Incorporating the time dimension in receiver operating characteristic curves: a case study of prostate cancer

Early diagnosis of disease has potential to reduce morbidity and mortality. Biomarkers may be useful for detecting disease at early stages before it becomes clinically apparent. Prostate-specific antigen (PSA) is one such marker for prostate cancer. This article is concerned with modeling receiver operating characteristic (ROC) curves associated with biomarkers at various times prior to the time at which the disease is detected clinically, by two methods. The first models the biomarkers statistically using mixed-effects regression models, and uses parameter estimates from these models to estimate the time-specific ROC curves. The second directly models the ROC curves as a function of time prior to diagnosis and may be implemented using software packages with binary regression or generalized linear model routines. The approaches are applied to data from 71 prostate cancer cases and 71 controls who participated in a lung cancer prevention trial. Two biomarkers for prostate cancer were considered: total serum PSA and the ratio of free to total PSA. Not surprisingly, both markers performed better as the interval between PSA measurement and clinical diagnosis decreased. Although the two markers performed similarly eight years prior to diagnosis, it appears that total PSA performed better than the ratio measure at times closer to diagnosis. The area under the ROC curve was consistently greater for total PSA than for the ratio four and two years prior to diagnosis and at the time of diagnosis.