Bimodality – the characteristic of a continuous variable having two distinct modes – is of widespread interest in data analysis. This is because, in some cases, we can use the presence or absence of bimodality to infer something about the underlying processes generating the distribution of a variable that we are interested in studying. In ecology, tests of bimodality have been used in many different contexts, such as to understand body size distributions, functional traits, and transitions among different ecosystem states. But a lack of evidence for bimodality has been reported in many studies. Our paper “Masting, mixtures and modes: are two models better than one?”, now shows that a widely-used statistical test of bimodality can fail to reject the null hypothesis that focal probability distributions are unimodal. We instead promote the use of mixture models as a theory oriented framework for testing hypotheses of bimodality.
Our interest in this problem arose with the publication of Allen et al. 2012 Oikos 121: 367–376. The paper impressive synthesised 43 years of seeding patterns in a New Zealand mountain beech Nothofagus forest. Seed production in these trees is interesting because a population can go several years without reproducing and then all the individuals in a population will do so. Such intermittent and synchronous reproduction is also known as mast seeding.
In their paper, Allen et al. tried to infer the importance of resource limitation in driving mast seeding patterns by describing various characteristics of seed production. A key finding was that they could not reject the null hypothesis that the distribution of annual seed production was unimodal using an empirical calculation known as Hartigan’s dip test. Allen et al. 2012 concluded that few studies could ‘robustly test for bimodality’ because they lacked ‘long time-series’ and used questionable methods.
Nothofagus solandri var. cliffortioides trees in Fiordland, New Zealand
The bimodality result inspired a lot of reflection. We were interested in why we might consider an annual count that takes values larger than zero at more than 2 year intervals to ever even present two modes. Populations should only have one mode because the single most frequently observed seed count will be relatively low, on average, in most years. Thus, plants can never be bimodal when a single distribution is fit to seed counts – even if they alternate annually, on average, between the absence and presence of seed production. This is because of the underlying statistical nature of seed counts, which we describe in further detail in our paper.
Stepping back further from the problem, we began to realize that there is a need to consider multiple probability models, each with at least one unique mode, where plants flower at >2 year intervals.We illustrate these ideas by analysing 37 years of data from five grass species in New Zealand. Critically, we found clear evidence for bimodality using mixture models that associate distinct probability distributions with medium- and high- versus non- and low-flowering years. We expect these patterns to be driven by different processes and hence, modelled by different probability distributions. We found no evidence for bimodality with Hartigan’s dip test that assumes a single probability distribution can be fitted to all the data. Our findings show the importance of coupling theoretical expectations with the appropriate statistical tools when predicting the responses of ecological processes.
The authors through Andrew J. Tanentzap