In his comments on our study introducing a procedure for estimating the seed dispersal kernel from categorical identification of source plants (Robledo-Arnuncio & García 2007), Ralph (2008) maintains that our method can be biased by variation in maternal fecundities. His assertion, which lacks a formal proof, seems to contradict the results presented in Robledo-Arnuncio & García (2007), showing that the method remains essentially unbiased under different dispersal and sampling scenarios, even in the presence of coefficients of variation for seed fecundity of as much as 200%. Here, I use Monte Carlo simulations to formally assess the statistical consequences of the modification to our method proposed by Ralph (2008), under different dispersal, demographic and sampling scenarios. I show that his modification to eliminate the putative bias of our method does not affect either its (low) bias or its variance except if the total number of maternal plants in the population is very small. If the number of maternal plants is very small, Ralph's (2008) modification will increase or decrease dispersal estimation errors depending on the seed collection strategy and the available information about the seed rain pattern, as detailed below.
The motivation in Robledo-Arnuncio & García (2007) was to devise an accurate method, based on categorical maternity analysis, to estimate the seed dispersal kernel independently of seed fecundity information. Such a method could circumvent the morphological measurements, allometric assumptions, and statistical problems of previous approaches that jointly estimate seed dispersal and fecundity parameters (see Clark et al. 2004). Inspired by more general methods for gene flow analysis, we proposed a maximum-likelihood estimation procedure based on the theoretically expected proportion (πij) of seeds in each sampling location j that have dispersed from each mother plant i within the study site. We modelled πij as a simple function of the assumed dispersal kernel and the relative spatial distribution of all potential mothers and seed collection sites (equation 2 in Robledo-Arnuncio & García 2007), showing through numerical simulation that disregarding the effect of individual seed fecundity variation on πij does not result in biased estimates of seed dispersal parameters.
Despite the results in Robledo-Arnuncio & García (2007), Ralph (2008) claims that our method ‘can be biased by variation in maternal fecundities’, his only argument being that ‘the assumption of equal fecundities could bias the results, for example, if the dispersal kernel is Gaussian with variance θ, then an extremely fecund individual farther away from the seed traps than most will cause the θ to be overestimated’. An obvious objection to this assertion is that, unless the total number of seed sources in the population is very small, it is likely that there will be an individual with ‘extremely’low fecundity, also ‘farther away from the seed traps’, which would cause the θ to be underestimated, compensating for the nearby one with extremely high fecundity. More generally, as already discussed in Robledo-Arnuncio & García (2007), ‘there is no obvious reason to expect a nonrandom distribution of individual fecundity values across the mother-trap distance distribution (i.e. the distribution of potential dispersal distances), which would explain the insensitivity of the dispersal estimates to fertility variation’. In fact, the results presented in Robledo-Arnuncio & García (2007) show unbiased estimates of the scale and shape parameters of the dispersal kernel in large populations under strong simulated levels of seed fecundity variation.
Ralph (2008) proposes a modification to our estimation method, aimed at reducing its putative bias under unequal female fecundity. Instead of deriving the theoretically expected probability that a seed is from mother i given that it was found in trap j (πij), he models, in a similar fashion, the probability that a seed is found in trap j given that it is from mother i and given that the total number of seeds dispersed from mother i into the sampling sites is Ni. He argues that conditioning on the Ni gives dispersal parameter estimates that are unbiased in the presence of fecundity variation. This approach is appealing, although it depends on a sampling requirement that is not needed to apply our original method: it is necessary to know the actual density of dispersed seeds (seed rain density) at every sampling location. It must be noted that reliable seed rain density estimates may sometimes not be available; for instance, seed traps may not be well protected from (potentially spatially heterogeneous) predation, and/or seeds may be collected individually and nonexhaustively from the ground, or a fixed number of seeds may have been collected for analysis from each sampling site (e.g. Jones et al. 2005). For these practical reasons, and since no formal investigation of the statistical consequences of the proposed modification was performed, it is unclear from the information in Ralph (2008) to what extent his modified estimator represents a useful advance.
In order to investigate the utility of the approach proposed by Ralph (2008), I used the Monte Carlo simulation scheme described in Robledo-Arnuncio & García (2007) to assess the bias and accuracy [root mean square error (RMSE)] of the estimates of the seed dispersal kernel parameters obtained with his modified procedure, comparing the estimation errors against those of the original method. I considered contrasting dispersal conditions and sampling sizes, and assumed female fecundity variation in all cases, with a reference coefficient of variation of 100% and maximum values of up to 200% (see Supplementary material). For a fixed total number of sampled seeds, two alternative seed collection strategies were investigated: (i) assuming that the actual seed rain density in each site was unknown, a fixed number of seeds was sampled at each collection site (as in Robledo-Arnuncio & García 2007); and (ii) assuming that the actual seed rain density in each site was known (a necessity to apply Ralph's approach), seeds were sampled proportionally to this quantity at each site. Since the modification proposed by Ralph (2008) is likely to make a difference only if the total number of maternal plants in the population is very small, I first considered a large population of 10 000 plants (as in Robledo-Arnuncio & García 2007) and then small populations of decreasing size, from 100 to 10 plants.
Looking first at the results for large populations, the simulations indicate that the modification proposed by Ralph (2008) does not significantly change the (low) biases and (variable) RMSE of our original method, which holds for increasing levels of maternal fecundity variation (Table S1, Supplementary material), different sample sizes (Table S2, Supplementary material), and different dispersal ranges and kurtosis (Table S3, Supplementary material). The statistical errors of the two approaches remain essentially equal to each other either if the number of seeds sampled per site is fixed (assuming seed rain densities are unknown) or if it is proportional to the seed rain density in the site. Interestingly, however, both methods exhibit reduced bias and RMSE when seed rain densities are known and accounted for (Tables S1–S3). Considering now the extreme case of a population comprising only a few tens of maternal plants (M ≤ 100; Table S4, Supplementary material), the simulation results indicate that: (i) if a fixed number of seeds is sampled at each site (assuming seed rain densities are unknown), then our original method is biased by seed fecundity variation, and Ralph's (2008) modification introduces further bias and increases the RMSE; and (ii) if the number of seeds sampled per site is proportional to the site's seed rain density, then both approaches are virtually unbiased, and Ralph's (2008) modification reduces the RMSE of the original method, especially under unequal female fecundity.
Some practical considerations can be drawn from Ralph's (2008) commentaries and the above results. First, if reliable seed rain density estimates are available for every sampling site, more accurate dispersal estimates will be obtained with either method by sampling seeds proportionally to the seed rain density in each site (or by correcting for this densities as Ralph (2008) indicates) than by ignoring the seed rain pattern and sampling a fixed number of seeds per site. Second, unless the number of maternal plants in the population is very small (roughly < 102), the modification to our method suggested by Ralph (2008) will not significantly affect dispersal parameter estimates. Last, if the number of maternal plants in the population is very small, Ralph's (2008) modification to our method will improve dispersal estimates if seed rain densities are known and accounted for in every seed sampling site, while it will increase estimation errors if seed rain densities are ignored and the number of seeds sampled per site is fixed.