This paper discusses regression analysis of multivariate current status failure time data (The Statistical Analysis of Interval-censoring Failure Time Data. Springer: New York, 2006), which occur quite often in, for example, tumorigenicity experiments and epidemiologic investigations of the natural history of a disease. For the problem, several marginal approaches have been proposed that model each failure time of interest individually (Biometrics 2000; 56:940-943; Statist. Med. 2002; 21:3715-3726). In this paper, we present a full likelihood approach based on the proportional hazards frailty model. For estimation, an Expectation Maximization (EM) algorithm is developed and simulation studies suggest that the presented approach performs well for practical situations. The approach is applied to a set of bivariate current status data arising from a tumorigenicity experiment.