On the Challenge of Fitting Tree Size Distributions in Ecology

Abstract

Patterns that resemble strongly skewed size distributions are frequently observed in ecology. A typical example represents tree size distributions of stem diameters. Empirical tests of ecological theories predicting their parameters have been conducted, but the results are difficult to interpret because the statistical methods that are applied to fit such decaying size distributions vary. In addition, binning of field data as well as measurement errors might potentially bias parameter estimates. Here, we compare three different methods for parameter estimation – the common maximum likelihood estimation (MLE) and two modified types of MLE correcting for binning of observations or random measurement errors. We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error. We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties. The modified MLE methods that consider such uncertainties (either binning or measurement error) are comparatively much more robust. We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions. In general, modified MLE methods that correct binning or measurement errors can be applied to ensure reliable results.

What the paper shows and why it matters

(AI-generated)

Strongly skewed size distributions, in which small values are common and large ones rare, occur frequently in ecology; tree stem diameters are a typical example. Theories such as metabolic scaling and self-organised criticality predict their form and parameters, so testing them means fitting the distributions to field data. Maximum likelihood estimation (MLE), which finds the parameter values that make the recorded data most probable, is regarded as the most accurate way to do this, and as better than regression on size classes. Field data, however, are rounded or sorted into classes (“binning”) or carry random errors, whereas standard MLE treats every recorded value as exact. To the authors’ knowledge, how this biases the fitted parameters and the choice between candidate distributions had not been examined systematically.

The paper shows that even maximum likelihood fitting of skewed size distributions becomes biased and often picks the wrong distribution once data are binned or carry random measurement error, and that likelihoods built to model this uncertainty largely repair the problem. The study used 1,000 simulated datasets per case (250 for the measurement-error correction), drawn from power-law, negative exponential and Weibull distributions. In the main runs, with 500 values each, standard MLE kept the mean parameter bias under 5% only for very small bin widths or errors. Beyond small thresholds it favoured the wrong distribution for power-law and exponential data when candidates were compared with Akaike weights, a score that rewards fit and penalises complexity, and larger samples did not remove the problem. Two modified likelihoods, one for binning and one for normally distributed measurement error, reduced the bias for most parameters and identified power-law and Weibull data correctly. For 207,105 stem diameters from Barro Colorado Island, where bin width and error are small, the MLE variants gave similar estimates, whereas regression fits differed markedly.

**Figure 1.** Outline of tree size measurements in forests. From Taubert et al. (2013), *PLoS ONE*, licensed under CC BY 4.0. Click the image to enlarge.

The paper reads these results as a warning: fitted parameters from different datasets, or from data and theory, are hard to compare unless binning and measurement error are handled. It advises reducing binning, quantifying observation accuracy and using the corrected likelihoods when uncertainty matters. Open questions include estimating the truncation limits and treating binning and error together. Moore et al. (2018) cite the paper for MLE working well on forest size distributions when binning is fine.

References

Moore et al. (2018). Equilibrium forest demography explains the distribution of tree sizes across North America. Environmental Research Letters 13, 084019. doi:10.1088/1748-9326/aad6d1