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The Series Hazard Model: An Alternative to Time Series for Event Data

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Abstract

An important pursuit by a body of criminological research is its endeavor to determine whether interventions or policy changes effectively achieve their intended goals. Because theories predict that interventions could either improve or worsen outcomes, estimators designed to improve the accuracy of identifying program or policy effects are in demand. This article introduces the series hazard model as an alternative to interrupted time series when testing for the effects of an intervention on event-based outcomes. It compares the two approaches through an example that examines the effects of two interventions on aerial hijacking. While series hazard modeling may not be appropriate for all event-based time series data or every context, it is a robust alternative that allows for greater flexibility in many contexts.

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Notes

  1. While different transfer functions can be used to model the intervention, all rely on an indicator marking the beginning of the intervention period (Cook and Campbell 1979).

  2. Kelly and Lim (2000) discuss three approaches to account for within subject correlation: conditional, marginal, and random effects. We use the conditional approach when we include time dependent independent variables to capture the structure of dependence across the recurrent failures. For example, the analyst may include a variable that measures the number of previous failures. The marginal approach, also referred to as variance-correction, models the data as if all failures are independent, and then calculates the robust standard errors by using a sandwich estimator (Kelly and Lim 2000; Box-Steffensmeier and Zorn 2002). The random effects approach, also referred to as frailty models, introduces a random covariate into the model which accommodates the dependence across repeated failures (Kelly and Lim 2000).

  3. Analysis of high frequency events would likely have much tied data. One strategy to avoid this problem would be to estimate the time until the next N events instead of the subsequent event.

  4. As it turns out, in this example the substantive findings are the same whether the (0, 0] cases are deleted or recoded and retained.

  5. The models were re-estimated after extending the end date by monthly increments for up to 36 months. Graphical displays of the estimates with 95% confidence bounds show that the findings were robust to any end date.

  6. Arey (1972) records the year of this event as 1930, however, current FAA records record it as August 31, 1931.

  7. See Brandt and Williams (2001) and Brandt et al. (2000) for introductions to the Poisson AR(p) model and the Poisson exponentially weighted moving average model, which better accommodate non-normal data in time series.

  8. Success density is measured by taking the current and six previous flights, and calculating the proportion of those flights that were successful over the number of days spanning the seven events and then calculating the natural logarithm after adding a value of 0.05 to avoid missing values. Thus, a large success density indicates that most events were successful over a relatively short period. This measure differs from the one used in Dugan et al. (2005), which only used the three most recent hijackings. I found that the one based on 7 flights better demonstrates the model and the natural logarithm is more linearly related to the Martingale residuals (Cleves et al. 2004). Both serve the same substantive purpose.

  9. The natural logarithm of the count of days plus 0.05 is used because it produces a more linear relationship with the Martingale residuals (Cleves et al. 2004).

  10. The results are similar when the intervention is lagged.

  11. The asymptotic change in hijackings was calculated using \( {\frac{\omega }{{\left( {1 - \delta } \right)}}} \), where ω is the coefficient estimate for Metal Detectors and δ is the estimated rate parameter (McDowall et al. 1980).

  12. For the restricted model, AIC = 5067.388 and BIC = 5100.361; and for the unrestricted model, AIC = 5054.063 and BIC = 5091.747.

  13. In a separate equation that includes the Metal Detector interaction with Monthly Count, the findings remain virtually the same. A likelihood ratio test concludes that model 5 is a better fit than the more flexible model that includes the Metal Detector interaction (p = 0.2204). Furthermore, when comparing model 5 to a more restrictive model that only includes the two policy interaction, model 5 is once again preferred (p = 0.0018).

  14. Temporal covariates can also be included in the series model. In another paper by LaFree et al. (2009), the authors model annual measures of the number of troops in Northern Ireland and the number of reported crimes. Because republican terrorists also combated loyalist terrorist groups we included the monthly total of loyalist attacks from the previous calendar month, which was found to be marginally significant.

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  1. University of Maryland, College Park, MD, USA

    Laura Dugan

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  1. Laura Dugan
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Correspondence to Laura Dugan.

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Dugan, L. The Series Hazard Model: An Alternative to Time Series for Event Data. J Quant Criminol 27, 379–402 (2011). https://doi.org/10.1007/s10940-010-9127-1

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