Pseudo-likelihood Estimation of Multivariate Normal Parameters in the Presence of Left-Censored Data

Abstract

Environmental data often include left-censored values reported to be less than some limit of detection (LOD). While simple imputation of a specific value such as LOD/2 is common, maximum likelihood methods accounting for censoring provide alternate ways of analyzing such data. Concentration levels of contaminants in water, for example, are typically modeled with normal or lognormal distributions. Corresponding maximum likelihood estimates (MLEs) of means and variances in univariate analyses can be obtained from standard software packages; however, multivariate analyses may be more appropriate when multiple measurements come from the same entity. For example, measures of several dissolved trace metals may be derived from freshwater stream samples. In less-polluted areas, one or more of these measures fall below the LOD. An index of overall contamination may be formed as a linear function of these measures. The desire to estimate this index led to the need to estimate the parameters in the presence of nondetects, which led to our proposed method. We propose a pseudo-likelihood method utilizing pairs of variables that provides MLEs of mean and unstructured covariance parameters corresponding to a multivariate normal or lognormal distribution in the presence of left-censored data. In conducting hypothesis tests and estimating functions of MLEs with standard errors, we apply this method to trace metal concentration data collected from freshwater streams across Virginia.