Assessment of the effect of low dose and low-dose-rate exposure depends critically on extrapolation from groups exposed at high dose and high-dose rates such as the Japanese atomic bomb survivor data, and has often been achieved via application of a dose and dose-rate effectiveness factor (DDREF). An important component of DDREF is the factor determining the effect of extrapolation of dose, the so-called low-dose extrapolation factor (LDEF). To assess LDEF models linear (or linear quadratic) in dose are often fitted. In this report LDEF is assessed via fitting relative rate models that are linear or linear quadratic in dose to the latest Japanese atomic bomb survivor data on solid cancer, leukemia and circulatory disease mortality (followed from 1950 through 2003) and to data on solid cancer, lung cancer and urinary tract cancer incidence. The uncertainties in LDEF are assessed using parametric bootstrap techniques. Analysis is restricted to survivors with <3 Gy dose. There is modest evidence for upward curvature in dose response in the mortality data. For leukemia and for all solid cancer excluding lung, stomach and breast cancer there is significant curvature (P < 0.05). There is no evidence of curvature for circulatory disease (P > 0.5). The estimate of LDEF for all solid cancer mortality is 1.273 [95% confidence intervals (CI) 0.913, 2.182], for all solid cancer mortality excluding lung cancer, stomach cancer and breast cancer is 2.183 (95% CI 1.090, >100) and for leukemia mortality is 11.447 (95% CI 2.390, >100). For stomach cancer mortality LDEF is modestly raised, 1.077 (95% CI 0.526, >100), while for lung cancer, female breast cancer and circulatory disease mortality the LDEF does not much exceed 1. LDEF for solid cancer incidence is 1.186 (95% CI 0.942, 1.626) and for urinary tract cancer is 1.298 (95% CI <0, 7.723), although for lung cancer LDEF is not elevated, 0.842 (95% CI 0.344, >100).