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 \( \mu_t \) |

0 | 100 | 0 | 20 | 0.2 |

1 | 100 | 20 | 40 | 0.2 |

2 | 120 | 40 | 84 | 0.2 |

3 | 180 | 64 | 56 | 0.2 |

4 | 200 | 100 | 120 | 0.2 |

5 | 280 | 140 | 10 | 0.2 |

6 | 234 | 196 | - | - |

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 \( \mu_t \) |

0 | 100 | 0 | 20 | 0.4 |

1 | 80 | 40 | 40 | 0.2 |

2 | 104 | 56 | 84 | 0.125 |

3 | 175 | 69 | 56 | 0.04 |

4 | 224 | 76 | 120 | 0.0625 |

5 | 330 | 90 | 10 | 0.1 |

6 | 307 | 123 | - | - |

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.