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.     

Estimating survival rates with time series of standing age-structure data

It has long been recognized that age-structure data contain useful information for assessing the status and dynamics of wildlife populations. For example, age-specific survival rates can be estimated with just a single sample from the age distribution of a stable, stationary population. For a population that is not stable, age-specific survival rates can be estimated using techniques such as inverse methods that combine time series of age-structure data with other demographic data. However, estimation of survival rates using these methods typically requires numerical optimization, a relatively long time series of data, and smoothing or other constraints to provide useful estimates. We developed general models for possibly unstable populations that combine time series of age-structure data with other demographic data to provide explicit maximum likelihood estimators of age-specific survival rates with as few as two years of data. As an example, we applied these methods to estimate survival rates for female bison (Bison bison) in Yellowstone National Park, USA. This approach provides a simple tool for monitoring survival rates based on age-structure data.     

Estimating the complexity of bird song by using capture-recapture approaches from community ecology

Repertoire size, the number of unique song or syllable types in the repertoire, is a widely used measure of song complexity in birds, but it is difficult to calculate this exactly in species with large repertoires. A new method of repertoire size estimation applies species richness estimation procedures from community ecology, but such capture-recapture approaches have not been much tested. Here, we establish standardized sampling schemes and estimation procedures using capture-recapture models for syllable repertoires from 18 bird species, and suggest how these may be used to tackle problems of repertoire estimation. Different models, with different assumptions regarding the heterogeneity of the use of syllable types, performed best for different species with different song organizations. For most species, models assuming heterogeneous probability of occurrence of syllables (so-called detection probability) were selected due to the presence of both rare and frequent syllables. Capture-recapture estimates of syllable repertoire size from our small sample did not differ significantly from previous estimates using larger samples of count data. However, the enumeration of syllables in 15 songs yielded significantly lower estimates than previous reports. Hence, heterogeneity in detection probability of syllables should be addressed when estimating repertoire size. This is neglected using simple enumeration procedures, but is taken into account when repertoire size is estimated by appropriate capture-recapture models adjusted for species-specific song organization characteristics. We suggest that such approaches, in combination with standardized sampling, should be applied in species with potentially large repertoire size. On the other hand, in species with small repertoire size and homogenous syllable usage, enumerations may be satisfactory. Although researchers often use repertoire size as a measure of song complexity, listeners to songs are unlikely to count entire repertoires and they may rely on other cues, such as syllable detection probability.Communicated by A. Cockburn     

Analysis of neighborhood dynamics of forest ecosystems using likelihood methods and modeling.

Advances in computing power in the past 20 years have led to a proliferation of spatially explicit, individual-based models of population and ecosystem dynamics. In forest ecosystems, the individual-based models encapsulate an emerging theory of "neighborhood" dynamics, in which fine-scale spatial interactions regulate the demography of component tree species. The spatial distribution of component species, in turn, regulates spatial variation in a whole host of community and ecosystem properties, with subsequent feedbacks on component species. The development of these models has been facilitated by development of new methods of analysis of field data, in which critical demographic rates and ecosystem processes are analyzed in terms of the spatial distributions of neighboring trees and physical environmental factors. The analyses are based on likelihood methods and information theory, and they allow a tight linkage between the models and explicit parameterization of the models from field data. Maximum likelihood methods have a long history of use for point and interval estimation in statistics. In contrast, likelihood principles have only more gradually emerged in ecology as the foundation for an alternative to traditional hypothesis testing. The alternative framework stresses the process of identifying and selecting among competing models, or in the simplest case, among competing point estimates of a parameter of a model. There are four general steps involved in a likelihood analysis: (1) model specification, (2) parameter estimation using maximum likelihood methods, (3) model comparison, and (4) model evaluation. Our goal in this paper is to review recent developments in the use of likelihood methods and modeling for the analysis of neighborhood processes in forest ecosystems. We will focus on a single class of processes, seed dispersal and seedling dispersion, because recent papers provide compelling evidence of the potential power of the approach, and illustrate some of the statistical challenges in applying the methods.     

An EM algorithm for capture-recapture estimation

Analysis of capture—recapture data often involves maximizing a complex likelihood function with many unknown parameters. Statistical inference based on selection of a proper model depends on successful attainment of this maximum. An EM algorithm is developed for obtaining maximum likelihood estimates of capture and survival probabilities conditional on first capture from standard capture—recapture data. The algorithm does not require the use of numerical derivatives which may improve precision and stability relative to other estimation schemes. The asymptotic covariance matrix of the estimated parameters can be obtained using the supplemented EM algorithm. The EM algorithm is compared to a more traditional Newton-Raphson algorithm with both a simulated and a real dataset. The two algorithms result in the same parameter estimates, but Newton-Raphson variance estimates depend on a numerically estimated Hessian matrix that is sensitive to step size choice.     

Stochastic gravity models for modeling lake invasions

Freshwater aquatic systems in North America are being invaded by many different species, ranging from fish, mollusks, cladocerans to various bacteria and viruses. These invasions have serious ecological and economic impacts. Human activities such as recreational boating are an important pathway for dispersal. Gravity models are used to quantify the dispersal effect of human activity. Gravity models currently used in ecology are deterministic. This paper proposes the use of stochastic gravity models in ecology, which provides new capabilities both in model building and in potential model applications. These models allow us to use standard statistical inference tools such as maximum likelihood estimation and model selection based on information criteria. To facilitate prediction, we use only those covariates that are easily available from common data sources and can be forecasted in future. This is important for forecasting the spread of invasive species in geographical and temporal domain. The proposed model is portable, that is it can be used for estimating relative boater traffic and hence relative propagule pressure for the lakes not covered by current boater surveys. This makes our results broadly applicable to various invasion prediction and management models.     

Nonparametric MLE incorporation of heterogeneity and model testing into premarked cohort studies

We utilize mixture models and nonparametric maximum likelihood estimation to both develop a likelihood ratio test (lrt) for a common simplifying assumption and to allow heterogeneity within premarked cohort studies. Our methods allow estimation of the entire probability model and thus one can not only estimate many parameters of interest but one can also bootstrap from the estimated model to predict many things, including the standard deviations of estimators. Simulations suggest that our lrt has the appropriate protection for Type I error and often has good power. In practice, our lrt is important for determining the appropriateness of estimators and in examining if a simple design with only one capture period could be utilized for a future similar study.     

An estimating function approach to the inference of catch-effort models

A class of catch-effort models, which allows for heterogeneous removal probabilities, is proposed for closed populations. The model includes three types of removal probabilities: multiplicative, Poisson and logistic. The usual removal and generalized removal models then become special cases. The equivalence of the proposed model and a special type of capture-recapture model is discussed. A unified estimating function approach is used to estimate the initial population size. For the homogeneous model, the resulting population size estimator based on optimal estimating functions is asymptotically equivalent to the maximum likelihood estimator. One advantage for our approach is that it can be extended to handle the heterogeneous populations in which the maximum likelihood estimators do not exist. The bootstrap method is applied to construct variance estimators and confidence intervals. We illustrate the method by two real data examples. Results of a simulation study investigating the performance of the proposed estimation procedure are presented.     

On parametric estimation of population abundance for line transect sampling

Despite recent advances in nonparametric methods for estimating animal abundance, parametric methods are still used widely among biometricians due to their simplicity. In this paper, we propose an optimal shrinkage-type estimator and an empirical Bayes estimator for estimating animal density from line transect sampling data. The performances of the proposed estimators are compared with those of the maximum likelihood estimator and a bias-corrected maximum likelihood estimator both theoretically and numerically. Simulation results show that the optimal shrinkage-type estimator works the best if the detection function has a very thin tail (for example, the half normal detection function), while the maximum likelihood estimator is the best estimator if the detection function has relatively thick tail (for example, the polynomial detection function).     

Estimating age-specific hazards from wildlife telemetry data

We illustrate 2 techniques for estimating age-specific hazards with wildlife telemetry data: Siler’s (Ecology 60:750–757, 1979) competing risk model fit using maximum likelihood and a penalized likelihood estimate that only assumes the hazard varies smoothly with age. In most telemetry studies, animals enter at different points in time (and at different ages), leading to data that are left-truncated. In addition, death times may only be known to occur within an interval of time (interval-censoring). Observations may also be right-censored (e.g., due to the end of the study, radio-collar failure, or emigration from the study area). It is important to consider the observation process, since the contribution of each individual’s data to the likelihood will depend on whether data are left-truncated or censored. We estimate age-specific hazards using telemetry data collected in two Phases during a 13-year study of white-tailed deer (Odocoileus virginianus) in northern Minnesota. The hazards estimated from the two methods were similar for the full data set that included 302 adults and 76 neonates (followed since or shortly after birth). However, estimated hazards for early-aged individuals differed considerably for subsets of the data that did not include neonates. We discuss the advantages and disadvantages of these two modeling approaches and also compare the estimators using a short simulation study.     

Bayes Estimation of a Distribution Function Using Ranked Set Samples

Aranked set sample (RSS), if not balanced, is simply a sample of independent order statistics gener- ated from the same underlying distribution F. Kvam and Samaniego (1994) derived maximum likelihood estimates of F for a general RSS. In many applications, including some in the environ- mental sciences, prior information about F is available to supplement the data-based inference. In such cases, Bayes estimators should be considered for improved estimation. Bayes estimation (using the squared error loss function) of the unknown distribution function F is investigated with such samples. Additionally, the Bayes generalized maximum likelihood estimator (GMLE) is derived. An iterative scheme based on the EM Algorithm is used to produce the GMLE of F. For the case of squared error loss, simple solutions are uncommon, and a procedure to find the solution to the Bayes estimate using the Gibbs sampler is illustrated. The methods are illustrated with data from the Natural Environmental Research Council of Great Britain (1975), representing water discharge of floods on the Nidd River in Yorkshire, England     

Estimation of the spatial covariance in Universal Kriging: Application to forest inventory

This paper presents a simple least squares procedure for estimating the spatial covariance and compares it with the numerically more difficult restricted maximum likelihood procedure. Thereafter, it compares design-based and kriging techniques for the estimation of spatial averages in the context of double sampling, as used in forest inventory, where terrestrial sample plots are combined with auxiliary information based on aerial photographs. A case study illustrates the theory.     

Weighted distributions and estimation of resource selection probability functions

Understanding how organisms selectively use resources is essential for designing wildlife management strategies. The probability that an individual uses a given resource, as characterized by environmental factors, can be quantified in terms of the resource selection probability function (RSPF). The present literature on the topic has claimed that, except when both used and unused sites are known, the RSPF is non-estimable and that only a function proportional to RSPF, namely, the resource selection function (RSF) can be estimated. This paper describes a close connection between the estimation of the RSPF and the estimation of the weight function in the theory of weighted distributions. This connection can be used to obtain fully efficient, maximum likelihood estimators of the resource selection probability function under commonly used survey designs in wildlife management. The method is illustrated using GPS collar data for mountain goats (Oreamnos americanus de Blainville 1816) in northwest British Columbia, Canada.     

Robust benchmark dose determination based on profile score methods

We investigate several methods commonly used to obtain a benchmark dose and show that those based on full likelihood or profile likelihood methods might have severe shortcomings. We propose two new profile likelihood-based approaches which overcome these problems. Another contribution is the extension of the benchmark dose determination to non full likelihood models, such as quasi-likelihood, generalized estimating equations, which are widely used in settings such as developmental toxicity where clustered data are encountered. This widening of the scope of application is possible by the use of (robust) score statistics. Benchmark dose methods are applied to a data set from a developmental toxicity study.     

Modelling species distributions with penalised logistic regressions: A comparison with maximum entropy models

An important aspect of species distribution modelling is the choice of the modelling method because a suboptimal method may have poor predictive performance. Previous comparisons have found that novel methods, such as Maxent models, outperform well-established modelling methods, such as the standard logistic regression. These comparisons used training samples with small numbers of occurrences per estimated model parameter, and this limited sample size may have caused poorer predictive performance due to overfitting. Our hypothesis is that Maxent models would outperform a standard logistic regression because Maxent models avoid overfitting by using regularisation techniques and a standard logistic regression does not. Regularisation can be applied to logistic regression models using penalised maximum likelihood estimation. This estimation procedure shrinks the regression coefficients towards zero, causing biased predictions if applied to the training sample but improving the accuracy of new predictions. We used Maxent and logistic regression (standard and penalised) to analyse presence/pseudo-absence data for 13 tree species and evaluated the predictive performance (discrimination) using presence-absence data. The penalised logistic regression outperformed standard logistic regression and equalled the performance of Maxent. The penalised logistic regression may be considered one of the best methods to develop species distribution models trained with presence/pseudo-absence data, as it is comparable to Maxent. Our results encourage further use of the penalised logistic regression for species distribution modelling, especially in those cases in which a complex model must be fitted to a sample with a limited size.     

Including model uncertainty in estimating variances in multiple capture studies

We present bootstrap-based methods which incorporate model uncertainty in estimating variances in multiple capture studies. Each of our three methods has a specific set of properties, and we discuss when each method should be used. Our first method can be used in any multiple capture setting, but it gives an estimate of the variance conditional on the number of observed animals. Our other two methods yield estimates of the unconditional variance; they require good estimates of part or all of the specific probability model, respectively. Smoothed estimated cell probabilities are utilized by the latter method. We contrast the three methods on a real-life data set, and then conduct simulations for a simple setting. Finally, we detail the use of our methodology for specific settings and discuss adaptations for tag-return studies.     

Measurement Error in Recreation Demand Models: The Joint Estimation of Participation, Site Choice, and Site Characteristics

In the demand for recreational fishing sites, an important explanatory variable differentiating sites is the unobserved expected catch rate. Since the observed catch rate is subject to sampling variability, using the average of a site's observed catch rates causes the parameter estimator on catch to be biased downward. We develop and demonstrate a solution to this errors-in-variables problem when there are repeated measurements on the catch rate. Consistent and efficient estimates of both the demand parameters and the expected catch rates are obtained by simultaneously estimating them by maximum likelihood. An empirical example demonstrates the importance of simultaneous estimation.     

Population viability analysis for several populations using multivariate state-space models

The International Union for the Conservation of Nature and Natural Resources (IUCN), the world's largest and most important global conservation network, has listed approximately 16,000 species worldwide as threatened. The most important tool for recognizing and listing species as threatened is population viability analysis (PVA), which estimates the probability of extinction of a population or species over a specified time horizon. The most common PVA approach is to apply it to single time series of population abundance. This approach to population viability analysis ignores covariability of local populations. Covariability can be important because high synchrony of local populations reduces the effective number of local populations and leads to greater extinction risk. Needed is a way of extending PVA to model correlation structure among multiple local populations. Multivariate state-space modeling is applied to this problem and alternative estimation methods are compared. The multivariate state-space technique is applied to endangered populations of pacific salmon, USA. Simulations demonstrated that the correlation structure can strongly influence population viability and is best estimated using restricted maximum likelihood instead of maximum likelihood.     

A generalized approach to modeling and estimating indirect effects in ecology

The need to model and test hypotheses about complex ecological systems has led to a steady increase in use of path analytical techniques, which allow the modeling of multiple multivariate dependencies reflecting hypothesized causation and mechanisms. The aim is to achieve the estimation of direct, indirect, and total effects of one variable on another and to assess the adequacy of whole models. Path analytical techniques based on maximum likelihood currently used in ecology are rarely adequate for ecological data, which are often sparse, multi-level, and may contain nonlinear relationships as well as nonnormal response data such as counts or proportion data. Here I introduce a more flexible approach in the form of the joint application of hierarchical Bayes, Markov chain Monte Carlo algorithms, Shipley's d-sep test, and the potential outcomes framework to fit path models as well as to decompose and estimate effects. An example based on the direct and indirect interactions between ants, two insect herbivores, and a plant species demonstrates the implementation of these techniques, using freely available software.     

Estimating Population Size in a Continuous-time Removal Experiment with a Known Sub-population Size Ratio

We study a continuous-time removal model for estimating the population size for a population in which a sub-population size ratio is known. The maximum likelihood estimate and the optimal martingale estimate of the population size are obtained; these are shown to be equivalent. A comparison between the proposed estimator and the maximum likelihood estimate which ignores the information on the known size ratio is made, using a simulation study. The asymptotic variances of these two estimators are also obtained, and a comparison between them is made. The sensitivity of mis-specification of the known size ratio is examined. We also apply the corresponding discrete-time model to the proposed continuous-time setting, and study the efficiency of the corresponding discrete-time type estimator relative to the proposed estimator.