Bias correction for estimates from linear excess relative risk models in small case-control studies.

Epidemiologic studies conducted to quantify the impact of radiation dose d on an outcome typically model the hazard ratio (HR) for association using a linear term, HR(d)=1+βd , via a linear excess relative risk (ERR) model, based on biological considerations. To study associations of risk of a second cancer with radiation treatment for a first cancer, several nested case-control designs to estimate β have been proposed that use refined doses received by different locations in the organ of interest. Here we first evaluated the small sample bias in maximum likelihood estimates of β for the linear ERR model using location-specific radiation doses in simulations. As we found substantial upward bias for studies of realistic sample sizes (more than 50% relative bias for studies with 75 cases), we also proposed and investigated several approaches to correct this bias. We studied first and second order jackknife bias corrections and we derived a modified set of score functions under retrospective case-control sampling, from which we directly obtained bias-corrected estimates. In simulations based on doses from a study of stomach cancer among testicular cancer survivors and synthetically generated data, neither the first nor second order jackknife bias correction performed well. Estimates based on the modified score equations corrected the bias much better, albeit not completely, and were numerically much more stable.