Elena N. Ieno

Zero-Truncated and Zero-Inflated Models for Count Data

In this chapter, we discuss models for zero-truncated and zero-inflated count data. Zero truncated means the response variable cannot have a value of 0. A typical example from the medical literature is the duration patients are in hospital. For ecological data, think of response variables like the time a whale is at the surface before re-submerging, counts of fin rays on fish (e.g. used for stock identification), dolphin group size, age of an animal in years or months, or the number of days that carcasses of road-killed animals (amphibians, owls, birds, snakes, carnivores, small mammals, etc.) remain on the road. These are all examples for which the response variable cannot take a value of 0.

A protocol for conducting and presenting results of regression-type analyses

Summary Scientific investigation is of value only insofar as relevant results are obtained and communicated, a task that requires organizing, evaluating, analysing and unambiguously communicating the significance of data. In this context, working with ecological data, reflecting the complexities and interactions of the natural world, can be a challenge. Recent innovations for statistical analysis of multifaceted interrelated data make obtaining more accurate and meaningful results possible, but key decisions of the analyses to use, and which components to present in a scientific paper or report, may be overwhelming. We offer a 10-step protocol to streamline analysis of data that will enhance understanding of the data, the statistical models and the results, and optimize communication with the reader with respect to both the procedure and the outcomes. The protocol takes the investigator from study design and organization of data (formulating relevant questions, visualizing data collection, data exploration, identifying dependency), through conducting analysis (presenting, fitting and validating the model) and presenting output (numerically and visually), to extending the model via simulation. Each step includes procedures to clarify aspects of the data that affect statistical analysis, as well as guidelines for written presentation. Steps are illustrated with examples using data from the literature. Following this protocol will reduce the organization, analysis and presentation of what may be an overwhelming information avalanche into sequential and, more to the point, manageable, steps. It provides guidelines for selecting optimal statistical tools to assess data relevance and significance, for choosing aspects of the analysis to include in a published report and for clearly communicating information.

Mixed Effects Modelling for Nested Data

In this chapter, we continue with Gaussian linear and additive mixed modelling methods and discuss their application on nested data. Nested data is also referred to as hierarchical data or multilevel data in other scientific fields (Snijders and Boskers, 1999; Raudenbush and Bryk, 2002).

Limitations of Linear Regression Applied on Ecological Data

This chapter revises the basic concepts of linear regression, shows how to apply linear regression in R, discusses model validation, and outlines the limitations of linear regression when applied to ecological data. Later chapters present methods to overcome some of these limitations; but as always before doing any complicated statistical analyses, we begin with a detailed data exploration. The key concepts to consider at this stage are outliers, collinearity, and the type of relationships between the variables. Failure to apply this initial data exploration may result in an inappropriate analysis forcing you to reanalyse your data and rewrite your paper, thesis, or report.

GLM and GAM for Count Data

A generalised linear model (GLM) or a generalised additive model (GAM) consists of three steps: (i) the distribution of the response variable, (ii) the specification of the systematic component in terms of explanatory variables, and (iii) the link between the mean of the response variable and the systematic part. In Chapter 8, we discussed several different distributions for the response variable: Normal, Poisson, negative binomial, geometric, gamma, Bernoulli, and binomial distributions. One of these distributions can be used for the first step mentioned above. In fact, later in Chapter 11, we see how you can also use a mixture of two distributions for the response variable; but in this chapter, we only work with one distribution at a time.

Long-term population trends of endangered Hawaiian waterbirds

We analyzed long-term winter survey data (1956–2007) for three endangered waterbirds endemic to the Hawaiian Islands, the Hawaiian moorhen (Gallinula chloropus sandvicensis), Hawaiian coot (Fulica alai), and Hawaiian stilt (Himantopus mexicanus knudseni). Time series were analyzed by species–island combinations using generalized additive models, with alternative models compared using Akaike information criterion (AIC). The best model included three smoothers, one for each species. Our analyses show that all three of the endangered Hawaiian waterbirds have increased in population size over the past three decades. The Hawaiian moorhen increase has been slower in more recent years than earlier in the survey period, but Hawaiian coot and stilt numbers still exhibit steep increases. The patterns of population size increase also varied by island, although this effect was less influential than that between species. In contrast to earlier studies, we found no evidence that rainfall affects counts of the target species. Significant population increases were found on islands where most wetland protection has occurred (Oahu, Kauai), while weak or no increases were found on islands with few wetlands or less protection (Hawaii, Maui). Increased protection and management, especially on Maui where potential is greatest, would likely result in continued population gains, increasing the potential for meeting population recovery goals.

GLMM and GAMM

In Chapters 2 and 3, we reviewed linear regression and additive modelling techniques. In Chapters 4–7, we showed how to extend these methods to allow for heterogeneity, nested data, and temporal or spatial correlation structures. The resulting methods were called linear mixed modelling and additive mixed modelling (see the left hand pathway of Fig. 13.1). In Chapter 9, we introduced generalised linear modelling (GLM) and generalised additive modelling (GAM), and applied them to absence–presence data, proportional data, and count data. We used the Bernoulli and binomial distributions for 0–1 data (the 0 stands for absence and the 1 for presence), and proportional data (Y successes out of n independent trials), and we used the Poisson distribution for count data. However, one of the underlying assumptions of theses approaches (GLM and GAM) is that the data are independent, which is not always the case. In this chapter, we take this into account and extend the GLM and GAM models to allow for correlation between the observations, and nested data structures. It should come as no surprise that these methods are called generalised linear mixed modelling (GLMM) and generalised additive mixed modelling (GAMM); see the right hand pathway of Fig. 13.1.

Dealing with Heterogeneity

This chapter, and the following three chapters, discuss solutions to the problems introduced in Chapters 2 and 3: heterogeneity, nested data, temporal correlation, and spatial correlation. We use both the linear regression model and the additive model as starting points. Figure 4.1 shows an overview of the methods we discuss in Chapters 4, 5, 6, and 7. In all these chapters, the model consists of a fixed term and a random term. The fixed term describes the response variable Y as a function of the explanatory variables via α + β 1 × X 1 + … + β q × X q in linear regression or α + f 1(X 1)+…+ f q (X q ) in additive modelling.

Things are not Always Linear; Additive Modelling

In the previous chapter, we looked at linear regression, and although the word linear implies modelling only linear relationships, this is not necessarily the case. A model of the form Y i = α + β 1 × X i + β 2 × X i 2 + ɛ i is a linear regression model, but the relationship between Y i and X i is modelled using a second-order polynomial function. The same holds if an interaction term is used. For example, in Chapter 2, we modelled the biomass of wedge clams as a function of length, month and the interaction between length and month. But a scatterplot between biomass and length may not necessarily show a linear pattern.

Analysing Forensic Entomology Data Using Additive Mixed Effects Modelling

Forensic pathologists and entomologists estimate the minimum post-mortem interval since a long time by describing the stage of succession and development of the necrophagous fauna (Amendt et al. 2004). From very simple calculations at the beginning, (Bergeret, see also Smith 1986) the discipline has evolved into a more mathematical one (e.g. Marchenko 2001; Grassberger and Reiter 2001, 2002) and tries to implement concepts like probabilities and confidence intervals (Lamotte and Wells 2000; Donovan et al. 2006; Tarone and Foran 2008, see also Villet et al. this book Chapter7). As pointed out by Tarone and Foran (2008) and Van Laerhoven (2008), the latter is one of the major tenets of the Daubert Standard (Daubert et al. v. Merrell Dow Pharmaceuticals (509 U.S. 579 (1993)).