The “model of the mean,” t-test, simple linear regression, and analysis of variance (ANOVA) are all just special cases of a very general and powerful statistical model, the general linear model. This model expresses a continuous response as a linear combination of the effects of discrete and/or continuous explanatory variables plus a single random contribution from a normal distribution, whose variance is estimated along with the coefficients of all discrete and continuous covariates and possible interactions. Models that contained both types of explanatory variables were usually treated as ANOVAs with typically a single continuous covariate to correct for preexisting variation among experimental units. These models were called analysis of covariance (ANCOVA) models. In WinBUGS, it is much easier to fit the means parameterization of the model, i.e., to specify three separate linear regressions for each mountain range. The effects are trivially easy to recover as derived parameters by just adding a few WinBUGS code lines. This allows for better comparison between input and output values.

A binomial analysis of covariance (ANCOVA) can be specified by adding discrete and continuous covariates to the linear predictor of a binomial generalized linear model (GLM). It has been hypothesized that the black color confers a thermal advantage, and therefore, the proportion of black individuals should be greater in cooler or wetter habitats. The value of the linear predictor is again obtained by matrix multiplication of the design matrix (Xmat) and the parameter vector (beta.vec). Moving from the normal and the Poisson to a binomial ANCOVA involves only minor changes in the code of WinBUGS (and also R). Similarly, the concepts of residuals and posterior predictive distributions carry over to this class of models.

Both probability theory and statistics are sciences that deal with uncertainty. Their subject is the description of stochastic systems, i.e., systems that are not fully predictable but include random processes that add a degree of chance—and therefore uncertainty—in their outcome. Stochastic systems are ubiquitous in nature; hence, probability and statistics are important not only in science but also to understand all facets of life. Both probability theory and statistics deal with the characteristics of a stochastic system and its outcomes, but these two fields represent different perspectives on stochastic systems. Probability theory specifies parameters and a model and then examines a variable outcome, whereas statistics takes the data, assumes a model, and then tries to infer the system properties, given the model. Parameters are key descriptors of the stochastic system. In statistics, there are two main views about how one should learn about the parameter values in a stochastic system: classical (also called conventional or frequentist) and Bayesian statistics. Both classical and Bayesian statistics view data as the observed realizations of stochastic systems that contain one or several random processes. However, in classical statistics, the quantities used to describe these random processes (parameters) are fixed and unknown constants, whereas in Bayesian statistics, parameters are themselves viewed as unobserved realizations of random processes. In Bayesian statistics, uncertainty is evaluated using the posterior distribution of a parameter, which is the conditional probability distribution of all unknown quantities, given the data, and the model.

A Poisson analysis of covariance (ANCOVA) can be called a Poisson regression with both discrete and continuous covariates. In most practical applications, Poisson models will have several covariates and of both types. To stress the similarity with the normal linear case, we only slightly alter the inferential setting. This chapter has generalized the general linear model from the normal to the Poisson case to model the effects on grouped counts of a continuous covariate. The changes involved in doing so in WinBUGS were minor, and the inclusion of further covariates is straightforward. The Poisson ANCOVA is an important intermediate step for one"s understanding of the Poisson generalized linear mixed model.

Modeling a binomial random variable in essence means modeling a series of coin flips. The crucial difference between binomial and Poisson random variables is the presence of a ceiling in the former: binomial counts cannot be greater than some upper limit. Data coming from coin flip-like processes are ubiquitous in nature and include survival or the occurrence of an organism. In coin flips, the binomial distribution describes the number of times r a coin shows heads among a number of N flips, where the coin has Pr(heads)= p. A binomial t-test in R suggests a significant difference in the perceived distribution of the Cross-leaved gentian and the Chiltern gentian. A special case of the binomial distribution with N=1, corresponding to a single flip of the coin, is called the Bernoulli distribution.

As in a Poisson generalized linear mixed model (GLMM), one can also add into a binomial generalized linear model (GLM) random variation beyond what is stipulated by the binomial distribution. The chapter examines the relationship between precipitation during the breeding season and reproductive success; wet springs are likely to depress the proportion of successful nests. One can assume that all shrike populations have the same relation- ship between breeding success and standardized spring precipitation, but at different levels, corresponding to a random-intercepts model. This means assuming that every shrike population has a specific response to precipitation but that both intercept and slope are “similar” among populations. As for the Poisson case, the introduction of random effects into a binomial GLM in WinBUGS is particularly straightforward and transparent. Fitting the resulting binomial GLMM in WinBUGS is very helpful for the general understanding of this class of models.

The BUGS language and program was developed by epidemiologists in Cambridge, UK in the 1990s. The acronym stands for Bayesian analysis using Gibbs sampling. WinBUGS is a groundbreaking program; it has made really flexible and powerful Bayesian statistical modeling available to a large range of users, especially for users who lack the experience in statistics and computing to fit such fully custom models by maximizing their likelihood in a frequentist mode of inference. WinBUGS lets one specify almost arbitrarily complex statistical models using a fairly simple model definition language that describes the stochastic and deterministic “local” relationships among all observable and unobservable quantities in a fully specified statistical model. The chapter provides examples that are analyzed with WinBUGS run from within program R by use of the R2WinBUGS package. In addition, the steps before and after an analysis in WinBUGS are greatly facilitated in R, e.g., data preparation as well as computations on the Markov chain Monte Carlo (MCMC) output and the presentation of results in graphs and tables. Importantly, after conducting an analysis in WinBUGS, R2WinBUGS will import back into R the results of the Bayesian analysis, which essentially consist of the Markov chains for each monitored parameter. Fitting statistical models in WinBUGS opens up a new world of modeling freedom to many ecologists. Writing WinBUGS code, or understanding and tailoring to one's own needs WinBUGS code written by others, is much more within the reach of typical ecologists than writing or adapting a similar analysis in some software that explicitly maximizes a likelihood function for the same problem.

The site-occupancy model is an extended logistic regression that can estimate true occurrence probability (ψ) and the factors affecting it, while correcting for imperfect detection. The extension is represented by the model component for detection probability (p):conventional logistic regression is a special case, when p = 1. Site-occupancy models are the only current framework for inference about species distributions that model true rather than apparent distributions; the latter is the product of occurrence and detection probability (i.e., ψ × p). It explains by showing example that not accounting for detection probability may lead to spectacularly wrong inferences about the distribution of a species under a conventional, naïve modeling approach. In contrast, the site-occupancy model applied to replicated “presence–absence” data was better able to estimate the true system state.

Two features specific to nonnormal generalized linear models (GLMs) are overdispersion and offsets. Zero-inflation can be called a specific form of overdispersion: there are more zeroes than expected. In both distributions commonly used to model counts (Poisson and binomial), the dispersion (the variability in the counts) is not a free parameter but instead is a function of the mean. The variance is equal to the mean (λ) for the Poisson and equal to the mean (N * p) times 1−p for the binomial distribution. This means that for a Poisson or binomial random variable, the models for the counts come with a “built-in” variability and the magnitude of that variability is known. In an analysis of deviance conducted in a classical statistical analysis of the model, the residual deviance of the model will be about the same magnitude as the residual degrees of freedom, i.e., the mean deviance ratio is about 1. The simplest way to correct for overdispersion in a classical analysis is by the quasi-likelihood and by using family=quasipoisson (or quasibinomial) in the Rfunction glm(). Using WinBUGS, there are several ways in which one can account for overdispersion. One is to specify a distribution that is overdispersed relative to the Poisson, such as the negative binomial. Overdispersion, zero-inflation, and offsets are important GLM topics. The specification of the associated models in WinBUGS is fairly easy and clarifies the actual meaning of these three topics. This is not usually the case when fitting these models in a canned routine in R or another software. Thus, this is another example of where the simple model speci- fication in the BUGS language enforces an understanding of the fitted model that is easily lost in other stats packages.

This chapter provides practical understanding for how Bayesian inference with vague priors works and why Bayesian inference is simply so useful. WinBUGS is the most widely used general-purpose Bayesian software. WinBUGS allows the average, somewhat numerate ecologist to conduct his or her own creative statistical modeling. WinBUGS has the potential to free the statistical modeler in anyone. Even without the benefits of the Bayesian paradigm of statistical inference, this would be enough to recommend WinBUGS to any ambitious quantitative ecologist. Implementation in WinBUGS is much more straightforward and arguably easier to understand for an ecologist. The reason for this is that in the BUGS language, a complex model is naturally broken apart into hierarchically linked submodels. Nonlinear models may be more realistic representations of a study system and may yield better predictions, especially outside of the observed range of covariates.