Bettina Almasi

Advanced Ecological Models

This chapter introduces five hierarchical ecological models and corresponding BUGS and Stan code. First, a hierarchical multinomial model is used to study the use of different roosting site types by an owl species based on telemetry data in relation to environmental variables. Second, we analyze the breeding success of the black stork using a zero-inflated Poisson model with random effects. Third, the distribution of a toad species is estimated from presence-absence data taking into account that the species may have gone undetected during a visit. To do so, an occupancy model is fitted to the data using Stan. Fourth, an extension of the occupancy model, the territory occupancy model, is used to estimate survival based on observations of unmarked birds. Finally, we introduce the Stan code to fit a Cormack–Jolly–Seber (CJS) model including random effects. The CJS model is widely used in ecology to estimate survival based on mark-recapture data.

Assessing Model Assumptions: Residual Analysis

Model assumptions need to be checked before drawing conclusions from any model. Usually, the residuals are independent and identically distributed (iid), when the model assumptions are met. We discuss several graphical methods for assessing independence and distribution of the residuals such as the Tukey–Anscombe or the QQ plot. We further discuss how to detect temporal and spatial autocorrelation as well as heteroscedasticity in the residuals.

Markov Chain Monte Carlo Simulation

Markov chain Monte Carlo (MCMC) is a simulation technique that can be used to find the posterior distribution and to sample from it. Thus, it is used to fit a model and to draw samples from the joint posterior distribution of the model parameters. The different MCMC algorithms differ in their performance in relation to speed and convergence depending on the model structure. Gibbs sampling, Metropolis–Hasting algorithm, and Hamiltonian Monte Carlo are briefly presented in a nontechnical way. The software OpenBUGS and Stan are MCMC samplers. Convergence of the chains is assessed graphically using traceplots and diagnostic statistics such as the -value or the Monte Carlo error.

Generalized Linear Mixed Models

Generalized linear mixed models (GLMM) are the combination of generalized linear models and linear mixed models, which means that the error distribution can be different from Gaussian and that a random factor is included in the model. We discuss the binomial model used to analyze proportions or binary outcome variables, and the Poisson model used to analyze counts or rates data. How we deal with residual analyses, overdispersion, zero-inflation, or modeling rates or densities using an offset are all analogous to previous chapters. We work step by step through a rather complex Poisson mixed model including an offset and variable transformations as well as a derived parameter.

Posterior Predictive Model Checking and Proportion of Explained Variance

We introduce a Bayesian method to assess goodness of fit and also discuss the use of the proportion of variance explained, R 2 , a classical measurement of goodness of fit. Posterior predictive model checking is the comparison of the simulated data from the model with the observed data. The simulated data can differ from the observed data in various aspects and to various degrees. The description of the discrepancies between the simulated and the observed data shows how well the model reflects different aspects of the data. The Bayesian p -value is the probability that a test statistic calculated from simulated data is larger than the test statistic calculated from the observed data. Bayesian p -values based on specific test statistics can be used to formally evaluate model fit, but the most important comparison between the observed and the simulated data is primarily done graphically. We show how R 2 values can be calculated for different levels of a mixed model.

Prerequisites and Vocabulary

This chapter starts with some general comments about the software used in this book, focusing on R, the statistical software package used in most chapters. R is complemented by OpenBUGS and Stan, which both allow fitting more complex models using Bayesian methods in a very flexible way ( Chapter 12 ). Then we provide some basic information regarding important statistical terms and link them to the use of R. The subsections touch on various topics: data sets, variables, observations, distributions and summary statistics, R objects, graphics, and writing R functions.

What Should I Report in a Paper

The results of the data analysis are the basis for your conclusions. They contain estimates of the effects that have been studied including a measurement of uncertainty. Often, derived parameters are more meaningful than model parameters. In addition, it is helpful to present the information needed to judge the biological (or other) relevance of the results. Such information may include a plot or a measure of the total variance or the residual variance in the data so that the effects can be compared to natural variation. Figures can be an informative and easily understandable tool to communicate results. The description of a linear model in the methods section consists of the error distribution, the link function, the linear predictor, the random structure, the prior distributions, the fitting method, and how conclusions are drawn from the joint posterior distribution of the model parameters.

Modeling Spatial Data Using GLMM

Spatial autocorrelation is a general property of most ecological data sets. When data are spatially autocorrelated, the value of a random variable characterizing a site can be partially predicted by the values at neighboring sites. Spatial autocorrelation can thus be described as one of the mechanisms leading to pseudoreplication. Uncertainty will be underestimated when this lack of independence is not properly accounted for. We discuss some methods of modeling such data using genersalized linear mixed models (GLMM), and we also present solutions for moderately large data sets.

Prior Influence and Parameter Estimability

Prior distributions, or “priors” are important parts of the model. Improper, flat, and weakly informative prior distributions are discussed. Informative or weakly informative prior distributions have computational as well as conceptual advantages over flat priors. Conjugate priors ease the calculation of posterior distributions by hand. The influence of the prior on the estimated slope of a regression line is assessed using a prior sensitivity analysis. We also use different prior distributions for the residual variance in a regression and the between-nest variance in a zero-inflated mixed model. As prior for the variance parameters, the uniform, inverse-gamma, half-Cauchy, and folded t -distributions are compared. Finally, the relative information in the data compared to the prior distribution is measured by the overlap between the prior and the posterior distribution.

Normal Linear Models

Normal linear models (LM) are used when the outcome variable can be modeled directly with predictor variables, that is, no link function is needed, and when the resulting residuals are approximately normally distributed. These models include widely used techniques: simple linear regression for a continuous predictor, one-way ANOVA for a categorical predictor (which includes comparing two samples as in a t -test), their analogs with more than one predictor (i.e., multiple regression and multiway ANOVA, respectively), and ANCOVA when continuous as well as categorical predictors are used. Thus, it becomes clear that all these techniques are special cases of one and the same technique: the normal linear model. The chapter covers fitting and interpreting such models and the related topics of collinearity, ordered factors, and polynomials. The chapter is fundamental for understanding the subsequent models covered in following chapters.