Handling data below the limit of quantification in mixed effect models

Background: Common approaches for handling of concentration measurements reported as below the limit of quantification (BLQ), such as discharging the information or substitution with the limit of quantification (LOQ) divided by two, have been shown to introduce bias to parameter estimates [1,2,4-6]. In 2001, Stuart Beal published an overview of ways to fit a PK model in the presence of BLQ data [1]. The method referred to as M2 applies conditional likelihood estimation to the observations above LOQ and the likelihood for the data being above LOQ are maximized with respect to the model parameters. This approach can be implemented in NONMEM VI by utilization of the YLO functionality [3]. By simultaneous modeling of continuous and categorical data where the BLQ data are treated as categorical, the likelihood for BLQ data to be indeed BLQ can be maximized with respect to the model parameters. The indicator variable F_FLAG can be used for to facilitate this approach in NONMEM VI [3]. This suggested method differs from the one referred to as M3 in the sense that the likelihood is only estimated for the BLQ data as opposed to all data.

Aim: Investigate the impact of BLQ observations occurring in three distinctly different ways and asses the best method to prevent bias parameter estimates.

Methods: Three typical ways in with BLQ can occur in a model was investigated with simulations from three different models. Model A was used to represent a case with BLQ observations in an absorption phase of a PK model whereas model B represented a case with BLQ observations in the elimination phase. The third model, C, is a indirect response model where variable of interest in some cases dips below the limit of quantification before returning towards baseline.

Five different approaches for handling of BLQ data was compared to estimation with the full data set from 100 simulated data sets following models A, B and C. The LOQ level was assumed where the average residual variability constituted 20% of the prediction. The residual variability model parameters were altered so that three different magnitudes of censored observations were investigated for each model and method.

Results and discussion: Censoring of data below the limit of quantification was associated with substantial bias in one or more parameter estimates for all tested models even for seemingly small amounts of censored data. The M2 method did not generate unbiased parameter estimates however bias was less pronounced than for omission. Best performance was seen with the M3 method implemented with F_FLAG. In the tested examples this method generated overall un-biased parameter estimates. In all cases estimation using the Laplacian method in NONMEM VI resulted in a lower rate of successful terminations compared to the FOCE method. However generally no significant difference could be detected in parameter estimates following successful or non-successful minimization. Results following substitution of BLQ observations with LOQ/2 was in some cases shown to introduce bias and was always suboptimal to the M3 method. These results are all in agreement with an earlier presented less thorough investigation [2].


  1. Beal SL, Ways to fit a PK model with some data below the quantification limit.  J Pharmacokinet Pharmacodyn, 2001. 28(5): p. 481-504.
  2. Bergstrand M, Plan E, Kjellsson M, Karlsson MO, A comparison of methods for handling of data below the limit of quantification in NONMEM VI.  PAGE 16 (2007) Abstr 1201 www.page-meeting.org/?abstract=1201.
  3. Boeckmann AJ, Sheiner LB, Beal SL. NONMEM Users Guide PartVIII. 1996–2006, NONMEM Project Group: San Francisco.
  4. Dartois C, Looby M, He H, Steimer J-L, and Pillai G., Impact of handling missing PK data on PD estimation – explicit modeling of BLQ data in WinBUGS® reduced bias in the PD predictions – a preclinical example. PAGE 16 (2007) Abstr 1101 www.page-meeting.org/?abstract=1101
  5. Duval V and Karlsson MO, Impact of omission or replacement of data below the limit of quantification on parameter estimates in a two-compartment model. Pharm Res, 2002. 19(12): p. 1835-40.
  6. Hing JP, et al., Analysis of toxicokinetic data using NONMEM: impact of quantification limit and replacement strategies for censored data. J Pharmacokinet Pharmacodyn, 2001. 28(5): p. 465-79.