Measures of precision for dissimilarity-based multivariate analysis of ecological communities

Abstract

Ecological studies require key decisions regarding the appropriate size and number of sampling units. No methods currently exist to measure precision for multivariate assemblage data when dissimilarity-based analyses are intended to follow. Here, we propose a pseudo multivariate dissimilarity-based standard error (MultSE) as a useful quantity for assessing sample-size adequacy in studies of ecological communities. Based on sums of squared dissimilarities, MultSE measures variability in the position of the centroid in the space of a chosen dissimilarity measure under repeated sampling for a given sample size. We describe a novel double resampling method to quantify uncertainty in MultSE values with increasing sample size. For more complex designs, values of MultSE can be calculated from the pseudo residual mean square of a permanova model, with the double resampling done within appropriate cells in the design. R code functions for implementing these techniques, along with ecological examples, are provided.

Keywords: Assemblage data; community ecology; dissimilarities; multivariate analysis; permanova; precision; replicates; sampling design; standard error; variability.

Figure 1

Multivariate pseudo standard error (MultSE) as a function of sample size on the basis of Bray–Curtis dissimilarities calculated on square-root transformed fish abundance data from Ningaloo (a) for an immersion time of 60 min only, showing results (means with 2.5 and 97.5 percentiles as error bars) from 10 000 resamples obtained using either the permutation or bootstrap approach and (b) for all three immersion times, using the double resampling method, with permutation-based means and bias-adjusted bootstrap-based error bars (with 10 000 resamples for each).

Figure 2

Multivariate pseudo standard error (MultSE) as a function of sample size (a) for each of 5 different areas on the basis of Jaccard dissimilarities for macrofaunal communities from the Norwegian continental shelf and (b) for each of three different times on the basis of Bray–Curtis dissimilarities calculated on log(x + 1)-transformed counts of fishes from the Poor Knights Islands, New Zealand. A double resampling scheme was used to generate means for each sample size using 10 000 permutations and error bars as bias-adjusted 2.5 and 97.5 percentiles from 10 000 bootstrap resamples.

Figure 3

Multivariate pseudo standard error (MultSE) calculated from the residual mean square of a one-way permanova model as a function of sample size on the basis of Bray–Curtis dissimilarities calculated on log(x + 1)-transformed counts of fishes from the Poor Knights Islands, New Zealand. A double resampling scheme was used to generate means for each sample size using 10 000 permutations and error bars as bias-adjusted 2.5 and 97.5 percentiles from 10 000 bootstrap resamples. Resampling was done separately within each of the three sampling times (i.e. within each of the three groups in the one-way permanova model design).