This chapter addresses the challenge of dealing with large MDP models by approximating the value function or state-action value function with linear and nonlinear lower-dimensional basis functions (also known as features). It shows how to use least squares methods to fit the basis functions to the value function and evaluate a fixed policy. An alternative approach using linear programming is also presented. Least squares value iteration, least squares policy iteration, least squares modified policy iteration, and linear programming methods are developed to identify good policies based on the approximations. Applications in advance appointment scheduling and golf are provided at the end of the chapter.

This chapter completes the analysis of community protection in the Philippines in two stages. First, a series of focused qualitative comparisons with non-Catholic denominations broadens the attention to other sectors of the Philippine religious landscape such as the Iglesia Ni Cristo, evangelical communities, and denominations of the National Council of Churches in the Philippines such as the United Methodist Church in the Philippines. While the Catholic Church possesses a blend of commitment and capacity that facilitates Drug War intervention, these other traditions offer differing configurations of these variables that, the authors show, shape their responses to the Drug War, which range from narrow self-preservation to internationally oriented protective efforts to, in some cases, religious endorsement of Drug War violence. Second, the chapter turns to an innovative dataset mapping Drug War violence to evaluate a key quantitative implication of the argument: that, all else being equal, neighborhoods with a Catholic church will feature fewer Drug War killings than those without. This community protection does not extend to infrastructure from other religious traditions.

How can regression-type studies, which try to get causal inferences by controlling for variables until whatever is left is plausibly a causal relationship, benefit as much as possible from qualitative research? This chapter offers three qualitative research designs that can help test and improve quantitative causal inferences based on control variables. First, we consider process-tracing designs that seek to discover previously unknown confounding variables that can be added as new control variables, thereby refining and improving the statistical model. Second, we discuss process tracing that seeks to confirm that control variables have a direct connection to the outcome variable, and therefore are relevant controls. Third, we discuss process tracing to check whether control variables might be posttreatment and therefore at least partially harmful. Finally, we provide a demonstration of the kind of iterated interaction that we have in mind between statistical efforts at causal inference and qualitative research to learn more about the assumptions behind the model, involving the relationship between labor union strength and the prevalence of employment law across U.S. states.

Regression is a foundational concept in applied mathematics and statistics, with applications in data analysis, inverse problems, control theory, and machine learning. Linear least-squares regression solves for the unknown parameters of a model that is written as a linear system of equations. This is a convex problem, where the unknowns are chosen to minimize the sum-squared error of the fit over the data. It is also a rare problem that has an analytic solution in terms of the pseudo-inverse of the matrix system. In practice, we solve large problems using the singular value decomposition (SVD), which is an optimal, low-rank matrix approximation. We consider the overdetermined case, with more equations than unknowns, and the underdetermined case, with more unknowns than equations. Nonlinear least squares generalizes regression to optimize the parameters of a nonlinear function, such as a neural network, which requires iterative methods, such as stochastic gradient descent or Gauss-Newton methods. We also address ill-conditioned least-squares by adding regularization, such as Tikhonov for ridge regression or sparsity for LASSO regression. We also discuss constrained least squares.

In linear regression, we approximate the regression function $\mu (x)$ with a linear function ${\beta }_0+{\beta }_1{x}_1+\cdots +{\beta }_d{x}_d$ and estimate the coefficients ${\beta }_0,\dots ,{\beta }_d$ using least squares. We construct confidence intervals, prediction bands, and show how residual plots help check the linear approximation. We also review some other regression tools, that are not as widely used as in the past.

In this chapter, we introduce the concept of regression, which is a way to quantify the relationship between an outcome Y and a vector of features X. This relationship is expressed by the regression function, which is the mean of Y given X. This chapter discusses the main ideas and goals of regression analysis.

In this chapter readers go through the creation of a reusable function. In Chapter 5 we created a data frame with some basic information on individual trees in a stand; here we create a function to calculate the biomass of those trees using allometric equations and introduce the idea of optional arguments and default values. Next, we learn to use one of R’s most powerful functions for linear modeling, allowing us to do things like statistical testing and linear regression with very little effort.

The chapter begins with the classical linear regression (OLS) assumptions for cross-sectional data. When lagged values of the regressand appear on the right-hand side of the model, OLS is no longer BLUE, but OLS estimates are still consistent under a weaker set of assumptions we introduce. The chapter lays out the details of the general-to-specific modeling strategy and describes the general model forms that are the starting points for the model-building process: the autoregressive distributed lag (ADL) model and the generalized error correction model (GECM). These two general models are widely used in the social sciences to model stationary outcome variables. We then discuss how to specify deterministic terms in these models and how to choose a lag structure for the general model. We discuss restrictions that may be imposed on the dynamic structure given in the general model and what those restrictions imply about dynamic relationships in the data and then move to interpreting relationships. We provide a practical guide to building and interpreting time series regression models for stationary data and provide an example using the presidential approval model discussed in Chapter 4.

We provide a practical guide to applied time series analysis using ordinary least squares. In our opening chapter we discuss the ways in which time series data are different than cross-sectional data and then introduce the statistical consequences for those differences. We provide guidelines for assembling a time series dataset – including discussions of sampling windows, sampling intervals, data length, and operationalizing variables. We then provide a brief outline of the remainder of the book.

Both qualitative and quantitative methods provide rigorous ways to establish and assess causality, though their understandings of causality and techniques for doing so differ. Qualitative and quantitative techniques are generally complementary rather than substitutes or opposites. Common quantitative tools include cross-tabulation, regression, and logit/probit models; which is appropriate typically depends on the level of measurement of the dependent variable. Common qualitative tools include two within-case techniques, process tracing and analytic narratives, and three between-case techniques, case control, structured focused comparison, and content analysis. “Intermediate-N” techniques exist for research questions whose number of cases is greater than that typically used for qualitative analysis but below the threshold for successful quantitative analysis, along with Big Data approaches for extreme-N analysis, content analysis techniques for corpora, and mixed methods approaches.

Psychoanalysis deals with individual and collective trauma, in relation to the unconscious reproduction of what has caused a psychic breakthrough for the subject and the group, with each aftermath re-semantising painful past experiences. Archaic fears and defence mechanisms are common to everyone before giving way to more evolved modes of functioning. Confrontation with major traumatic experiences can undo the symbolic capacities acquired during human development and plunge the subject and the group into archaic modes of functioning: projection of the bad object outside the group, splitting between an all-good self-image and an all-bad image of the other, and denial of the suffering of the outside group. This regressive logic traps the actors in a victimisation likely to lead to further violence or to maintain a climate of permanent threat. Working on the dual recognition of trauma means that the experiences are not hidden, that the point of view of the other (whether inside or outside the group) is integrated and that a narrative of the past that leaves room for internal and external polyphony is passed on to future generations. This leads to the expression of internal conflict within oneself and one’s community, rejecting a monolithic vision of identity and history.

This chapter statistically tests the relationship between American hierarchy, property rights, and state capacity using mediation analysis. It finds that American economic hierarchy enhances property rights in partner states, indirectly strengthening state capacity. The analysis explores scope conditions and the interaction between security and economic hierarchy, highlighting the contrasting effects on state-building. The chapter discusses the implications of the quantitative results for cases like Afghanistan.

This chapter begins by defining an averaging procedure for random variables, known as the mean. We show that the mean is linear, and also that the mean of the product of independent variables equals the product of their means. Then, we derive the mean of popular parametric distributions. Next, we caution that the mean can be severely distorted by extreme values, as illustrated by an analysis of NBA salaries. In addition, we define the mean square, which is the average squared value of a random variable, and the variance, which is the mean square deviation from the mean. We explain how to estimate the variance from data and use it to describe temperature variability at different geographic locations. Then, we define the conditional mean, a quantity that represents the average of a variable when other variables are fixed. We prove that the conditional mean is an optimal solution to the problem of regression, where the goal is to estimate a quantity of interest as a function of other variables. We end the chapter by studying how to estimate average causal effects.