Extrapolated Species Richness in a Species Pool

The functions estimate the extrapolated species richness in a species pool, or the number of unobserved species. Function specpool is based on incidences in sample sites, and gives a single estimate for a collection of sample sites (matrix). Function estimateR is based on abundances (counts) on single sample site.

Usage

specpool(x, pool, smallsample = TRUE)
estimateR(x, ...)
specpool2vect(X, index = c("jack1","jack2", "chao", "boot","Species"))
poolaccum(x, permutations = 100, minsize = 3)
estaccumR(x, permutations = 100, parallel = getOption("mc.cores"))
# S3 method for class 'poolaccum'
summary(object, display, alpha = 0.05, ...)
# S3 method for class 'poolaccum'
plot(x, alpha = 0.05, type = c("l","g"), ...)

Arguments

x: Data frame or matrix with species data or the analysis result for plot function.
pool: A vector giving a classification for pooling the sites in the species data. If missing, all sites are pooled together.
smallsample: Use small sample correction \((N-1)/N\), where \(N\) is the number of sites within the pool.
X, object: A specpool result object.
index: The selected index of extrapolated richness.
permutations: Usually an integer giving the number permutations, but can also be a list of control values for the permutations as returned by the function how, or a permutation matrix where each row gives the permuted indices.
minsize: Smallest number of sampling units reported.
parallel: Number of parallel processes or a predefined socket cluster. With parallel = 1 uses ordinary, non-parallel processing. The parallel processing is done with parallel package.
display: Indices to be displayed.
alpha: Level of quantiles shown. This proportion will be left outside symmetric limits.
type: Type of graph produced in xyplot.
...: Other parameters (not used).

Details

Many species will always remain unseen or undetected in a collection of sample plots. The function uses some popular ways of estimating the number of these unseen species and adding them to the observed species richness (Palmer 1990, Colwell & Coddington 1994).

The incidence-based estimates in specpool use the frequencies of species in a collection of sites. In the following, \(S_P\) is the extrapolated richness in a pool, \(S_0\) is the observed number of species in the collection, \(a_1\) and \(a_2\) are the number of species occurring only in one or only in two sites in the collection, \(p_i\) is the frequency of species \(i\), and \(N\) is the number of sites in the collection. The variants of extrapolated richness in specpool are:

Chao	\(S_P = S_0 + \frac{a_1^2}{2 a_2}\frac{N-1}{N}\)
Chao bias-corrected	\(S_P = S_0 + \frac{a_1(a_1-1)}{2(a_2+1)} \frac{N-1}{N}\)
First order jackknife	\(S_P = S_0 + a_1 \frac{N-1}{N}\)
Second order jackknife	\(S_P = S_0 + a_1 \frac{2N - 3}{N} - a_2 \frac{(N-2)^2}{N (N-1)}\)
Bootstrap	\(S_P = S_0 + \sum_{i=1}^{S_0} (1 - p_i)^N\)

specpool normally uses basic Chao equation, but when there are no doubletons (\(a2=0\)) it switches to bias-corrected version. In that case the Chao equation simplifies to \(S_0 + \frac{1}{2} a_1 (a_1-1) \frac{N-1}{N}\).

The abundance-based estimates in estimateR use counts (numbers of individuals) of species in a single site. If called for a matrix or data frame, the function will give separate estimates for each site. The two variants of extrapolated richness in estimateR are bias-corrected Chao and ACE (O'Hara 2005, Chiu et al. 2014). The Chao estimate is similar as the bias corrected one above, but \(a_i\) refers to the number of species with abundance \(i\) instead of number of sites, and the small-sample correction is not used. The ACE estimate is defined as:

ACE	\(S_P = S_{abund} + \frac{S_{rare}}{C_{ace}}+ \frac{a_1}{C_{ace}} \gamma^2_{ace}\)
where	\(C_{ace} = 1 - \frac{a_1}{N_{rare}}\)
	\(\gamma^2_{ace} = \max \left[ \frac{S_{rare} \sum_{i=1}^{10} i(i-1)a_i}{C_{ace} N_{rare} (N_{rare} - 1)}-1, 0 \right]\)

Here \(a_i\) refers to number of species with abundance \(i\) and \(S_{rare}\) is the number of rare species, \(S_{abund}\) is the number of abundant species, with an arbitrary threshold of abundance 10 for rare species, and \(N_{rare}\) is the number of individuals in rare species.

Functions estimate the standard errors of the estimates. These only concern the number of added species, and assume that there is no variance in the observed richness. The equations of standard errors are too complicated to be reproduced in this help page, but they can be studied in the R source code of the function and are discussed in the vignette that can be read with the browseVignettes("vegan"). The standard error are based on the following sources: Chiu et al. (2014) for the Chao estimates and Smith and van Belle (1984) for the first-order Jackknife and the bootstrap (second-order jackknife is still missing). For the variance estimator of \(S_{ace}\) see O'Hara (2005).

Functions poolaccum and estaccumR are similar to specaccum, but estimate extrapolated richness indices of specpool or estimateR in addition to number of species for random ordering of sampling units. Function specpool uses presence data and estaccumR count data. The functions share summary and plot methods. The summary returns quantile envelopes of permutations corresponding the given level of alpha and standard deviation of permutations for each sample size. NB., these are not based on standard deviations estimated within specpool or estimateR, but they are based on permutations. The plot function shows the mean and envelope of permutations with given alpha for models. The selection of models can be restricted and order changes using the display argument in summary or plot. For configuration of plot command, see xyplot.

Value

Function specpool returns a data frame with entries for observed richness and each of the indices for each class in pool vector. The utility function specpool2vect maps the pooled values into a vector giving the value of selected index for each original site. Function estimateR returns the estimates and their standard errors for each site. Functions poolaccum and estimateR return matrices of permutation results for each richness estimator, the vector of sample sizes and a table of means of permutations for each estimator.

References

Chao, A. (1987). Estimating the population size for capture-recapture data with unequal catchability. Biometrics 43, 783–791.

Chiu, C.H., Wang, Y.T., Walther, B.A. & Chao, A. (2014). Improved nonparametric lower bound of species richness via a modified Good-Turing frequency formula. Biometrics 70, 671–682.

Colwell, R.K. & Coddington, J.A. (1994). Estimating terrestrial biodiversity through extrapolation. Phil. Trans. Roy. Soc. London B 345, 101–118.

O'Hara, R.B. (2005). Species richness estimators: how many species can dance on the head of a pin? J. Anim. Ecol. 74, 375–386.

Palmer, M.W. (1990). The estimation of species richness by extrapolation. Ecology 71, 1195–1198.

Smith, E.P & van Belle, G. (1984). Nonparametric estimation of species richness. Biometrics 40, 119–129.

Author

Bob O'Hara (estimateR) and Jari Oksanen.

Note

The functions are based on assumption that there is a species pool: The community is closed so that there is a fixed pool size \(S_P\). In general, the functions give only the lower limit of species richness: the real richness is \(S >= S_P\), and there is a consistent bias in the estimates. Even the bias-correction in Chao only reduces the bias, but does not remove it completely (Chiu et al. 2014).

Optional small sample correction was added to specpool in vegan 2.2-0. It was not used in the older literature (Chao 1987), but it is recommended recently (Chiu et al. 2014).

Examples

data(dune)
data(dune.env)
pool <- with(dune.env, specpool(dune, Management))
pool
#>    Species     chao   chao.se    jack1 jack1.se    jack2     boot  boot.se n
#> BF      16 17.19048 1.5895675 19.33333 2.211083 19.83333 17.74074 1.646379 3
#> HF      21 21.51429 0.9511693 23.40000 1.876166 22.05000 22.56864 1.821518 5
#> NM      21 22.87500 2.1582871 26.00000 3.291403 25.73333 23.77696 2.300982 6
#> SF      21 29.88889 8.6447967 27.66667 3.496029 31.40000 23.99496 1.850288 6
op <- par(mfrow=c(1,2))
boxplot(specnumber(dune) ~ Management, data = dune.env,
        col = "hotpink", border = "cyan3")
boxplot(specnumber(dune)/specpool2vect(pool) ~ Management,
        data = dune.env, col = "hotpink", border = "cyan3")

par(op)
data(BCI)
## Accumulation model
pool <- poolaccum(BCI)
summary(pool, display = "chao")
#> $chao
#>        N     Chao     2.5%    97.5%   Std.Dev
#>  [1,]  3 162.7319 143.9393 187.5332 11.945606
#>  [2,]  4 177.5652 158.5016 196.0885 11.192608
#>  [3,]  5 185.6759 164.2696 213.3850 12.877291
#>  [4,]  6 191.2801 171.6428 214.8908 12.385474
#>  [5,]  7 195.6433 177.0344 226.4467 12.600388
#>  [6,]  8 199.8243 180.8158 227.8784 11.678016
#>  [7,]  9 204.2709 181.3366 227.9547 11.778603
#>  [8,] 10 208.0971 188.7984 237.1717 12.850721
#>  [9,] 11 210.7514 190.4101 237.2609 12.710824
#> [10,] 12 212.5455 191.8712 237.9534 12.000344
#> [11,] 13 215.2654 194.8808 239.1741 11.975056
#> [12,] 14 217.4286 199.0328 240.5277 12.123952
#> [13,] 15 219.6550 199.9609 245.6360 11.804799
#> [14,] 16 220.6682 200.4314 241.3243 11.867461
#> [15,] 17 222.6279 202.5397 244.1830 13.137527
#> [16,] 18 224.3636 205.5863 250.2521 12.541136
#> [17,] 19 224.9883 206.7775 251.9407 11.451247
#> [18,] 20 225.9783 208.8084 248.7393 10.281435
#> [19,] 21 227.4685 211.4459 252.0768 11.094347
#> [20,] 22 229.0373 212.1119 250.7617 10.978382
#> [21,] 23 230.4335 213.6481 255.7681 10.703026
#> [22,] 24 231.6720 213.5777 258.8306 11.200376
#> [23,] 25 233.1331 213.8453 262.3925 11.376831
#> [24,] 26 233.1843 214.4064 255.9835 10.419627
#> [25,] 27 234.2276 217.3700 253.9270  9.917935
#> [26,] 28 235.3133 220.2375 254.6652  9.717874
#> [27,] 29 236.2590 221.7516 254.6852  9.836369
#> [28,] 30 236.4171 221.4532 256.8216 10.115269
#> [29,] 31 236.6669 219.9412 264.3588 10.140717
#> [30,] 32 237.3542 221.2701 262.8832 11.005182
#> [31,] 33 237.3841 220.7957 260.9730 11.093535
#> [32,] 34 237.2035 221.3186 263.1168 11.107173
#> [33,] 35 237.7958 222.1835 261.7541 10.943907
#> [34,] 36 237.7037 223.2374 257.9742  9.503201
#> [35,] 37 237.5417 222.5612 257.0841  8.756801
#> [36,] 38 237.8074 222.3610 254.0359  8.615232
#> [37,] 39 237.6095 223.2400 253.6776  8.366448
#> [38,] 40 237.6945 224.5364 252.3392  7.884032
#> [39,] 41 237.8858 225.0594 250.9073  7.342289
#> [40,] 42 238.0273 225.0633 250.0096  7.027352
#> [41,] 43 238.1064 224.7337 249.9578  6.890723
#> [42,] 44 238.0102 225.6088 249.4941  6.454997
#> [43,] 45 237.3579 225.9317 248.5996  5.794307
#> [44,] 46 236.7545 227.3723 248.6117  5.217047
#> [45,] 47 236.9764 227.7084 245.3901  4.497045
#> [46,] 48 237.0667 229.0456 245.3993  3.644490
#> [47,] 49 236.9449 231.3403 245.4082  3.201099
#> [48,] 50 236.3732 236.3732 236.3732  0.000000
#> 
#> attr(,"class")
#> [1] "summary.poolaccum"
plot(pool)

## Quantitative model
estimateR(BCI[1:5,])
#>                   1          2          3          4          5
#> S.obs     93.000000  84.000000  90.000000  94.000000 101.000000
#> S.chao1  117.473684 117.214286 141.230769 111.550000 136.000000
#> se.chao1  11.583785  15.918953  23.001405   8.919663  15.467344
#> S.ACE    122.848959 117.317307 134.669844 118.729941 137.114088
#> se.ACE     5.736054   5.571998   6.191618   5.367571   5.848474