In Chapter 12 we discussed the modeling and fitting of a logistic regression equation with a dependent variable with three or more ordered categories. In this chapter we discuss the modelling and fitting of a logistic regression equation with a multi-categorical dependent variable, but here the dependent variable will have response categories that are not ordered, that is, they are nominal. The most frequently used method for estimating a nominal categorical dependent variable is the multinomial logistic regression model, the subject of this chapter. This model is a natural extension of logistic regression for a binary dependent variable.

Many of the dependent variables analyzed in the social sciences involve a time period of nonoccurrence prior to their occurrence. Demographers study death; but one cannot die without being born. Thus, one’s death is preceded by a time period after the person has been born during which time they do not die. Such a dependent variable is referred to as a time-to-event variable because there must be a time period of nonoccurrence before the event occurs. Such analyses have several names. The broadest ones are survival analysis or hazard analysis, owing to their early development in biostatistics and epidemiology, where researchers modeled the occurrence of death. The event of death was referred to as a hazard. Persons over a time interval not experiencing the hazard, that is, not dying, were referred to as surviving the hazard. There are two main types of survival models, continuous-time models and discrete-time methods. We direct most of our attention in this chapter to continuous-time models of survival analysis, and specifically to the Cox proportional hazard model. In the last section of the chapter, we focus on discrete-time survival models.

Many dependent variables analyzed in the social sciences are not continuous, but are dichotomous, with a yes/no response. A dichotomous dependent variable takes on only two values; the value 1 represents yes, and the value 0, no. The independent variables in the regression model are then used to predict whether the subjects fall into one of the two dependent variable categories. In this chapter we discuss the modeling of a dichotomous dependent variable and show why ordinary least squares regression is not appropriate. We discuss the logistic regression model. We fit a logistic regression equation and address several statistical concepts and issues: log likelihoods, the likelihood ratio chi-squared statistic, Pseudo R2, model adequacy, and statistical significance. We then discuss the interpretation of logit coefficients, odds ratios, standardized logit coefficients, and standardized odds ratios. We show how to use “margins” in the interpretation of logit models with predicted probabilities. The last sections deal with testing and evaluating nested logit models, and with comparing logit models with probit models.

In this chapter we present brief discussions of a few statistical topics not covered in earlier chapters. We first cover structural equation models, factor analysis, and path analysis. In future work fitting regression models in the social sciences, we frequently see reference to one or more of them. In the second section of the chapter, we address in summary form a few topics already discussed but which we believe require some additional attention. For instance, as part of our discussion of ordinary least squares regression, we covered in Chapter 8 the topic of regression diagnostics. But regression diagnostics is not an issue applicable only to OLS regression; so we present here a further discussion. Similarly, we expand with some additional commentary our earlier discussions of addressing issues of survey design (covered in Chapter 10) and multilevel models (covered in Chapter 16).

This chapter covers the two topics of descriptive statistics and the normal distribution. We first discuss the role of descriptive statistics and the measures of central tendency, variance, and standard deviation. We also provide examples of the kinds of graphs often used in descriptive statistics. We next discuss the normal distribution, its properties and its role in descriptive and inferential statistical analysis.

This chapter is an introduction to Stata. We note the essential features and commands of the Stata statistical software package. Our objective is to familiarize the reader with the skills that will allow them to understand and complete the examples in the later chapters of our book. We describe the main components of the interface, followed by those of each Stata file option, e.g., do file, log file, and graphs. The third section of this chapter gives examples that readers can use to practice the most commonly used commands. Last, we summarize best practices for data management.

When we use ordinary least squares (OLS) regression with data sampled from a larger population, there are several assumptions that need to be met for the results to be reliably extended to the larger population. In the first part of this chapter, we discuss each of these assumptions. We note some of the problems that will occur if one or more of them are violated. In the second part of the chapter, we turn to issues of regression diagnostics, that is, methods and approaches for determining whether the assumptions are met in the sample data. In the third and last section of the chapter, we discuss the topic of robust regression. We note that, under ideal conditions, OLS regression is preferred over other regression methods. But sometimes when some of the OLS regression assumptions are not met, the OLS regression breaks down and should not be used for the analysis. In such situations, regression methods less demanding than OLS may be introduced. Robust regression is one such method. It sometimes performs in a more satisfactory manner than OLS when some of the OLS assumptions are not met and when there are other statistical problems in the analysis.

In this chapter we cover the modeling of a dependent variable that is neither continuous nor categorical, but is a count of the number of events. Dependent count variables measure the number of times an event has occurred. An example from demography is the number of children ever born to a woman or man in their lifetimes. Frequently, count variables are treated as though they are continuous and unbounded, and ordinary least squares (OLS) models are used to estimate the effects of independent variables on their occurrence. But if the OLS assumptions we discussed in Chapter 8 are not met, then the use of OLS for count outcomes may result in inefficient, inconsistent, and biased estimates. There are many kinds or classes of models that may be used to estimate count dependent variables. In this chapter we consider five models: (1) the Poisson regression model; (2) the negative binomial regression model; (3) the zero-inflated count model; (4) the zero-truncated count model; and (5) the hurdle regression model.

A major concern in the social sciences is understanding and explaining the relationship between two variables. We showed in Chapter 5 how to address this issue using tabular presentations. In this chapter we show how to address the issue statistically via regression and correlation. We first cover the two concepts of regression and correlation. We then turn to the issue of statistical inference and ways of evaluating the statistical significance of our results. Since most social science research is undertaken using sample data, we need to determine whether the regression and correlation coefficients we calculate using the sample data are statistically significant in the larger population from which the sample data were drawn.

In this chapter we explore the statistical software program called R along with the integrated development environment known as RStudio. We provide a brief review of the history of R followed by guidance on how to download and install R and RStudio. Next, we explore features of the program including a basic review of the graphical interface. We then turn to working with basic commands, loading and saving data, and provide examples of working with packages in R.

This chapter extends to the multivariate context our discussions of ordinary least squares (OLS) regression in the Chapter 6. We first address the logic of multiple regression. We next cover the interpretation of the multiple regression coefficient intercept and slopes, paying particularly close attention to the interpretation of the b slopes. We then address model fit in the multivariate context and extend our discussions of the F-test and the coefficient of multiple determination (R2) by including the standard error of estimate and the Bayesian information criterion (BIC).

Most social science research analyzes data from samples drawn from larger populations. However, most of the standard statistical methods used for analyzing the data are based on the assumption that the sample data have been drawn with simple random samples. But few probability samples are completely random. Some sample respondents may be more heavily weighted than other respondents, and some respondents may be included in the sample by virtue of their membership in groups based on race, sex, age, and other characteristics. Nonetheless, many investigators treat their samples as random where each person in the larger population has an equal chance or probability of being included in the sample. We discuss in this chapter the methods that need to be followed to enable researchers to make correct inferences to the larger population with sample data that are not completely random. We review the three main types of probability samples. Then we discuss how and why researchers need to address and take into account the design of their samples.