Unbiased Linear Property Estimation For Spheres From Sections Exhibiting Overprojection And Truncation

In the biological sciences, stereological techniques are frequently used to infer changes in structural parameters (volume fraction, for example) between samples from different populations or subject to differing treatment regimes. Non-homogeneity of these parameters is virtually guaranteed, both between experimental animals and within the organ under consideration. A two-stage strategy is then desirable, the first stage involving unbiased estimation of the required parameter, separately for each experimental unit, the latter being defined as a subset of the organ for which homogeneity can reasonably be assumed. In the second stage, these point estimates are used as data inputs to a hierarchical analysis of variance, to distinguish treatment effects from variability between animals, for example. Techniques are therefore required for unbiased estimation of parameters from potentially small numbers of sample profiles. This paper derives unbiased estimates of linear properties in one special case—the sampling of spherical particles by transmission microscopy, when the section thickness is not negligible and the resulting circular profiles are subject to lower truncation. The derivation uses the general integral equation formulation of Nicholson (1970); the resulting formulae are simplified, algebraically, and their efficient computation discussed. Bias arising from variability in slice thickness is shown to be negligible in typical cases. The strategy is illustrated for data examining the effects, on the secondary lysosomes in the digestive cells, of exposure of the common mussel to hydrocarbons. Prolonged exposure, at 30 μg 1−1 total oil-derived hydrocarbons, is seen to increase the average volume of a lysosome, and the volume fraction that lysosomes occupy, but to reduce their number.

Differences in performance among four indices used to evaluate diversity in planktonic ecosystems

The performance of four common estimators of diversity are investigated using calanoid copepod data from the Continuous Plankton Recorder (CPR) survey. The region of the North Atlantic and the North Sea was divided into squares of 400 nautical miles for each 2-month period. For each 144 possible cases, Pielou's pooled quadrat method was performed with the aims of determining asymptotic diversity and investigating the CPR sample-size dependence of diversity estimators. It is shown that the performance of diversity indices may greatly vary in space and time (at a seasonal scale). This dependence is more pronounced in higher diverse environments and when the sample size is small. Despite results showing that all estimators underestimate the `actual' diversity, comparison of sites remained reliable from a few pooled CPR samples. Using more than one CPR sample, the Gini coefficient appears to be a better diversity estimator than any other indices and spatial or temporal comparisons are highly satisfactory. In situations where comparative studies are needed but only one CPR sample is available, taxonomic richness was the preferred method of estimating diversity. Recommendations are proposed to maximise the efficiency of diversity estimations with the CPR data.

Comparison of Zooplankton Sampling Performance of Longhurst-Hardy Plankton Recorder and Bongo Nets

Data on the abundance and biomass of zooplankton off the northwestern Portuguese coast, separately estimated with a Longhurst-Hardy Plankton Recorder (LHPR) and a Bongo net, were analysed to assess the comparative performance of the samplers. Zooplankton was collected along four transects perpendicular to the coast, deployments alternating between samplers. Total zooplankton biomass measured using the LHPR was significantly higher than that using the Bongo net. Apart from Appendicularia and Cladocera, abundances of other taxa (Copepoda, Mysidacea, Euphausiacea, Decapoda larvae, Amphipoda, Siphonophora, Hydromedusae, Chaetognatha and Fish eggs) were also consistently higher in the LHPR. Some of these differences were probably due to avoidance by the zooplankton of the Bongo net. This was supported by a comparative analysis of prosome length of the copepod Calanus helgolandicus sampled by the two nets that showed that Calanus in the LHPR samples were on average significantly larger, particularly in day samples. A ratio estimator was used to produce a factor to convert Bongo net biomass and abundance estimates to equate them with those taken with the LHPR. This method demonstrates how results from complementary zooplankton sampling strategies can be made more equivalent.