Environmental and Ecological Statistics with R (2nd edition)

The second edition of EESwithR is coming in fall 2016.  I added one new chapter to the book and it is posted as an example chapter on github, along with R code and data sets.  The other main change is the replacement of the term statistical significant with something like "statistically different from 0."  The second edition also includes a large number of exercises, many of them have been used as homework assignments in my classes over the last ten years.  I am working on a solution pamphlet, as well as additional problems.  One unique feature of these exercises is the lack of a unique solution to almost each of these questions.  There are always multiple interpretations of a problem.  When grading homework, I look for student's thought process.  I welcome suggestions and recommendations on additional exercise problems.

Hypothesis testing and the Raven Paradox

I was going over my old reading logs the other day and the saw my notes on the Raven paradox (a.k.a. Hempel's paradox).  The statement that "all ravens are black" is apparently straightforward and entirely true.  The logical contrapositive "everything that is not black is not a raven" is also obviously true and uncontroversial.  In mathematics, proof by contrapositive is a legitimate inference method.  That is, you can show "If not B then not A" to support "If A then B."  The raven paradox is apparently paradoxical because it suggests that observing a white shoe (I. J. Good) is evidence supporting the claim that all ravens are black.  I.J. Good proposed a Bayesian explanation (or solution) of the paradox. The weight of evidence provided by seeing a white shoe (or any none black object that is not a raven) is positive, but small if the number of raven is small compared to all non-black objects.   But how is the paradox relevant to statistical hypothesis testing?

Statistical Hypothesis Inference and Testing is relevant to discussing the Raven paradox because we show support to our theory (the alternative hypothesis) by showing that a non-white object is not a raven (the null hypothesis).  If we are interested in showing that a treatment has an effect, we start by setting the null hypothesis as the treatment of no effect.  Using statistics, we show that data do not support the null hypothesis; hence the logic of contrapositive leads to the conclusion that the treatment is effective.  I have no problem with this thought process, as long as we are only interested in a yes/no answer about the effectiveness of the treatment.  How effective is of no interest.  But if we are interested in quantifying the treatment effect, hypothesis testing is almost always not appropriate.  When we are interested in quantifying the effect, we are interested in a specific alternative.  For example, when discussing the effectiveness of agricultural conservation practices on reducing nutrient loss, we want to know the magnitude of the effect, not whether or not the effect exists.  Showing that the effect is not zero gives some support to the claim that the effect is X, but not much.  This is why we often advise our students that statistical significance is not always practically useful, especially when the null hypothesis itself is irrelevant to the hypothesis of interest (the alternative hypothesis).  

A "threshold" model known as TITAN is a perfect example of the Raven paradox.  

The basic building block of TITAN is a series of permutation tests. Although TITAN's authors never clearly stated the null and alternative hypothesis, it is not difficult to derive these hypotheses using the basic characteristics of a permutation test.  The hypothesis of interest (the alternative) is that changes of a taxon's abundance along an environmental gradient can be approximated by a threshold model (specifically, a step function model).  The null hypothesis is that the taxon's abundance is a constant along the same gradient.  We can rephrase the alternative hypothesis as: the pattern of change of a taxon's abundance is a threshold model.  The null hypothesis is that the pattern of change is flat.  When we reject the null, we say that the pattern of change is not flat.  The rejection can be seen as evidence supporting the alternative, but the weight of evidence is small if the number of non-flat and non-threshold patterns of change is large.

The Everglades wetland's phosphorus retention capacity

In 1997, I and Curt Richardson published a paper on using a piecewise linear regression model for estimating the phosphorus retention capacity in the Everglades.  At the time, fitting a piecewise linear model is not a simple task.  As I was up to date on Bayesian computation, I used the Gibbs sampler.  It was an interesting exercise to derive the full set of conditional probability distribution function.  The process is tedious but not hard.  When applied to the Everglades data, we concluded that the Everglades' phosphorus retention capacity is about 1 gram of phosphorus per year per square meter (the median is 1.15), with a 90% credible interval of (0.61, 1.47) (Table 2 in Qian and Richardson, 1997).  The posterior distribution of the retention capacity is skewed to the left.  In subsequent papers, Curt Richardson name the result as "the 1 gram rule".  The South Florida Water Management District (SFWMD) never believed our work and often claimed that the retention rate would be much higher.

Since then, SFWMD has constructed several Stormwater Treatment Areas (STAs) -- wetlands for removing phosphorus and has been monitoring the performances.  The latest results (Chen, et al, 2015) showed that the retention capacity of these STAs is 1.1  +/- 0.5 grams per square meter per year.

I was satisfied that finally SFWMD agreed with my finding, even if the agreement took them nearly 20 years (and hundreds of millions of dollars).

Chen, H., Ivanoff, D., and Pietro, K. (2015) Long-term phosphorus removal in the Everglades stormwater treatment areas of South Florida in the United States.  Ecological Engineering, 29:158-168.

Qian, S.S. and C.J. Richardson (1997) Estimating the long-term phosphorus accretion rate in the Everglades: a Bayesian approach with risk assessment.  Water Resources Research, 33(7): 1681-1688.