CVEN9407 Chap.11 Survival Analysis of Time Between Crashes
Survival Analysis of Time Between Crashes
Survival analysis asks how long until the next crash rather than how many crashes occur. This chapter shows how to build the gaps between consecutive crashes, defines the survivor and hazard functions, and compares the constant hazard of the exponential model with the rising or falling hazards of the Weibull.
It introduces the baseline hazard and the Cox proportional hazards model, works median gaps and a hazard ratio for an invented area, and covers censoring and the proportional hazards check.
The brief places survival analysis in Weeks 10 to 11 and asks for a short report of 4 to 6 pages that is folded into the final submission.
It is the method most students meet for the first time in this course, so a clear explanation of the hazard idea, in words a council officer would follow, earns as much as the estimation itself.
What this chapter covers
- 01
Building gaps between consecutive crashes
- 02
Survivor and hazard functions
- 03
Exponential and Weibull models
- 04
Baseline hazard and the Cox model
- 05
Hazard ratios
- 06
Censoring and the proportional hazards check
Worked example · free
Survival probability and median gap at a busy corridor
- 1Hazard: lambda = 1 / 5 = 0.2 crashes per day, constant over the gap.
- 1S(3) = e to the power -0.2 x 3 = e to the power -0.6 = 0.549.
- 1Median gap = ln 2 / 0.2 = 0.693 / 0.2 = 3.47 days.
Key terms
- Survivor function
- The survivor function gives the probability that the waiting time to the next event exceeds a given value.
- Hazard function
- The hazard function is the instantaneous rate of an event at a given time, given that it has not yet occurred.
- Baseline hazard
- The baseline hazard is the hazard for a case with all covariates at zero, left unspecified in the Cox model.
- Censoring
- Censoring occurs when a waiting time is only partly observed, such as a gap still open when the data end.
- Weibull model
- The Weibull model is a duration model whose shape parameter lets the hazard rise, stay constant or fall over time.
Survival Analysis of Time Between Crashes FAQ
What does a hazard ratio greater than one mean for crashes?
It means the condition raises the instantaneous chance of the next crash at every point in a gap. A ratio of 1.5 implies a 50% higher hazard, so gaps between crashes tend to be shorter while that condition holds.
Exponential, Weibull or Cox: which should I fit?
Fit the exponential as the simple benchmark, the Weibull to test whether risk rises or falls with time since the last crash, and the Cox model when you want covariate effects without committing to a hazard shape. Compare the parametric fits by likelihood or BIC.
How do I handle the last gap in my data?
Treat it as censored: the study period ended before the next crash, so you know only that the gap was at least as long as observed. Survival software takes a censoring indicator for exactly this case.
Why model gaps between crashes instead of counts?
Gaps reveal timing that counts hide. Two areas with the same yearly total can differ in whether crashes cluster or arrive evenly, and a hazard model shows how conditions such as season or weather change the waiting time to the next crash.
What does the proportional hazards assumption mean?
It assumes that a covariate multiplies the hazard by the same factor at every point in the gap, so hazard curves for different groups keep a constant ratio over time. If the ratio changes, for example strongly early in a gap and weakly later, a single hazard ratio misleads.
Assessment move
Build the gap file carefully from sorted crash timestamps and plot its histogram first. Fit the exponential and Weibull models, read the Weibull shape parameter, and then fit a Cox model with two or three conditions. Translate each hazard ratio into a statement about waiting times in your area. Keep the Weibull shape estimate in your model log, since it tells you whether risk builds or fades after a crash.
Working through Survival Analysis of Time Between Crashes in CVEN9407? Sia is AskSia’s AI Engineering tutor — ask any CVEN9407 Survival Analysis of Time Between Crashes question and get a clear, step-by-step explanation grounded in how CVEN9407 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
