[Update: thoughts in this post contributed to Kristensen et al. (2020).]

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.

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.