In this chapter, we present an alternative modeling strategy to the fully parametric methods discussed in the previous chapter. Specifically, we consider the Cox proportional hazards model (Cox 1972, 1975). The Cox model is an attractive alternative to fully parametric methods because the particular distributional form of the duration times is left unspecified, although estimates of the baseline hazard and baseline survivor functions can be retrieved.

Problems with Parameterizing the Baseline Hazard

The parametric models discussed in Chapter 3 are desirable if one has a good reason to expect the duration dependency to exhibit some particular form. With the exception of the restrictive exponential model, any of the distribution functions discussed in the previous chapter are “flexible” inasmuch as the hazard rate may assume a wide variety of shapes, given the constraints of the model, i.e., the Weibull or Gompertz must yield monotonic hazards. However, most theories and hypotheses of behavior are less focused on the notion of time-dependency, and more focused on the relationship between some outcome (the dependent variable) and covariates of theoretical interest. In our view, most research questions in social science should be chiefly concerned with getting the appropriate theoretical relationship “right” and less concerned with the specific form of the duration dependency, which can be sensitive to the form of the posited model.

Moreover, ascribing substantive interpretations to ancillary parameters (for example the p, σ, or γ terms) in fully parametric models can, in our view, be tenuous.

In this chapter, we consider some important issues regarding model selection, assessment, and diagnostic methods through the use of residuals. The issues discussed in this chapter have a direct analog to methods of model selection and to model diagnostics in the context of the traditional linear model. For example, issues pertaining to functional form, influential observations, and similar other topics are directly relevant to the duration model. Because most of the methods of specification analysis discussed in this chapter make use of residuals, in the next section, we consider the different kinds of residuals that are retrievable from a typical duration model. Following this, we present several illustrations using residual analysis to assess various facets of the duration model. Most of the discussion in this chapter is presented in terms of the Cox model; however, diagnostic methods for parametric models are considered at the end of the chapter.

Residuals in Event History Models

The basic idea of a residual is to compare predicted and observed durations. In OLS regression, residuals are deviations of the observed values of the dependent variable from the values estimated or predicted value under the regression model, that is yi – ŷi. In event history analysis, defining a residual is more difficult because of censoring and because of issues relevant to estimation methods like maximum likelihood (in the case of parametric models) or maximum partial likelihood (in the case of the Cox model).

Our work on event history began in graduate school. We met as graduate students attending the Political Methodology Society's annual meeting in 1993 at Florida State University in Tallahassee, Florida. A small group of us at the meeting were interested in event history modeling and we saw its great potential for unlocking new answers to old questions and for revealing new questions in political science. We are indebted to the Political Methodology Group for bringing us together, providing a forum for us to present subsequent work, and providing ready and constructive critics and supporters. We are also indebted to our home departments for surrounding us with highly talented graduate students and interesting, stimulating colleagues. Meetings subsequent to our initial one in 1993, collaborations, and prodding from students and colleagues across the country who were interested in event history methodology, led to this manuscript.

This work has several goals. Our first goal in writing this book was to connect the methodology of event history to a core interest that social scientists, and indeed many scientists in fields as diverse as biostatistics and engineering, are interested in, namely understanding the causes and consequences of change over time. Scholars are commonly interested in “events.” For example, political scientists who study international relations might investigate the occurrence of a militarized dispute or criminologists might study instances of victimization. Events such as these connote change and frequently, this concern with events is concomitantly tied to an interest in the “history” preceding the event.

The applications of event history methods discussed to this point have all presumed that the event history process is absolutely continuous. This assumes change can occur anywhere in time. Nevertheless, continuity is often belied by the data: measures of time are frequently imprecise, or are made out of practical concerns (and out of convenience). For example, although cabinet governments may fall presumably at any time, the data used in our examples treat the termination point as occurring within a month. This implies that although we have data for processes that are continuous in nature, the data themselves are discrete. As event occurrences amass at discrete intervals, it may be more practical, and perhaps substantively natural, to consider models for discretetime processes. In this chapter, we consider some approaches for modeling event history processes where events only occur (or are only observed) at discrete intervals.

Discrete-Time Data

Event history data for discrete-time processes generally record the dependent variable as a series of binary outcomes denoting whether or not the event of interest occurred at the observation point. To illustrate, consider the public policy data in Table 5.1. These data are from a study of state adoption of restrictive abortion policy (Brace, Hall, and Langer 1999). The event of interest is whether or not a state adopted legislation that placed restrictions on abortion rights. The starting point of the analysis is the first legislative session after the Roe v. Wade decision (1973).

One of the strengths of the event history model over the traditional regression model is the ability of the event history model to account for covariates that change values across the span of the observation period. In principle, inclusion of TVCs in the event history model is straightforward. However, the use of TVCs can raise special problems, both substantively and statistically, for event history analysis. And while these problems are not exclusive to duration models, the problems sometimes manifest themselves differently in the context of duration analysis. In this chapter, we consider some of these problems and note that the “solutions” to these problems are in large part theoretical, and not statistical. Additionally, we illustrate how TVCs can be readily included in each of the models discussed to this point.

Incorporating Exogenous TVCs into the Duration Model

Several researchers have variously categorized the different kinds of TVCs used in event history analysis. Kalbfleisch and Prentice (1980) provide the most thorough and widely used categorization scheme. They distinguish TVCs as either being external or internal. Further, they subcategorize external covariates as being “fixed,” “defined,” or “ancillary” (Kalbfleisch and Prentice 1980, 123). A fixed external covariate is equivalent to a time-independent covariate; i.e., one having values that are known in advance and do not change over the course of the study. Defined covariates have values that can change over time, but the time path is known in advance.

The basic logic underlying parametric event history models is to directly model the time dependency exhibited in event history data. This is easily done by specifying a distribution function for the failure times. If the researcher suspects that the risk of an event occurrence is increasing, or “rising” over time, for example, then one may specify a distribution function that accounts for such a relationship. Social scientists, and in particular, political scientists, have made use of parametric methods to understand such phenomena as coalition durations (King et al. 1990; Warwick 1992), the survival of political leaders (Bueno de Mesquita and Siverson 1995), and the duration of military conflicts (Bennett and Stam 1996). Parametric models for political analysis would seem most reasonable when there exists a strong theoretical expectation regarding the “shape” of the hazard rate (or by extension, survival times), conditional on the covariates included in the model. Under such conditions, correctly specifying the distribution function will yield slightly more precise estimates of the time dependency in the data as well as more precise estimates of covariate parameters than nonparametric approaches in small samples (Collett 1994); however, if the distribution of failure times is parameterized incorrectly (for any size sample) then the nice interpretations afforded parametric models may not hold (Bergström and Edin 1992).

We make this cautionary note primarily because parametric methods directly specify the shape of the hazard rate. This will happen even if the parameterization is wrong.

The models discussed to this point have all involved so-called “single-state” processes, or equivalently, one-way transition models. In such models, there is a singular event of interest and once the event is experienced—or once an observation fails—the observation leaves the risk set and is assumed to be no longer at risk of returning to the previously occupied state. Concomitantly, in a single-state process, we assume that an observation is only at risk of experiencing a single event; that is, the observation is not at risk of making a transition to another state. Is this a reasonable assumption? Often it is not, and, at a minimum, it is an assumption that should be tested. In this chapter, we consider some models that account for repeatable events.

Additionally, in previous applications, we did not attempt to account for the different kinds of events that could occur. Some research problems, however, may lead one to consider how observations are at risk of experiencing one of several kinds of events. Problems of this kind are sometimes referred to as multi-state processes or competing risks processes because survival times may terminate in a variety of substantively interesting ways. In this chapter, we consider event history approaches to deal with the issue of competing risks.

The issues of repeatable events and competing risks serve to highlight the greater concern of this chapter: how does one employ event history models in the face of complicated social processes?

The lexicon of event history analysis stems from its historical roots in biostatistics. Terms like “death,” “failure,” and “termination” are natural for analyses of medical survival data, but may seem awkward for social science analysis. In the context of medical research, survival data usually consist of longitudinal records indicating the duration of time individuals survive until death (if death is observed). In analyzing survival data, medical researchers are commonly interested in how long subjects survive before they die. The “event” is death, while the duration of time leading up to the death, the “history,” is the observed survival time. Analysts working with survival data may be interested in assessing the relationship between survival times and covariates of interest such as drug treatments.

Likewise, social scientists frequently work with “survival data,” although such data are generally not thought of in terms of survival and death. Nevertheless, much of the data social scientists use are generated from the same kinds of processes producing survival data. Concepts like “survival,” “risk,” and “failure” are directly analogous to concepts with which social scientists work. Thus, the concept of survival and the notion of survival and failure times are useful starting points to motivate event history analysis.

Event history data are, as Petersen (1995) notes, generated from failure-time processes. A failure-time process consists of units (individuals, governments, countries, dyads) observed at some natural starting point or time-of-origin.