The key term of this book is, admittedly, a definitional morass. To refer to a work as a “case study” might mean: (a) that its method is qualitative, small-N, (b) that the research is holistic, thick (a more or less comprehensive examination of a phenomenon), (c) that it utilizes a particular type of evidence (e.g., ethnographic, clinical, nonexperimental, non-survey-based, participant-observation, process-tracing, historical, textual, or field research), (d) that its method of evidence gathering is naturalistic (a “real-life context”), (e) that the topic is diffuse (case and context are difficult to distinguish), (f) that it employs triangulation (“multiple sources of evidence”), (g) that the research investigates the properties of a single observation, or (h) that the research investigates the properties of a single phenomenon, instance, or example.

Evidently, researchers have many things in mind when they talk about case study research. Confusion is compounded by the existence of a large number of near-synonyms – single unit, single subject, single case, N=1, case-based, case-control, case history, case method, case record, case work, within-case, clinical research, and so forth. As a result of this profusion of terms and meanings, proponents and opponents of the case study marshal a wide range of arguments but do not seem any closer to agreement than when this debate was first broached several decades ago. Jennifer Platt notes that “much case study theorizing has been conceptually confused, because too many different themes have been packed into the idea ‘case study.’”

Case study analysis focuses on a small number of cases that are expected to provide insight into a causal relationship across a larger population of cases. This presents the researcher with a formidable problem of case selection. Which cases should be chosen?

In large-sample research, case selection is usually handled by some version of randomization. If a sample consists of a large enough number of independent random draws, the selected cases are likely to be fairly representative of the overall population on any given variable. Furthermore, if cases in the population are distributed homogeneously across the ranges of the key variables, then it is probable that some cases will be included from each important segment of those ranges, thus providing sufficient leverage for causal analysis. (For situations in which cases with theoretically relevant values of the variables are rare, a stratified sample that oversamples some subset of the population may be employed.)

A demonstration of the fact that random sampling is likely to produce a representative sample is shown in Figure 5.1, a histogram of the mean values of 500 random samples, each consisting of 1,000 cases. For each case, one variable has been measured: a continuous variable that falls somewhere between zero and one. In the population, the mean value of this variable is 0.5. How representative are the random samples? One good way of judging this is to compare the means of each of the 500 random samples to the population mean.

Before entering into a discussion of specific research design techniques, it is important to insert a preliminary discussion of several factors that overshadow all research design issues in case study work. These include evidence-gathering techniques, the formulation of a hypothesis, degrees of falsifiability, the tension between particularizing and generalizing objectives in a case study, the identification of a population that the case study purports to represent, and the importance of cross-level research. Although these six issues affect all empirical work in the social sciences, they are particularly confusing in the context of case study work, and consequently merit our close attention.

The Evidence

The case study, I argued in Chapter Two, should not be defined by a distinctive method of data collection but rather by the goals of the research relative to the scope of the research terrain. Evidence for a case study may be drawn from an existing dataset or set of texts or may be the product of original research by the investigator. Written sources may be primary or secondary. Evidence may be quantitative, qualitative, or a mixture of both – as when qualitative observations are coded numerically so as to create a quantitative variable. Evidence may be drawn from experiments (discussed in Chapter Six), from “ethnographic” field research, from unstructured interviews, or from highly structured surveys.

Let us suppose that one has chosen one's case (or cases) according to one of the techniques (or some combination of techniques) described in the previous chapter. And let us further suppose that one has refined one's research question into a specific (X1/Y) hypothesis. One then faces a problem of internal validity. How does one construct a research design that might illuminate the causal relationship of interest?

The fundamental problem of causal inference is that one cannot rerun history to see what effects X1 actually had on Y in a particular case. At an ontological level, this problem is unsolvable. There are no time machines. However, there are various ways of reducing uncertainty so that causal inference is possible, and indeed quite plausible. The argument of this chapter is that the various methods of doing so are all quasi-experimental in nature. This is because the true experiment is the closest approximation we have at our disposal to a time machine. Through this technique, and others modeled on it, one can imagine what it would be like to go back in time, alter a “treatment,” and observe its true causal effect.

This chapter thus calls into question some of the usual assumptions applied to case study research. Most case study researchers perceive only a distant and tenuous connection between their work and the laboratory experiment, with a manipulated treatment and randomized control.

Objectives

This chapter describes the means by which we label and treat known and unknown values. Basically there are two types of observable data, and the abstract terminology for yet-to-be observed values should also reflect this distinction. We first talk here about the levels of measurement for observed values where the primary distinction is discrete versus continuous. We will then see that the probability functions used to describe the distribution of such variables preserves this distinction. Many of the topics here lead to the use of statistical analysis in the social sciences.

Levels of Measurement

It is important to classify data by the precision of measurement. Usually in the social sciences this is an inflexible condition because many times we must take data “as is” from some collecting source. The key distinction is between discrete data, which take on a set of categorical values, and continuous data, which take on values over the real number line (or some bounded subset of it). The difference can be subtle. While discreteness requires countability, it can be infinitely countable, such as the set of positive integers. In contrast, a continuous random variable takes on uncountably infinite values, even if only in some range of the real number line, like [0 :1], because any interval of the real line, finitely bounded or otherwise, contains an infinite number of rational and irrational numbers.

To see why this is an uncountably infinite set, consider any two points on the real number line. It is always possible to find a third point between them. Now consider finding a point that lies between the first point and this new point; another easy task.

Objectives

This chapter introduces more theoretical and abstract properties of vectors and matrices. We already (by now!) know the mechanics of manipulating these forms, and it is important to carry on to a deeper understanding of the properties asserted by specific row and column formations. The last chapter gave some of the algebraic basics of matrix manipulation, but this is really insufficient for understanding the full scope of linear algebra. Importantly, there are characteristics of a matrix that are not immediately obvious from just looking at its elements and dimension. The structure of a given matrix depends not only on the arrangement of numbers within its rectangular arrangement, but also on the relationship between these elements and the “size” of the matrix. The idea of size is left vague for the moment, but we will shortly see that there are some very specific ways to claim size for matrices, and these have important theoretical properties that define how a matrix works with other structures. This chapter demonstrates some of these properties by providing information about the internal dynamics of matrix structure. Some of these topics are a bit more abstract than those in the last chapter.

Space and Time

We have already discussed basic Euclidean geometric systems in Chapter 1. Recall that Cartesian coordinate systems define real-measured axes whereby points are uniquely defined in the subsequent space. So in a Cartesian plane defined by ℜ2, points define an ordered pair designating a unique position on this 2-space. Similarly, an ordered triple defines a unique point in ℜ3 3-space. Examples of these are given in Figure 4.1.

Objectives

This chapter introduces an underappreciated topic in the social sciences that lies directly in the intersection of matrix algebra (Chapters 3 and 4) and probability (Chapter 7). It is an interesting area in its own right with many applications in the social sciences, but it is also a nice reinforcement of important principles we have already covered. Essentially the idea is relevant to the things we study in the social sciences because Markov chains specifically condition on the current status of events. Researchers find that this is a nice way to describe individual human decision making and collective human determinations.

So Markov chains are very practical and useful. They model how social and physical phenomena move from one state to another. The first part of this chapter introduces the mechanics of Markov chains through the kernel. This is the defining mechanism that “moves” the Markov chain around. The second part of the chapter describes various properties of Markov chains and how such chains can differ in how they behave. The first few properties are elementary, and the last few properties are noticeably more advanced and may be skipped by the reader who only wants an introduction.

Defining Stochastic Processes and Markov Chains

Markov chains sound like an exotic mathematical idea, but actually the principle is quite simple. Suppose that your decision-making process is based only on a current state of affairs. For example, in a casino wagering decisions are usually dictated only by the current state of the gambler: the immediate confronting decision (which number to bet on, whether to take another card, etc.) and the available amount of money.

Objectives (the Width of a Circle)

This chapter introduces the basic principles of analytic geometry and trigonometry specifically. These subjects come up in social science research in seemingly surprising ways. Even if one is not studying some spatial phenomenon, such functions and rules can still be relevant. We will also expand beyond Cartesian coordinates and look at polar coordinate systems. At the end of the day, understanding trigonometric functions comes down to understanding their basis in triangles.

Radian Measurement and Polar Coordinates

So far we have only used Cartesian coordinates when discussing coordinate systems. There is a second system that can be employed when it is convenient to think in terms of a movement around a circle. Radian measurement treats the angular distance around the center of a circle (also called the pole or origin for obvious reasons) in the counterclockwise direction as a proportion of 2π.

Most people are comfortable with another measure of angles, degrees, which are measured from 0 to 360. However, this systemis arbitrary (although ancient) whereas radian measurement is based on the formula for the circumference of a circle: c = 2πr, where r is the radius. If we assume a unit radius (r = 1), then the linkage is obvious. That is, from a starting point, moving 2π around the circle (a complete revolution) returns us to the radial point where we began. So 2π is equal to 360° in this context (more specifically for the unit circle described below).