SIGNAL DETECTION THEORY

The first topic we will cover in this chapter is signal detection theory (SDT). Let's begin by considering an area of psychology closely associated with SDT – psychophysics. The first psychological experiments ever conducted were in psychophysics. Psychophysics describes how physical stimuli, such as light or sound waves, are translated into subjective impressions – what we see and hear. For example, we know that the wavelength of light determines the color we perceive.

But our subjective impressions do not map onto physical stimuli in a one-to-one fashion. There are several variables that can affect our subjective impression. Take the physical environment. Imagine that you are listening to another person at a cocktail party, and the party grows so loud that you have a difficult time hearing the person. This is an example of the physical environment interfering with your perception. The neural structure of the perceiver can make a difference as well; if the person has some sort of neural damage, it can interfere with or alter his or her perceptions of a physical stimulus. People who are red-green color-blind do not have the subjective impressions of green or red. Finally, the motivation of the perceiver can make a difference, especially when a task is difficult. Imagine that you are looking in a Where's Waldo puzzle.

Let's begin by discussing the exercise in the previous chapter. As you learned if you read the Festinger and Carlsmith (1959) article, cognitive dissonance theory more accurately predicted the results of the experiment. And you may have noticed that in order to evaluate these theories and their predictions, you had to make assumptions about the difference between asking the subject to do something and requiring him or her to do it.

WORKING WITH THE THEORY OF BEHAVIORISM

First, what does reinforcement theory say about the difference between asking and requiring? Answer: Nothing in reinforcement theory says anything about the difference between asking and requiring a subject to do something. You might think it's reasonable to assume that if you're asked, you'll write a higher-quality essay than if you're required to do it – that being asked provides a greater incentive. That's a reasonable assumption, though it is certainly not a part of reinforcement theory. Reinforcement theory just says that when a behavior is observed and then reinforced, there will be a larger effect if the reinforcement is larger, repeated, or more intense. There are different qualities of reinforcement that will increase or decrease its impact, but the theory only refers to the quality of reinforcement given the behavior. The difference between being asked and being required concerns the behavior itself prior to reinforcement. What you'd say about the behavior itself is not part of reinforcement theory, though it could be part of other kinds of learning theories.

QUANTIFICATION AND THE RISE OF THE WEST

The origins of quantification in social thought reach far back into history. Numbers are found in the classical Greek texts as well as in traditional Indian, Chinese, and Islamic texts, but the sort of quantitative analysis that we are particularly interested in is a European invention that has flourished in the past few centuries and whose roots go back to the Middle Ages. I'll be referring to an interesting work by Alfred Crosby (1997), who asks, why do we count, and do Europeans count differently from everybody else? Did they start at a different time than everybody else, and if so, why? Crosby studies quantification in Western society from 1250 to 1600, and he incidentally sheds light on another of the great debates in history: Why did Europe conquer or expand over the rest of the world? It should not have occurred. Europe in 1500 was not the wealthiest area in the world, nor was it the most advanced area economically, nor did it have the most advanced technology at the time. It was in fact a relatively backward zone of world civilization. Nonetheless, Europeans conquered the world's sea lanes and became the dominant world powers until the twentieth century. Europeans did not merely touch on areas of the world unknown to them and then move on. After all, the Chinese had junks trading on the East African coast long before the Portuguese reached the West African coast.

OVERVIEW

What is a good theory, and what are its characteristics? In this part of the book, we'll discuss several theories in psychology, along with methods for using and evaluating formal theories to understand experimental results. We'll be focusing in large part on the theories rather than on specific methods of data analysis. If you want to understand research methodology, you can't really succeed without first understanding what you're trying to accomplish by using these methods, and this requires an understanding of theories and how they work.

Whether the theory is implicit or explicit, as you're conducting research you have something in mind that you're trying to observe or demonstrate. In other words, you're typically trying to find a pattern in the data that you can make sense of. Your purpose, if possible, is to make some more general inferences from your specific data. So, you hope that your end result will be some general rule or description. In a way, you can think about statistics as a method for finding some order in your data, but that's not what comes first. You begin the process of conducting research by formulating a question that you want to answer, and this question will be grounded in a particular theory (either your own or someone else's). This part on psychology begins with a discussion of theory in an abstract sense. What are theories? What makes a theory good or bad?

We conclude the first part of this book with a critical overview of an application of mathematical theory to a problem in history – a game-theoretic model that was developed more than twenty years ago to describe some surprising behavior in the trenches in the First World War. As we shall see, the model is appealing because it seems to bring some sense to a scary and confusing subject. However, ultimately, I find this particular analysis unconvincing, and I'll talk about why such a model, if wrong, was so persuasive. This is an important step in social science analysis, inside or outside the academic setting: If you're going to claim that you're right and other people are wrong, you'd better also explain why those other people (who were generally not fools) got it wrong. Scholarly journals are littered with proudly “counterintuitive” findings that are counterintuitive for the simple reason that they are wrong and most people's intuition is right.

THE PRISONER'S DILEMMA: A GAME-THEORETIC MODEL APPLIED TO SOCIAL INTERACTIONS

Mathematical theory can be a powerful tool for understanding social phenomena but can also mislead, as I'll illustrate with an example from my undergraduate thesis, which was in political science. At the beginning of my senior year in college, I reviewed a list of potential advisors and found one who worked on game theory – I was a physics and mathematics major, minoring in political science, and game theory seemed like a good topic.

PROPENSITY-SCORE MATCHING

In order to apply the potential-outcome framework to get causal estimates that don't depend too strongly on untestable assumptions, we first need to make sure that the distributions of the treatment and control groups are balanced. This means, in other words, that we need to make sure that we are comparing apples with apples. To do so, we need to match those units that receive the treatment and those that do not receive the treatment, using a number of covariates (X). Going back to our example in Chapter 21, we need to find households that are identical in all possible, pre-treatment aspects (income, education, health, number of siblings, geographical region of origin, etc.) but that differ in their migratory experience. This procedure would create a smaller dataset with only the matched households. Once we accomplish this, we just need to estimate the average difference in means (E(Yγ − Yγ′) = E(Yγ) − E(Yγ′)) to find the impact of migration on children's emotional state. The life of an applied researcher, however, is not that easy. The introduction of a significant number of covariates (X) such as income, education, health, number of siblings, geographical region, and so on, makes it very difficult to match treated and control households. For example, if we match two households on income, then we are probably going to unmatch them on another dimension, such as number of siblings. Therefore, matching on a large number of covariates creates a high-dimensionality problem (Dehejia 2004).

My studies of the riots left me wanting to investigate fluctuations over time in the rate of collective violence. As noted in Chapter 12, this could not be done with the riot data; I therefore chose to examine an altogether different kind of collective violence, one that occurred at the turn of the twentieth century: lynchings of black Americans. The phenomenon of lynching speaks to the brutal dark side of American southern culture; some 5,000 persons were murdered in this manner. The data on lynchings have a rich history in social science research, in large part because they provide a long time series of a rather infamous nature. There are published records covering the period 1882 to 1966. Moreover, when we look at the time series, we see that it is much more complicated than the one for the riots of the 1960s. This complexity makes it easier to choose among competing explanations of the lynching rate. With these data, our focus is not on the idiosyncratic causes of particular lynchings, but on accounts of the varying rate over time. If we're willing to work with explanations that are likely to hold over a lengthy interval, then our time-series study might yield findings of value. To put it another way, with data covering a long period of time, there should be few plausible explanations that are consistent with the complexity of the time series.

I want now to turn to a very different sort of issue. To this point, we have considered the behavior of individuals in social settings that are not constrained by formal rules. That is, there are no established rules about how one should behave in a riot or as part of a lynching party. Sociologists, however, spend much of their time examining the structures of formal organizations – schools, churches, workplaces, and the like – and studying the behavior of individuals in these settings. In this chapter I discuss such an investigation, one that examined the structural features of a large corporation and tried to assess how its structural arrangements have differentially influenced the advancement prospects and attainments of male and female employees. The data come from an American insurance company that, a few years ago, gave me access to the records of its employees, which numbered around 16,000 at any point in time. These data cover an eight-year period during the 1970s. Because of employee turnover, I have career history information for some 25,000 workers.

We need, first, to define what we mean by advancement. It could mean a salary increase. That's certainly one measure of getting ahead. It could also mean promotion or a combination of promotion and salary increase. I decided to focus on promotion since I was interested in questions of organizational structure, and promotion is a concept that is linked to movement across structural units. But we still have to specify what constitutes a promotion.

In this chapter, I return to economic time series with two examples in detail. First, we'll look at an article by Cumby and Mishkin (1986), “The International Linkage of Real Interest Rates: The European–U.S. Connection.” It makes some very important points and uses some of the methods we have talked about. What is the motivation for this article? There is a very important variable in the economics of the financial markets that our theories indicate should have great influence on the economy, but it is a variable that we cannot directly observe. The variable is the expected or ex-ante real rate of interest (denoted hereafter by rrth, that is, the expectation at time t of what the real rate of return to holding an h-period bond from time t to time t + h will be). Economists have shown that the rate of interest that is most relevant for borrowing and lending decisions is the expected real rate of interest on the loan. What is the expected real rate of interest? Basically, it is the cost of borrowing money once we take into account the rate of inflation that will prevail over the life of the loan. Inflation is just the percentage rise in the money price of goods and services while the funds are out on loan. In many loans, the stated or nominal rate of interest is known up front.

OFFICIAL GOVERNMENT STATISTICS

Chapter 4 laid out the evolution of the use of numerical data by historians up to our time. Let me stress that presently only a small minority of historians uses quantitative methods of some kind, and I would guess that fewer than half of those use them with any sophistication. Historians by and large blanch at mathematical equations and have now largely left to economists and sociologists the application of more advanced quantitative methods to social data. As I mentioned in Chapter 4, economic history is no longer done in history departments but in economics departments and in business schools, where some fine institutional histories have been produced. Likewise, as mentioned, the sociology department has taken over from historians most quantitative social history except for demographic history, which is still done by a small band of specially trained historians. This is where we left off in Chapter 4, where I described what quantitative history was in earlier times and what it has become today, when history is without a doubt the weakest of the social sciences in using quantitative methods. There are very few areas where historians have been innovative in the use of quantitative methods. Some original programming has been done. For instance, the Soundex Index for linking records was developed by historians. This is a rather complex computer program that lets you link similar-sounding names from different lists, say, “Smythe” and “Smith.” It then can tell you the probability that these are the same person.

In the previous chapter, we concluded that manipulating the media environment is a central concern for strategic players in the “game” of Supreme Court nominations. From this perspective, the president and his allies must try to establish a favorable media environment for the nominee. Individuals opposed to the nominee and the president must do the reverse: establish a hostile environment. Of course, a favorable environment may include flying under the radar – no attention in the media, or very little.

How can strategic actors manipulate the information environment? A useful distinction is between direct and indirect manipulation. By “direct manipulation,” I mean, for example, the president going public (making speeches) on behalf of the nominee, interest groups staging protests and demonstrations, and members of the Judiciary Committee shaking their finger at the nominee and asking “gotcha” questions about judicial doctrine. These are direct manipulations because the media may deem them newsworthy and broadcast them or report on them.

Indirect manipulation is somewhat more subtle. The idea is that you engage in actions that make direct manipulations (your own or others) more or less effective, or encourage reporters to write the right kinds of stories on their own. Consider, for example, the president's choice of a moderate versus an extreme nominee or a highly qualified one versus a marginally qualified one. These choices are indirect manipulations of the media because they affect how the media portray the nominee.

WHAT THIS BOOK IS ABOUT

The seven parts of this book cover different examples of quantitative reasoning in social science applications, along with interspersed discussions on the role of quantitative theories and methods in understanding the social world. We are trying to show how researchers in different social sciences think about and use quantitative ideas. These approaches vary across the different disciplines, and we think that the best way for a student to get a broad overview is to see how quantitative ideas are applied in a variety of settings.

Hence, we focus on applications, including combat in the First World War, demographics in nineteenth-century Mexico, forecasts of economic conditions by the Federal Reserve, racial disturbances in the 1960s, voting in congressional committees, Pavlovian conditioning, and the effect of migration on solidarity. Each part of the book has several examples, with discussion of the theories used to explain each phenomenon under study, along with the data and models used to describe what is happening. We thus hope to give a general perspective of how these models and methods are used in the social sciences. It is not the usual straightforward story of the form, “Here are some data, here's the model, and look how well the method works.” Rather, we explore the strengths but also the limitations of the models and methods in the context of the real problems being studied.