As part of the STOR-i MRes programme, I have been taking part in a series of masterclasses which have covered a wide range of modern topics in both statistics and operational research. The first masterclass of the year was given by Professor Marian Scott from Glasgow University, who presented a series of talks on Digital Earth, data and analytic challenges for earth systems. This masterclass focussed on some of the statistical and data analytic challenges that are becoming increasingly important in environmental and ecological sciences. This included some common statistical features such as dealing with spatio-temporal data, fusing multiple data streams, developing indicators and targets and handling uncertainty.
During the course of the week, we were asked to choose an environmental investigation, of current scientific debate, and investigate the scientific evidence surrounding this topic. For this short case study, I chose to investigate how statistical methods can be used to identify early warnings of problems in populations. That is, how can we tell if a species is at risk of short-term extinction?
A short case study
A recent paper commented that there is controversy regarding the current status and trends, and therefore short-term extinction risk of two particular species of forest birds. Namely the Hawaii Creeper and Hawaii Akepa.
“There is debate about the current population trends and predicted short-term fates of the
No evidence of critical slowing down in two endangered Hawaiian honeycreepers.
endangered forest birds, Hawaii Creeper and Hawaii Akepa . Using long-term population size estimates, some studies report forest bird populations as stable or increasing, while other studies report signs of population decline or impending extinction. This is predominantly a statistical issue, assuming that sufficient data have been collected.”
The current threat to this endangered species includes habitat loss and degradation, as well as introduced diseases such as avian malaria and pox. Furthermore, it is thought that climate change aught to further exacerbate extinction risk.
Early Warning Signal Analyses
Time series data are available for two species of endangered bird, the Hawaii Creeper and Hawaii Akepa, over a 25-year period (1987-2012). These data can be evaluated by looking for evidence of consistent increases in three early warning signals of critical slowing down:
- Autocorrelation at lag-1
- Sample variance
- Linear skewness
The first two early warning signals are standard statistical moments that can be easily calculated in the statistical software package R. For those who are not aware, the autocorrelation at lag 1 is a measure of correlation between values that are one time period apart, and the sample variance gives an indication as to how close the values lie to the average value. The third indicator is arguably less intuitive. Note that we do not use ordinary skewness (which describes the shape of the distribution) because it is biased, particularly for small sample sizes (n\leq100).
It is suggested that each of the early warning indicators are calculated from sequential overlapping subsets of the data in a rolling window through the time series (e.g., values from 1 to s, 2 to s+1, 3 to s+2, etc). It is important to note here that there is a trade-off between choosing a longer window length to obtain reliable baseline estimates, which will result in fewer data points to analyse, and hence increases the likelihood of missing a “transition window”.
Kendall’s \tau
In order to determine if there is evidence of a system approaching a critical transition (i.e. if a population of endangered birds is at risk of extinction) Kendall’s \tau can be calculated for each early warning signal. Kendall’s \tau gives a quantitative measure of early warning signal trend, and in particular
- A value of Kendall’s \tau close to 1 indicates strong support of an increasing non-zero slope. More specifically, significant positive values are expected when a system is approaching a critical transition
- A value of Kendall’s \tau near 0 is indicative of a stable system, and we therefore expect there to be no trend in early warning signals.
Application and Results
Early warning signal analyses were conducted for the Hawaii Creeper and Hawaii Akepa for trends using a rolling window of two lengths: 50\% (which provided metrics for 1999-2012) and 25\% (which provided metrics for 1992-2012). Kendall’s \tau was also calculated for each indicator in the two windows.
It was found that there were no consistent increases in autocorrelation, variance or linear skewness over the entire time series in either rolling window length. More specifically, Kendall’s \tau never exceeded 0.43 for any indicator in any window. This clearly suggests that there are no strongly increasing trends in the putative early warning metrics.
So there you have it, looks like the Hawaii Creeper and Hawaii Akepa are ok after all!
Further Considerations and Reading
Whilst the results obtained in the study are somewhat reassuring, it is important to note that there are two scenarios that could possibly lead us to this result
- A signal was not detected because there was no change—or impending change—in state for either species and their current state is optimistic or
- A signal that is present went undetected (a Type II error).
There are conditions that could prevent detecting a signal that is present, and some types of critical transitions do not exhibit critical slowing down. One possibility is that early warning signals could be masked by environmental noise (e.g. annual stochastic variation in food availability) and by noise caused by observation error. Therefore, we need to be careful when drawing conclusions from analyses such as these.
Check out the link below for more information:
No evidence of critical slowing down in two endangered Hawaiian honeycreepers