We review two foundations of statistical inference, the theory of likelihood and the Bayesian paradigm. We begin by applying principles of likelihood to generate point estimators (maximum likelihood estimators) and hypothesis tests (likelihood ratio tests). We then describe the Bayesian approach, focusing on two controversial aspects: the use of prior information and subjective probability. We illustrate these analyses using simple examples.

This chapter reviews major types of statistical resampling approaches used in paleontology. They are an increasingly popular alternative to the classic parametric approach because they can approximate behaviors of parameters that are not understood theoretically. The primary goal of most resampling methods is an empirical approximation of a sampling distribution of a statistic of interest, whether simple (mean or standard error) or more complicated (median, kurtosis, or eigenvalue). This chapter focuses on the conceptual and practical aspects of resampling methods that a user is likely to face when designing them, rather than the relevant mathematical derivations and intricate details of the statistical theory. The chapter reviews the concept of sampling distributions, outlines a generalized methodology for designing resampling methods, summarizes major types of resampling strategies, highlights some commonly used resampling protocols, and addresses various practical decisions involved in designing algorithm details. A particular emphasis has been placed here on bootstrapping, a resampling strategy used extensively in quantitative paleontological analyses, but other resampling techniques are also reviewed in detail. In addition, ad hoc and literature-based case examples are provided to illustrate virtues, limitations, and potential pitfalls of resampling methods.

Paleobiologists are reaching a consensus that biases in diversity curves, origination rates, and extinction rates need to be removed using statistical estimation methods. Diversity estimates are biased both by methods of counting and by variation in the amount of fossil data. Traditional counts are essentially tallies of age ranges. Because these counts are distorted by interrelated factors such as the Pull of the Recent and the Signor-Lipps effect, counts of taxa actually sampled within intervals should be used instead. Sampling intensity biases can be addressed with randomized subsampling of data records such as individual taxonomic occurrences or entire fossil collections. Fair subsampling would yield taxon counts that track changes in the species pool size, i.e., the diversity of all taxa that could ever be sampled. Most of the literature has overlooked this point, having instead focused on making sample sizes uniform through methods such as rarefaction. These methods flatten the data, undersampling when true diversity is high. A good solution to this problem involves the concept of frequency distribution coverage: a taxon's underlying frequency is said to be “covered” when it is represented by at least one fossil in a data set. A fair subsample, but not a uniform one, can be created by drawing collections until estimated coverage reaches a fixed target (i.e., until a “shareholder quorum” is attained). Origination and extinction rates present other challenges. For many years they were thought of in terms of simple counts or ratios, but they are now treated as exponential decay coefficients of the kind featuring in simple birth-death models. Unfortunately, these instantaneous rates also suffer from counting method biases (e.g., the Pull of the Recent). Such biases can be removed by only examining taxa sampled twice consecutively, three times consecutively, or in the first and third of three intervals but not the second (i.e., two timers, three timers, and part timers). Two similar equations involving these counts can be used. Alternative methods of estimating diversity and turnover through extrapolation share some of the advantages of quorum subsampling and two-timer family equations, but it remains to be shown whether they produce precise and accurate estimates when applied to fossil data.

We rely on observations of occurrences of fossils to infer the rates and timings of origination and extinction of taxa. These estimates can then be used to shed light on questions such as whether extinction and origination rates have been higher or lower at different times in earth history or in different geographical regions, etc. and to investigate the possible underlying causes of varying rates. An inherent problem in inference using occurrence data is one of incompleteness of sampling. Even if a taxon is present at a given time and place, we are guaranteed to detect or sample it less than 100% of the time we search in a random outcrop or sediment sample that should contain it, either because it was not preserved, it was preserved but then eroded, or because we simply did not find it. Capture-mark-recapture (CMR) methods rely on replicate sampling to allow for the simultaneous estimation of sampling probability and the parameters of interest (e.g. extinction, origination, occupancy, diversity). Here, we introduce the philosophy of CMR approaches especially as applicable to paleontological data and questions. The use of CMR is in its infancy in paleobiological applications, but the handful of studies that have used it demonstrate its utility and generality. We discuss why the use of CMR has not matched its development in other fields, such as in population ecology, as well as the importance of modelling the sampling process and estimating sampling probabilities. In addition, we suggest some potential avenues for the development of CMR applications in paleobiology.

Diversity (the variety of different types of organisms) of an ecological or paleoecological system reflects processes and history operating across a range of hierarchically related scales. For example, the diversity of a biofacies is the sum of the diversity in all the local patches composing the biofacies, the diversity of a depositional system is composed of all the biofacies composing the depositional system, and the diversity of a biotic province is composed of all the landscapes composing the province. Diversity at a larger scale (γ-diversity) incorporates both the average inventory diversity of units of the next smaller scale (α-diversity) and the compositional differences, or differentiation diversity, among the smaller units (β-diversity). Many familiar means of measuring diversity can be mathematically partitioned to determine the relative contribution of different diversity components at any hierarchical level. When using richness (the number of taxa in an ecological system) as a measurement of diversity, it is necessary to use rarefaction to correct for differences in sample size. The divergence between sample-based and individual-based rarefaction curves of a composite collection (γ-diversity) incorporating all the samples (α-diversity) contributing to a given hierarchical level reflects the degree of non-random compositional difference among the smaller scale units (β-diversity). Alternatively, Shannon's entropy can be partitioned additively: β-entropy equals γ-entropy (based on a composite sample) minus average α-entropy of the constituent samples. A useful property of entropy is that it can be converted to effective richness, the number of taxa that would result in the same entropy value if all were equally abundant. Effective richness can be thought of as a unit conversion from non-intuitive entropy units to more easily understood richness units. Effective richness derived from Shannon's entropy partitions diversity multiplicatively – i.e., β-diversity is the number of compositionally distinct smaller units that contribute to the total diversity at the higher level. Diversity partitioning is rapidly becoming adopted as a tool for directly addressing how the structure of higher-level ecological and paleoecological systems reflects interactions among lower-level units in response to environmental and evolutionary changes.

Macroecology is a rapidly growing sub-discipline within ecology that is concerned with characterizing statistical patterns of species' abundance, distribution and diversity at spatial and temporal scales typically ignored by traditional ecology. Both macroecology and paleoecology are concerned with answering similar questions (e.g., understanding the factors that influence geographic ranges, or the way that species assemble into communities). As such, macroecological methods easily lend themselves to many paleoecological questions. Moreover, it is possible to estimate the variables of interest to macroecologists (e.g., body size, geographic range size, abundance, diversity) using fossil data. Here we describe the measurement and estimation of the variables used in macroecological studies and potential biases introduced by using fossil data. Next we describe the methods used to analyze macroecological patterns and briefly discuss the current understanding of these patterns. This chapter is by no means an exhaustive review of macroecology and its methods. Instead, it is an introduction to macroecology that we hope will spur innovation in the application of macroecology to the study of the fossil record.

Food webs represent trophic interactions among species in communities. Those interactions both structure and are structured by species richness, ecological diversity, and evolutionary processes. Geological and macroevolutionary timescales are therefore important to the understanding of food web dynamics, and there is a need for the consideration of paleocommunity food webs. The fossil record presents challenges in this regard, but the problem can be approached with combinatoric analysis and network theory. This paper is an introduction to the aspects of those disciplines relevant to the study of paleo-food webs, and explores a probabilistic and numerical approach.

Landmark-based geometric morphometrics is a powerful approach to quantifying biological shape, shape variation, and covariation of shape with other biotic or abiotic variables or factors. The resulting graphical representations of shape differences are visually appealing and intuitive. This paper serves as an introduction to common exploratory and confirmatory techniques in landmark-based geometric morphometrics. The issues most frequently faced by (paleo)biologists conducting studies of comparative morphology are covered. Acquisition of landmark and semilandmark data is discussed. There are several methods for superimposing landmark configurations, differing in how and in the degree to which among-configuration differences in location, scale, and size are removed. Partial Procrustes superimposition is the most widely used superimposition method and forms the basis for many subsequent operations in geometric morphometrics. Shape variation among superimposed configurations can be visualized as a scatter plot of landmark coordinates, as vectors of landmark displacement, as a thin-plate spline deformation grid, or through a principal components analysis of landmark coordinates or warp scores. The amount of difference in shape between two configurations can be quantified as the partial Procrustes distance; and shape variation within a sample can be quantified as the average partial Procrustes distance from the sample mean. Statistical testing of difference in mean shape between samples using warp scores as variables can be achieved through a standard Hotelling's T2 test, MANOVA, or canonical variates analysis (CVA). A nonparametric equivalent to MANOVA or Goodall's F-test can be used in analysis of Procrustes coordinates or Procrustes distance, respectively. CVA can also be used to determine the confidence with which a priori specimen classification is supported by shape data, and to assign unclassified specimens to pre-defined groups (assuming that the specimen actually belongs in one of the pre-defined groups).

Examples involving Cambrian olenelloid trilobites are used to illustrate how the various techniques work and their practical application to data. Mathematical details of the techniques are provided as supplemental online material. A guide to conducting the analyses in the free Integrated Morphometrics Package software is provided in the appendix.

Quantitative phylogenetic inference estimates the probability of observed character distributions given trees and rates. Most available programs for doing this assume (tacitly or explicitly) that the sampled taxa are contemporaneous. However, paleontologists usually sample taxa over a clade's history. Thus, we must estimate the probability of observed character-state distributions over time given trees and rates. When we include information about sampling intensity, then we really are estimating the probability of the observed record given trees and rates. Some additional problems that should be issues for neontologists, but which are much more obvious concerns for paleontologists include: 1) ancestor-descendant relationships; 2) punctuated versus continuous morphological change; and, 3) the effects of extinction and speciation rates on prior probabilities of trees. Future goals of paleosystematists include incorporating these and other “nuisance” parameters so that, ultimately, our tests of phylogeny are really tests of evolutionary histories.

Morphological integration and modularity are closely related concepts about how different traits of an organism are correlated. Integration is the overall pattern of intercorrelation; modularity is the partitioning of integration into evolutionarily or developmentally independent blocks of traits. Modularity and integration are usually studied using quantitative phenotypic data, which can be obtained either from extant or fossil organisms. Many methods are now available to study integration and modularity, all of which involve the analysis of patterns found in trait correlation or covariance matrices. We review matrix correlation, random skewers, fluctuating asymmetry, cluster analysis, Euclidean distance matrix analysis (EDMA), graphical modelling, two-block partial least squares, RV coefficients, and theoretical matrix modelling and discuss their similarities and differences. We also review different coefficients that are used to measure correlations. We apply all the methods to cranial landmark data from and ontogenetic series of Japanese macaques, Macaca fuscata to illustrate the methods and their individual strengths and weaknesses. We conclude that the exploratory approaches (cluster analyses of various sorts) were less informative and less consistent with one another than were the results of model testing or comparative approaches. Nevertheless, we found that competing models of modularity and integration are often similar enough that they are not statistically distinguishable; we expect, therefore, that several models will often be significantly correlated with observed data.

In this chapter we discuss methods for analyzing continuous traits, with an emphasis on those approaches that rely on explicit statistical models of evolution and incorporate genealogical information (ancestor–descendant or phylogenetic relationships). After discussing the roles of models and genealogy in evolutionary inference, we summarize the properties of commonly used models including random walks (Brownian motion), directional evolution, and stasis. These models can be used to devise null-hypothesis tests about evolutionary patterns, but it is often better to fit and compare models equally using information criteria and related approaches. We apply these methods to a published data set of dental measurements in a sequence of ancestor–descendant populations in the early primate Cantius, with the particular goal of determining the best-supported mode of evolutionary change in this lineage. We also assess a series of questions about the evolution of femoral dimensions in several clades of dinosaurs, including testing for a trend of increasing body size (Cope's Rule), testing for correlations among characters, and reconstructing ancestral states. Finally, we list briefly some additional models, approaches, and issues that arise in genealogically informed analyses of phenotypic evolution.

Computers excel at applying simple, logical rules to prodigious amounts of information. Such is the nature of biochronology. Range charts of first- and last-occurrences of fossil species must be combined from many locations to compensate for local incompleteness of the fossil record. Enlarging the geographic scope adds the complications of faunal migration and provinciality – for which the remedy is yet more information. Expert biostratigraphers have managed to divide Phanerozoic time into hundreds of biozones by limiting the amount of information they consider. A set of biozones specifies the sequence of only a fraction of available species, typically from a single clade in a particular province across a limited time interval. Once the human expertise applied to this task is rendered into logical algorithms, computers can extend the exercise to huge data sets of otherwise unmanageable scope.

Two factors make this computer-assisted sequencing of first- and last-appearance events easy to understand and implement: the biostratigraphers' ground rules are straightforward and the computations, although tediously repetitive, proceed by simply analogy rather than esoteric mathematics. Two other factors force the outcome to be a set of time lines that fit the field data equally well: there is rarely enough information to identify a unique best-fit solution and there is more than one set of expert ground-rules for measuring the fitness of paleobiologic time-lines. A set of equally well-fit time lines serves as an appropriate statement of uncertainty in the order of events. Mapping the local ranges back into a best-fit composite range can reveal biogeographic and taxonomic complications; quality control and interpretation advance together.

One of the many contributions paleontology makes to our understanding of the biosphere and its evolution is a direct temporal record of biotic events. However, assuming fossils have been correctly identified and accurately dated, stratigraphic ranges underestimate true temporal ranges: observed first occurrences are too young, and observed last occurrences are too old. Here I introduce the techniques developed for placing confidence intervals on the end-points of stratigraphic ranges. I begin with the analysis of single taxa in local sections – with the simplest of assumptions – random fossilization. This is followed by a discussion of the methods developed to handle the fact that the recovery of fossils is often non-random in space and time. After discussion of how confidence intervals can be used to test for simultaneous origination and extinctions, I conclude with an example application of confidence intervals to unravel the relative importance of background extinction, environmental change and mass extinction of ammonite species at the end of the Cretaceous in western Tethys.