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On the challenge of fitting tree size distributions in ecology.

Taubert F, Hartig F, Dobner HJ, Huth A - PLoS ONE (2013)

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecological Modelling, Helmholtz Centre for Environmental Research, Leipzig, Saxony, Germany. franziska.taubert@ufz.de

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.

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Related in: MedlinePlus

Effects of binning and random measurement errors on parameter estimation using different MLE methods.(a) MLE including binning (multinomial MLE) and (b) MLE accounting for measurement errors (Gaussian MLE). We evaluate virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Solid lines represent the mean estimates and shaded areas show the standard deviation (of (a) 1000 values and (b) 250 values). (a) Effect of binning on parameter estimates. Virtual data are classified into classes of certain bin width (x-axis in cm). (b) Effect of random measurement errors on parameter estimates. An error value generated from a Gaussian distribution with mean  cm and an assumed standard deviation  (x-axis in cm) is added to each virtual data value.
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pone-0058036-g005: Effects of binning and random measurement errors on parameter estimation using different MLE methods.(a) MLE including binning (multinomial MLE) and (b) MLE accounting for measurement errors (Gaussian MLE). We evaluate virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Solid lines represent the mean estimates and shaded areas show the standard deviation (of (a) 1000 values and (b) 250 values). (a) Effect of binning on parameter estimates. Virtual data are classified into classes of certain bin width (x-axis in cm). (b) Effect of random measurement errors on parameter estimates. An error value generated from a Gaussian distribution with mean cm and an assumed standard deviation (x-axis in cm) is added to each virtual data value.

Mentions: Using multinomial MLE, the negative effects can be reduced to a large extent (Fig. 5a, Table S1). For the entire range of investigated bin widths, a significantly lower mean bias of -, - and -parameter estimates can be observed not exceeding a mean bias of 9% of the corresponding true parameter value (Table S1). For -estimates binning correction fails only for high widths (>11 cm, Fig. 5a). However, it reaches a maximum absolute mean bias of 59% of the true -value, which is still smaller than for employing standard MLE (Table S1). Standard deviations of the parameter estimates increase with increasing bin width for nearly all parameters, except for, which decreases (Fig. 5a).


On the challenge of fitting tree size distributions in ecology.

Taubert F, Hartig F, Dobner HJ, Huth A - PLoS ONE (2013)

Effects of binning and random measurement errors on parameter estimation using different MLE methods.(a) MLE including binning (multinomial MLE) and (b) MLE accounting for measurement errors (Gaussian MLE). We evaluate virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Solid lines represent the mean estimates and shaded areas show the standard deviation (of (a) 1000 values and (b) 250 values). (a) Effect of binning on parameter estimates. Virtual data are classified into classes of certain bin width (x-axis in cm). (b) Effect of random measurement errors on parameter estimates. An error value generated from a Gaussian distribution with mean  cm and an assumed standard deviation  (x-axis in cm) is added to each virtual data value.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3585190&req=5

pone-0058036-g005: Effects of binning and random measurement errors on parameter estimation using different MLE methods.(a) MLE including binning (multinomial MLE) and (b) MLE accounting for measurement errors (Gaussian MLE). We evaluate virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Solid lines represent the mean estimates and shaded areas show the standard deviation (of (a) 1000 values and (b) 250 values). (a) Effect of binning on parameter estimates. Virtual data are classified into classes of certain bin width (x-axis in cm). (b) Effect of random measurement errors on parameter estimates. An error value generated from a Gaussian distribution with mean cm and an assumed standard deviation (x-axis in cm) is added to each virtual data value.
Mentions: Using multinomial MLE, the negative effects can be reduced to a large extent (Fig. 5a, Table S1). For the entire range of investigated bin widths, a significantly lower mean bias of -, - and -parameter estimates can be observed not exceeding a mean bias of 9% of the corresponding true parameter value (Table S1). For -estimates binning correction fails only for high widths (>11 cm, Fig. 5a). However, it reaches a maximum absolute mean bias of 59% of the true -value, which is still smaller than for employing standard MLE (Table S1). Standard deviations of the parameter estimates increase with increasing bin width for nearly all parameters, except for, which decreases (Fig. 5a).

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecological Modelling, Helmholtz Centre for Environmental Research, Leipzig, Saxony, Germany. franziska.taubert@ufz.de

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.

Show MeSH
Related in: MedlinePlus