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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 virtual data including different levels of measurement errors.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. 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 point before applying standard MLE. (a) Effect of random measurement errors on parameter estimates of the three investigated distributions. (b)–(d) Effect of random measurement errors 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-g004: Analyses of virtual data including different levels of measurement errors.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. 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 point before applying standard MLE. (a) Effect of random measurement errors on parameter estimates of the three investigated distributions. (b)–(d) Effect of random measurement errors 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: Random measurement errors included in the virtual data sets with 500 values also have substantial effects on parameter estimates (Fig. 4a). For -, - and -estimates the mean parameter value is underestimated (again, except for the parameter). Significant effects already start at a small measurement error of cm with a mean bias of approximately 5% of the true parameter value (Fig. 4a, Table S1). Absolute mean biases reach their maximum in the range between 37% (-estimates) and 110% (-estimates) (Table S1). Standard deviations of parameter estimates show similar trends as was observed for binned data.


On the challenge of fitting tree size distributions in ecology.

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

Analyses of virtual data including different levels of measurement errors.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. 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 point before applying standard MLE. (a) Effect of random measurement errors on parameter estimates of the three investigated distributions. (b)–(d) Effect of random measurement errors 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-g004: Analyses of virtual data including different levels of measurement errors.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. 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 point before applying standard MLE. (a) Effect of random measurement errors on parameter estimates of the three investigated distributions. (b)–(d) Effect of random measurement errors 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: Random measurement errors included in the virtual data sets with 500 values also have substantial effects on parameter estimates (Fig. 4a). For -, - and -estimates the mean parameter value is underestimated (again, except for the parameter). Significant effects already start at a small measurement error of cm with a mean bias of approximately 5% of the true parameter value (Fig. 4a, Table S1). Absolute mean biases reach their maximum in the range between 37% (-estimates) and 110% (-estimates) (Table S1). Standard deviations of parameter estimates show similar trends as was observed for binned data.

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