A joint hazard and time scaling model to compare survival curves

To provide a more general method for comparing survival experience, we propose a model that independently scales both hazard and time dimensions. To test the curve shape similarity of two time-dependent hazards, h(t)(t) and h(2)(t), we apply the proposed hazard relationship, h(12)(tK(t))/h(1)(t) = K-h, to h(1). This relationship doubly scales h(1) by the constant hazard and time scale factors, K-h and K-t, producing a transformed hazard, h(12), with the same underlying curve shape as h(1). We optimize the match of h(12) to h(2) by adjusting K-h and K-t. The corresponding survival relationship S-12(tK(t)) = [S-1(t)]KtKh transforms S-1 into a new curve S-12 of the same underlying shape that can be matched to the original S-2. We apply this model to the curves for regional and local breast cancer contained in the National Cancer Institute's End Results Registry (1950-1973), Scaling the original regional curves, h(1) and S-1 with K-t = 1.769 and K-h = 0.263 produces transformed curves h(12) and S-12 that display congruence with the respective local curves, h(2) and S-2. This similarity of curve shapes suggests the application of the more complete curve shapes for regional disease as templates to predict the longterm survival pattern for local disease, By extension, this similarity raises the possibility of scaling early data for clinical trial curves according to templates of registry or previous trial curves, projecting long-term outcomes and reducing costs, The proposed model includes as special cases the widely used proportional hazards (K-t = 1) and accelerated life (KtKh = 1) models.