Weibull regression is a parametric survival analysis method that models time-to-event data. It assumes survival times follow a Weibull distribution and estimates a shape parameter (k) and scale parameter (λ). A regression coefficient (β) captures the effect of a predictor on survival time. The model can be expressed as an Accelerated Failure Time (AFT) model, where predictors directly accelerate or decelerate the time scale.

Cox regression is semi-parametric and does not assume a specific distribution for survival times, making it more flexible. Weibull regression is fully parametric — it assumes survival times follow a Weibull distribution. This assumption makes Weibull regression more efficient and allows extrapolation beyond the observation period. Cox regression is preferred when the distributional form is unknown; Weibull is preferred when the assumption can be verified and the shape parameter has scientific meaning.

The shape parameter k (also written as α or p) determines how the hazard rate changes over time. When k < 1, hazard decreases — this is called the "infant mortality" or "early failure" pattern and is common in quality defects or ecological stress responses. When k = 1, hazard is constant (the exponential distribution). When k > 1, hazard increases — indicating aging, accumulating damage, or wear-out failure. Biologically, k > 1 in aging studies means subjects become more vulnerable as they get older.

The time ratio is the most direct effect size in an AFT model. TR = exp(β) tells you how many times longer (or shorter) survival time is in the exposed group compared to the reference group. TR = 1.5 means the exposed group survives 50% longer. TR = 0.7 means the exposed group experiences the event 30% sooner (30% shorter survival). TR = 1.0 means no difference. If the 95% confidence interval for TR excludes 1.0, the effect is statistically significant.

Censoring occurs when the event has not been observed by the end of the study period (right-censoring) or the subject dropped out. Censored observations contribute partial information — we know the event had not occurred by the censoring time, even if we don't know when it will occur. Weibull regression handles censoring correctly through the likelihood function, which uses the survival function S(t) for censored observations and the PDF f(t) for observed events. This ensures unbiased parameter estimates even with substantial censoring.

The Weibull probability plot is the primary visual check — plot log(−log(S(t))) versus log(t). If the points fall on a roughly straight line, the Weibull distribution is appropriate. The slope of the line estimates the shape parameter k. Non-linearity suggests a different distribution (log-normal, log-logistic, gamma). You can also compare AIC/BIC across different parametric distributions (exponential, log-normal, log-logistic) to select the best-fitting model.

Yes. Multiple Weibull regression extends the AFT model to include two or more covariates: log(λᵢ) = μ + β₁x₁ + β₂x₂ + … Each βⱼ coefficient represents the effect of predictor j on log-survival time, holding all other predictors constant. You can include continuous predictors (age, dose, temperature), categorical predictors (treatment group, sex), and interaction terms. This calculator handles a single binary predictor (group comparison); for multiple regression, use R's survreg() function with the Weibull distribution.

AIC (Akaike Information Criterion) = −2ℓ + 2p, where ℓ is the maximised log-likelihood and p is the number of parameters. Lower AIC indicates better model fit, penalised for complexity. To compare models, fit the same data with different distributions (Weibull, exponential, log-normal) and choose the one with the lowest AIC. A ΔAIC > 2 is typically considered a meaningful difference. BIC applies a stronger penalty for model complexity and is preferred for large samples.

Report: shape parameter (k), scale parameter (λ), coefficient (β) with SE, Wald z-statistic, p-value, time ratio (TR) with 95% CI, and sample size with event count. Example: "A Weibull AFT regression model revealed that group membership significantly predicted time to event (β = 0.34, SE = 0.12, z = 2.83, p = .005; TR = 1.40, 95% CI [1.11, 1.77]). The shape parameter k = 1.85 indicated increasing hazard over time (n = 40, events = 34, AIC = 214.3)."

The scale parameter λ (lambda) is also called the "characteristic life" or "characteristic time." It is the time at which approximately 63.2% of subjects have experienced the event, regardless of the shape parameter k. Larger λ indicates longer survival or greater reliability. In an AFT model, the predictor shifts λ multiplicatively: λ for the exposed group = λ₀ × exp(β), where λ₀ is the scale for the reference group. It is the primary measure of location (central tendency) for the Weibull survival distribution.

A common rule of thumb is at least 10 events per parameter estimated (EPP rule). For a simple two-group Weibull regression (3 parameters: k, μ, β), you need at least 30 events. With high censoring rates (> 40%), larger samples are needed. Very small event counts (< 15) can produce unstable parameter estimates and unreliable confidence intervals. For reliable AIC model comparison, at least 50 events per distribution being compared is recommended.

Both are valid representations of the same Weibull model — they are mathematically equivalent for the Weibull distribution (the only distribution that is simultaneously a PH and AFT model). In the AFT parameterization: exp(β) = time ratio — directly multiplies survival time. In the PH parameterization: exp(−β/σ) = hazard ratio — multiplies the hazard rate. This dual nature makes the Weibull model particularly interpretable. This calculator uses the AFT parameterization as it is most commonly used in biological and medical research.