logminuslogTransformation

logminuslogTransformation is a statistical transformation defined as y = log(-log(S(t))), where S(t) is the survival function P(T > t). It is also referred to as the complementary log-log transformation (cloglog) when applied to probabilities p in (0,1). In survival contexts, p typically plays the role of S(t), so the transformation becomes y = log(-log(S(t))). This function maps survival probabilities from (0,1] to the real numbers and is monotone in t.

Under a Weibull model, S(t) = exp(-(t/α)^β). Applying the logminuslogTransformation yields log(-log(S(t))) = β log t − β log α, which is linear in log t with slope β and intercept −β log α. This linearity enables graphical diagnostics and straightforward linear regression to estimate Weibull shape and scale parameters. The transformation is also useful for comparing survival curves and assessing goodness-of-fit for Weibull or related hazard structures.

Computation and cautions: with an estimated survival function Ŝ(t) (for example from Kaplan-Meier estimates), compute y_i = log(-log(Ŝ(t_i))) and examine its relationship with log t_i or fit a linear model. The transform is undefined when S(t) = 0 and can be unstable near extreme survival probabilities (near 0 or 1), particularly with limited data or heavy censoring. It is most appropriate when a Weibull-like hazard is plausible; otherwise, alternative transformations or nonparametric methods may be preferable.