.13524 14.1438     0.05 *
#> gamma     14.1603    0.000 14.16027 14.16027 14.16027 14.1603     1.00  
#> beta.1     1.3568   35.732  1.15706  1.14782  1.15762  1.1661     0.05 *
#> beta.2     1.0710   33.869  1.00363  0.99971  1.00400  1.0068     0.05 *
#> beta.3     1.1794  304.537  1.00186  1.00116  1.00177  1.0032     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
multipart(mite ~ ., levsm, index="renyi", scales=1, nsimul=19, relative=TRUE)
#> multipart object
#> 
#> Call: multipart(formula = mite ~ ., data = levsm, index = "renyi",
#> scales = 1, relative = TRUE, nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> options:  index renyi, scales 1, global FALSE
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic      SES      mean      2.5%       50%   97.5% Pr(sim.)  
#> alpha.1  8.055481  -50.313 12.222701 12.042490 12.246283 12.3173     0.05 *
#> alpha.2 11.235261  -94.151 14.069192 14.018070 14.074128 14.1149     0.05 *
#> alpha.3 12.006443 -366.472 14.136866 14.127636 14.135888 14.1486     0.05 *
#> gamma   14.160271    0.000 14.160271 14.160271 14.160271 14.1603     1.00  
#> beta.1   0.078594   18.416  0.068115  0.067460  0.067972  0.0693     0.05 *
#> beta.2   0.535514   31.199  0.502410  0.500907  0.502089  0.5042     0.05 *
#> beta.3   0.589695  431.589  0.500828  0.500411  0.500862  0.5012     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
multipart(mite ~ ., levsm, index="renyi", scales=1, nsimul=19, global=TRUE)
#> multipart object
#> 
#> Call: multipart(formula = mite ~ ., data = levsm, index = "renyi",
#> scales = 1, global = TRUE, nsimul = 19)
#> 
#> nullmodel method ‘r2dtable’ with 19 simulations
#> options:  index renyi, scales 1, global TRUE
#> alternative hypothesis: statistic is less or greater than simulated values
#> 
#>         statistic      SES    mean    2.5%     50%   97.5% Pr(sim.)  
#> alpha.1    8.0555  -56.225 12.1877 12.0713 12.1978 12.2979     0.05 *
#> alpha.2   11.2353  -94.896 14.0895 14.0396 14.0877 14.1383     0.05 *
#> alpha.3   12.0064 -298.657 14.1371 14.1248 14.1372 14.1472     0.05 *
#> gamma     14.1603    0.000 14.1603 14.1603 14.1603 14.1603     1.00  
#> beta.1     1.7578   84.986  1.1619  1.1514  1.1609  1.1731     0.05 *
#> beta.2     1.2603  119.017  1.0050  1.0016  1.0052  1.0086     0.05 *
#> beta.3     1.1794  351.607  1.0016  1.0009  1.0016  1.0025     0.05 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Multiplicative Diversity Partitioning

Arguments

Details

Value

References

See also

Examples

Contents

Author