Model-based imputation approach for data analysis in the presence of non-detects

A model-based multiple imputation approach for analyzing sample data with non-detects is proposed. The imputation approach involves randomly generating observations below the detection limit using the detected sample values and then analyzing the data using complete sample techniques, along with suitable adjustments to account for the imputation. The method is described for the normal case and is illustrated for making inferences for constructing prediction limits, tolerance limits, for setting an upper bound for an exceedance probability and for interval estimation of a log-normal mean. Two imputation approaches are investigated in the paper: one uses approximate maximum likelihood estimates (MLEs) of the parameters and a second approach uses simple ad hoc estimates that were developed for the specific purpose of imputations. The accuracy of the approaches is verified using Monte Carlo simulation. Simulation studies show that both approaches are very satisfactory for small to moderately large sample sizes, but only the MLE-based approach is satisfactory for large sample sizes. The MLE-based approach can be calibrated to perform very well for large samples. Applicability of the method to the log-normal distribution and the gamma distribution (via a cube root transformation) is outlined. Simulation studies also show that the imputation approach works well for constructing tolerance limits and prediction limits for a gamma distribution. The approach is illustrated using a few practical examples.