title: Optimal methods for fitting probability distributions to propagule retention time in studies of zoochorous dispersal.
abstract: BACKGROUND
  Propagule retention time is a key factor in determining propagule dispersal distance and the shape of "seed shadows". Propagules dispersed by animal vectors are either ingested and retained in the gut until defecation or attached externally to the body until detachment. Retention time is a continuous variable, but it is commonly measured at discrete time points, according to pre-established sampling time-intervals. Although parametric continuous distributions have been widely fitted to these interval-censored data, the performance of different fitting methods has not been evaluated. To investigate the performance of five different fitting methods, we fitted parametric probability distributions to typical discretized retention-time data with known distribution using as data-points either the lower, mid or upper bounds of sampling intervals, as well as the cumulative distribution of observed values (using either maximum likelihood or non-linear least squares for parameter estimation); then compared the estimated and original distributions to assess the accuracy of each method. We also assessed the robustness of these methods to variations in the sampling procedure (sample size and length of sampling time-intervals).
  
  
  RESULTS
  Fittings to the cumulative distribution performed better for all types of parametric distributions (lognormal, gamma and Weibull distributions) and were more robust to variations in sample size and sampling time-intervals. These estimated distributions had negligible deviations of up to 0.045 in cumulative probability of retention times (according to the Kolmogorov-Smirnov statistic) in relation to original distributions from which propagule retention time was simulated, supporting the overall accuracy of this fitting method. In contrast, fitting the sampling-interval bounds resulted in greater deviations that ranged from 0.058 to 0.273 in cumulative probability of retention times, which may introduce considerable biases in parameter estimates.
  
  
  CONCLUSIONS
  We recommend the use of cumulative probability to fit parametric probability distributions to propagule retention time, specifically using maximum likelihood for parameter estimation. Furthermore, the experimental design for an optimal characterization of unimodal propagule retention time should contemplate at least 500 recovered propagules and sampling time-intervals not larger than the time peak of propagule retrieval, except in the tail of the distribution where broader sampling time-intervals may also produce accurate fits.
authors.affiliation: Estación Biológica de Doñana (EBD-CSIC), C/Américo Vespucio, s/n, 41092, Seville, Spain. dviana@ebd.csic.es.
authors.forename: Duarte S
authors.initials: DS
authors.lastname: Viana
authors.affiliation: Estación Biológica de Doñana (EBD-CSIC), C/Américo Vespucio, s/n, 41092, Seville, Spain. luis.santamaria@ebd.csic.es.
authors.forename: Luis
authors.initials: L
authors.lastname: Santamaría
authors.affiliation: Estación Biológica de Doñana (EBD-CSIC), C/Américo Vespucio, s/n, 41092, Seville, Spain. jordi@ebd.csic.es.|CIBER Epidemiología y Salud Pública (CIBERESP), Seville, Spain. jordi@ebd.csic.es.
authors.forename: Jordi
authors.initials: J
authors.lastname: Figuerola
mesh_terms: D000818:Animals; D029785:Cyperaceae / Q000502:physiology; D004372:Ducks; D018521:Plant Physiological Phenomena; D031821:Potamogetonaceae / Q000502:physiology; D011336:Probability; D018401:Sample Size; D058614:Seed Dispersal; D013997:Time Factors
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