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This chapter introduces the main concepts of statistical inference, or drawing conclusions from data. There are three main types of inference: point estimation, confidence estimation, and hypothesis testing. There are two major statistical paradigms which address the statistical inference questions: the classical, or frequentist paradigm, and the Bayesian paradigm. While most of statistics and machine learning is based on the classical paradigm, Bayesian techniques are being embraced by the statistical and scientific communities at an ever-increasing pace. The chapter begins with a short comparison of classical and Bayesian paradigms, and then discusses the three main types of statistical inference from the classical point of view.

This chapter introduces the most important aspects of Bayesian statistical inference and techniques for performing such calculations in practice. It first reviews the basic steps in Bayesian inference in early sections of the chapter, and then illustrates them with several examples in sections that follow. Numerical techniques for solving complex problems are next discussed, and the final section provides a summary of pros and cons for classical and Bayesian method. It argues that most users of Bayesian estimation methods are likely to use a mix of Bayesian and frequentist tools. The reverse is also true—frequentist data analysts, even if they stay formally within the frequentist framework, are often influenced by “Bayesian thinking,” referring to “priors” and “posteriors.” The most advisable position is to know both paradigms well, in order to make informed judgments about which tools to apply in which situations.

This chapter explores the relationship between scientific hypotheses and the customary procedures of classical statistical inference. It argues that classical significance tests are violently biased against the null hypothesis. Thus, a conservative theorist will associate his theory with the null hypothesis, while an enthusiast will not—and they may often reach conflicting conclusions, whether or not the theory is correct. No procedure can satisfactorily test the goodness of fit of a single model to data. The remedy is to compare the fit of several models to the same data. Such procedures do not compare null with alternative hypotheses, and so are in this respect unbiased.

There are two main classes of interpretations of probability. The first are those that rely on a measure of frequency. The other is those that take a logical or subjective view of a unique event, independent of past or future events. The interpretation of probability which is used in the book is then defined as evidential probability, a function based on a set of known statements based on frequency or measure. The properties of probability are then enumerated and explained. Probabilities can also be based on statistical inference. Statistical inferences are statistical properties with high probability based on statistical evidence.

This chapter extends chapter 5's formal proof into a general discussion of how we should assess evidence — and, in particular, how we should test scientific theories — given the possibility that Everettian quantum mechanics is correct. The chapter proves — initially with informal arguments, but in due course with full mathematical rigor — that rational scientists should treat the branch weights assigned by quantum theory to quantum-mechanical branches exactly as if they were probabilities; as such, the chapter argues, probability is actually on firmer conceptual footing in quantum theory than in classical physics. After a brief digression to consider probability in approaches to quantum theory other than the Everett interpretation, the chapter concludes by placing the work in the broader context of recent work on Everett probability.

This chapter champions the elicitation and use of probabilistic measures of uncertainty. It compares two different views (roughly corresponding, respectively, to the Bayesian and frequentist approaches to statistical inference) as to how probabilities should be used as evidence. It argues that, while the former may be a logical ideal, the latter may be more appropriate in the special circumstances of the courtroom.

This chapter discusses the often-acknowledged notion that, for practical decision-making, definite probabilities are required instead of indefinite ones. Theories that take indefinite probabilities as basic need a way of deriving definite probabilities from them. Theories of how to do this are theories of direct inference. Theories of objective indefinite probability propose that statistical inference gives us knowledge of indefinite probabilities, and then direct inference gives us knowledge of definite probabilities. Reichenbach pioneered the theory of direct inference. Kyburg was the first to attempt to provide firm logical foundations for direct inference, and Pollock took this as a starting point, constructing a modified theory with a more epistemological orientation. The chapter builds upon some of the basic ideas of this latter theory.

Error-statistical methods in science have been the subject of enormous criticism, giving rise to the popular statistical “reform” movement and bolstering subjective Bayesian philosophy of science. Is it possible to have a general account of scientific evidence and inference that shows how we learn from experiment despite uncertainty and error? One way that philosophers have attempted to affirmatively answer this question is to erect accounts of scientific inference or testing where appealing to probabilistic or statistical ideas would accommodate the uncertainties and error. Leading attempts take the form of rules or logics relating evidence (or evidence statements) and hypotheses by measures of confirmation, support, or probability. We can call such accounts logics of evidential relationship (or E-R logics). This chapter reflects on these logics of evidence and compares them with error statistics. It then considers measures of fit vs. fit combined with error probabilities, what we really need in a philosophy of evidence, criticisms of Neyman-Pearson statistics and their sources, the behavioral-decision model of Neyman-Pearson tests, and the roles of statistical models and methods in statistical inference.

Those who criticize the Everett interpretation on the grounds that it makes no sense of probability apply a double-standard, for no other physical theory of probability does any better in explaining probability or in deriving its link with decision theory. In fact, others do worse, for in any one world theory it is a mystery as to why, given that only a single outcome of a chance process occurs, we should nevertheless act so as to maximize expected utilities, which involves all possible outcomes of a chance process. This difficulty does not apply to the Everett interpretation, in which all outcomes happen.

Induction is the inference from a sample to a population, regardless of the possible existence of exceptions. Induction is used in the practice of science and engineering based on knowledge that can be accepted as evidence. There are two bodies of knowledge: evidential corpus, a set of propositions acceptable as evidence in a certain context; and practical corpus, a set of propositions counting as “practically certain” in that context. There are five kinds of induction described: statistical, universal, nomic, theoretical, and instantial.

This chapter takes on the problem of model adequacy and makes an argument for reformulating the way model-based statistical inference is carried out. In the new formulation, it does not treat the model as “truth.” It is instead an approximation to truth. Rather than testing for model fit, an integral part of the proposed statistical analysis is to assess the degree to which the model provides adequate answers to the statistical questions being posed. One method for doing so is to create a single overall measure of inadequacy that evaluates the degree of departure between the model and truth. The chapter argues that there are two components of errors in any statistical analysis. One component is due to model misspecification; that is, the working model is different from the true data-generating process. The chapter compares confidence intervals on model misspecification error with external knowledge of the scientific relevance of prediction variability to address the issue of scientific significance. The chapter also analyzes several familiar measures of statistical distances in terms of their possible use as inadequacy measures.

This chapter demonstrates the types of questions one could ask about a continuous random variable of interest and answer using statistical inference. It provides conceptual preparation for understanding statistical inference, demonstrates how to get data ready for analysis in R, and then illustrates how to conduct two types of statistical inferences—null hypothesis testing and confidence interval construction—regarding the population attributes of a continuous random variable, using sample data. Both the one-sample t-test and the difference-of-means test are presented. Two key points in this chapter are worth noting. First, statistical inference is primarily concerned about figuring out population attributes using sample data. Hence, it is not the same as causal inference. Second, statistical inference can help to answer various questions of substantive interest. This chapter focuses on statistical inferences regarding one continuous random outcome variable.

The questioning of science and the scientific method continues within the science of ecology. The use of Bayesian statistical analysis has recently been advocated in ecology, supposedly to aid decision makers and enhance the pace of progress. Bayesian statistics provides conclusions in the face of incomplete information. However, Bayesian statistics represents a much different approach to science than the frequentist statistics studied by most ecologists. This chapter discusses the influence of postmodernism and relativism on the scientific process and in particular its implications, through the use of subjective Bayesian approach, in statistical inference. It argues that subjective Bayesianism is “tobacco science” and that its use in ecological analysis and environmental policy making can be dangerous. It claims that science works through replicability and skepticism, with methods considered ineffective until they have proven their worth. It proposes the use of a frequentist approach to statistical analysis because it corresponds to the skeptical worldview of scientists.

Decision theory requires the assignment of probabilities for the different possible states of nature. Bayesian inference provides such probabilities, but at the cost of requiring prior probabilities for the states of nature. In this century, the justification for prior probabilities has often rested on subjective theories of probability. Subjective probability can lead to internally consistent systems relating belief and action for a single individual; but severe difficulties emerge in trying to extend this model to justify public decisions. Objective probability represents probability as a literal frequency that can be communicated as a matter of fact and that can be verified by independent observers confronting the same information. This chapter argues that the Bayesian approach is best for making decisions and that one needs to put probabilities on various hypotheses. It proposes an interpretation of statistical inference for decision making, but disapproves of the subjective aspects of Bayesianism and suggests, as an alternative, using related data to create “objective” priors. The chapter also considers a compound sampling perspective and presents a concrete example of compound sampling.

Inferring the probability density function (pdf) from a sample of data is known as density estimation. The same methodology is often called data smoothing. Density estimation in the one-dimensional case has been discussed in the previous chapters. This chapter extends it to multidimensional cases. Density estimation is one of the most critical components of extracting knowledge from data. For example, given a pdf estimated from point data, we can generate simulated distributions of data and compare them against observations. If we can identify regions of low probability within the pdf, we have a mechanism for the detection of unusual or anomalous sources. If our point data can be separated into subsamples using provided class labels, we can estimate the pdf for each subsample and use the resulting set of pdfs to classify new points: the probability that a new point belongs to each subsample/class is proportional to the pdf of each class evaluated at the position of the point.

This chapter provides an introduction and overview of the three statistical components of the meta-analysis: (1) the statistical model that describes how the study-specific estimates of interest will be combined; (2) the key statistical approaches for meta-analysis; and (3) the corresponding estimates, inferences, and decisions that arise from a meta-analysis. First, it describes common statistical models used in ecological meta-analyses and the relationships between these models, showing how they are all variations of the same general structure. It then discusses the three main approaches to analysis and inference, again with the aim of providing a general understanding of these methods. Finally, it briefly considers a number of statistical considerations which arise in meta-analysis. In order to illustrate the concepts described, the chapter considers the Lepidoptera mating example described in Appendix 8.1. This is a meta-analysis of 25 studies of the association between male mating history and female fecundity in Lepidoptera.

This introductory chapter first outlines the book's main objectives. These are to provide students with a basic understanding of the role that statistics can play in scientific research; to introduce students to the core ideas in experimental design, statistical inference, and statistical modelling; and to prepare students for further reading, or for more specialized courses appropriate to their particular areas of research. It then explains the role of statistics in scientific method followed by an overview of the subsequent chapters.

In many settings, a random sample may not be cost‐inefficient, so that a two‐stage stratified random sample is adopted in contingent valuation (CV) studies. The first stage comprises identifying a number of enumeration areas from which to sample households in the second stage. The Monte Carlo simulation, from a CV case study, suggests that increasing the second‐stage sample size within a limited number of enumeration areas selected at the first stage of sampling will result in a greater return in statistical and sampling design efficiency than will increasing the number of enumeration areas with a limited second‐stage sampling size. In both approaches, the marginal return diminishes as the number of sampling units increases.

Ecological studies are often hampered by insufficient data on the quantity of interest. Limited data usually lead to a relatively flat likelihood surface that is not very informative. One solution is to augment the available data by incorporating other possible sources of information. A wealth of information in the form of “soft” data, such as expert opinion about whether pollutant concentration exceeds a certain threshold, may be available. This chapter proposes a mechanism to incorporate such soft information and expert opinion in the process of inference. A commonly used approach for incorporating expert opinion in statistical inference is via the Bayesian paradigm. This chapter discusses various difficulties associated with the Bayesian approach. It introduces the idea of eliciting data instead of priors and examines its practicality. It then describes a general hierarchical model setup for combining elicited data and the observed data. It illustrates the effectiveness of this method for presence-absence data using simulations. The chapter demonstrates that incorporating expert opinion via elicited data substantially improves the estimation, prediction, and design aspects of statistical inference for spatial data.