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# Predictions from data

Predictions from data

## The error curve shown in

The error curve shown in Figure 5.4 is an optimum fit to the error data shown in Tables 5.5 to5.8, using a function of the form y = k/ %x, where x is the sample size. This is a statisticallystandard application, error declining as the inverse square root of sample size.Figure 5.4. richness estimation error as a function of sample sizeAs will be seen in the next section, the percentage contribution to variance by the sourcecommunity is relatively low for small sample sizes, with (pure) sample variance predominating.The resulting error curve is therefore the weighted sum of two functions, the weights graduallyshifting from predominantly sample-influenced to community-influenced, as sample sizeincreases. The approximation curve is reasonably close to the error data and, particularly forlarger samples, may overestimate the error.As can be seen from the relative positions of open (one-step) and closed (two-step) circles, thereis little to choose between the two methods in respect of accuracy. The entries in table 5.9 wereread directly from the curve, in effect. The table, although useful in a limited way, serves only asan example until further experiments sharpen and extend these results.5.6 Analysis of the two sources of varianceAs was pointed out earlier in this chapter, both the community and the sample contribute somevariance to the final estimate of richness. The point of the Series One tests was to tease out thevariance due to the sample alone. The results of Series Two incorporate both sources of variationand they are, of course larger. Corresponding entries of Tables 5.7 and 5.8 may be compared to18

analyse the respective contributions at two intensities or, in this case, sample sizes.Idealized communitiesratio r samp. size # teststwo-step:varianceone-step:variance0.005681049790.010136151216Perturbed communitiesratio r samp. size # teststwo-step:varianceone-step:variance0.00568101081180.0101361594106Table 5.10. Contributions to variances in methods from smooth vs ragged communitiesSince the two sources of variation are completely independent, the variance in the sample may becalculated as a percentage of overall variance:For the two-step method the proportion (percentage) of independent sample variance is about45% at sample size 68, dropping to roughly 13% at sample size 136. For the one-step method,the proportion falls from approximately 68% to 15%. It makes sense that the proportion shoulddrop since larger samples tend to have less relative variation than smaller ones for the samesource community, whatever its state. On the other hand, the one step method produces largerpercentages across the board, the difference diminishing as the sample size increases.There is a clear tradeoff between the two methods. The one step method is easier to carry outand enjoys much the same accuracy as the two-step method, but it produces a more diffuse kindof prediction, for which the error term is significantly larger than that of the two-step method.19

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