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This chapter introduces and applies the concept of parallel spatial and temporal unmixing with group independent component analysis (ICA) for concurrent electroencephalography-functional magnetic resonance imaging (EEG-fMRI). Hemodynamic response function (HRF) deconvolution and single-trial estimation in the fMRI data were employed, and the single-trial weights were used as predictors for the amplitude modulation in the EEG. For illustration, data from a previously published performance-monitoring experiment were analyzed, in order to identify error-preceding activity in the EEG modality. EEG components that displayed such slow trends, and which were coupled to the corresponding fMRI components, are described. Parallel ICA for analysis of concurrent EEG-fMRI on a trial-by-trial basis is a very useful addition to the toolbelt of researchers interested in multimodal integration.

Variability is a ubiquitous characteristic in even highly skilled performance, and it can serve as a useful window into the determinants of skill acquisition and control. Variability is specifically informative when a task is redundant (i.e., the same result can be obtained in many different ways). This chapter discusses a novel analysis technique which has been developed for redundant tasks that parses observed variability into three components: tolerance, noise, and covariation. Three experiments addressed the following questions: What aspects of variability decrease with practice? Are actors sensitive to their intrinsic noise in selecting strategies? For all experiments, a throwing task served as the model system. Using a virtual set-up, subjects threw a pendular projectile in a simulated concentric force field to hit a target. The movement was experimentally constrained such that only two variables, angle and velocity of ball release, fully determined the projectile's trajectory and thereby the accuracy of the throw. While leaving the task redundant, this simplification facilitated analysis and decomposition of variability.

This chapter reviews evidence showing that people are remarkably sensitive to the rarity of events when making inferences about them. Indeed, people are so attuned to event rarity that their implicit assumptions about rarity guide their thinking even in laboratory tasks where experimenters have implicitly assumed that rarity would not matter. Lack of awareness of this problem has led many experimenters to misinterpret people’s adaptive responses as irrational. Indeed, focusing on data that are rare leads people to behave in a qualitatively Bayesian manner. These points are illustrated using tasks that involve assessing the covariation between variables, evaluating hypotheses after passively receiving data, and actively searching for data to test hypotheses. Participants’ sensitivity to, and assumptions about, rarity have important implications for understanding lay inference.

Causal induction has two components: learning about the structure of causal models and learning about causal strength and other quantitative parameters. This chapter argues for several interconnected theses. First, people represent causal knowledge qualitatively, in terms of causal structure; quantitative knowledge is derivative. Second, people use a variety of cues to infer causal structure aside from statistical data (e.g. temporal order, intervention, coherence with prior knowledge). Third, once a structural model is hypothesized, subsequent statistical data are used to confirm, refute, or elaborate the model. Fourth, people are limited in the number and complexity of causal models that they can hold in mind to test, but they can separately learn and then integrate simple models, and revise models by adding and removing single links. Finally, current computational models of learning need further development before they can be applied to human learning.

Many models of causal induction are based on covariation information, which depicts whether the presence or absence of an event co-occurs with the presence or absence of another event. In all covariation-based models of causal induction, events that are classified as the same type play an identical role throughout learning. This chapter reviews three sets of studies demonstrating that people treat the same type of evidence differently depending on at what point during learning the evidence is presented. The major thesis is that people develop a hypothesis about causal relations based on a few pieces of initial evidence and interpret the subsequent data in light of this hypothesis. Thus, depending on what the initial hypothesis is and when the data are presented, the identical data can play different roles. Such dynamic interpretations of data result in the primacy effect, varying inferences about unobserved, alternative causes, and the context-dependent interpretations of ambiguous stimuli.

This chapter provides the first empirical test of the claim that people recover meaning from covariation alone. It examines two theories that feature covariation as an important component of learning the meaning of concepts — Launder and Dumais 1997; Rogers et al. 2004 — followed by reasons to question a reliance on covariation alone. It then presents the results of three experiments that demonstrate limits on how much meaning can be recovered from covariation alone. Given these limits, it discusses an issue central to the debate — what kind of data supports the notion that arbitrary, abstract, amodal (AAA) symbols play a role in cognition.

In this chapter, the Scientific Reasoning subtest is described. The reason for each of the task-types chosen for this subtest is discussed. The skills tapped by this subtest include: covariation detection; falsification tendencies in the four-card selection task; understanding the logic of converging evidence; the ability to avoid drawing causal inferences from correlation evidence; the tendency to accurately assess the likelihood ratio by processing P(D/~H); and the tendency to use control-group reasoning. A large study of this subtest is described and correlations with cognitive ability and thinking dispositions are presented, as well as correlations with some other subtests of the CART.

This chapter focuses on dispersal syndromes and how they describe patterns of covariation of morphological, behavioural, and/or life-history traits associated with dispersal. Covariation is a continuous measure of statistical association between traits composing a complex phenotype. There are four main reasons why this chapter is interested in covariation: firstly, syndromes may help us predict a priori, from the observation of dispersal phenotypes, the intensity, nature, and modalities of movement. Secondly, they have the potential to provide information regarding the mechanistic determinants of dispersal and the constraints associated with movement. Thirdly, they may provide information about the proximate motivations and ultimate causes of dispersal. Lastly, and most convincingly, the patterns of covariation between traits associated with dispersal will critically affect both the demographic and genetic consequences of movement.

A number of authors have observed that definite descriptions can havecovarying interpretations. In fact they can appear in positions, andyield up interpretations, associated with both c-commanded boundvariable pronouns and donkey pronouns. This chapter shows how thesemantic system introduced in Chapter Two accounts for these readingswith no further ado. Both donkey sentences formed with relative clausesand donkey sentences formed with conditionals are analysed. In contrastto previous work in this area, both c-commanded bound variable readingsand donkey anaphoric readings are accounted for by having the situationpronouns associated with definite descriptions bound by operators in thesyntax analogous to the syntactic lambda operator of Heim and Kratzer.

The chapter begins with a brief overview of life-history trait variability among species at a coarse resolution of phylogenetic affinity before drilling down into variability between classes within a single subphylum (vertebrates). The chapter then unfolds with examples of how life histories can be strikingly variable among populations within the same species. Natural selection plays a dominant role in generating life-history variability within and among populations of the same species. But among species and higher-level taxonomic ranks, a considerable amount of life-history variation can be attributed to constraints. These are developmental, structural, physiological, or genetic boundaries that hinder or limit life-history expression. Evidence of one type of constraint emerges when species are unable to express trait values common in other species. A second type of constraint is evident because of the nature of trait covariation and what it can potentially say about constancy; these are termed life-history invariants. The chapter concludes with a consideration of how patterns of life-history trait covariation might evolve. The question here is whether traits covary with one another in ways that are reasonably predictable, empirically defensible, and plausibly adaptive. It is these patterns of covariation that have driven efforts to classify trait combinations in accordance with various continuums of divergence, a well-known one being that which distinguishes r- from K-selection.

Brownian motion and concepts of the Itôs calculus are explained, including total variation, quadratic variation, Levy’s characterization of Brownian motion, the Itô integral, the difference between martingales and local martingales, the martingale (predictable) representation theorem , Itô’s formula (Itô’s lemma), geometric Brownian motion, covariation (joint variation) processes, the relationship between variance and expected quadratic variation, the relationship between covariance and expected covariation, and rotations of Brownian motions.

Interests in how variables may relate to each other and how systems of relationships among variables may be at play often underlie the questions music education researchers pose. This chapter describes basic design and analysis considerations in research that involves the systematic investigation of whether and how variables are related; in other words, correlational research. The chapter poses correlational research as an extension of the book’s previous discussion of descriptive research. The chapter briefly describes the role of correlational studies in advancing theory, presents several issues to consider when designing studies, and provides an introduction to correlation as a statistical concept.

Miles Hewstone describes his most underappreciated work: a paper with Jos Jaspars entitled “Covariation and causal attribution: A logical model of the intuitive analysis of variance.” Published in 1987 in the Journal of Personality and Social Psychology, the paper explored how lay perceivers combine different aspects of information in reaching a “causal attribution”; that is, how they decided to attribute a known effect to one or more possible causes. The model proposed by Hewstone and Jaspars was the first to emphasize that it was patterns of consensus, consistency, and distinctiveness that mattered, not main effects. Hewstone reflects on the lessons that may be learned from his story.