Introduction

Several statistical methods not previously discussed are briefly described in this Chapter. Our goal is to introduce the methods and indicate where more information on them may be found rather than to present in-depth discussions of the techniques. The first three Sections discuss relatively new methods that have not been widely used in behavioral ecology, or at least some of its subdisciplines, but that may be useful in a variety of applications. The subsequent three Sections discuss other branches of statistics with well-developed methods that we have not had space in this book to cover in detail.

Adaptive sampling

In conventional survey sampling plans, such as those discussed in Chapter Four, the sampling plan and sample size are determined before collecting the data and, in theory, all of the population units to be included in the sample could be identified before data collection begins. This approach, however, is sometimes unsatisfactory when the population units are uncommon and clumped in space and/or time. For example, suppose we are estimating the proportion of trees in an orchard damaged by rodents, and the damage occurs in widely scattered patches. We select rows of trees to inspect. Most trees are undamaged but occasionally we encounter an area in which most trees are damaged. Under conventional sampling plans we can only include trees in the selected rows in our sample. Yet we may be able to see that the damage extends to nearby rows and feel that some way should exist to include those trees in the sample as well.

This book describes the sampling and statistical methods used most often by behavioral ecologists. We define behavioral ecology broadly to include behavior, ecology and such related disciplines as fisheries, wildlife, and environmental physiology. Most researchers in these areas have studied basic statistical methods, but frequently have trouble solving their design or analysis problems despite having taken these courses. The general reason for these problems is probably that introductory statistics courses are intended for workers in many fields, and each field presents a special, and to some extent unique, set of problems. A course tailored for behavioral ecologists would necessarily contain much material of little interest to students in other fields.

The statistical problems that seem to cause behavioral ecologists the most difficulty can be divided into several categories.

1. Some of the most difficult problems faced by behavioral ecologists attempting to design a study or analyze the resulting data fall between statistics – as it is usually taught – and biology. Examples include how to define the sampled and target populations, the nature and purpose of statistical analysis when samples are collected nonrandomly, and how to avoid pseudoreplication.

2. Some methods used frequently by behavioral ecologists are not covered in most introductory texts. Examples include survey sampling, capture–recapture, and distance sampling.

3. Certain concepts in statistics seem to need reinforcement even though they are well covered in many texts. Examples include the rationale of statistical tests, the meaning of confidence intervals, and the interpretation of regression coefficients.

4. Behavioral ecologists encounter special statistical problems in certain areas including index methods, detecting habitat ‘preferences’, and sampling behavior.

5. […]

Introduction

This Chapter reviews methods for studying the relationship between two or more variables. We begin with a brief description of scatterplots and simple summary statistics commonly calculated from them. Emphasis is given to examining the effect of outliers and influential points and to recognizing that measures of association such as the correlation coefficient only describe the linear (i.e., straight line) relationship. Simple linear regression is then described including the meaning of the slope, the basic assumptions needed, and the effects of violating the assumptions. Multiple regression is then introduced, again with emphasis on quantities of direct value to behavioral ecologists rather than on the analytical methods used to obtain these results.

We make a slight notational change in this Chapter. In prior Chapters, we have used lower-case letters for quantities associated with the sample and upper-case letters for quantities associated with the population. In discussions of regression, upper-case letters are generally used for individual values and their means regardless of whether they are associated with the sample or population. Regression coefficients for the population are generally identified by the symbols β (e.g., β0, β1 …) and the corresponding sample estimates are denoted b0, b1 and so on. These practices are so well established in the statistical literature that we follow them in this Chapter even though it introduces some inconsistency with other Chapters.

Scatterplots and correlation

Among the simplest and most commonly used methods for studying the relationship between two quantitative variables are scatterplots and correlations.

Introduction

This Chapter provides an overview of how statistical problems are formulated in behavioral ecology. We begin by identifying some of the difficulties that behavioral ecologists face in deciding what population to study. This decision is usually made largely on nonstatistical grounds but a few statistical considerations are worth discussing. We then introduce the subject of making inferences about the population, describing objectives in statistical terms and discussing accuracy and the general ways used to measure it. Finally, we note that statistical inferences do not necessarily apply beyond the population sampled and emphasize the value of drawing a sharp distinction between the sampled population and larger populations of interest.

Specifying the population

Several conflicting goals influence decisions about how large and variable the study population should be. The issues are largely nonstatistical and thus outside the scope of this book, but a brief summary, emphasizing statistical issues insofar as they do occur, may be helpful.

One issue of fundamental importance is whether the population of interest is well defined. Populations are often well defined in wildlife monitoring studies. The agencies carrying out such studies are usually concerned with a specific area such as a State and clearly wish to survey as much of the area as possible. In observational studies, we would often like to collect the data throughout the daylight hours – or some portion of them – and throughout the season we are studying.

Sampling throughout the population of interest, however, may be difficult for practical reasons. For example, restricting surveys to roads and observations to one period of the day may permit the collection of a larger sample size.

Introduction

Survivorship, the proportion of individuals surviving throughout a given period, may be estimated simply as p = x/n where n = the number alive at the start of the period and x equals the number alive at the end of the period, or it may be estimated using capture–recapture methods as discussed in Chapter Nine. Two additional issues, however, often arise in behavioral ecology. One is that in many studies a measure of overall survival across several periods, each having a separate survival estimate, may be desired. The second issue is that in studies of nesting birds or other animals, information is often incomplete because many nests are not discovered until well after they have been initiated and may not be followed to completion. In this Chapter we discuss methods developed to handle both of these cases. We focus on telemetry studies, which raise the first issue, and studies of nesting success which raise both issues.

Telemetry studies

In telemetry studies, transmitters are attached to animals which are then monitored for various purposes, including the estimation of survival rates. The simplest case arises when all transmitters are attached at about the same time and animals are checked periodically at about the same times until they die or the study ends. Cohorts based on age, sex, or other factors may be defined.

Data of this type are basically binomial (White and Garrott 1990; Samuel and Fuller 1994). On any sampling occasion, t, the proportion of animals still alive is the appropriate estimator of survivorship to that time.

Introduction

We use the phrase resource selection to mean the process that results in animals using some areas, or consuming some food items, and not consuming others. In some studies, resources are defined using mutually exclusive categories such as ‘wooded/nonwooded’ in a habitat study or ‘invertebrate/ plant/bird/mammal’ in a diet study. In other studies, resources are defined using variables that are not mutually exclusive and that define different aspects of the resource such as elevation, aspect, distance to water, and cover type. In some studies, only use is measured. In many others, availability of the resources is also measured and analyses are conducted to determine whether the resources are used in proportion to their availability or whether some are used more – and some less – than would be expected if resources were selected independently of the categories defined by the investigator. In this Chapter, we concentrate on methods in which resources are assigned to mutually exclusive categories because this approach has been by far the most common in behavioral ecology. However, studies in which resources are defined using multivariate approaches are discussed briefly, and we urge readers to learn more about this approach since it can be formulated to include the simpler approach but offers considerably more flexibility.

The investigator has great flexibility in defining use. In the case of habitat studies use may mean that an area is used at least once during the study or (less often in practice) used more than some threshold number of times, or used for a specific activity (e.g., nesting). When the population units are possible prey items, used items might be those attacked, consumed, partly consumed, etc.

Introduction

The term pseudoreplication was introduced by Hurlbert (1984) to describe analyses in which ‘treatments are not replicated (though samples may be) or replicates are not statistically independent’. In survey sampling terms, the problem arises when a multistage design is employed but the data are treated as a one-stage sample in the analysis. For example suppose m secondary units are selected in each of n primary units. In most cases the nm observations do not provide as much information as nm observations selected by simple random sampling. The correct approach is to base the analysis on the means per primary unit.

The initial point made by Hurlbert and soon thereafter by Machlis et al. (1985) was incontrovertible. When multistage sampling is employed, ignoring the sampling plan and treating the data set as though it is a simple random sample can lead to gross errors during interval estimation and testing. This point is emphasized in survey sampling books, and calling attention to the error, which was quite common at the time, was a service to the discipline. Subsequently, however, cases began to appear in which the proper analytical approach was more difficult to agree on. These cases often did not involve large, clearly described populations, and well-defined sampling plans. Instead, they usually involved some combination of incompletely specified populations, nonrandom sampling, nonindependent selection, and small sample sizes. As noted in Chapter Four these issues are inherently difficult and full agreement on the best analytical approach and most appropriate interpretation cannot always be attained. Nonetheless, we believe that progress may be attainable in some cases by applying the ideas developed in Chapter Four.

Introduction

This Chapter describes some of the statistical methods for developing point and interval estimators. Most statistical problems encountered by behavioral ecologists can be solved without the use of these methods so readers who prefer to avoid mathematical discussions may skip this Chapter without compromising their ability to understand the rest of the book. On the other hand, the material may be useful in several ways. First, we believe that study of the methods in this Chapter will increase the reader's understanding of the rationale of statistical analysis. Second, behavioral ecologists do encounter problems frequently that cannot be solved with ‘off the shelf’ methods. The material in this Chapter, once understood, will permit behavioral ecologists to solve many of these problems. Third, in other cases, consultation with a statistician is recommended but readers who have studied this Chapter will be able to ask more relevant questions and may be able to carry out a first attempt on the analysis which the statistician can then review. Finally, many behavioral ecologists are interested in how estimators are derived even if they just use the results. This Chapter will help satisfy the curiosity of these readers.

The first few sections describe notation and some common probability distributions widely used in behavioral ecology. Next we explain ‘expected value’ and describe some of the most useful rules regarding expectation. The next few Sections discuss variance, covariance, and standard errors, defining each term, and discussing a few miscellaneous topics such as why we sometimes use ‘n’ and sometimes ‘n – 1’ in the formulas. Section 2.10 discusses linear transformations, providing a summary of the rules developed earlier regarding the expected value of functions of random variables.

Introduction

Monitoring abundance means estimating trends in abundance, usually through time though occasionally across space or with respect to some other variable. Estimating temporal trends in abundance of animal populations is a common objective in both applied and theoretical research. The most common design involves surveys in the same locations run once or more per year for several years. The data are often collected using ‘index methods’ in which the counts are not restricted to well-defined plots or if they are the animals present are not all detected and the fraction detected is not known. Results are usually summarized by calculating the mean number of animals detected per plot, route, or some other measure of effort, during each period. These means are then plotted against time (Fig. 8.1). When the counts come from complete surveys of well-defined plots, then the Y-axis is in density units. In the more common case of index data, the Y-axis shows the number recorded per survey route or some other measure of effort. We assume (or at least hope) that a 5% change in survey results indicates a 5% change in population size but we have no direct measure of absolute density.

The analysis of trend data raises several difficult issues. First, ‘the trend’ can be defined in different ways, and the choice among them may be difficult and subjective. Second, statistical difficulties arise in estimating precision because the same routes are surveyed each year so the annual means are not independent. Third, use of index methods, rather than complete counts on well-defined plots, means that change in survey efficiency may cause spurious trends in the data.

Introduction

The Macropodoidea (kangaroos, wallabies and rat-kangaroos) are the most distinctive and widely recognised of Australia's unique native fauna. All macropodoids are fundamentally similar in body form (Flannery 1984) but they nevertheless display an amazing diversity of species, habitat preferences and lifestyles. There are over 60 extant species [new species are still being discovered (Flannery et al. 1995) and active speciation in rock-wallabies (Petrogale) has led to considerable flux in their taxonomy (Eldridge & Close 1992)] divided into two families: the Potoroidae (rat-kangaroos, bettongs and potoroos) and the more derived Macropodidae (‘true’ wallabies and kangaroos). There are small (< 1 kg), solitary, homomorphic rainforest species (e.g. Musky Rat-kangaroo, Hypsiprymnodon moschatus) and large (>90 kg), gregarious, hetero-morphic species of the open arid and semi-arid plains (Red Kangaroo, Macropus rufus). There are arboreal tree-kangaroos (Dendrolagus) in the rainforests of northern Australia and New Guinea, a bettong that burrows like a rabbit (Burrowing Bettong, Bettongia lesueur) and species adapted to rocky outcrops and escarpments (rock-wallabies, Petrogale and Peradorcas, and Common Wallaroo, Macropus robustus). The range of adaptations in macropodoids was considered by Flannery (1989) to be greater than that of any placental family or superfamily with the exception of murids.

The closest placental equivalent to macropodoids are the Artiodactyla, particularly the Cervidae and Bovidae. Despite some 130 million years of phyletic separation, macropodoids and artiodactyls display striking behavioural and ecological convergence as comparisons of Jarman (1974) and Kaufmann (1974a, b) demonstrate. Macropodoids and artiodactyls represent alternative evolutionary pathways for large-bodied herbivorous mammals and this offers excellent opportunities for phylo-genetic comparisons of play.

Social play: evolution, pretense, and the cognitive turn

Pierre Huber (1810, p. 148), in his book about the behavior of ants, claims that if one were not accustomed to treating insects as machines one would have trouble explaining the social behavior of ants and bees without attributing emotions to them. Although we shall skirt the issue of emotion, many observers would agree that animals play because it is fun for them to do so. But even if the issue of emotions is set aside, readers conditioned by the scruples of modern psychology are likely to be skeptical of Darwin's ready acceptance that Huber observed ants playing. Play, as the quotation above indicates, seems to involve pretense, and pretense is commonly thought to require more sophisticated intentions than are usually attributed to ants. How could Huber have seen or inferred pretense from the behavior of the ants? And how could he be sure that the observed behavior was not, in fact, directed toward some very specific and immediate function? These questions raise the difficult issue of what play is, or, as biologists are wont to put it, how to define ‘play’.

Among birds, the most common types of play (locomotor, object and social) are found in those orders (Psittaciformes and Passeriformes) with the most developed forebrains (Ortega & Bekoff 1987). Among the Passeriformes, the corvids are considered to have the most complex play behavior (Ficken 1977). The raven, Corvus corax is the largest passerine and probably has the largest brain volume of any corvid. It also inhabits the greatest geographical range and the most diverse habitats. The raven may therefore be expected to show tremendous behavioral flexibility, perhaps acquired in part through play (Gwinner 1966, Ficken 1977, Ortega & Bekoff 1987). Given these considerations, ravens provide a particularly useful ‘outgroup’ for comparison with mammals, as well as other birds.

In putting together this review of raven play behavior however, we were faced immediately with the problem of defining ‘play’. ‘Play’ is notoriously difficult to define (Fagen 1981, Bekoff & Byers 1981, Bekoff 1984, Martin & Caro 1985; see also Bekoff & Allen, Chapter 5). We all recognize it at the extremes, but cannot define it clearly enough to fit it into an exclusive and objectively defined category of behavior. Ficken (1977) points out, and we agree, that play is even more difficult to identify in birds than in mammalian species. Perhaps the most widely accepted definition of play is as follows: ‘…all motor activity performed postnatally that appears purposeless, in which motor patterns from other contexts may often be used in modified forms or altered sequencing’ (Bekoff 1984). This definition seems problematic in that it is not clear what is meant by ‘appears purposeless,’ and is therefore very difficult to apply.

Introduction

There is no shortage of descriptive literature on play behavior, nor is there a lack of theories as to the putative functions and evolutionary origins of play. But despite this wealth of information and speculation, very little is known about how the brain is involved in playfulness. We can presume that those animals which play do so because their brains evolved in a way that favored the presence of certain types of neural circuitry, the activation of which results in behaviors that we can readily identify as being playful. Since it is now fairly well established that play is a distinct behavioral entity and not simply the juvenile form of more adult-typical behaviors (e.g., Fagen 1981), it can also be assumed that play would be represented by a distinct neural topography. This is not to say that overlap doesn't exist between brain areas involved in play and brain areas involved in other behaviors. For example, it would be expected that the pleasure derived from engaging in playful interactions taps into neural systems which code pleasure derived from other behaviors. Similarly, those systems which are involved in the actual patterning of movements exhibited during play would not be expected to differ greatly from those involved in motor patterning of other behaviors. The neural uniqueness of play may be in how these individual neural units are combined, and which types of stimuli initiate them.

Play has often been thought of as a distinctly mammalian behavior pattern (e.g., MacLean 1990). Such an assumption would make the search for neural systems involved in play relatively straightforward in that the search could be limited to parts of the brain unique to mammals.

The current status of identifying nonavian reptile play

The origins of vertebrate play are obscure, but more understanding of these origins would aid greatly in clarifying and evaluating hypotheses about play (Bekoff & Byers 1981; Burghardt 1984, 1998). It has been clearly shown that some birds and most if not all groups of mammals show behavior currently classified as play (Ficken 1977; Fagen 1981; Ortega & Bekoff 1987). Is play behavior restricted to endothermic vertebrates with extensive parental care? Since the nineteenth century, during which claims for play were made for many invertebrates and vertebrates (Fagen 1981; Burghardt in press; Bekoff & Allen, Chapter 5), the generally accepted phyletic scope of play has become narrowed to the extent that it is generally limited to mammals and birds. If credible evidence for play outside of the mammalian and avian radiations is to be sought, key groups are found in the nonavian reptiles. Although I will use the term reptiles from here on, many authorities hold that reptiles are not a monophyletic group, that share a common ancestor. However, even if reptiles are monophyletic, birds would be part of that group, related most closely to crocodilians. Crocodilians are in many physiological, paleontological, and life history characteristics more similar to birds than other traditional reptile groups such as turtles. For example, crocodilians have a four-chambered heart and all show postnatal parental care, complete with a complex vocal communication system that includes ‘contact’ and ‘distress’ calls (Herzog & Burghardt 1977). As archosaurs, they share a more recent common ancestor with birds than with squamate reptiles or turtles.

Introduction

By taking the reader along the play-path of my personal and professional life, I encourage readers to examine their own play experiences, attitudes, and observations. By reviewing my experiences sequentially, a gradually evolving broad view of the importance of, and speculations about what play is and what playfulness does, emerges. Play is seen as a broad category of behavior, as basic in its phenomenology to smart complex animals as sleep and dreams, and as scientifically enigmatic. Its healthy presence seems necessary for the maintenance of flexibility and adaptability.

I am a physician-psychiatrist by training and practice and more recently have engaged in independent scholarship, educational film production and popular writing (Brown 1987, 1988, 1995; Brown & Cousineau 1990; Brown & Moses 1995). As a method useful to me in making better sense of the world and its parts, I have generally relied on clinical observation, first to demonstrate a phenomenon and then have gone searching for ‘explanations’ which best explain and characterize it. This has worked well for me as a physician, and has enriched explorations in other areas. The general subject of play in animals and humans has gradually emerged as a broad category of behavior which warrants fresh exploration (Brownlee 1997). Thus the views given here about this complex and slippery subject will reflect my current efforts to place it in context as I have encountered it. My goals for this article are that it will offer the play enthusiast and student fresh ways of viewing the subject and stimulate personal examination of the cultures and biology of playfulness. I look upon this effort as if I was engaged in telling my story of play.