Most of this book is devoted to examples and tools for the practical use and understanding of regression models, starting with linear regression with a single predictor and moving to multiple predictors, nonlinear models, and applications in prediction and causal inference. In this chapter, we lay out some of the mathematical structure of inference for regression models and some algebra to help you understand estimation for linear regression. We also explain the rationale for the use of the Bayesian fitting routine stan_glm and its connection to classical linear regression. This chapter thus provides background and motivation for the mathematical and computational tools used in the rest of the book.

Chapter 7 discussed the use and logic of time-series designs and how they can provide tentative causal inferences about the effectiveness of a program or policy. It also mentioned single-case designs, which are a type of time-series design that can be used not only to evaluate the outcomes of programs, but also by direct service practitioners to evaluate their own practice. This chapter will examine alternative single-case deigns, their logic for making causal inferences, and how to use them to evaluate practice and programs.

If you have taken a course on research methods you probably learned that the gold standard (ideal) design for outcome studies on the effectiveness of interventions involves randomly assigning clients to treatment versus control groups to control for threats to internal validity like history, passage of time, selectivity bias, and regression to the mean. It’s good that you learned that because such designs are the best way to determine whether a tested intervention appears to be the real cause of any observed outcomes.

The present chapter considers two sorts of operations that are done as adjuncts to fitting a regression. In poststratification, the outputs from a fitted model are combined to make predictions about a new population that can differ systematically from the data. The model allows us to adjust for differences between sample and population–as long as the relevant adjustment variables are included as predictors in the regression, and as long as their distribution is known in the target population. Poststratification is a form of post-processing of inferences that is important in survey research and also arises in causal inference for varying treatment effects, as discussed in subsequent chapters. In contrast, missing-data analysis is a pre-processing step in which data are cleaned or imputed in some ways so as to allow them to be used more easily in a statistical analysis. This chapter introduces the basic ideas of poststratification and missing-data imputation using a mix of real and simulated-data examples.

We can apply the principle of logistic regression–taking a linear “link function“ y = a + bx and extending it through a nonlinear transformation and a probability model–to allow it to predict bounded or discrete data of different forms. This chapter presents this generalized linear modeling framework and goes through several important special cases, including Poisson or negative binomial regression for count data, the logistic-binomial and probit models, ordered logistic regression, robust regression, and some extensions. As always, we explain these models with a variety of examples, with graphs of data and fitted models along with associated R code, with the goal that you should be able to build, fit, understand, and evaluate these models on new problems.

In this chapter you’ll learn about the ethical and cultural issues that influence program evaluation. You’ll learn about Institutional Review Boards (IRBs) that often have to approve of program evaluations that use research methods with human participants. After you learn about the kinds of questions you’ll have to answer should you ever need the approval of an IRB, the chapter will turn to cultural issues that bear on recruiting and retaining evaluation participants and collecting and analyzing data in a culturally sensitive fashion. Finally, you’ll examine some key concepts that pertain to developing your own cultural competence.

In this chapter we turn to the assumptions of the regression model, along with diagnostics that can be used to assess whether some of these assumptions are reasonable. Some of the most important assumptions rely on the researcher’s knowledge of the subject area and may not be directly testable from the available data alone. Hence, it is good to understand the ideas underlying the model, while recognizing that there is no substitute for engagement with data and the purposes for which they are being used. We show different sorts of plots of data, fitted models, and residuals, developing these methods in the context of real and simulated-data examples. We consider diagnostics based on predictive simulation from the fitted model, along with numerical summaries of fit, including residual error, explained variance, external validation, and cross validation. The goal is to develop a set of tools that you can use in constructing, interpreting, and evaluating regression models with multiple predictors.