Alternatives to statistical hypothesis testing in ecology: a guide to self teaching.

Statistical methods emphasizing formal hypothesis testing have dominated the analyses used by ecologists to gain insight from data. Here, we review alternatives to hypothesis testing including techniques for parameter estimation and model selection using likelihood and Bayesian techniques. These methods emphasize evaluation of weight of evidence for multiple hypotheses, multimodel inference, and use of prior information in analysis. We provide a tutorial for maximum likelihood estimation of model parameters and model selection using information theoretics, including a brief treatment of procedures for model comparison, model averaging, and use of data from multiple sources. We discuss the advantages of likelihood estimation, Bayesian analysis, and meta-analysis as ways to accumulate understanding across multiple studies. These statistical methods hold promise for new insight in ecology by encouraging thoughtful model building as part of inquiry, providing a unified framework for the empirical analysis of theoretical models, and by facilitating the formal accumulation of evidence bearing on fundamental questions.     

Testing Hypotheses of Bird Extinctions at Rio Palenque,Ecuador, with Informal Species Lists

Abstract: Informally gathered species lists are a potential source of data for conservation biology, but most remain unused because of questions of reliability and statistical issues. We applied two alternative analytical methods (contingency tests and occupancy modeling) to a 35‐year data set (1973–2007) to test hypotheses about local bird extinction. We compiled data from bird lists collected by expert amateurs and professional scientists in a 2‐km² fragment of lowland tropical forest in coastal Ecuador. We tested the effects of the following on local extinction: trophic level, sociality, foraging specialization, light tolerance, geographical range area, and biogeographic source. First we assessed extinction on the basis of the number of years in which a species was not detected on the site and used contingency tests with each factor to compare the frequency of expected and observed extinction events among different species categories. Then we defined four multiyear periods that reflected different stages of deforestation and isolation of the study site and used occupancy modeling to test extinction hypotheses singly and in combination. Both types of analyses supported the biogeographic source hypothesis and the species‐range hypothesis as causes of extinction; however, occupancy modeling indicated the model incorporating all factors except foraging specialization best fit the data.     

Analyzing or explaining beta diversity? Understanding the targets of different methods of analysis

It has been actively discussed recently what statistical methods are appropriate when one is interested in testing hypotheses about the origin of beta diversity, especially whether one should use the raw-data approach (e.g., canonical analysis such as RDA and CCA) or the distance approach (e.g., Mantel test and multiple regression on distance matrices). Most of the confusion seems to stem from uncertainty as to what is the response variable in the different approaches. Here our aim is to clarify this issue. We also show that, although both the raw-data approach and the distance approach can often be used to address the same ecological hypothesis, they target fundamentally different predictions of those hypotheses. As the two approaches shed light on different aspects of the ecological hypotheses, they should be viewed as complementary rather than alternative ways of analyzing data. However, in some cases only one of the approaches may be appropriate. We argue that S. P. Hubbell's neutral theory can only be tested using the distance approach, because its testable predictions are stated in terms of distances, not in terms of raw data. In all cases, the decision on which method is chosen must be based on which addresses the question at hand, it cannot be based on which provides the highest proportion of explained variance in simulation studies.     

The Use of Bayesian Model Averaging to Better Represent Uncertainty in Ecological Models

Abstract:   In conservation biology, uncertainty about the choice of a statistical model is rarely considered. Model-selection uncertainty occurs whenever one model is chosen over plausible alternative models to represent understanding about a process and to make predictions about future observations. The standard approach to representing prediction uncertainty involves the calculation of prediction (or confidence) intervals that incorporate uncertainty about parameter estimates contingent on the choice of a "best" model chosen to represent truth. However, this approach to prediction based on statistical models tends to ignore model-selection uncertainty, resulting in overconfident predictions. Bayesian model averaging (BMA) has been promoted in a range of disciplines as a simple means of incorporating model-selection uncertainty into statistical inference and prediction. Bayesian model averaging also provides a formal framework for incorporating prior knowledge about the process being modeled. We provide an example of the application of BMA in modeling and predicting the spatial distribution of an arboreal marsupial in the Eden region of southeastern Australia. Other approaches to estimating prediction uncertainty are discussed.     

Bayesian areal wombling via adjacency modeling

Recently, public health professionals and other geostatistical researchers have shown increasing interest in boundary analysis, the detection or testing of zones or boundaries that reveal sharp changes in the values of spatially oriented variables. For areal data (i.e., data which consist only of sums or averages over geopolitical regions), Lu and Carlin (Geogr Anal 37: 265–285, 2005) suggested a fully model-based framework for areal wombling using Bayesian hierarchical models with posterior summaries computed using Markov chain Monte Carlo (MCMC) methods, and showed the approach to have advantages over existing non-stochastic alternatives. In this paper, we develop Bayesian areal boundary analysis methods that estimate the spatial neighborhood structure using the value of the process in each region and other variables that indicate how similar two regions are. Boundaries may then be determined by the posterior distribution of either this estimated neighborhood structure or the regional mean response differences themselves. Our methods do require several assumptions (including an appropriate prior distribution, a normal spatial random effect distribution, and a Bernoulli distribution for a set of spatial weights), but also deliver more in terms of full posterior inference for the boundary segments (e.g., direct probability statements regarding the probability that a particular border segment is part of the boundary). We illustrate three different remedies for the computing difficulties encountered in implementing our method. We use simulation to compare among existing purely algorithmic approaches, the Lu and Carlin (2005) method, and our new adjacency modeling methods. We also illustrate more practical modeling issues (e.g., covariate selection) in the context of a breast cancer late detection data set collected at the county level in the state of Minnesota.     

Model selection and model averaging in behavioural ecology: the utility of the IT-AIC framework

Behavioural ecologists often study complex systems in which multiple hypotheses could be proposed to explain observed phenomena. For some systems, simple controlled experiments can be employed to reveal part of the complexity; often, however, observational studies that incorporate a multitude of causal factors may be the only (or preferred) avenue of study. We assess the value of recently advocated approaches to inference in both contexts. Specifically, we examine the use of information theoretic (IT) model selection using Akaike’s information criterion (AIC). We find that, for simple analyses, the advantages of switching to an IT-AIC approach are likely to be slight, especially given recent emphasis on biological rather than statistical significance. By contrast, the model selection approach embodied by IT approaches offers significant advantages when applied to problems of more complex causality. Model averaging is an intuitively appealing extension to model selection. However, we were unable to demonstrate consistent improvements in prediction accuracy when using model averaging with IT-AIC; our equivocal results suggest that more research is needed on its utility. We illustrate our arguments with worked examples from behavioural experiments.     

Variation in size at sex-change among natural populations of the protandrous hermaphrodite, <Emphasis Type="Italic">Crepidula fornicata</Emphasis> (Gastropoda,Calyptraeidae)

Protandrous hermaphrodites are predicted to change sex from male to female when relative reproductive fitness of females surpasses that of males. How size at sex transition varies with population, mating group and individual parameters was investigated for five populations of the protandrous hermaphrodite slipper snail, Crepidula fornicata. The populations varied for density, size distribution, average mating group size and sex ratio. Size at sex-change was correlated with the population sex ratio. Comparisons of multiple hypotheses revealed that variables predicting the sex of a snail vary among positions in the mating group. The variables included body size, the relative size of the snail sitting atop the focal snail and population density. Our data support the conclusions that size at sex-change (and by inference, the size at which one sex has relatively greater fitness) is not fixed for these hermaphrodites and that individual size, social conditions and population differences all influence variation in relative fitness.     

Phylogeography of the calanoid copepods <Emphasis Type="Italic">Calanus helgolandicus</Emphasis> and <Emphasis Type="Italic">C. euxinus</Emphasis> suggests Pleistocene divergences between Atlantic,Mediterranean, and Black Sea populations

Molecular systematic analyses of marine taxa are crucial for recording ocean biodiversity, so too are elucidation of the history of population divergence and the dynamics of speciation. In this paper we present the joined phylogeography of the calanoid copepod Calanus helgolandicus (Claus 1863) from the North East (NE) Atlantic and the Adriatic Sea and the closely related C. euxinus (Hulsemann 1991) from the Black Sea based on sequences of a mitochondrial Cytochrome Oxidase subunit I (COI) fragment. Coalescent-based Bayesian methods and minimum spanning networks are used to reconstruct the history of population divergence. Our results reveal that copepod populations from all three basins share a great number of haplotypes and demonstrate a close genetic affinity of C. euxinus with C. helgolandicus. The data do not support significant genetic structuring among samples within seas. Coalescent analyses suggest divergences between NE Atlantic, Mediterranean, and Black Sea populations dating back to the middle Pleistocene, with the NE Atlantic–Mediterranean divergence being the earliest and the Mediterranean–Black Sea divergence the most recent. These middle Pleistocene dates are much older than the estimated dates of colonisation of the Mediterranean and Black Seas based on paleoclimatic scenarios. Our results do not rule out that the assumed colonisations took place but they indicate that the populations colonising the Mediterranean and the Black Sea were already, and have since remained, diverged. The chaetognath Sagitta setosa, which has a comparable distribution pattern and feeds upon the copepods, provides a unique opportunity to compare phylogeographic patterns and distinguish among alternative hypotheses. The dates produced in this paper are in agreement with those estimated elsewhere for S. setosa. We propose that a great deal of the genetic make-up of marine planktonic populations comprises divergences that date back to long before the last glacial maximum. We consider questions on the taxonomic status of C. euxinus to remain open. However, its high genetic affinity to the C. helgolandicus calls for further investigation.     

Methods for joint inference from panel survey and demographic data

A number of methods for joint inference from animal abundance and demographic data have been proposed in recent years, each with its own advantages. A new approach to analyzing panel survey and demographic data simultaneously is described. The approach fits population-dynamics models to the survey data, rather than to a single index of abundance derived from them and thus avoids disadvantages inherent in analyzing such an index. The methodology is developed and illustrated with British Lapwing data, and the results are compared with those obtained from existing approaches. The estimates of demographic parameters and population indices are similar for all methods. The results of a simulation study show that the new method performs well in terms of mean squared error.     

Evaluation of variance approximation techniques for non-linear photosynthesis—irradiance models

Parameters derived from photosynthesis-irradiance (P-I) models, although often empirical in nature, are useful indicators of the photoadaptive state of phytoplankton in culture and in situ. However objective criteria for determining significant changes in P-I curves are rarely provided, because confidence intervals for parameters of non-linear models are not estimated easily. Examination of least-squares residuals in parameter space and Monte Carlo approaches have been used to estimate confidence regions around parameter values, but the computationally intensive nature of these methods has prevented their routine application. We present an alternative method of estimating confidence intervals for parameters of P-I curves that runs quickly on a microcomputer and is easily combined with common parameter-estimation routines. This algorithm was tested using a 3-parameter P-I model and curves describing a wide range of photoadaptive states, with different numbers of observations and different amounts of inherent variability. The method produced results comparable to the Monte Carlo technique. This analysis makes it possible to specify the sample size required to define parameters with acceptable confidence as a function of data variance and photoadaptive state. In most reasonable situations, 25 observations are sufficient.     

Cluster detection using Bayes factors from overparameterized cluster models

In this paper, we consider the use of a partition model to estimate regional disease rates and to detect spatial clusters. Formal inference regarding the number of partitions (or clusters) can be obtained with a reversible jump Markov chain Monte Carlo algorithm. As an alternative, we consider the ability of models with a fixed, but overly large, number of partitions to estimate regional disease rates and to provide informal inferences about the number and locations of clusters using local Bayes factors. We illustrate and compare these two approaches using data on leukemia incidence in upstate New York and data on breast cancer incidence in Wisconsin.     

Introduction to the special issue of Ecological Modelling: “Advances in Modeling Estuarine and Coastal Ecosystems: Approaches, Validation, and Applications”

The development and application of ecosystem models in estuarine and coastal systems has grown exponentially over the past four decades. Models have become ensconced as major tools for both heuristic study of ecosystem structure and function as well as for informing management decisions, particularly with respect to cultural eutrophication. In recent years an ever-expanding toolbox of modeling approaches is being offered to complement traditional methods. This expansion of modeling in estuarine and coastal science was exemplified by four sessions devoted to modeling at the 2007 biennial conference of the Estuarine Research Federation in Providence, RI. We felt the time was right to propose a special session of Ecological Modelling to synthesize talks from these sessions to present the state of the art in coastal and estuarine modeling. The collection of papers contained in this special issue presents a diversity of traditional and novel modeling approaches, methods for assessing model validity and predictability, and the utility of models in management applications. We believe that together these papers provide an excellent overview of current approaches to modeling estuarine hydrodynamics, water quality, and ecosystem/food web dynamics, applications of complex and relatively simple modeling approaches, applications in both deep and shallow coastal systems, goals relevant for both heuristic and management applications, and perspectives based on traditional mechanistic model development as well as more recent alternative approaches.     

Model averaging,missing data and multiple imputation: a case study for behavioural ecology

Model averaging, specifically information theoretic approaches based on Akaike’s information criterion (IT-AIC approaches), has had a major influence on statistical practices in the field of ecology and evolution. However, a neglected issue is that in common with most other model fitting approaches, IT-AIC methods are sensitive to the presence of missing observations. The commonest way of handling missing data is the complete-case analysis (the complete deletion from the dataset of cases containing any missing values). It is well-known that this results in reduced estimation precision (or reduced statistical power), biased parameter estimates; however, the implications for model selection have not been explored. Here we employ an example from behavioural ecology to illustrate how missing data can affect the conclusions drawn from model selection or based on hypothesis testing. We show how missing observations can be recovered to give accurate estimates for IT-related indices (e.g. AIC and Akaike weight) as well as parameters (and their standard errors) by utilizing ‘multiple imputation’. We use this paper to illustrate key concepts from missing data theory and as a basis for discussing available methods for handling missing data. The example is intended to serve as a practically oriented case study for behavioural ecologists deciding on how to handle missing data in their own datasets and also as a first attempt to consider the problems of conducting model selection and averaging in the presence of missing observations.     

Random denominators and the analysis of ratio data

Ratio data, observations in which one random value is divided by another random value, present unique analytical challenges. The best statistical technique varies depending on the unit on which the inference is based. We present three environmental case studies where ratios are used to compare two groups, and we provide three parametric models from which to simulate ratio data. The models describe situations in which (1) the numerator variance and mean are proportional to the denominator, (2) the numerator mean is proportional to the denominator but its variance is proportional to a quadratic function of the denominator and (3) the numerator and denominator are independent. We compared standard approaches for drawing inference about differences between two distributions of ratios: t-tests, t-tests with transformations, permutation tests, the Wilcoxon rank test, and ANCOVA-based tests. Comparisons between tests were based both on achieving the specified alpha-level and on statistical power. The tests performed comparably with a few notable exceptions. We developed simple guidelines for choosing a test based on the unit of inference and relationship between the numerator and denominator.     

Comparing discriminant analysis, neural networks and logistic regression for predicting species distributions: a case study with a Himalayan river bird

We assessed the occurrence of a common river bird, the Plumbeous Redstart Rhyacornis fuliginosus, along 180 independent streams in the Indian and Nepali Himalaya. We then compared the performance of multiple discrimant analysis (MDA), logistic regression (LR) and artificial neural networks (ANN) in predicting this species’ presence or absence from 32 variables describing stream altitude, slope, habitat structure, chemistry and invertebrate abundance. Using the entire data (=training set) and a threshold for accepting presence in ANN and LR set to P≥0.5, ANN correctly classified marginally more cases (88%) than either LR (83%) or MDA (84%). Model performance was assessed from two methods of data partitioning. In a ‘leave-one-out’ approach, LR correctly predicted more cases (82%) than MDA (73%) or ANN (69%). However, in a holdout procedure, all the methods performed similarly (73–75%). All methods predicted true absence (i.e. specificity in holdout: 81–85%) better than true presence (i.e. sensitivity: 57–60%). These effects reflect species’ prevalence (=frequency of occurrence), but are seldom considered in distribution modelling. Despite occurring at only 36% of the sites, Plumbeous Redstarts are one of the most common Himalayan river birds, and problems will be greater with less common species. Both LR and ANN require an arbitrary threshold probability (often P=0.5) at which to accept species presence from model prediction. Simulations involving varied prevalence revealed that LR was particularly sensitive to threshold effects. ROC plots (received operating characteristic) were therefore used to compare model performance on test data at a range of thresholds; LR always outperformed ANN. This case study supports the need to test species’ distribution models with independent data, and to use a range of criteria in assessing model performance. ANN do not yet have major advantages over conventional multivariate methods for assessing bird distributions. LR and MDA were both more efficient in the use of computer time than ANN, and also more straightforward in providing testable hypotheses about environmental effects on occurrence. However, LR was apparently subject to chance significant effects from explanatory variables, emphasising the well-known risks of models based purely on correlative data.     

Modelling spatial zero-inflated continuous data with an exponentially compound Poisson process

A parsimonious model is presented as an alternative to delta approaches to modelling zero-inflated continuous data. The data model relies on an exponentially compound Poisson process, also called the law of leaks (LOL). It represents the process of sampling resources that are spatially distributed as Poisson distributed patches, each containing a certain quantity of biomass drawn from an exponential distribution. In an application of the LOL, two latent structures are proposed to account for spatial dependencies between zero values at different scales within a hierarchical Bayesian framework. The LOL is compared to the delta-gamma (ΔΓ) distribution using bottom-trawl survey data. Results of this case study emphasize that the LOL provides slightly better fits to learning samples with a very high proportion of zero values and small strictly positive abundance data. Additionally, it offers better predictions of validation samples.     

Polyandry in the genus Apis, particularly Apis andreniformis

Using four polymorphic microsatellite loci, we found that four Apis andreniformis queens collected in Thailand each mated at least 10–20 times, producing an average relatedness, g _ww, of workers of 0.30 ± 0.007, and an average effective number of matings of 9.1 ± 2.2. The degrees of polyandry and intra-colonial genetic relatedness in A. andreniformis are similar to those in A. mellifera, slightly more than in A. florea, and up to 6 times less than in A. dorsata. We argue that while presently favoured hypotheses for the evolution of polyandry in monogynous social insects may adequately explain the evolution of up to five or six matings, they are inadequate to explain the extreme polyandry (10–60 matings) observed in Apis. One alternative possibility is that colony fitness is a non-additive function of the fitness of individual subfamilies. Such behavioral over-dominance may mean that queen fitness is increased by high levels of polyandry, which increase the probability of desirable combinations of worker genotypes occurring in one colony. The special attributes of honey bees which may lead to behavioral over-dominance include colony aggregation (which may increase the incidence of disease), and frequent long-distance migration. Received: 8 May 1996/Accepted after revision: 9 August 1996     

Estimating species occurrence, abundance, and detection probability using zero-inflated distributions

Researchers have developed methods to account for imperfect detection of species with either occupancy (presence absence) or count data using replicated sampling. We show how these approaches can be combined to simultaneously estimate occurrence, abundance, and detection probability by specifying a zero-inflated distribution for abundance. This approach may be particularly appropriate when patterns of occurrence and abundance arise from distinct processes operating at differing spatial or temporal scales. We apply the model to two data sets: (1) previously published data for a species of duck, Anas platyrhynchos, and (2) data for a stream fish species, Etheostoma scotti. We show that in these cases, an incomplete-detection zero-inflated modeling approach yields a superior fit to the data than other models. We propose that zero-inflated abundance models accounting for incomplete detection be considered when replicate count data are available.     

Evaluating animal personalities: do observer assessments and experimental tests measure the same thing?

The animal personality literature uses three approaches to assess personality. However, two of these methods, personality ratings and experimentation, have been little compared in captivity and never compared in the wild. We assessed the boldness of wild chacma baboons Papio ursinus using both ratings and experimental methods. Boldness was experimentally assessed when individuals were presented with a novel food item during natural foraging. The boldness of the same individuals was rated on a five-point scale by experienced observers. The ratings and experimental assessments of boldness were found to correlate positively and in a linear fashion. When considered categorically the two approaches showed variable agreement depending on the number of categories assigned and the cut-off criteria adopted. We suggest that the variation between approaches arises because each method captures different aspects of personality; ratings consider personality in absolute terms (using predefined criteria) and multiple contexts, while experimental assessments consider personality in relative terms (using experimental scores relative to the population average) and in limited contexts. We encourage animal personality researchers to consider adopting both methodologies in future studies. We also propose that future studies restrict their analyses to continuous data, since the greatest comparability between methods was found with these data. However, if individuals must be categorised, we suggest that researchers either (a) analyse only those individuals categorised as bold or shy by both ratings and experimental approaches or, if these methods cannot be employed simultaneously, (b) do not use approach-specific criteria but choose a cut-off that can be compared by both approaches.     

Interactions among females in polygynous yellow-headed blackbirds

Summary Two alternative hypotheses which attempt to predict how females within harems in polygynous species should behave toward one another were examined in a population of yellow-headed blackbirds (Xanthocephalus xanthocephalus). The hypothesis that females within a harem are co-operative was not supported. Females in large harems did not enjoy higher reproductive success than those in smaller harems and females within a territory did not initiate their nests close in time or space. The alternative hypothesis that females within a harem are competitive was also not supported. Nest density did not have a negative effect on female reproductive success, females within a territory did not initiate their nests far apart in time or space, and resident females were not very aggressive toward intruding female mounts during the settling period. The data support the idea that females settled independently of one another and had no effect on one another's reproductive success.