Calculator Inputs
Generated Data Preview
This preview shows the first rows of the active result set. Use CSV export for the complete dataset.
Example Data Table
Example values below use an exponential model with rate λ = 0.12.
| Time | Hazard h(t) | Survival S(t) | Cumulative Hazard H(t) |
|---|---|---|---|
| 0 | 0.120000 | 1.000000 | 0.000000 |
| 2 | 0.120000 | 0.786628 | 0.240000 |
| 4 | 0.120000 | 0.618783 | 0.480000 |
| 6 | 0.120000 | 0.486752 | 0.720000 |
| 8 | 0.120000 | 0.382893 | 0.960000 |
Formula Used
The calculator generates hazard, survival, density, and cumulative hazard values from the selected time-to-event model.
General relationships
h(t) = f(t) / S(t)
H(t) = ∫ h(u) du from 0 to t
S(t) = exp(-H(t))
F(t) = 1 - S(t)
Exponential
h(t) = λ, H(t) = λt, S(t) = exp(-λt)
Weibull
h(t) = (k / λ) (t / λ)^(k-1)
H(t) = (t / λ)^k, S(t) = exp(-(t / λ)^k)
Gompertz
h(t) = b exp(ηt)
H(t) = (b / η) (exp(ηt) - 1)
Log-logistic
h(t) = (β / α)(t / α)^(β-1) / (1 + (t / α)^β)
H(t) = ln(1 + (t / α)^β), S(t) = 1 / (1 + (t / α)^β)
How to Use This Calculator
- Select a distribution that matches your event process.
- Enter the relevant parameters for that distribution.
- Set the time start, time end, and time step.
- Enter an observed time to inspect point estimates.
- Press Calculate and Plot to generate results.
- Review summary cards, the plot, and the data table.
- Download CSV for raw values or PDF for reporting.
FAQs
1. What does the hazard function represent?
It shows the instantaneous event risk at time t, assuming the subject or unit has survived up to that same time.
2. Why is hazard different from probability?
Probability measures chance over an interval. Hazard measures the immediate event rate at a specific time, conditional on survival just before that time.
3. When should I use the Weibull model?
Use Weibull when hazard may increase, decrease, or stay constant. Its shape parameter makes it flexible for reliability and survival analysis.
4. What does a constant hazard mean?
A constant hazard means the instantaneous event rate does not change over time. The exponential model is the standard constant-hazard case.
5. Why can the hazard be very high near time zero?
Some parameter choices create a steep early failure pattern. In those cases, the hazard may approach infinity at the lower boundary.
6. What is cumulative hazard used for?
Cumulative hazard summarizes the total accumulated event intensity up to time t. It is often easier to model and interpret alongside survival.
7. What does the observed time field do?
It evaluates hazard, survival, density, and cumulative hazard at one chosen time point, giving a compact interpretation summary.
8. Can I use the exported CSV in other tools?
Yes. The CSV contains time, hazard, survival, density, and cumulative hazard columns, making it easy to import into spreadsheets or analysis software.