Calculator Inputs
Example Data Table
This sample table shows how different lifetime models can produce different survival and hazard patterns for the same observation time.
| Model | Parameters | Time t | S(t) | F(t) | h(t) |
|---|---|---|---|---|---|
| Exponential | λ = 0.18 | 5.00 | 0.4066 | 0.5934 | 0.1800 |
| Weibull | Scale λ = 8, Shape k = 1.6 | 5.00 | 0.6241 | 0.3759 | 0.1509 |
| Log-Logistic | Scale α = 7, Shape β = 2.4 | 5.00 | 0.6916 | 0.3084 | 0.1480 |
Formula Used
S(t) = P(T > t) = 1 - F(t)
S(t) = e-λt
F(t) = 1 - e-λt
h(t) = λ
H(t) = λt
S(t) = e-(t/λ)k
F(t) = 1 - e-(t/λ)k
h(t) = (k/λ)(t/λ)k-1
H(t) = (t/λ)k
S(t) = 1 / (1 + (t/α)β)
F(t) = (t/α)β / (1 + (t/α)β)
h(t) = (β/α)(t/α)β-1 / (1 + (t/α)β)
H(t) = ln(1 + (t/α)β)
Median survival is the time where S(t) = 0.5. The requested failure-time quantile solves F(t) = p, where p is your target failure probability.
How to Use This Calculator
- Select a distribution that matches your time-to-event assumption.
- Enter the model parameters shown for that distribution.
- Provide the evaluation time, graph range, and number of plot points.
- Set a target failure percentage to estimate a quantile time.
- Press the calculate button to view summary metrics, the curve table, and the interactive graph.
FAQs
1. What does the survival function measure?
It measures the probability that the event time T remains greater than a chosen time t. In practical terms, it shows the chance a unit, person, or process survives beyond that moment.
2. How is survival different from cumulative failure?
Survival is S(t), the probability of lasting beyond time t. Cumulative failure is F(t), the probability the event has already happened by time t. They always add to one.
3. When should I use the exponential model?
Use it when the hazard rate is constant over time. It is common for simple reliability models, quick benchmarks, and situations where memoryless behavior is a reasonable assumption.
4. Why would I choose a Weibull model?
Weibull is flexible. With different shape values, it can represent decreasing, constant, or increasing hazard. That makes it useful for aging products, wear-out patterns, and general life-data analysis.
5. What is special about the log-logistic model?
It can represent non-monotonic hazard behavior in some settings and often appears in medical and economic duration analysis. Its survival curve also has a closed-form expression that is easy to interpret.
6. What does the hazard value tell me?
Hazard is the instantaneous event rate at time t, conditional on surviving just before t. It is not a probability by itself, but it shows the local intensity of risk.
7. What does the failure quantile output mean?
It is the time at which the chosen percentage of the population is expected to have failed or experienced the event. For example, a 50% failure quantile is the median time.
8. Can this replace Kaplan-Meier analysis with censored data?
No. This page evaluates parametric survival models from supplied parameters. Kaplan-Meier estimates survival directly from observed event and censoring data rather than from a chosen distribution.