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Many decisions that people are called on to make can be thought of as involving thresholds for action. In each case, we can understand the decision maker to be answering two questions: (1) How strong are the arguments in favor of taking this action? (2) How strong must the arguments be in order for me to take the action? Decision makers in court cases, whether judges or jurors, are commonly required to make this kind of decision. The aim of this chapter is to set out a framework for analyzing decisions to take action in a judicial context. We begin by outlining a general model, continue with a description of several studies of mock-juror decision making, and conclude with implications for studying judges.

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.

This chapter introduces some key concepts of statistical inference and shows their use to investigate the statistical significance of the (linear) relationships modelled through regression analysis, or to investigate the validity of the classical assumptions in simple and multiple linear regression models. The discussions cover statistical hypothesis testing in simple and multiple regression models; testing linear restrictions on regression coefficients; joint tests of linear restrictions; testing general linear restrictions; the relationship between the F test and the coefficient of multiple correlation; the joint confidence region; multicollinearity and the prediction problem; implications of mis-specification of the regression model on hypothesis testing; Jarque-Bera's test of the normality of regression residuals; the predictive failure test; the Chow test; and non-parametric estimation of the density function. Exercises are provided at the end of the chapter.

The ANPRM seeks to shift scarce regulatory resources from “low risk” studies, where they unnecessarily burden research, to studies that “pose risks of serious physical or psychological harm,” which currently suffer from insufficiently rigorous review, thereby endangering participant welfare. This chapterthree challenges for governing human subjects research by such “risk-based regulation.” First, because a study’s riskiness does not depend on the research method, discipline, or type of risk it involves, it is difficult to designate low- and high-risk studies in advance, at the level of statute or regulation. This means that most risk-based research regulation will occur by IRBs reviewing individual protocols. A second challenge, however, is that IRBs suffer from various biases in assessing risks and potential benefits, including asymmetric incentives to avoid Type I and II errors and double risk aversion. Third, even if IRBs could be rid of their biases, because prospective research participants are heterogeneous in their preferences and other circumstances, the same protocol will offer a different risk-benefit profile for different participants, further frustrating attempts at risk-based regulation. The chapter concludes by suggesting an alternative way to redistribute scarce regulatory resources that embraces, rather than ignores, all three challenges.

This chapter builds on the previous chapter by elaborating from theories of causal knowledge presented earlier to practical considerations for the design, execution, and analysis of randomized experiments and randomized controlled trials in music education research. The straightforward statistical analysis of the two-group experimental designs is explained through the t test. The analysis of variance technique is explained for the analysis of experimental and quasi-experimental data involving more than two groups. The chapter closes with a discussion of the analysis of data arising from experiments where additional data, beyond group membership and the score on an outcome measure, is known about the participants (i.e., analysis of covariance).

Having recorded a signal, it is necessary to interpret its functional significance. The way in which this is done is to relate the signal to a psychological condition. As in other branches of science, an experimental condition is contrasted with a control condition. The interpretation is clearest when these differ in just one respect, though this can be difficult to achieve. Standard statistics are used to evaluate the significance of the difference. However, the analysis of imaging data can be onerous and many methods have been developed to avoid false-positive and false-negative results. These include robust correction for the number of statistical comparisons that are made, as the image is made up of thousands of voxels across many regions. Researchers also use targeted region-of-interest analysis; in this case the region must be specified beforehand. One must also study enough subjects: if small groups are used, the study may be underpowered.