We propose a new methodological framework for the analysis of hierarchical functional data when the functions at the lowest level of the hierarchy are correlated. For small data sets, our methodology leads to a computational algorithm that is orders of magnitude more efficient than its closest competitor (seconds versus hours). For large data sets, our algorithm remains fast and has no current competitors. Thus, in contrast to published methods, we can now conduct routine simulations, leave-one-out analyses, and nonparametric bootstrap sampling. Our methods are inspired by and applied to data obtained from a state-of-the-art colon carcinogenesis scientific experiment. However, our models are general and will be relevant to many new data sets where the object of inference are functions or images that remain dependent even after conditioning on the subject on which they are measured. Supplementary materials are available at Biostatistics online.

SUMMARY

For many diseases, it is difficult or impossible to establish a definitive diagnosis because a perfect “gold standard” may not exist or may be too costly to obtain. In this paper, we propose a method to use continuous test results to estimate prevalence of disease in a given population and to estimate the effects of factors that may influence prevalence. Motivated by a study of human herpesvirus 8 among children with sickle-cell anemia in Uganda, where 2 enzyme immunoassays were used to assess infection status, we fit 2-component multivariate mixture models. We model the component densities using parametric densities that include data transformation as well as flexible transformed models. In addition, we model the mixing proportion, the probability of a latent variable corresponding to the true unknown infection status, via a logistic regression to incorporate covariates. This model includes mixtures of multivariate normal densities as a special case and is able to accommodate unusual shapes and skewness in the data. We assess model performance in simulations and present results from applying various parameterizations of the model to the Ugandan study.