In this chapter we use our favored technique of fake-data simulation to understand a simple regression model, use a real-data example of height and earnings to warn against unwarranted causal interpretations, and discuss the historical origins of regression as it relates to comparisons and statistical adjustment.

The previous chapters described causal inference strategies that assume ignorability of exposure or treatment assignment. It is reasonable to be concerned about this assumption, however. After all, when are we really confident that we have measured all confounders? This chapter explores several alternative causal inference strategies that rely on slightly different sets of assumptions that may be more plausible in certain settings. We also discuss the relation between statistical causal inference (estimates of the effects of specified treatments or exposures) and causal explorations or searches for causes of patterns in observed data.

In the usual regression context, predictive inference relates to comparisons between units, whereas causal inference addresses comparisons of different treatments if applied to the same units. More generally, causal inference can be viewed as a special case of prediction in which the goal is to predict what would have happened under different treatment options. Causal interpretations of regression coefficients can only be justified by relying on much stronger assumptions than are needed for predictive inference. As discussed in the previous chapter, controlled experiments are ideal settings for using regression to estimate a treatment effect because the design of data collection guarantees that treatment assignment is independent of the potential outcomes, conditional on the information used in the design. This chapter illustrates the use of regression in the setting of controlled experiments, going through issues of adjustment for pre-treatment predictors, interactions, and pitfalls that can arise when building a regression using experimental data and interpreting coefficients causally.

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.