Quantitative analysis in political science entails assigning numbers to observed events in the political world, and then using the rules of mathematics, probability, and statistics to make statements about the world we live in. Quantitative analysis enables not just precise identification of the relationships between events, but also statements about the uncertainty surrounding these relationships. As apparent from the term event, this type of analysis relies on some level of abstraction, and is based on the notion that unique events are considered instances of more generic variables. For example, individual transactions can be lumped into imports, voters’ opinions can be compared based on their responses to survey questions, and violent conflict between two groups in two different places and times can be treated as two instances of “war.” With the rapid growth of computing power and development of large-scale data sets over the last sixty years, quantitative analysis has migrated from being the domain of a few scholars working in specific areas (e.g., voter behavior or public opinion) to its current position at the heart of empirical political science in all major subfields.
Quantitative analysis is distinct from formal modeling. Formal models are self-contained exercises designed to identify the consequences of a set of assumptions using the rules of mathematics and formal logic. Formal models and quantitative analysis frequently complement one another and coexist in specific scholarly works. Both use the language of mathematics but only quantitative analysis involves the creation and analysis of numerical data from the observable world.
Quantitative analysis is frequently contrasted with qualitative analysis. While the differences between them often appear obvious, on closer consideration, it becomes harder to distinguish the two. Analysts using both sets of tools often have similar goals. Works considered qualitative commonly report numerical data, while quantitative analysis is useless without substantive interpretation. Items such as words in speeches, while appearing qualitative, are amenable to analysis using advanced graphical and statistical tools. The logic of quantitative research can profitably apply in qualitative settings, while qualitative or historical knowledge can incorporate directly into quantitative analysis, especially in a Bayesian setting: The Bayesian paradigm combines prior beliefs with our newly analyzed data to provide a formal method for knowledge accumulation. Our priors are a function of historical and qualitative information.
Measurement And Data
Levels Of Measurement
Assigning numerical values to observed events can take different levels of precision. Conventionally, these levels are grouped into four categories representing the extent of numerical content implied in numerical measurements. The level of measurement affects the numerical manipulations applicable to statistical modeling choices.
Nominal data contain numeral representing pure names; the actual value assigned has no meaning. For example, a democratic country can be coded with the numeral 1 and a nondemocratic country with a 0, or vice versa. Or, democracies could be coded as 5498 and no democracies as -3.14. It makes no difference; what matters is that the numerical values denote difference of kind. For nominal measurement to be useful, the extent of the categorization must be exhaustive—all observations are assigned a value—and mutually exclusive—no observation gets more than one value. In describing a distribution of a nominal variable, only frequencies can be used. It makes no sense, for example, to discuss the “mean of democracy.”
Ordinal data measurements convey a sense of magnitude using the ordered nature of numbers, that is, the binary relations of greater than (>) and less than (<).The actual distance between the numbers, however, has no meaning. For example, a survey question may ask respondents whether they agree with a particular statement “a lot,” “somewhat,” or “not at all.” An analyst might code these responses as 2, 1, and 0 respectively. Equally valid, however, is -5, -5.2, and -223. For data measured on this scale, conclusions can be made only that one respondent agrees more than another, and not how much more. The difference between the numbers 2 and 1 is not meaningful. In describing ordinal data, modes, medians, and percentiles can be used but the mean and standard deviation are undefined. Similar to ordinal data, interval data holds meaningful differences between values are meaningful. Equal distances between measurements represent equal intervals. Time is a commonly used interval-level variable.
Ratio data are interval data that also have a meaningful 0 point. In this way, statements such as, “There were twice as many battle deaths in Iraq last year compared to this year,” can be made. In other words, researchers can interpret ratios. In describing both ratio and interval data, the full suite of tools is available, including the arithmetic and geometric means, standard deviation, coefficient of variation, and so on.
Reliability, Validity, And Error
All measurement occurs in the presence of error. The tools used for measurement vary in their reliability—the extent to which an instrument gives the same answer under the same conditions—and their validity—the degree to which an instrument returns the true value on average. A related problem is one of bias, since some measurements may be reliable but biased away from the true value. Selection bias affects the ability to observe a representative sample of possible outcomes. A canonical example is women’s wages: only wages of the women who chose to work are observed, so the estimate of women’s wages is therefore likely to be biased.
Data Structures
Quantitative data are frequently described by their structure. The simplest is cross-sectional data. These data are simply a collection of observations at a single point in time. For example, survey data from a random sample records the answers of N people to K questions at the time of the survey. These data are simple in that they are independent samples from a population. More complicated data structures have increasingly complicated dependencies among observations. Longitudinal data measure one unit over time. For example, the U.S. unemployment rate is measured monthly. Clearly, one month’s unemployment rate depends on the prior month’s rate. Spatial data are data in which the units of study are distributed in space, either as regions or as point processes, for example, whether the level of democracy in one country is affected by the regime type of its neighbors. Dyadic data measure the links between two other units. For example, trade between pairs of countr ies is dyadic. It seems likely that the trade between the United States and Canada, the United States and Japan, and Canada and Japan might be correlated. Dyadic data, taken together, describes a network.
Quite common in some subfields of political science, panel data and time series cross-section (TSCS) both involve the repeated observation of a set of units over time. Panel studies typically observe a large number of units over a small number of intervals (e.g., a survey administered to the same set of respondents each year for three years). TSCS data describe data on a smaller set of units over a longer period, such as the rich Western democracies since World War II (1939–1945). These data structures require the analyst to address both the longitudinal dependence within subjects and the spatial or network dependence within time periods. TSCS are one type of hierarchical data. Other examples include cities within counties within states or children within schools. Hierarchical models attempt to account for the dependence among observations (e.g., children within the same school).
Goals Of Quantitative Analysis
Quantitative data analysis has at least one of four major objectives: description, prediction, theory testing, and causal inference. While not mutually exclusive, the bread-and-butter of quantitative analysis, the regression model, can be deployed in the service of any of these goals.
Description
Description is probably the most common goal of any quantitative analysis. A variable observed repeatedly generates a sample. Description then begins by characterizing the distribution of the observations with easily understood summary statistics such as the range, mode, median, mean, and standard deviation (average deviation from the mean) as well as higher moments. Rarely, however, is an analyst only concerned with a single variable, but more typically cares about the relationship between variables. Correlation is the most basic description of a bivariate relationship. It describes the extent to which bigger values of one variable associate with bigger values of another.
Correlation is only a description of association, not a sufficient condition for a causal relationship. Strong correlations often form the basis of interesting puzzles, such as the strong negative correlation between pairs of democratic countries and the likelihood of war. Comparativists have long attempted to explain the strong positive correlation between the proportionality of the electoral system and the number of political parties in a country.
The most powerful tool in descriptive quantitative analysis lies in the ability to present large quantities of data in compact, intuitive, visual form. Simple scatterplots can present enormous amounts of information efficiently. By creative use of perspective, contour plots, and even color, visual displays can offer insight into the relationships between three and four variables far beyond what summary statistics can provide. Visualization can also describe the outputs from statistical and simulation analysis in understandable ways for audiences who may not have background in the complexities of the models themselves. While visual displays are increasingly common in political science, more rigorous and advanced visualization techniques are a major research frontier.
Because the political world is complex, understanding any relationship usually requires accounting for other factors, and regression is the tool of choice. A simple regression is a procedure for drawing a line through a cloud of points such that the average distance from the line is as small as possible. For example, to describe the relationship between a variable y and x1 while accounting for the relationship between y and x2, y could be a voter’s level of approval for the American president, x1 is the voter’s age and x2 is the voter’s income. This is an example of what some call multiple regression. Such a model might take the form
y = a + b1x1 + b2x2 + e
In this example, y is referred to as the dependent variable or response, while x1 and x2 are called independent variables or covariates. The parameters a, b1, and b2 are estimates, and e represents random error. Here b1 describes the relationship between age and approval for the president holding income constant. The values of a, b1, and b2 are chosen to make e as small as possible, on average. In general, in order to estimate this model, there must be at least as many observations as parameters to estimate. In this example, we need at least three observations. The difference between the number of observations and the number of parameters is called the degrees of freedom. A major strength of regression analysis—and statistical analysis more generally—is the ability to quantify the amount of uncertainty about an estimate of a particular relationship. Discussions of uncertainty ultimately derive from statistical assumptions often couched in terms of standard errors and statistical significance.
Originally developed with ratio-level data in mind, the regression framework extends in numerous ways to accommodate different types of data and for different scientific questions. The generalized linear model (GLM) is among the most widely used amplifications of linear regression. Different assumptions about the data-generating process distinguish one form of the GLM from another. Logit and probit models assume binary nominal responses. Ordered logit and probit assume ordinal response variables. More generally, polytonomous nominal data are often modeled as emerging from a multinomial process. The Poisson and negative binomial models treat the response as integer counts. Political scientists have applied these and other members of the GLM family in a wide range of empirical contexts.
Descriptive statistics and visual displays of data, so widespread in political science, make citation of any particular study unnecessary. Regression analysis is equally as common but analysts usually interpret regression analysis as representing something more than just description (e.g., as a test of a theory). Even if regression is deployed for more aggressive goals, it provides a geometric simplification of a set of relationships between variables.
Prediction
A common goal of quantitative analysis is predicting outcomes. Election prediction probably receives the most attention. Political scientists have also spent considerable effort developing predictive models in other areas, such as the onset of war or the collapse of states. Analysts usually derive their predictions from some form of regression model, though other types of quantitative models are used, including dynamical systems and local simulation.
Typically, prediction connotes forecasting (i.e., predicting unrealized future events). Other types of predictive exercises exist, however. In-sample prediction is a way to evaluate a model, helping determine the effectiveness of different models at predicting the data the analyst used to fit the model. The frequently used R2 statistic is a summary of in-sample prediction. Out of sample prediction evaluates how well a model for one set of data predicts another set of data not used to estimate the model. In the earlier example, out of sample prediction would involve using estimates of the b derived from one survey to predict someone else’s level of presidential approval, given that new person’s age and income.
Accurate prediction does not presuppose a true model of the world or even that there is any causal relationship between the predictor variables and the outcome. A correct casual model will give great predictions, but an analyst whose goal is only predictive is purely concerned with identifying the variables that improve the ability to predict accurately. In this way, exercises such as data mining are justified. With prediction, true or parsimonious models are neither necessary nor of paramount importance, while strong and exhaustive correlations do hold significance.
Causal Inference And Theory Testing
Ultimately, science is concerned with identifying and quantifying causal relationships. Causal relationships rely on the notion of counterfactuals. For example, what would have been the economic growth rate in Ecuador had it not participated in an International Monetary Fund (IMF) structural adjustment program? It is impossible to simultaneously observe Ecuador’s economic performance both with and without its experience with the IMF. This problem is known as the fundamental problem of causal inference. To estimate a causal relationship, analysts typically rely on repeated measures. By divorcing the level of the independent variable of interest, or treatment, from all other possible confounding variables, one can estimate a causal effect. Randomized treatment is the most effective way this is achieved in practice.
The identification of a causal effect is difficult in political science since researcher-controlled randomized experiments are often impossible. Analysts must therefore be cautious in asserting the existence of a causal relationship. Matching models are one set of quantitative tools recently developed to assist in this exercise. These models pair observations that are as similar as possible along all dimensions, save the variable of interest and then use the differences in outcomes to estimate causal effects. Sandy Gordon and Greg Huber, in their article “The Effect of Electoral Competitiveness on Incumbent Behavior,” use matching to examine the effect of electoral competition on judges’ sentencing behavior. In another article, titled “Do UN Interventions Cause Peace? Using Matching to Improve Causal Inference” Michael Gilligan and Ernest Sergenti use matching to examine the effectiveness of UN peacekeeping missions.
Theory testing, related to causal inference, frequently allows political scientists to describe a set of plausible causal relationships based on some assumptions and understanding of the political world. The analyst then asserts a series of hypotheses, typically relationships between variables, that the theory would lead analysts to observe. Quantitative analysis permits analysts to describe whether the relationships estimable from available data are consistent with their theoretical expectations.
Analysts do not directly observe or estimate a causal relationship. Rather they assert one and then confront it with data. Analysis consistent with the assertions leads to increased confidence that the theoretical story may describe an actual causal relationship. Thus, political science is largely populated with statements of causal relationships that, while not demonstrated through randomized experimentation, have yet to be convincingly disproven.
Quantitative Analysis And The Future Of Political Science
Quantitative analysis has transformed political science over the last eighty years, expanding the understanding of numerous facets of the world—from public opinion to the relationship between democracy and economic development. But this expansion of knowledge has generated at least as many questions as it answered. As data sets proliferate, researchers will need to pay closer attention to where their data are originating and the incentives for governments, survey respondents, and interest groups in acquiring and reporting truthful values. Vast new sources of political information are soon to be available for quantitative analysis—including Internet traffic patterns, satellite images, and high-precision spatial locations—opening up whole new research frontiers.
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