Several natural history studies of the Acquired Immunodeficiency Syndrome (AIDS) have been based on the follow-up of individuals infected with the AIDS virus prior to enrollment. The natural time scale for studying the preclinical course of AIDS is the time since first infection; however, in these studies the time at infection was a random unknown quantity. The biases inherent in using follow-up time instead of time from infection are investigated for estimation of both the cumulative distribution function and the hazard ratio for proportional hazards models with both fixed and time-dependent covariates. Although the magnitudes of the biases depend on the shape of the epidemic curve, a number of bounds on the biases are established. These results are useful for interpreting prevalent cohort studies and then comparing them with studies on newly infected individuals in order to assess consistency of results across studies.