Simulation of random variables is important in applied statistics for several reasons. First, we use probability models to mimic variation in the world, and the tools of simulation can help us better understand how this variation plays out. Second, we can use simulation to approximate the sampling distribution of data and propagate this to the sampling distribution of statistical estimates and procedures. Third, regression models are not deterministic; they produce probabilistic predictions. Simulation is the most convenient and general way to represent uncertainties in forecasts.

Chapter 3 reviews fundamental physical concepts that contribute to understanding the development of geologic structures. We begin by defining the units and dimensions of physical quantities encountered in structural geology. We point out that equations composed of these quantities must have consistent units and dimensions to be part of a valid explanation of a tectonic process. Next, we introduce the concept of a material continuum, and describe displacement and stress fields that demonstrate the continuum is an effective way to idealize rock at length scales from nanometers to tens of kilometers. Then, we consider the conservation laws for mass, momentum, and energy. We use them to derive the fundamental equations of continuity, motion, and heat transport in a material continuum. These equations underlie the three different styles of rock deformation and the canonical models for the five categories of geologic structures.

As discussed in Chapter 1, regression is fundamentally a technology for predicting an outcome y from inputs x1, x2, . . . . In this chapter we introduce regression in the simple (but not trivial) case of a linear model predicting a continuous y from a single continuous x, thus fitting the model yi = a+bxi +errortodata(xi,yi), i=1, ..., n. We demonstrate with an applied example that includes the steps of fitting the model, displaying the data and fitted line, and interpreting the fit. We then show how to check the fitting procedure using fake-data simulation, and the chapter concludes with an explanation of how linear regression includes simple comparison as a special case.

Most of this book is devoted to examples and tools for the practical use and understanding of regression models, starting with linear regression with a single predictor and moving to multiple predictors, nonlinear models, and applications in prediction and causal inference. In this chapter, we lay out some of the mathematical structure of inference for regression models and some algebra to help you understand estimation for linear regression. We also explain the rationale for the use of the Bayesian fitting routine stan_glm and its connection to classical linear regression. This chapter thus provides background and motivation for the mathematical and computational tools used in the rest of the book.

Chapter 7 discussed the use and logic of time-series designs and how they can provide tentative causal inferences about the effectiveness of a program or policy. It also mentioned single-case designs, which are a type of time-series design that can be used not only to evaluate the outcomes of programs, but also by direct service practitioners to evaluate their own practice. This chapter will examine alternative single-case deigns, their logic for making causal inferences, and how to use them to evaluate practice and programs.

If you have taken a course on research methods you probably learned that the gold standard (ideal) design for outcome studies on the effectiveness of interventions involves randomly assigning clients to treatment versus control groups to control for threats to internal validity like history, passage of time, selectivity bias, and regression to the mean. It’s good that you learned that because such designs are the best way to determine whether a tested intervention appears to be the real cause of any observed outcomes.