Observations below multiple lower limits of quantification: How to estimate the mean and variance.

Multiple lower limits of quantification (MLOQs) result if various laboratories are involved in the analysis of concentration data and some observations are too low to be quantified. For normally distributed data under MLOQs there exists only the multiple regression method of Helsel to estimate the mean and variance. We propose a simple imputation method and two new maximum likelihood estimation methods: the multiple truncated sample method and the multiple censored sample method. A simulation study is conducted to compare the performances of the newly introduced methods to Helsel's via the criteria root mean squared error (RMSE) and bias of the parameter estimates. Two and four lower limits of quantification (LLOQs), various amounts of unquantifiable observations and two sample sizes are studied. Furthermore, the robustness is investigated under model misspecification. The methods perform with decreasing accuracy for increasing rates of unquantified observations. Increasing sample sizes lead to smaller bias. There is almost no change in the performance between two and four LLOQs. The magnitude of the variance impairs the performance of all methods. For a smaller variance, the multiple censored sample method leads to superior estimates regarding the RMSE and bias, whereas Helsel's method is superior regarding the bias for a larger variance. Under model misspecification, Helsel's method was inferior to the other methods. Estimating the mean, the multiple censored sample method performed better, whereas the multiple truncated sample method performs best in estimating the variance. Summarizing, for a large sample size and normally distributed data we recommend to use Helsel's method. Otherwise, the multiple censored sample method should be used to obtain estimates of the mean and variance of data including MLOQs.