Comparing E/MSY and Chisholm method

Historical data (e.g. sighting records) can be used to estimate historical extinction rates in a variety of ways. The Chisholm et al. (2016) method (earlier post) uses the data to estimate yearly extinction probabilities. The extinctions per million species-years (E/MSY) approach (Pimm et al., 2014) estimates they extinction probability averaged over species-years. Below, I use two small examples to illustrate the similarities and differences between them.

Chisholm \( = \) E/SY when extinction constant

Consider the example in the table below, where the extinction probability \( \mu_t \) is the same in every year.

discovered
year \( t \)extant \( S_t \)extinct \(E_t\)discoveries \( \delta_t \)extinction rate \( \mu_t \)
01000200.2
110020400.2
212040840.2
318064560.2
42001001200.2
5280140100.2
6234196--

The Chisholm et al. (2016) method calculates the total extinction probability \( p \) as described in a previous blog post. The total extinction probability over the observation period is 1 – the cumulative probability of survival
\[
\begin{align}
p &= 1 – \text{probability of survival} \\
& = 1 – \prod_{t=0}^5 1 – \mu_t \\
& = 1 – (1-0.2)^6 \\
& = 0.74.
\end{align}
\]
The year-averaged extinction probability is obviously 0.2, which we can obtain from \( p \) using the geometric mean of survival probabilities
\[
\begin{align}
\text{year-averaged extinction rate} & = 1 – (1-p)^{1/6} \\
& = 0.2
\end{align}
\]

The E/SY is calculated as follows. Each species’ number of species-years is calculated from the time of its discovery to the time of its extinction, so the total number of species-years is
\[
\begin{align}
\text{species-years} &= \sum_{t=0}^5 S_t \\
& = 980.
\end{align}
\]
Therefore
\[
\begin{align}
\text{species-year averaged extinction rate} & = \frac{196}{980} \\
& = 0.2,
\end{align}
\]
which is the same as the Chisholm et al. (2016) method above.

Chisholm \( \neq \) E/SY when extinction probability varies

Consider the example in the table below, where the extinction probability \( \mu_t \) varies year to year.

discovered
year \( t \)extant \( S_t \)extinct \( E_t \)discoveries \( \delta-t \)extinction rate \( \mu_t \)
01000200.4
18040400.2
210456840.125
317569560.04
4224761200.0625
533090100.1
6307123--

The total extinction probability over the observation period is
\[
\begin{align}
p &= 1 – (0.6)(0.8)(0.875)(0.96)(0.9375)(0.9)\\
&= 0.66,
\end{align}
\]
giving
\[
\begin{align}
\text{year-averaged extinction rate} &= 1 – (1-0.66)^{1/6} \\
&= 0.16.
\end{align}
\]

The total number of species-years
\[
\begin{align}
\text{species-years} &= \sum_{t=0}^5 S_t \\
&= 1013,
\end{align}
\]
so the E/SY is
\[
\begin{align}
\text{species-year averaged extinction rate} &= \frac{123}{1013} \\
&= 0.12,
\end{align}
\]
which is different to the Chisholm et al. (2016) method.

References:

Chisholm, R.A., Giam, X., Sadanandan, K.R., Fung, T. and Rheindt, F.E. (2016), A robust nonparametric method for quantifying undetected extinctions. Conservation Biology, 30: 610-617. doi:10.1111/cobi.12640.

Pimm, S. L., Jenkins, C. N., Abell, R., Brooks, T. M., Gittleman, J. L., Joppa, L. N., Raven, P. H., Roberts, C. M. and Sexton, J. O. (2014). The biodiversity of species and their rates of extinction, distribution, and protection, Science 344(6187): 1246752.

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