A method of estimating failure time distributions for data collected from independent groups of correlated individuals is developed. This method is appropriate for commonly interval censored data (e.g., when individuals are inspected at regular common intervals) or exact time data, including possibly right censored data. The technique improves upon previously published methods by allowing for large and variable group sizes, heterogeneous correlation structures, and the incorporation of explanatory variable information. Both parametric and nonparametric failure time models can be estimated, and correlations may be modeled as well;The outcome for any individual at risk during a specific time interval is modeled as a conditional binary random variable indicating the failure or success of the individual during a specific time interval given success in the previous intervals. For each time interval, a separate vector of binary responses is constructed for each group, consisting of the responses for individuals belonging to the group who are at risk at the beginning of the interval and not censored during the interval. The elements of the corresponding mean vector are hazard probabilities and are thus functions of the failure time distribution parameters. The covariance matrix for each binary response vector is a function of the corresponding mean vector and parameters describing the correlations among the elements of the observed response vector. A Gauss-Newton algorithm is used to implement multivariate nonlinear least squares estimation of the parameters for the failure time distribution. These estimators are shown to have a joint asymptotic normal distribution under mild regularity conditions when the Gauss-Newton iterations are initiated with consistent estimates of the mean model and correlation parameters;The new methodology is applied to commonly interval censored data from a study comparing the effectiveness of three smoking cessation programs, and to an exact time data set assessing the developmental responses of rat pups in relation to prenatal doses of methylmercuric chloride. Research is also presented on the properties and performance of estimators of correlation coefficients for clustered binary data.