Sampling in ecology and evolution – bridging the gap between theory and practice

Sampling is a key issue for answering most ecological and evolutionary questions. The importance of developing a rigorous sampling design tailored to specific questions has already been discussed in the ecological and sampling literature and has provided useful tools and recommendations to sample and analyse ecological data. However, sampling issues are often difficult to overcome in ecological studies due to apparent inconsistencies between theory and practice, often leading to the implementation of simplified sampling designs that suffer from unknown biases. Moreover, we believe that classical sampling principles which are based on estimation of means and variances are insufficient to fully address many ecological questions that rely on estimating relationships between a response and a set of predictor variables over time and space. Our objective is thus to highlight the importance of selecting an appropriate sampling space and an appropriate sampling design. We also emphasize the importance of using prior knowledge of the study system to estimate models or complex parameters and thus better understand ecological patterns and processes generating these patterns. Using a semi‐virtual simulation study as an illustration we reveal how the selection of the space (e.g. geographic, climatic), in which the sampling is designed, influences the patterns that can be ultimately detected. We also demonstrate the inefficiency of common sampling designs to reveal response curves between ecological variables and climatic gradients. Further, we show that response‐surface methodology, which has rarely been used in ecology, is much more efficient than more traditional methods. Finally, we discuss the use of prior knowledge, simulation studies and model‐based designs in defining appropriate sampling designs. We conclude by a call for development of methods to unbiasedly estimate nonlinear ecologically relevant parameters, in order to make inferences while fulfilling requirements of both sampling theory and field work logistics.     

Using Inverse Probability Bootstrap Sampling to Eliminate Sample Induced Bias in Model Based Analysis of Unequal Probability Samples

In ecology, as in other research fields, efficient sampling for population estimation often drives sample designs toward unequal probability sampling, such as in stratified sampling. Design based statistical analysis tools are appropriate for seamless integration of sample design into the statistical analysis. However, it is also common and necessary, after a sampling design has been implemented, to use datasets to address questions that, in many cases, were not considered during the sampling design phase. Questions may arise requiring the use of model based statistical tools such as multiple regression, quantile regression, or regression tree analysis. However, such model based tools may require, for ensuring unbiased estimation, data from simple random samples, which can be problematic when analyzing data from unequal probability designs. Despite numerous method specific tools available to properly account for sampling design, too often in the analysis of ecological data, sample design is ignored and consequences are not properly considered. We demonstrate here that violation of this assumption can lead to biased parameter estimates in ecological research. In addition, to the set of tools available for researchers to properly account for sampling design in model based analysis, we introduce inverse probability bootstrapping (IPB). Inverse probability bootstrapping is an easily implemented method for obtaining equal probability re-samples from a probability sample, from which unbiased model based estimates can be made. We demonstrate the potential for bias in model-based analyses that ignore sample inclusion probabilities, and the effectiveness of IPB sampling in eliminating this bias, using both simulated and actual ecological data. For illustration, we considered three model based analysis tools—linear regression, quantile regression, and boosted regression tree analysis. In all models, using both simulated and actual ecological data, we found inferences to be biased, sometimes severely, when sample inclusion probabilities were ignored, while IPB sampling effectively produced unbiased parameter estimates.     

Zur Versuchsplanung für Schätzungen im gemischten Modell der linearen Regression1

The covariance matrix of the least-squares-estimator for the coefficients of the mixed model of linear regression is deduced. This serves as a base to work out procedures experimental design for point and confidence estimations of the regression coefficients and the regression function. So it was shown, that the C-, A-, D- and G-optimal designs in the mixed model are the same as in model I. Further an assertion for sample size determination is proved especially for point estimation of the regression function.     

Optimizing noninvasive sampling of a zoonotic bat virus

Outbreaks of infectious viruses resulting from spillover events from bats have brought much attention to bat‐borne zoonoses, which has motivated increased ecological and epidemiological studies on bat populations. Field sampling methods often collect pooled samples of bat excreta from plastic sheets placed under‐roosts. However, positive bias is introduced because multiple individuals may contribute to pooled samples, making studies of viral dynamics difficult. Here, we explore the general issue of bias in spatial sample pooling using Hendra virus in Australian bats as a case study. We assessed the accuracy of different under‐roost sampling designs using generalized additive models and field data from individually captured bats and pooled urine samples. We then used theoretical simulation models of bat density and under‐roost sampling to understand the mechanistic drivers of bias. The most commonly used sampling design estimated viral prevalence 3.2 times higher than individual‐level data, with positive bias 5–7 times higher than other designs due to spatial autocorrelation among sampling sheets and clustering of bats in roosts. Simulation results indicate using a stratified random design to collect 30–40 pooled urine samples from 80 to 100 sheets, each with an area of 0.75–1 m², and would allow estimation of true prevalence with minimum sampling bias and false negatives. These results show that widely used under‐roost sampling techniques are highly sensitive to viral presence, but lack specificity, providing limited information regarding viral dynamics. Improved estimation of true prevalence can be attained with minor changes to existing designs such as reducing sheet size, increasing sheet number, and spreading sheets out within the roost area. Our findings provide insight into how spatial sample pooling is vulnerable to bias for a wide range of systems in disease ecology, where optimal sampling design is influenced by pathogen prevalence, host population density, and patterns of aggregation.     

Covariance density estimation for autoregressive spectral modelling of point processes

The use of autoregressive modelling has acquired great importance in time series analysis and in principle it may also be applicable in the spectral analysis of point processes with similar advantages over the nonparametric approach. Most of the methods used for autoregressive spectral analysis require positive semidefinite estimates for the covariance function, while current methods for the estimation of the covariance density function of a point process given a realization over the interval [0,T] do not guarantee a positive semidefinite estimate. This paper discusses methods for the estimation of the covariance density and conditional intensity function of point processes and present alternative computational efficient estimation algorithms leading always to positive semidefinite estimates, therefore adequate for autoregressive spectral analysis. Autoregressive spectral modelling of point processes from Yule-Walker type equations and Levinson recursion combined with the minimum AIC or CAT principle is illustrated with neurobiological data.     

Spatial pattern of diversity in a tropical rain forest in Malaysia

The diversity of trees (species richness, abundance and Shannon diversity) in a tropical rain forest of Malaysia has been studied from the point of view of its spatial organization in order to formulate hypotheses about the origin of the observed spatial patterns. The question that motivated this study is whether tropical forests communities are in a state of equilibrium or non-equilibrium. Three aspects have been examined: (1) changes in diversity were studied with respect to sampling area and sampling designs. A minimum area of 5–10 ha is recommended by the species–area curves, while 2–5 ha seem appropriate based on the Shannon diversity–area curves. Different sampling designs significantly affect the species–area curves. The power function, which can be derived under the equilibrium assumption, is not appropriate to fit the observed diversity–area curves. (2) The spatial features of diversity variables were then studied. Variograms showed that there are dominant short-range effects (around 150 m), obvious anisotropic distribution, and high random variation in the diversity data. (3) Partitioning the variation of the diversity measures into environmental (topographic) and spatial components indicated that the spatial organisation of that community was mostly unpredictable. There may be many processes controlling the formation of the spatial patterns in the tropical rain forest. Unidentified causes, affecting mainly the small-scale processes (<20 m), seem responsible for the large amount of undetermined variation in the diversity data sets. The study suggests that the Pasoh forest of Malaysia may not be in a state of equilibrium.     

Proportional hazards regression for cancer studies

Summary.   There has been some recent work in the statistical literature for modeling the relationship between the size of cancers and probability of detecting metastasis, i.e., aggressive disease. Methods for assessing covariate effects in these studies are limited. In this article, we formulate the problem as assessing covariate effects on a right-censored variable subject to two types of sampling bias. The first is the length-biased sampling that is inherent in screening studies; the second is the two-phase design in which a fraction of tumors are measured. We construct estimation procedures for the proportional hazards model that account for these two sampling issues. In addition, a Nelson–Aalen type estimator is proposed as a summary statistic. Asymptotic results for the regression methodology are provided. The methods are illustrated by application to data from an observational cancer study as well as to simulated data.     

Tests and model selection for the general growth curve model

The model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitution. It is defined as Yp x N = Xp x m tau m x r Ar x N + epsilon p x N, where tau is unknown, and X and A are known design matrices of ranks m less than p and r less than N, respectively. Furthermore, the columns of epsilon are independent p-variate normal with mean vector 0 and common covariance matrix sigma. In general, p is the number of time (or spatial) points observed on each of the N cases, (m - 1) is the degree of polynomial in time, and r is the number of groups. The main focus of this paper is the selection of models for the general growth curve model with regard to the covariance matrix sigma. Likelihood ratio tests and selection procedures based on sample reuse and predictions are proposed. Special emphasis is on the serial covariance structure for sigma, which has been shown to be quite important in the prediction of biological data and technology substitution data. One-population and K-population problems are considered. Some of the results are illustrated with two sets of biological data.     

Scales of association: hierarchical linear models and the measurement of ecological systems

A fundamental challenge to understanding patterns in ecological systems lies in employing methods that can analyse, test and draw inference from measured associations between variables across scales. Hierarchical linear models (HLM) use advanced estimation algorithms to measure regression relationships and variance–covariance parameters in hierarchically structured data. Although hierarchical models have occasionally been used in the analysis of ecological data, their full potential to describe scales of association, diagnose variance explained, and to partition uncertainty has not been employed. In this paper we argue that the use of the HLM framework can enable significantly improved inference about ecological processes across levels of organization. After briefly describing the principals behind HLM, we give two examples that demonstrate a protocol for building hierarchical models and answering questions about the relationships between variables at multiple scales. The first example employs maximum likelihood methods to construct a two-level linear model predicting herbivore damage to a perennial plant at the individual- and patch-scale; the second example uses Bayesian estimation techniques to develop a three-level logistic model of plant flowering probability across individual plants, microsites and populations. HLM model development and diagnostics illustrate the importance of incorporating scale when modelling associations in ecological systems and offer a sophisticated yet accessible method for studies of populations, communities and ecosystems. We suggest that a greater coupling of hierarchical study designs and hierarchical analysis will yield significant insights on how ecological processes operate across scales.     

Analysis of cytoplasmic and maternal effects I. A genetic model for diploid plant seeds and animals

A genetic model for modified diallel crosses is proposed for estimating variance and covariance components of cytoplasmic, maternal additive and dominance effects, as well as direct additive and dominance effects. Monte Carlo simulations were conducted to compare the efficiencies of minimum norm quadratic unbiased estimation (MINQUE) methods. For both balanced and unbalanced mating designs, MINQUE (0/1), which has 0 for all the prior covariances and 1 for all the prior variances, has similar efficiency to MINQUE(), which has parameter values for the prior values. Unbiased estimates of variance and covariance components and their sampling variances could be obtained with MINQUE(0/1) and jackknifing. A t-test following jackknifing is applicable to test hypotheses for zero variance and covariance components. The genetic model is robust for estimating variance and covariance components under several situations of no specific effects. A MINQUE(0/1) procedure is suggested for unbiased estimation of covariance components between two traits with equal design matrices. Methods of unbiased prediction for random genetic effects are discussed. A linear unbiased prediction (LUP) method is shown to be efficient for the genetic model. An example is given for a demonstration of estimating variance and covariance components and predicting genetic effects.     

Disentangling Intervention and Temporal Effects in Longitudinal Designs Using Latent Curve Analysis

This article is concerned with an approach to disconfounding effects in repeated measure designs with multiple groups. An extension of latent curve analysis (MEREDITH and TISAK, 1990; RAO, 1958, 1965) is described. The method permits estimation and testing of intervention effects separately from temporal effects. It is based on restricted factor analysis of individual change profiles and utilizes basis curves representing group patterns of change. The approach is illustrated on data from a two-group intervention study.     

Efficiency of cohort sampling designs: some surprising results.

Cohort sampling designs are proposed which one would intuitively expect to be more efficient than nested case-control sampling. Two of these designs start with a nested case-control sample and distribute controls to sampled risk sets other than those for which they were picked. The third design has the goal of maximizing the number of distinct persons in a nested case-control sample. Simulation results show surprisingly little gain, and more often a loss in efficiency of these new designs relative to nested case-control sampling. This is due to the sampling-induced covariance between score terms. We conclude that the often stated intuition that nested case-control sampling does not make good use of sampled individuals' covariate histories is false.     

Sampling design in large-scale vegetation studies: Do not sacrifice ecological thinking to statistical purism!

Most of the historical phytosociological data on vegetation composition have been sampled preferentially and thus belong to those ecological data that do not fulfill the statistical assumption of independence of observations, necessary for valid statistical testing and inference. Nevertheless, phytosociological data have been recently used for various ecological meta-analyses, especially in studies of large-scale vegetation patterns. For this reason, we focus on the comparison of preferential sampling with other sampling designs that have been recommended as more convenient alternatives from the point of view of statistical theory. We discuss that while simple random sampling, systematic sampling and stratified random sampling better meet some of the statistical assumptions, preferential sampling yields data sets that cover a broader range of vegetation variability. Moreover, today’s large phytosociological databases provide huge amounts of vegetation data with unrivalled geographic extent and density. We conclude that in the near future ecologists will not be able to replace the preferentially sampled phytosociological data in large-scale studies. At the same time, phytosociological databases have to be complemented with relevés of vegetation composed mostly of common and generalist species, which are under-represented in historical data. Stratified random sampling seems to be a suitable tool for doing this. Nevertheless, a methodology and input data for stratification have to be developed to make stratified random sampling an ecologically more relevant and practical method.     

On sampling procedures in population and community ecology

In this paper we emphasize that sampling decisions in population and community ecology are context dependent. Thus, the selection of an appropriate sampling procedure should follow directly from considerations of the objectives of an investigation. We recognize eight sampling alternatives, which arise as a result of three basic dichotomies: parameter estimation versus pattern detection, univariate versus multivariate, and a discrete versus continuous sampling universe. These eight alternative sampling procedures are discussed as they relate to decisions regarding the required empirical sample size, the selection or arrangement of sampling units, and plot size and shape. Our results indicate that the decision-making process in sampling must be viewed as a flexible exercise, dictated not by generalized recommendations but by specific objectives: there is no panacea in ecological sampling. We also point to a number of unresolved sampling problems in ecology.     

Good–Turing frequency estimation in a finite population

Good–Turing frequency estimation (Good, 1953 ) is a simple, effective method for predicting detection probabilities of objects of both observed and unobserved classes based on observed frequencies of classes in a sample. The method has been used widely in several disciplines, such as information retrieval, computational linguistics, text recognition, and ecological diversity estimation. Nevertheless, existing studies assume sampling with replacement or sampling from an infinite population, which might be inappropriate for many practical applications. In light of this limitation, this article presents a modification of the Good–Turing estimation method to account for finite population sampling. We provide three practical extensions of the modified method, and we examine performance of the modified method and its extensions in simulation experiments.     

Estimating the species accumulation curve using mixtures

As a significant tool in ecological studies, the species accumulation curve or the collector's curve is the graph of the expected number of detected species as a function of sampling effort. The problem of estimating the species accumulation curve based on an empirical data set arising from quadrat sampling is studied in a nonparametric binomial mixture model. It will be shown that estimating the species accumulation curve not only is independent of the unknown number of species but also includes estimating the number of species as a limiting case. For the purpose of interpolation, moment-based estimators, associated with asymptotic confidence intervals, are developed from several points of view. A likelihood-based procedure is developed for the purpose of extrapolation, associated with bootstrap confidence intervals. The proposed methods are illustrated by ecological data sets.     

MAXIMUM-LIKELIHOOD APPROACHES APPLIED TO QUANTITATIVE GENETICS OF NATURAL POPULATIONS

Growing interest in adaptive evolution in natural populations has spurred efforts to infer genetic components of variance and covariance of quantitative characters. Here, I review difficulties inherent in the usual least-squares methods of estimation. A useful alternative approach is that of maximum likelihood (ML). Its particular advantage over least squares is that estimation and testing procedures are well defined, regardless of the design of the data. A modified version of ML, REML, eliminates the bias of ML estimates of variance components. Expressions for the expected bias and variance of estimates obtained from balanced, fully hierarchical designs are presented for ML and REML. Analyses of data simulated from balanced, hierarchical designs reveal differences in the properties of ML, REML, and F-ratio tests of significance. A second simulation study compares properties of REML estimates obtained from a balanced, fully hierarchical design (within-generation analysis) with those from a sampling design including phenotypic data on parents and multiple progeny. It also illustrates the effects of imposing nonnegativity constraints on the estimates. Finally, it reveals that predictions of the behavior of significance tests based on asymptotic theory are not accurate when sample size is small and that constraining the estimates seriously affects properties of the tests. Because of their great flexibility, likelihood methods can serve as a useful tool for estimation of quantitative-genetic parameters in natural populations. Difficulties involved in hypothesis testing remain to be solved.     

Scale as a lurking factor: incorporating scale-dependence in experimental ecology

Ecologists have recognized for decades the importance of spatial scale in ecological processes and patterns, as well as the complications scale poses for understanding ecological mechanisms. Here we highlight the opportunity attention to scale offers experimental ecology. Despite many advantages to considering scale, a review of the literature indicates that multi-scale experimental studies are rare. Although much work has focused on scale as a primary factor (e.g. island size), we draw attention to scale as a 'lurking' variable: one which influences the relationship between two or more variables that are not usually understood to be scale-dependent.
We highlight three basic observations from which scale-dependence arises: abundance increases with area, environmental conditions vary across space, and the effect of an organism on its environment is spatially limited. From these arise first-order scale-dependence, which relates an ecological variable of interest to a measure of scale. Combining first-order relationships together, we can produce second-order scale-dependencies, which occur when the relationship between two or more variables is mediated by scale. It is these relationships that are of particular interest, as they have the potential to confound experimental results.
Most ecological experiments have incorporated scale either implicitly or not at all. We suggest that an explicit consideration of scale could help resolve some long-standing debates when scale is turned from a lurking variable into a working variable. Finally, we review and evaluate four different experimental sampling designs and corresponding statistical analyses that can be used to address the effects of scale in ecological experiments.     

How to obtain a precise and representative estimate for parameters in LCA

Background, Aim and Scope  Quite often there is need for precise and representative parameters in LCA studies. Probably the most relevant have direct influence on the functional unit, whose definition is crucial in the conduct of any LCA. Changes in the functional unit show directly in LCI and LCIA results. In comparative assertions, a bias in the functional unit may lead to a bias in the overall conclusions. Since quantitative data for the functional unit, such as geometric dimensions and specific weight, often vary, the question arises how to determine the functional unit, especially if a comparative assertion shall be representative for a region or market. Aim and scope of the study is to develop and apply methods for obtaining precise and representative estimates for the functional unit as one important parameter in an LCA study. Materials and Methods  Statistical sampling is applied in order to get empirical estimates for the weight of yoghurt cups, as a typical parameter for the functional unit. We used a two-stage sampling design, with stratified sampling in the first stage and three different sampling designs in the second stage, namely stratified, clustered, and a posteriori sampling. Sampling designs are motivated and described. In a case study, they are each used to determined a representative weight for 150 g yoghurt cups in Berlin, at the point of sale and within a specific time. In the first sampling stage, food markets are randomly selected, while in the second stage, yoghurt cups in these food markets are sampled. The sampling methods are applicable due to newly available internet data. These data sources and their shortcomings are described. Results  The random sampling procedure yields representative estimates, which are compared to figures for market leaders, i.e. yoghurt cups with very high occurrence in the supermarkets. While single types of yoghurt cups showed moderate uncertainty, representative estimates were highly precise. Discussion results show, for one, the performance of the applied statistical estimation procedures, and they show further that adding more information in the estimation procedure (on the shape of the cup, on the type of plastic, on the specific brand) helps reducing uncertainty. Conclusions  As conclusions, estimates and their uncertainty depend on the measurement procedure in a sensitive manner; any uncertainty information should be coupled with information on the measurement procedure, and it is recommended to use statistical sampling in order to reduce uncertainty for important parameters of an LCA study. Recommendations and Perspectives  Results for market leaders differed considerably from representative estimates. This implies to not use market leader data, or data with a high market share, as substitute for representative data in LCA studies. Statistical sampling has been barely used for Life Cycle Assessment. It turned out to be a feasible means for obtaining highly precise and representative estimates for the weight of yoghurt cups in the case study, based on empirical analysis. Further research is recommended in order to detect which parameters should best be investigated in LCA case studies; which data sources are available and recommended, and which sampling designs are appropriate for different application cases. ESS-Submission Editor: Seungdo Kim. PhD (kimseun@msu.edu)     

Fishermen's knowledge as background information in tropical fish ecology: a quantitative comparison with fish sampling results

An investigation of fishermen's knowledge of fish occurrence patterns on various spatio-temporal scales has been realized in the Fatala Estuary (Guinea, West Africa), accompanied by a one-year survey with standardized gill-net sets. Seventy one fishermen distributed in four zones corresponding to gill-net sampling sites were questioned about seasonal variations of species' relative abundances. Longitudinal and seasonal patterns of fish relative abundances were described with correspondence analysis and ANOVA for both approaches. Comparison of results showed a good coherence between fishermen's answers and gill-net sampling results. Thus, it is proposed that investigation of fishermen's ecological knowledge should be used as a preliminary study to help defining fish sampling designs in tropical rivers and estuaries.