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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

Analyses of binned virtual data using different bin widths.We evaluate 1,000 virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Virtual data are classified into classes of certain bin width (x-axis in cm) before applying standard MLE. (a) Effect of binning on parameter estimates of the three investigated distributions. (b)–(d) Effect of binning on Akaike weights supposing three distributions (power-law, negative exponential and Weibull distribution) for (b) power-law distributed virtual data, (c) negative exponentially distributed virtual data and (d) Weibull distributed virtual data. The highest Akaike weight determines the best fit of a frequency distribution to the data. Solid lines represent the mean values and shaded areas show the standard deviation (of 1,000 calculated values).
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pone-0058036-g003: Analyses of binned virtual data using different bin widths.We evaluate 1,000 virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Virtual data are classified into classes of certain bin width (x-axis in cm) before applying standard MLE. (a) Effect of binning on parameter estimates of the three investigated distributions. (b)–(d) Effect of binning on Akaike weights supposing three distributions (power-law, negative exponential and Weibull distribution) for (b) power-law distributed virtual data, (c) negative exponentially distributed virtual data and (d) Weibull distributed virtual data. The highest Akaike weight determines the best fit of a frequency distribution to the data. Solid lines represent the mean values and shaded areas show the standard deviation (of 1,000 calculated values).

Mentions: Increasing bin widths generally affects the parameter estimates of all three considered distributions, thus creating remarkable biases (Fig. 3a). Based on representative virtual data of sample size  = 500, only small bin widths of approximately cm ensure a mean bias of less than 5% of the true parameter of the corresponding distributions (Table S1). With incrementing widths of cm, nearly all parameters are on average underestimated, except the parameter of the Weibull distribution, which is highly overestimated (Fig. 3a). Maximum absolute values of the mean bias range from 48% (-estimates) to 280% (-estimates) (Table S1). Standard deviations of -, - and -parameter estimates decrease with bin width, whereas the standard deviation of -values increases (Fig. 3a).


On the challenge of fitting tree size distributions in ecology.

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

Analyses of binned virtual data using different bin widths.We evaluate 1,000 virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Virtual data are classified into classes of certain bin width (x-axis in cm) before applying standard MLE. (a) Effect of binning on parameter estimates of the three investigated distributions. (b)–(d) Effect of binning on Akaike weights supposing three distributions (power-law, negative exponential and Weibull distribution) for (b) power-law distributed virtual data, (c) negative exponentially distributed virtual data and (d) Weibull distributed virtual data. The highest Akaike weight determines the best fit of a frequency distribution to the data. Solid lines represent the mean values and shaded areas show the standard deviation (of 1,000 calculated values).
© Copyright Policy
Related In: Results  -  Collection

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

pone-0058036-g003: Analyses of binned virtual data using different bin widths.We evaluate 1,000 virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Virtual data are classified into classes of certain bin width (x-axis in cm) before applying standard MLE. (a) Effect of binning on parameter estimates of the three investigated distributions. (b)–(d) Effect of binning on Akaike weights supposing three distributions (power-law, negative exponential and Weibull distribution) for (b) power-law distributed virtual data, (c) negative exponentially distributed virtual data and (d) Weibull distributed virtual data. The highest Akaike weight determines the best fit of a frequency distribution to the data. Solid lines represent the mean values and shaded areas show the standard deviation (of 1,000 calculated values).
Mentions: Increasing bin widths generally affects the parameter estimates of all three considered distributions, thus creating remarkable biases (Fig. 3a). Based on representative virtual data of sample size  = 500, only small bin widths of approximately cm ensure a mean bias of less than 5% of the true parameter of the corresponding distributions (Table S1). With incrementing widths of cm, nearly all parameters are on average underestimated, except the parameter of the Weibull distribution, which is highly overestimated (Fig. 3a). Maximum absolute values of the mean bias range from 48% (-estimates) to 280% (-estimates) (Table S1). Standard deviations of -, - and -parameter estimates decrease with bin width, whereas the standard deviation of -values increases (Fig. 3a).

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