Modeling observations with a detection limit using a truncated normal distribution with censoring

When data are collected subject to a detection limit, observations below the detection limit may be considered censored. In addition, the domain of such observations may be restricted; for example, values may be required to be non-negative.

We propose a method for estimating population mean and variance from censored observations that accounts for known domain restriction. The method finds maximum likelihood estimates assuming an underlying truncated normal distribution.

We show that our method, tcensReg, has lower bias, Type I error rates, and mean squared error than other methods commonly used for data with detection limits such as Tobit regression and single imputation under a range of simulation settings from mild to heavy censoring and truncation. We further demonstrate the consistency of the maximum likelihood estimators. We apply our method to analyze vision quality data collected from ophthalmology clinical trials comparing different types of intraocular lenses implanted during cataract surgery. All of the methods yield similar conclusions regarding non-inferiority, but estimates from the tcensReg method suggest that there may be greater mean differences and overall variability.

In the presence of detection limits, our new method tcensReg provides a way to incorporate known domain restrictions in dependent variables that substantially improves inferences.