Methods and Software for Analysis of Population Structure
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Detecting Bimodality in a Frequency Distribution

Sometimes there is the need to examine whether a frequency distribution is composed of more than one mode.   Detection of multiple modes in frequency distributions by fitting mixtures of normal distributions has been used in analysis of a number of biological problems, e.g., Pearson (1894) in distinguishing families of crabs by their carapace measurements, Doehlert et al. (2004) defining oat kernel uniformity, Fan et al. (2005) detecting prevalence of type 2 diabetes from blood sugar measures, Hosenfeld et al (1997) in analogical reasoning performance of elementary school children.  

In our research we are interested in occurrence, or otherwise, of bimodal distributions in size of plants undergoing competition.  If a bimodal distribution does occur then the implication is that there are two groups of plants , dominants (large) and suppressed (small).  The suggestion is that such a distribution can be produced by large plants having a distinctly different relative growth rate (RGR) from small plants (Ford 1975), i.e., there is some form of non-linearity in the increase of RGR with plant size and this informs about the type of competition occurring and particularly whether it is largely one-sided, i.e., where large plants affect small ones but not vice-versa.

A number of models in which competition is represented as one-sided or asymmetric in favour of large plants predict that bimodality develops in a stand's population structure (Diggle 1976; Gates 1978; Aikman & Watkinson 1980; Ford & Diggle 1981, Franc 2001, Turley & Ford 2009).   However, these judgements about occurrence of bimodality have been made by visual inspection of histograms.   This may be satisfactory where the modes are distinct but not where there is doubt about there occurrence or where estimates of the location of modes and their relative sizes is required.

In the following pages we provide:

              Introduction to the basics of fitting multiple modes to frequency distributions.     arrow gif

              Discussion of the particular problem of assessing modes in size frequency distributions of plants undergoing competition.     arrow gif

              Software and methods for examining whether multiple modes occur in bivariate (plant weight:plant height) frequency distributions.     arrow gif



Aikman, D.P. & Watkinson, A.R. (1980) A model for growth and self-thinning in even-aged monocultures of plants. Annals of Botany, 45, 419-427.


Diggle, P.J. (1976) A spatial stochastic model of inter-plant competition. Journal of Applied Probability, 13, 662-671.


Doehlert D.C, McMullen, M.S, Jannink J.L, Panigrahi, S., Gu, H., Riveland, N.R. (2004) Evaluation of oat kernel size uniformity. Crop Science, 44, 1178-1186.


Fan, J., May S.J, Zhou Y., Barrett-Connor, E. (2005) Bimodality of 2-h plasma glucose distributions in whites. Diabetes Care, 28, 1451-1456.


Ford E.D. (1975) Competition and stand structure in some even-aged plant monocultures. Journal of Ecology, 63, 311-333.


Ford E.D. & Diggle P.J. (1981) Competition for light in a plant monoculture modelled as a spatial stochastic process. Annals of Botany, 48, 481-500.


Franc, A. (2001) Bimodality for plant sizes and spatial pattern in cohorts: the role of competition and site conditions. Theoretical Population Biology, 60, 117-132


Gates, D.J. (1978) Bimodality in even-aged plant monocultures. Journal of Theoretical Biology, 71, 525-540.


Hosenfeld, B., van der Maas, H.L.J., van den Boom, D.C. (1997) Detecting bimodality in the analogical reasoning performance of elementary schoolchildren. International Journal of Behavioral Development, 20, 529-547.


Pearson, K. (1894) III. Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London,A. 185, 71-110.


Turley,M.C. & Ford, E.D. (2009) Definition and calculation of uncertainty in ecological process models. Ecological Modelling, 220, 1968-1983.