Overview Event history analysis (also called survival analysis, duration analysis, or transition analysis) is an umbrella term for a set of procedures for time series analysis. Coleman (1981: 1) defined event history analysis in terms of three attributes: (1) data units (ex., individuals or organizations) move along a finite series of states; (2) at any time point, changes (events) may occur, not just at certain time points; and (3) factors influencing events are of two types, time-constant and time-dependent. Event history models focus on the hazard function, which has to do with the probabilities that an event will occur after any given duration. Duration to the hazard of death was the classic example in medical research, but the hazard may have a positive meaning also, such as duration until the event of adoption of an innovation in diffusion research. Event history models fall in three broad classes: Nonparametric. Nonparametric models make no assumptions about the shape of the hazard function or about how covariates affect it. Instead the hazard function is estimated based on the empirical data, showing change over time. The effect of covariate variables is show only by stratifying the data into groups (ex., by gender) to plot and contrast separate hazard functions for each group. Nonparametric models can only compare a very limited number of groups, cannot handle continuous data, and cannot support multivariate control for other explanatory variables. Kaplan-Meier survival analysis, discussed separately, is the primary example of the nonparametric approach to event history analysis. See also the life tables procedure, used for descriptive, actuarial studies of duration where time is the only salient variable and censored and uncensored cases do not differ. Semi-parametric Semi-parametric models also make no assumption about the shape of the hazard function in relation to time but do make strong assumptions about how covariates affect the hazard function. Specifically, they assume that hazard rates are proportional between groups over time. While estimates of the shape of the hazard function may be derived empirically, these estimates are data-driven and may be considered overfitted, with the result that semi-parametric models are not considered appropriate for testing hypotheses about time dependence (about the shape of the hazard function in relation to time). Also, estimates of the parameters for explanatory variables are less precise than in parametric models, discussed below. However, because semi-parametric models support multivariate analysis yielding results usually identical in substance to parametric models without requiring the often questionable assumptions of parametric models, semi-parametric models are often the method of choice in event history analysis. Cox regression, discussed separately, is the primary example of the semi-parametric approach to event history analysis. Parametric. Parametric models are the type discussed in this section. Parametric models require the researcher to posit in advance the shape of the hazard function as well as how covariates affect the hazard function. This type of model supports multivariate analysis of discrete and continuous explanatory variables and yields precise parameter estimates, provided the correct model assumptions are made. Estimates are derived using maximum likelihood methods. If the wrong shape of the hazard function is specified, parameter estimates can be seriously biased. Parametric models are preferred when time is itself considered a meaningful independent variable and the researcher wants to be able to describe the nature of time dependence. Also, because parametric models specify the shape of the baseline hazard function, that function can be extrapolated into the future, making it useful for predictive modeling. Weibull models, discussed below, are the primary example of the parametric approach to event history analysis, but there are several other parametric types also discussed below. Event history analysis models are duration models. In contrast, event count models analyze the number of events since some starting time but not duration-to-event. Poisson regression is a leading type of event count model, but logit, probit, and logistic models are also used. All these are also discussed separately. Event history analysis is prominent in the field of international relations, where it has been used to analyze time series of international conflict and diplomatic events. It is also used in diffusion-of-innovation studies, in biostatistics, in the study of demographic changes such as marriage, studies of social mobility, labor market studies of becoming unemployed, and a variety of other fields. Event history analysis can be a form of panel study in which the periods of observation are not arbitrarily spaced but instead measurement is taken at each stage of a sequence of events. The timing and spacing of observations thus becomes a critical variable in its own right. Moreover, often in event history studies the data may not be interviews of individuals as in panel studies, but rather measurements pertaining to organizations or even governments.

Contents Key concepts and terms Modeling Model fit Assumptions Frequently asked questions Bibliography




Key Concepts and Terms

• States are discrete classes of statuses in which the person, organization, or other unit of analysis may be found. Often there a just two states, such as alive/dead in a mortality study, or unemployed/employed in a labor force study. However, in multistate models there may be multiple states, such as one state represented by each school type attended in a study of educational careers, or each rank in a study of military careers. State space is the set of all pertinent states (ex., all school types in a study of educational careers)

• Time variable. Event history methods assess change over time, and it makes a difference what time units are selected and if the time axis is continuous or not (continuous is the norm). The time axis should be set based on theory, not statistics.
  • Start time. The start time ("time of origin") for the time series in event history analysis is critical. Coefficients will differ for different start times. Start times must be selected on a theoretical basis, not statistical. Sometimes there is a natural and obvious start time (time of manufacture of an engine in industrial studies of engine failure times; birth in medical studies of cancer). At other times, the choice of a start time is less obvious. In a study of duration of members of Congress in office, start time might be first election to office. Time would be number of time units (ex., years) since start time, even though the start time might vary from one member of Congress to another.

  • Discrete vs. continuous time. In some studies, time may be continuous in the sense that the event of interest may occur literally at any time. In other studies, the event may occur only at certain discrete times - for instance, passage of a given type of bill may only occur during legislative sessions. With discrete data, there will be a series of discrete time blocks (ex., 2004 legislative session, 2005 legislative session, 2006 legislative session, etc.) and the event (ex., passage of an enterprise zone act) will be coded "0" for each time block until it occurs, when it is coded "1". Because the event is thus a series of 0's and 1's, it may be modeled as a logit or probit model (ex., the Weibull model is a form of logit model, discussed below). Continuous time studies, in contrast, may be parameterized as conditional logit (fixed effects) models in Cox regression. However, as Steffensmeier & Jones (2004: 83, 87-88) note, this is something of a distinction without a difference since Cox models can be used even with highly discretized data and would yield quite similar substantive conclusions without the need to make strong assumptions about the shape of the baseline hazard function.

• Covariates are additional variables which may influence the failure-time process being studied. There are two types:
  1. Time independent covariates do not vary over the course of the study. For instance, in a state-level study of diffusion of a particular governmental innovation, contiguity of the state with an ocean would be time independent.

  2. Time dependent covariates vary over the course of the study. For instance, in a state-level study of diffiusion, the majority party in the state legislature could change over time. It is up to the researcher, not software, to obtain measurements for each time period for each time-dependent variable, whether through actual measurement in each time period, linear interpolation between measurement times, or by assuming the last measurement carries forward until the next one. Software algorithms are also needed to assign the value of the time-dependent variable at the moment of the event, with SPSS using the "programming method," Stata using the "episode splitting method," and SAS offering a choice. However, this aspect of estimation is handled in software, not requiring researcher choice, and the methods yield equivalent results (Allison, 2004: 377-378).

• Episodes. The number of time units an individual spends in a given state is called an episode. Synonyms include duration, waiting time, and spell.

• Events. An event is normally what terminates an episode (ex., death in a mortality study; adopting an innovation in a diffusion study). That is, it is the change which causes the subject to transition from one state to another. In some research contexts, there may be multiple episodes and multiple events (ex., multiple ranks in a study of military careers). In summary, failure events are the events which mark the end of episodes, such as death in a medical study or the burning out of a motor in an industrial study. Note that failure events may be "positive" in nature, such as adoption of an innovation in diffusion studies. Therefore some prefer to label these simply "events."

• The dependent. Usually event history analysis models (1) the probability of an event, and (2) the duration of an episode. Accordingly, event history analysis may be termed "failure-time analysis" or "duration studies." In addition, accelerated failure time models (AFT), discussed below, model time.

• Censored data. Data are censored if episodes started before (left-censored) or ended after (right-censored) the period of observation. Censoring creates special problems of data analysis and interpretation. Event history methods handle right-censoring on a built-in basis (unlike OLS and logistic regression). Ordinarily, left-censored cases are dropped from analysis.

• Survival function, S(t). The survival or survivor function, usually expressed as S(t), is the a cumulative frequency of the proportion of the sample not experiencing the event by time t (hence, in some studies, those remaining "alive" or "surviving"). Put another way, the survival function is the probability of the event will not occur until time t. One may also say that the survival function is the probability that the episode's duration is at least t (the event occurs later than t). This also implies one may say the survival function reflects the proportion of subjects surviving beyond any given time, t. By plotting the survival function one can compare the survival rates of two or more groups. There are also statistical tests of the significance of group differences in survival functions.

  The survival function is represented graphically by a downward-sloping curve which might reach 0% probability of survival by a certain final time. When based on sample data, the survival function is a step function, with steps of descent at each time period when an observation encounters the hazard and drops out of the risk pool. This sample stepped survival function is interpreted as the best sample-based estimate of the true underlying continuous (non-stepped) survival function.

• Cumulative probability function, F(t) may be thought of as a table which for every time t, lists the probability that the duration to event is t or less. It is the opposite of the survival function. F(t) = 1 - S(t); also S(t) = 1 - F(t).

• Probability density function, f(t). A familiar density function is the bell-shaped normal curve, where the area under the curve at any point on the x axis gives the probability. Similarly, in the probability density function, which is directly derived from the cumulative probability function, the area under the curve corresponds to the probability that the event has occurred by time t, where t is time plotted on the x axis. The density function, f(t), thus describes the unconditional (not conditioned on covariates) instantaneous (at any given instant t) probability of the event (that is, the failure rate). Put another way, the density function reflects the probability that the event will occur at any given time interval (t' - t), where that interval approaches zero duration, thus the probability at any given instant. Graphically, the density function would be represented with time on the x axis and probability of the event on the y axis.

• Hazard rates. The hazard rate, usually expressed as h(t), is the probability of the event occuring in time t + 1, given survival to time t. That is, the hazard rate is the conditional failure rate: the rate of failure per time unit at a given time t, given survival to time t. It is the risk that a unit will fail (the episode will end) at a given time, given survival up to that time. It is the propensity to change states at time t. The hazard rate is defined in relation time and covariate measures in the "risk set," which is that part of the sample still at risk because they have not yet experienced the event. As subjects experience the event, they drop out of the risk set.
  • Synonyms for hazard rate include failure rate, risk function, transition rate, intensity rate, transition intensity, and mortality rate. The hazard rate answers questions like, "given electoral success up to now, what is the likelihood a member of Congress will fail to be reelected in the next time period?"; "Given drug addiction up to now, what is the probability the subject will end his or her addiction in the next time period?".

  • Relation to survival and density functions. The survival function, S(t), reflects the proportion of subjects surviving at time t. The hazard rate, h(t), is the ratio of the density function to the survival function: h(t) = f(t)/S(t). The numerator reflects the unconditional probability of survival to time t. The denominator makes this probability condition on survival to time t. The resulting hazard function, h(t) is the conditional probability of survival to time t, given survival to time t.

• Hazard function, also called duration dependence, is a specification of the shape of the hazard rate over time. Hazard rate and hazard function are closely connected and often used interchangeably. The shape of the hazard function must be specified by parametric models but not by Cox regression models, which are semi-parametric. Many scholars, following the pioneering work of Berry & Berry (1990, 1992) have assumed a flat hazard rate, conditional on model covariates. That is, it is assumed that duration to event is unchanging over time, and since covariates are all fixed rather than time-varying, the shape of the hazard rate is flat. As this may not be a realistic assumption, the following methods of relaxing this assumption are recommended:
  • Dummy variables for time. A common and recommended method for modeling the effects of time, relaxing the assumption of a flat hazard rate, is to add dummy variables to the model, with each dummy coded 0 or 1 according to whether it applies to the given observation. Thus one adds (t - 1) dummy variables for each observation. This method works regardless of the specification of the shape of the hazard rate over time. However, when the number of degrees of freedom in the model is already small, this approach can lead to lack of convergence in the solution.

  • Regressor variable for time. An alternative approach is to include a counter variable as a covariate representing time of the observation, or some transform (ex., natural log, natural cubic smoothing spline) of it. If the decision of how to model time (to transform or not, and if so which transform) is purely data-driven, there is a danger of overfitting the model. Cross-validation of the model as developed in one sample and then validated with another sample is a method of guarding against overfitting. Buckley & Westerland (2004) have shown how the original substantive conclusions of Berry & Berry (1990) are significantly altered by addition of a regressor variable for time, as well as being affected by different transforms of it (they found the cubic spline transform best).

• Cumulative hazard function. The cumulative hazard function, denoted H(t), is the integral over 0 of t to h(t), the hazard rate.

• Hazard ratios are the ratios of two hazard rates. Classically in medical research, this was the hazard rate for the treatment group in ratio to the hazard rate for the placebo group. Hazard ratios tend to be the focus of EHA analysis. Discussion of interpretation of hazard rates and hazard ratios is given in more detail in the section on Cox regression. Note that the hazard ratio has to do with the gap in rates between two groups formed by a coded covariate (ex., placebo=0, treatment=1). For a continuous covariate, the hazard ratio has to do with the gap in rates between the covariate with no increase and the rate with the covariate having a 1 unit increase. The hazard ratio does not itself contain any information about the shape of the hazard function for either group, but rather in proportional hazards models (ex., Cox models) the gap (ratio) is simply assumed to be the same throughout time.
  • Baseline hazard function: This is the form of the hazard ratio over time before the effects of covariates are taken into account. The function may be flat (the same for all time periods), rising, declining, or some combination.

  • Hazard rate graphs. One may use graphs to visually convey information about hazard ratios. In a typical graph, the y axis is the hazard rate and the x axis is the time duration. Curves within the graph depict the shape of the hazard function over time for various groups defined by the covariates (ex., for men if gender is a covariate). If the assumption of proportional hazards is met, curves will have the same shape but different heights on the y axis.

  
  

• Modeling
  • Event-count models, such as Poisson regression, are a largely earlier approach to event data, not discussed in this section. See sections on the generalized linear model and generalized estimating equations, and on Poisson regression. Event count data, as the name suggests, record data on the count of events by time period but do not record duration times. OLS regression was inappropriate for event-count data for a variety of reasons, including biased estimates and the possibility of predicting negative counts, which are out of range. Poisson regression was a response to these problems, designed for the analysis of rare events (in a large number of trials where the probability of an event is small in any one trial, the total number of events follows a Poisson distribution). When the number of trials without events (i.e., the number of 0's) is large, modifications such as zero-inflated Poisson regression, and alternatives, such as negative binomial regression (NBR models), have been developed and are now recommended for event-count data. In spite of these developments, analyzing pooled time-series cross-section data with event count models may not be able to account for unobserved heterogeneity (different subjects, countries, etc., may vary in rate of event occurrence), for diffusion/contagion processes, or for overdispersion (variance > mean).

  • Duration models, often equated with event history models, are based on data containing information on start/stop times (hence duration) as well as counts of events. Where event count data measure if an event occured, event history data measure if and when an event occurred. The unit of analysis is the event, not the subject (though in some settings this is equivalent). Information is not lost when, as in event-count models, data are aggregated by unit of analysis (ex., by country). Because duration data are retained, duration models can account for time-varying covariates, unlike event count models, and this may be more realistic since in real life it is common for covariates to vary over time.

  • Assumptions about how covariates affect the hazard rate for parametric duration models. Parametric models require two strong assumptions: (1) positing in advance the shape of the baseline hazard function, discussed below; and (2) positing in advance how the covariates affect the hazard rate. With regard to (2), there are two main possibilities: proportional hazards (PH) or accelerated failure time (AFT) assumptions, but not every choice in (1) can support every choice in (2).
    • Proportional hazard models. In PH models, covariates are assumed to raise or lower the hazard function in a multiplicative fashion. In a PH model the dependent variable is the hazard for the event. PH parameterization is available for exponential, Weibull, and Gompertz models. The proportional hazards assumption is that all groups of observations have the same shape hazard function, but that function is moved up or down in parallel with the others according to the influence of covariates in the model. For instance, in a medical study, smokers may have a higher hazard function than non-smokers, but for both groups, the shape of the function is assumed to be the same. Most parametric models are proportional hazard models.

    • Accelerated failure time models. In AFT models, covariates are assumed to multiply the time scale. In an AFT model, the dependent variable is event time. AFT parameterization is available for exponential, Weibull, gamma, log-normal, log-logistic models. The AFT assumption is that all observations have the same shape hazard function, but the time axis varies such that some groups pass through stages of the hazard curve faster than others. An analogy would be people-years vs. dog-years with respect to the hazard of death, where one might hypothesize the same shape hazard function but with one people-year equaling seven dog years. The same hazard function would mean that if 90% of people survive to age 63, then 90% of dogs would survive to age 9 (=63/7). Parameter estimates in AFT models have the opposite sign from corresponding estimates in PH models because PH models predict hazard and AFT models predict time, and when hazard is high in PH parameterization, time to event is low in AFT parameterization.Also, the significance of parameter estimates for the same covariate need not be the same in both parameterizations.

  • Assumptions about the baseline hazard function shape for parametric duration models. Different parametric models make different assumptions about the shape of the baseline hazard function. Selection of an inappropriate parametric model may render interpretation of resulting coefficients problematic or misleading (Bergström & Edin, 1992; Larsen & Vaupel, 1993: 96).
    • Single-episode parametric models
      • Exponential models. In exponential models, the baseline hazard function is assumed to be flat (constant). This means that exponential models cannot display time dependence (are not appropriate for testing hypotheses about how the hazard rate increases or decreases with time). Put another way, it is assumed that no matter how long the duration for an observation, it still has the same chance of the event in the current time period. The risk of the event occurring is the same for all time periods, contingent on covariates. That is, any departure of the hazard function from a flat line is assumed to be due to covariate effects. The proportional hazards assumption is usually used for parameterization: the ratio of hazards for any two observations is the same across time periods. For instance, the ratio of hazards for persons a and b should be the same this year as in the period 10 years from now. Also, the exponential distribution of survival times is not contingent on knowing survival duration to the given time of interest, yet in real life this is often important. The assumptions of exponential models are not often reflected in real life and may lead the researcher to consider other models.
        • Log-linear models are a common type of exponential model. The log of survival times, T, is a function of the vector of covariates times their respective regression coefficients plus a regression constant equal to the baseline hazard function. The expected duration time for subject i at time t, E(ti) = exp(ß0 + ß1xi1 + ß2xi2 + ... +ßkxik) where ß is the covariate regression weights, except ß0 is the regression constant; k is the number of covariates; and exp() is the function which raises the natural log, e, to the power of the vector in the parentheses.

        • Piecewise exponential models are similar but assume that time is divided into periods and that the hazard rate is constant within any time period, making the hazard rate a step function of T, time. The model has a time dummy variable for each time period and the parameter estimates for each dummy reflect hazard function changes for the given time period.

      • Weibull models are perhaps the most common type of parametric EHA model in social science. Weibull models assume a baseline hazard function that may be flat like exponential models, but which also may be monotonically increasing or decreasing. If flat, the Weibull and exponential models are the same (or rather, the exponential model is thus a special case of a Weibull model). If monotonically increasing or decreasing, the researcher is positing that hazard rates are always increasing or decreasing. Although this assumption may not reflect real life, Weibull models are nonetheless common in social science, if only because they are more flexible than exponential models.

        Weibull models have a "scale parameter" (alpha) and a "shape parameter" (p). Covariates affect the scale but not the shape of the distribution, which is always proportional from one subject to another in the usual PH parameterization. If the scale parameter, alpha>1, then the hazard function is increasing over time. If alpha<1, the hazard function is decreasing. If alpha=1, the hazard function is flat, equivalent to an exponential model. The shape parameter, p, when 1 corresponds to a flat line; when 1.5 increases quickly and first and then is slowly increasing; when 2 is linearly increasing; 3 or higher is exponentially increasing; .5 is exponentially decreasing. Various combinations of the scale and shape parameters mean the hazard function in a Weibull model may assume a very wide number of shapes. The mathematical model is h(t) = apt^(p-1), where a is the alpha scale parameter, p is the shape parameter, and t is time.

        There are two types of Weibull model parameterizations, AFT and PH:

        • Weibull PH models are proportional hazards models, also called "relative hazard models". Coefficients are parameterized in terms of the hazard rates. That is, in PH parameterization, if a covariate coefficient is significant in a positive direction, then for that covariate the duration to event is shortened and the hazard rate is increased, compared to the baseline hazard. Covariates are assumed to have a multiplicative effect on the hazard rate. PH models are much more common than AFT models.

        • Weibull AFT models are accelerated failure time models, also called "log expected failure time models" or "accelerated life models." Covariates are assumed to have a multiplicative effect on the time scale (not the hazard rate as PH models). That is, covariates accelerate time by some factor. Coefficients are parameterized in terms of the log of survival times. Regression parameters for a covariate is positive to the extent that the covariate increases survival times. That is, if in an AFT parameterization, a covariate coeeficient is significant in a positive direction, then for that covariate the duration to event is lengthened and the hazard rate is decreased, compared to the reference (usually 0) category of the covariate. Exp(b) for the covariate gives the odds ration, which shows the odds that case with a given covariate with a positive parameter coefficient will survive longer than a case in the reference category. Note that coefficients in a Weibull PH model will have the opposite sign from their counterparts in a Weibull AFT model for the same covariates and data.

        • Testing Weibull models as exponential models. If the scale parameter for a Weibull model is not significantly different from 1, then that model is equivalent to an exponential model, discussed below. Rather than select in advance between a Weibull model and an exponential model, standard practice is to run a Weibull model and test for alpha=1. If this test is nonsignificant, then the researcher would run an exponential model, preferring it on parsimony grounds since fewer parameters are in the model. The test for alpha=1, where a is the scale parameter in a Weibull model and s(a) is its standard deviation, is z = (a - 1)/s(a). A significant z score corresponds to rejecting the null hypothesis that alpha=1 (Blossfeld, Hamerle, & Mayer, 1989: 240).

      • Generalized gamma models often yield coefficients similar to Weibull models. In fact, because the hazard function assumes so many different forms depending on the value of their shape and scale parameters, gamma models may be equivalent to Weibull, exponential, log-normal, or other models. Generalized gamma models may be used to assess model fit using Wald (or z) tests for this reason, as described above. These tests take advantage of the fact that generalized gamma models estimate a scale parameter as well as a shape parameter, whereas other parametric models only estimate the latter.

      • Log-logistic models permit non-monotonic hazard functions. That is, the hazard function might increase, then decrease, or vice versa. Log-logistic models in log-linear form are similar to Weibull models, making the log of survival times a function of the vector of covariate effects (including the baseline hazard rate) plus a stochastic disturbance term. In log-logistic models, the distribution of the disturbance term is specified to be logistic. One effect is that log-logistic models may take on a greater variety of hazard curve shapes than Weibull models, and the shape of curves for different groups formed by the covariates may differ (that is, the proportional hazards assumption does not apply). A positive regression coefficient for a covariate means that covariate is associated with increases in expected duration times and with a lower hazard rate.

      • Log-normal models is a similar type of non-monotonic parametric model whose covariate coefficients are apt to be very similar to those from log-logistic models. In log-normal models, the distribution of the disturbance term is specified to be standard normal. Log-normal models also do not make the proportional hazards assumption.

      • Logit and probit models are similar to log-logistic and log-normal models respectively, but are used when the time variable is discrete. The logit function has thicker tails than the probit function and it approaches 0 and 1 more slowly. In practice, both types of model are very likely to yield equivalent substantive results. However, when the research hinges critically on understanding of the minority of cases in the tails of a distribution, choice of functional form could make a difference in substantive conclusions.

      • Cloglog models are similar to logit models but use the complementary log-log function and are characterized by a fatter tail than logit or probit models as the function approaches 0, and by more quickly approaching 1. This implies that "1" responses are rare events. Buckley & Westmoreland (2004) recommend cloglog for study of discrete rare events.

      • Gompertz models make the hazard rate an exponential function of duration times. Gompertz models assume proportional hazards. The exponential growth assumption fits with certain topics in demographic research (ex., mortality), but often does not describe real life in other social science domains.

    • Multiple episode and multiple state EHA models, Multiple EHA models cover the situations where the person, state, or other unit of analysis is at risk for more than one kind of event ("multiple states") and/or the same kind more than once ("multiple events").
      • Multiple destinations vs. multiple episodes. Multiple destinations refers to the "multiple states" situation where an episode may end in two or more different outcomes (ex., unemployment episodes may end by getting a job or by giving up looking and dropping out of the job market). Multiple episodes refers to the "multiple events" situation where the subject may experience the event more than once (ex., multiple episodes of unemployment over time).

      • Competing risks/Multiple destination models
        • Analysis. In multiple destination models, the occurrence of one type of event for an individual removes them from the risk set for other events. Thus in a competing risks model, when running models for each type of risk, cases are treated as right-censored if the failure event is not of the type being analyzed in the given run of the model. This makes the analysis episode-specific ("uniepisodal"), in contrast to multiple episode models discussed below. Alternatively, if the researcher can assume that the different types of risks are independent of one another, then multiple destination (multiple state or competing risks) models may be approached by separating the sample by end-state and running the EHA model (ex., an exponential model) separately for each of the states. See Blossfeld, Golsch, & Rohwer (2007: 101-109); Allison (2004: 379-381).

        • Output centers on comparing the hazard ratios between one state and another, and between any state and the hazard ratios computed for the pooled dataset as a whole.

        • Significance. Note that multiple destination models raise a special issue for significance. Significance depends in part on number of events (with higher n, a finding of significance becomes more likely). Since different types of events will have different numbers of episodes, as Blossfeld, Golsch, & Rohwer (2007: 108) note, "if one wants to compare the statistical significance of effects of covariates across competing transitions, one should always be aware that different event numbers might have an effect on statistical tests." That is, a given covariate may be significant for one type of event and not for another type because the two events have different n's.

      • Multiple episode models
        • Analysis. A focus of multiple episode models is understanding how covariate effects change across episodes, not just comparing effects between covariates as in uniepisodal models.
          • Number of transitions. Comparing across episodes requires that the n for each transition be large enough. For instance, in a study of spells of unemployment, the range may be from those with 0 to those with 14 spells, but the great majority may have only 4 or fewer spells. Because of low n, significance testing of covariate effects between transitions greater than 4 may be suspect. That is, analysis may need to be restricted to the first 4 transitions.

          • Output. By running an EHA model by transition (episode, spell), the output will generate hazard ratios for each covariate for each transition (ex., each spell of unemployment). By looking at these coefficients, one can see if they are time-constant (time-stationary) or time-dependent (time-varying). If the change process lacks time stationarity, some covariate effects my start out significant and become insignificant for later transitions, or vice versa. Note, though, that the number of subjects will decline with each transition, and this will make a finding of non-significance more likely simply for that reason.

        • Cox models. For Cox regression, see the separate discussion of frailty models and multiple events.

        • Multiple event models are discussed further in Blossfeld, Golsch, & Rohwer (2007: 109-115) and in Box-Steffensmeier & Jones (2004: chapter 10).

  
  
• Model fit should be assessed to assure that the most appropriate model has been selected. Model fit can be assessed through likelihood ratio tests, the Akaike information criterion, and Wald tests:
  • Log-likelihood and likelihood ratio tests (LR). Assuming maximum likelihood (ML) estimation has been used to generate log likelihoods, the closer the log likelihood (LL) to zero, the better the model fit. However, because LL is very sensitive to sample size, it is not used directly for comparison across models. For nested models, the likelihood ratio (LR) test is used. When LL is multiplied by -2, it assumes a chi-square distribution. The -2LL from the null model (the parameters for all covariates are 0) is subtracted from the -2LL for the researcher's full model and the resulting chi-square value is assessed, with degrees of freedom equal to the difference in the number of parameters for the two models. This likelihood ratio tests the null hypothesis that covariate coefficients are not different from zero. If the LR test is significant, the researcher concludes that the covariates are contributing to explanation.

  • AIC. The Akaike information criterion is the appropriate model fit test for non-nested models as well as nested. It penalizes a model for lack of parsimony (for overparameterization). The model with the lower AIC is the best fit.

  • Wald tests. The Wald test may be used to assess model fit for some parametric models. In general, the LR and AIC tests are preferred over the Wald test. Following Box-Steffensmeier and Jones (2004: 42), if one runs a generalize gamma model, discussed below, one obtains a shape parameter (sigma) and a scale parameter (kappa). If the scale parameter is not significantly different from 1 and the shape parameter is not significantly different from 1 by the Wald (or z) test, an exponential model is appropriate. If the scale parameter is not significantly different from 1 by the Wald (or z) test, a Wiebull model is appropriate. If the shape parameter is not significantly different from 1 by the Wald (or z) test, a gamma model is appropriate. If the scale parameter is not significantly different from 1 by the Wald (or z) test, a log-normal model is appropriate.

  • Analysis of residuals. As in any procedure, residuals reflect model error. Analysis of residuals is one criterion useful when assessing which is the most appropriate parametric model. For a discussion of analysis of residuals in EHA models, see Steffensmeier & Jones (2004: chapter 8), where five types of residuals are discussed, each based on a different mathematical definition of "residual":
    1. Cox-Snell residuals are the most common type. In a well-fitting model, Cox-Snell residuals will follow a standard exponential distribution. The model which best reproduces a standard exponential distribution of residuals is the best fitting model. If the researcher's model is correct, Cox-Snell residuals will have a unit exponential distribution with a hazard ratio of 1.0. A unit exponential distribution is demonstrated in a plot of Cox-Snell residuals against an estimate of the integrated hazard rate based on Cox-Snell residuals. To the extent that this plot forms a 45-degree straight line through the origin, the model is correct (Box-Steffensmeier & Jones, 2004: 120).

    2. Schoenfeld residuals are used to test the assumption of proportional hazards. Schoenfeld residuals "can essentially be thought of as the observed minus the expected values of the covariates at each failure time" (Steffensmeier & Jones, 2004: 121). There is a Schoenfeld residual for each subject for each covariate. The plot of Schoenfeld residuals against time for any covariate should not show a pattern of changing residuals for that covariate. If there is a pattern, that covariate is time-dependent.

    3. Martingale residuals are random among observations and have an expected value of 0. They can be used to assess the functional form (should it be linear, quadratic, cubic, etc.) of a covariate. If one plots martingale residuals on the Y axis and units of the covariate on the X axis, the regression line when residuals are predicted by the covariate should have a slope and intercept of 0 if the current functional form is acceptable (that is, no need to apply a square root, log, quadratic, or other transformation to the covariate). If many observations have high martingale residual values, the interpretation of hazard ratios is less reliable. Also, if martingale residuals on Y are plotted against the linear predictor (the right-hand side of the model equation) on X, there should be no pattern of correlation if the proportional hazards assumption is met. SPSS does not save Martingale residuals directly but they may be computed as mresid = event-haz_1, where event is the event variable and haz_1 is the variable saved under the Save option for the cumulative hazard function.

    4. Deviance residuals normalize martingale residuals to be symmetric around 0 rather than varying from minus infinity to plus 1. Using deviance residuals rather than martingale residuals when assessing covariate functional form makes plots easier and more intuitive to interpret. Negative deviance residuals mean predicted failure times are less than actual times, meaning that estimated hazard rates are overestimated. One can plot deviance residuals on the Y axis against duration time on the X axis; if a trend toward higher negative deviance residuals exists as one has longer duration times, then the Cox model is has a systematic bias, possibly due to violation of the proportional hazards assumption.

    5. Score residuals are covariate specific, with a score residual for each observation for each covariate. They are used to identify outliers and influential observations. A high absolute score residual means that observation has high influence on the regression coefficient for that covariate. The matrix of score residuals can be scaled by multiplying it times the variance-covariance matrix, resulting in scaled influence coefficients analogous to dfbeta coefficients used in regression models to identify influential cases. One may then plot scaled influence on the Y axis against ascending covariate values on the X axis, with spikes up or down representing influential cases for that covariate.

  • Graphical methods. See Blossfeld & Rohwer (1995).
    • Exponential models. A graph of -ln(S(t) against t should be a straight line if the distribution is consistent with an exponential model. Here, ln is the natural log function, S(t) is the sample survival function, and t is time.

    • Weibull models. A graph of ln[-ln(S(t))] against ln(t) should be a straight line if the distribution is consistent with a Weibull model.

    • Log-logistic models. A graph of ln[(1-S(t))/S(t)] against ln(t) should be a straight line if the distribution is consistent with a log-logistic model.

    • Log-normal models . A graph of Φ^-1[1 - S(t)] against ln(t) should be a straight line if the distribution is consistent with a log-normal model. Φ is phi, a standard normal function.




Assumptions

• Parametric models assume positing a particular shape of the hazard function. As noted by Buckley & Westerland (2004), in the past there has been over-reliance on parametric EHA models such as logit and probit which, unlike Cox models, require the researcher to specify duration dependence (the shape of the hazard rate over time), misspecification of which can lead to computed standard errors which are too large or too small, leading to errors of inference.

• Parametric models assume positing a relation of the covariates to the hazard function. While Cox regression models are widely termed "proportional hazard models," in fact most parametric models are PH models as well. The researcher must correctly posit whether the model is best parameterized as a PH model, and AFT model, or rarely, some other type.

• Unobserved heterogeneity is the EHA term referring to model specification. In EHA, as in other regression models, parameter estimates may be strongly biased by omission of important covariates from the model or by inclusion in the model of variables correlated with the causally important covariates but which themselves are only spuriously related to the event. When causally significant covariates are omitted from the model, hazard functions tend to spuriously decline with time. This is because observations in the high-risk subgroup of the unobserved covariate experience the event earlier, leaving in the risk pool the observations which are low-risk on the unobserved covariate. While there are methods of attempting to address unobserved heterogeneity, such as frailty models (which use a random error term to reflect observation-specific unknown variables) , no method approaches having the correct variables in the model to begin with.

• See the assumptions section for Cox regression. In particular, note the assumption of proportional hazards, which applies not just to Cox models but also to most widely-used paramtetric EHA models, such as the Weibull.

• See also time series analysis.




Frequently Asked Questions

• Can't I use regular time series methods such as OLS or logistic regression as long as I adjust for autocorrelation?
  This is answered in the section on Cox regression.

• Should I use a parametric EHA model or a semiparametric Cox model?
  Cox is usually preferred because its assumptions are less restrictive, not requiring the researcher to posit a shape for the distribution of the baseline hazard function. Also, if time-varying covariates are present, these can be handled easily in Cox models. However, if a parametric model (ex., the Weibull) has substantially lower standard errors for the covariate parameters, then the parametric model would be preferred.

• How are parametric models run in Stata?
  The general format in Stata is: streg (varlist), dist(distribution) time. For ex.: streg score age income, dist(weib) time; or dist(gamma); etc.

• Shouldn't censored cases just be dropped from analysis, since for these cases we do not know when the event will occur?
  No. To do so would risk bias in sample selection. Event history methods adequately handle right censoring. The uncensored cases define the failure density function. Both the censored and uncensored cases can be used to define the survival function. See further discussion by Box-Steffensmeier and Jones (2004: 18-19).

• How are data organized for event history analysis?
  Each row is an episode. One column, usually the first, is an id variable to identify all the episodes that are associated with one individual or other unit of analysis such as organization. If there are multiple episodes per unit of analysis, there is also an episode id variable numbered 1 to n. Then there are time-start and time-finish variables containing the start and finish times for each episode (not duration, but number of time units from origin for the start-time and finish-time). Then there will be additional columns for other variables (ex., gender) thought to affect the duration of episodes. Data organization specific to cox regression is discussed in that section. Details of data entry into Stata, including do file programs, are discussed by Blossfeld, Golsch, and Rohwer (2007) in their Chapter 2.

• I have data on a certain number of events. Is it acceptable to fill in the other periods with zeros, to indicate non-events?
  This is acceptable if the zeros indicate true absence of activity. If, however, the zeros reflect irregular media coverage (notorious for African events, for instance) then one has non-random measurement error and valid time series analysis may be impossible.




Bibliography

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Copyright 2007, 2008 by G. David Garson.
Last update 3/29/08.