# On Log-Log Regressions

Log-log regressions are commonly used in ecological papers, and my attention to their limitations was twigged by a recent paper by Hatton et al. (2015) in Science. I want to look at just one example of a log-log regression from this paper as an illustration of what I think might be some pitfalls of this approach. The regression under discussion is Figure 1 in the Hatton paper, a plot of predator biomass (Y) on prey biomass (X) for a variety of African large mammal ecosystems. I emphasize that this is a critique of log-log regression problems, not a detailed critique of this paper.

Figure 1 shows the raw data reported in the Hatton et al. (2015) paper but plotted in arithmetic space. It is clear that the variance increases with the mean and the data are highly variable, as well as slightly curvilinear, so a transformation is clearly desirable for statistical analysis. Unfortunately we are given no error bars on each of the point estimates, so it is not possible to plot confidence limits for each estimate.

We log both the axes and get Figure 2 which is identical to that plotted as Figure 1 in Hatton et al. (2015). Clearly the regression fit is better that that of Figure 1 and yet there is still considerable variation around the line of best fit.

The variation around this log-log line is the main issue I wish to discuss here. Much depends on the reason for the regression line. Mac Nally (2000) made the point that regressions are often used for predictive purposes but sometimes used only as explanations. I assume here one wishes this to be a predictive regression.

So the next question is if the Figure 2 regression is predictive, how wide are the confidence limits? In this case we will adopt the usual 95% confidence predictions for a single data point. The result is shown in Figure 3, which did not appear in the Science article. The red lines define the 95% confidence belt.

Now comes the main point of my concerns with log-log regressions. What do these error limits really mean when they are translated back to the original measurements that define the graph?

The table given below gives the prediction intervals for a hypothetical set of 8 prey abundances scattered along the span of prey densities reported.

 Prey abundance (kg/km2) Estimated predator abundance (kg/km2) Predicted lower 95% confidence limit Predicted upper 95% confidence limit Width of lower confidence interval (%) Width of upper confidence interval (%) 200 4.4 2.46 7.74 -44% +76% 1000 14.1 8.16 24.6 -42% +74% 1500 19.0 11.0 33.2 -42% +70% 2000 23.4 13.2 41.0 -44% +75% 4000 38.7 22.4 69.0 -42% +78% 8000 64.0 35.4 113.6 -45% +78% 10000 75.2 43.6 134.4 -42% +79% 12000 85.8 49.0 147.6 -43% +72%

The overall average confidence limits for this log-log regression are -43% to +75%, given that the SE of the predictions varies little across the range of values used in the regression. These are very broad confidence limits for any prediction from a regression line.

The bottom line is that log-log regressions can camouflage a great deal of variation, which may or may not be acceptable depending on the use of the regression. These plots always visually look much better than they are. You probably already knew this but I worry that it is a point that can be easily overlooked.

Lastly, a minor quibble with this regression. Some authors (e.g. Ricker 1983, Smith 2009) have discussed the issue of using the reduced major axis (or geometric mean regression) when the X variable is measured with error instead of the standard regression method. One could argue for this particular data set that the X variable is measured with error, so that I have used a reduced major axis regression in this discussion. The overall conclusions are not changed if standard regression methods are used.

Hatton, I.A., McCann, K.S., Fryxell, J.M., Davies, T.J., Smerlak, M., Sinclair, A.R.E. & Loreau, M. (2015) The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes. Science 349 (6252). doi: 10.1126/science.aac6284

Mac Nally, R. (2000) Regression and model-building in conservation biology, biogeography and ecology: The distinction between – and reconciliation of – ‘predictive’ and ‘explanatory’ models. Biodiversity & Conservation, 9, 655-671. doi: 10.1023/A:1008985925162

Ricker, W.E. (1984) Computation and uses of central trend lines. Canadian Journal of Zoology 62 (10), 1897-1905.doi: 10.1139/z84-279

Smith, R.J. (2009) Use and misuse of the reduced major axis for line-fitting. American Journal of Physical Anthropology, 140, 476-486. doi: 10.1002/ajpa.21090

# On Philanthropic Investment in Biodiversity Conservation

In the holiday season there is much talk and recommendations about donations to worthy causes, and this raises an interesting conundrum in biodiversity conservation. The question is relatively simple to answer if you have little money, but any reading of the business pages of our newspapers or a walk around the shopping centers of our large cities makes you realize that there are a great many people with more than a little money. What should you do with your excess cash?

Some people (but not all) will want to ‘make a difference’ with their accumulated wealth, at least until medical science can overcome the universal belief that “you can’t take it with you”. Peter Singer (2015) has addressed this question of how to spend your money most effectively when you donate. It comes down in the first instance of your time frame. If you wish to make a difference in the short term of a few years, your choices may differ fundamentally from those taken to make a difference in the long term of 100-500 years. The bulk of philanthropic donations now are in the short-term camp. We have poor people living on the street in most of our cities, people with curable diseases in less developed countries but no medical aid, and victims of wars, earthquakes and tsunamis who must rebuild their lives. So we must start with what I think is the biggest decision regarding philanthropy – do we worry only about people, or do we worry about the biological world as well? Most donations are directly related to improving the human condition, locally or globally.

But there is hope because more and more people are realizing that we cannot separate people from biodiversity because of ecosystem services. Without well-functioning ecosystems on Earth, all the medical advances of our time are for naught. This is an important message to convey to potential donors.

Conservation philanthropy is a curious mix of short term and long term goals. Many endangered species need action now to survive. But ecologists typically look at both the shorter and the longer term goals of conservation. The simplest goal is to set aside land for protection. Without habitat all is lost. But this goal must be paired with long term funding to hire rangers to protect the area from poachers and to monitor the status of the species within the protected zone. Relying on the government to do this by itself is not adequate and never has been. But beyond this primary goal of land protection, the conservation movement fractionates. There are arguments that without effective human population stabilization biodiversity loss must continue. So does this mean that effective donations should be earmarked for agencies that empower women and offer reproductive services? But this points out that we must not fall into the trap of thinking we can do only one thing at a time. Pandas or population – why not “both and”? Climate change is a similar ‘elephant in the room’ problem.

What are the long-term goals of conservation biology that would benefit from philanthropic investment? Start with pest control. Biological control of pests is a long-term issue par excellence (Goldson et al. 2015, Myers et al. 2009, Wyckhuys et al. 2013). But biological control programs are underfunded by governments and obtain little private philanthropy. Weed control, insect pest control, vertebrate pest control all fit in the same problem basket – long term problem supported only by short term funding. Invasive pest eradication on islands is one area of pest control in which both governments and private funding have been joining forces (http://www.islandconservation.org/ ) with good results.

Two other areas of conservation biology that are classically underfunded are taxonomy and monitoring. In many taxonomic groups the majority of the species on Earth are not yet identified and described with a scientific name. The nearest analogy would be having a bank with tons of coins of different sizes and shapes, but only a few of which had any engraving on them. Taxonomy which is so vital to biology suffers because physical scientists consider it “stamp collecting” and unworthy of scientific funding. Monitoring of ecological communities faces the same problem. Monitoring ecological communities is similar to monitoring weather, yet we support meteorological stations around the world but provide little support for ecological monitoring. At present ecological monitoring is done ad hoc by dedicated people but with little systematic organization. Monitoring of changes in the arctic is being coordinated globally (http://www.amap.no/ ) and specific programs have been outlined for example for northern Canada (https://www.ec.gc.ca/faunescience-wildlifescience/, but the funding levels are low considering the size of the areas under consideration. Tropical ecosystem monitoring is even less well funded, yet that is where much of global biodiversity is located (c.f. for example, Cardoso et al. 2011, Burton 2012).

So what can you do about this? Talk up the necessity and the advantages of conservation biodiversity. Imagine what would happen to any of these biodiversity problems if a foundation the size of the Bill & Melinda Gates Foundation devoted a large amount of its donations to conservation. Environmental stewardship is the key to the Earth’s survival, and a combination of problem solving of current biodiversity problems combined with a strong research component on how species interact and ecosystems operate to sustain themselves would be a legacy for future generations and a flagship for the next 100 years.

Burton, A.C. (2012) Critical evaluation of a long-term, locally-based wildlife monitoring program in West Africa. Biodiversity and Conservation, 21, 3079-3094. doi: 10.1007/s10531-012-0355-6

Cardoso, P., Erwin, T.L., Borges, P.A.V. & New, T.R. (2011) The seven impediments in invertebrate conservation and how to overcome them. Biological Conservation, 144, 2647-2655. doi: 10.1016/j.biocon.2011.07.024

Glen, A.S., Atkinson, R., Campbell, K.J., Hagen, E., Holmes, N.D., Keitt, B.S., Parkes, J.P., Saunders, A., Sawyer, J. & Torres, H. (2013) Eradicating multiple invasive species on inhabited islands: the next big step in island restoration? Biological Invasions, 15, 2589-2603. doi: 10.1007/s10530-013-0495-y

Goldson, S.L., Bourdôt, G.W., Brockerhoff, E.G., Byrom, A.E., Clout, M.N., McGlone, M.S., Nelson, W.A., Popay, A.J., Suckling, D.M. & Templeton, M.D. (2015) New Zealand pest management: current and future challenges. Journal of the Royal Society of New Zealand, 45, 31-58. doi: 10.1080/03036758.2014.1000343

Myers, J.H., Jackson, C., Quinn, H., White, S.R. & Cory, J.S. (2009) Successful biological control of diffuse knapweed, Centaurea diffusa, in British Columbia, Canada. Biological Control, 50, 66-72. doi: 10.1016/j.biocontrol.2009.02.008

Singer, P. (2015) The Most Good You Can Do. Yale University Press, New Haven. ISBN: 978-0-300-18027-5

Wyckhuys, K.A.G., Lu, Y., Morales, H., Vazquez, L.L., Legaspi, J.C., Eliopoulos, P.A. & Hernandez, L.M. (2013) Current status and potential of conservation biological control for agriculture in the developing world. Biological Control, 65, 152-167. doi: 10.1016/j.biocontrol.2012.11.010 http://www.islandconservation.org/where-we-work/