In vaccine clinical trials, immunologic responses sometimes can not be accurately measured by bioassays. For example, a serial dilution assay usually reports the range of the response instead of the exact value. In some other assays, the measurement is not available if the response is lower than the assay's detection limit. In both cases, the measurements are censored. We are interested in computing the confidence interval for the correlation coefficient of two assay measurements that are subject to censoring. We propose using the maximum likelihood method to estimate the correlation coefficient, and constructing its confidence interval based on the second-order Taylor's expansion of the Fisher Z transformation. The method can be viewed as an extension of the Fisher Z transformation to the case of censored data. Extensive simulations show that the proposed method provides satisfactory coverage probabilities under finite sample sizes. The proposed method performs well compared with existing methods, but it is computationally much simpler. In addition, the proposed method works with many types of censored data in a similar way. Furthermore, we proposed an Monte Carlo exact test to assess the goodness-of-fit of the model.