This article compares the estimates produced by a number of solutions to the identifiability problem in age-period-cohort models using a series of disease rates with known structure. The results suggest that only those methods that are based on the estimable functions such as curvatures can be recommended for use in all circumstances. The other common approaches that give parameter estimates that are easier to interpret all have induced bias in the estimates. In particular methods based on the minimization of a penalty function to achieve identifiability are only of use if there is no change in the rates with time. Any drift in the rates tends to be expressed as a cohort-based trend. The methods based on individual records introduce a bias if there is a strong age effect in the direction of a decreasing cohort trend and a compensating increase based on period effects. The nonparametric testing method has little power to detect trends in the rates in small tables but ascribes a strong drift in the rates to both period and cohort trends. With careful interpretation, all methods estimate nonlinear components correctly.