The ghosts of departed quantities: approaches to dealing with observations below the limit of quantitation

Abstract

A common but not necessarily logical requirement in drug development is that a 'limit of quantitation' be set for chemical assays and that observations that fall below the limit should not be treated as real data but should be labelled as below the limit and set aside for special treatment. We examine five of seven approaches to analysing such data considered by Beal in 2001, concentrating in particular on two: one that treats the data as a truncated sample and another that treats them as a censored sample. In fact, using a pattern-mixture framework, one can show that the former consists of using the conditional distribution of the 'acceptable values' and the latter adds the information from the marginal mixing distribution. We illustrate these approaches with a real example, concentrating in particular on the two likelihood-based methods, provide various formulae that may be used to compare these and other approaches, check these formulae using simulations and make some recommendations as to which approach one should use.

A note regarding meta‐analysis of sequential trials with stopping for efficacy

“…For simplicity, the standard deviation is assumed known. The expected means for stopped and continued trials are calculated using formulae for the truncated Normal distribution . For the stopped trial, the expected mean is and for the continued trial, it is where φ (.)…”

Evaluating the Pharmacodynamic Effect of Antimalarial Drugs in Clinical Trials by Quantitative PCR

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The present and future of withdrawal period calculations for milk in the European Union: dealing with data below the limit of quantification

“…(ii) SAEM and omission of BLQ data; (iii) imputation of LOQ/2 on the first BLQ data and omission of the following ones (Thi ebaut et al, 2006) Dataset: Rich simulated and real datasets with 12% of BLQ data Structural model: HCV dynamics-system of 3 differential equations (PD) Estimation methods: MLE with left-censored data vs MLE with omission Specificity: Efficacy of the treatment assessed (Byon et al, 2008) Dataset: Rich simulated and real datasets with different levels of BLQ data from 10.2% to 49.1% Structural model: One-and two-compartment IV bolus models (PK) Estimation methods: Only MLE with left-censored data. No comparison with omission or substitution methods Specificity: Impact of the analytical CV (10, 50 and 100%) at LOQ assessed (Ahn et al, 2008) Dataset: Rich simulated dataset with 10, 20, 30, or 40% of BLQ data Structural model: Two-compartment model with first-order absorption (PK) Estimation methods: Beal methods 1, 2, 3, and 4 (Bergstrand & Karlsson, 2009) Dataset: Rich simulated dataset with different proportions of BLQ data Structural model: 2 PK models and an indirect response PD model Estimation methods: 8 methods including variants of omission, substitution, and MLE with left-censored data Specificity: Studied the impact of the location of BLQ data (absorption and elimination phases in PK and rebounds in PD) (Yang & Roger, 2010) Dataset: Rich (4 per subject) simulated dataset with 5 levels of BLQ data, from 25 to 75% Structural model: One-compartment model with first-order elimination (PK) Estimation methods: (i) MLE (Laplacian method) with left-censored data; (ii) omission; (iii) substitution with LOQ/2 (Xu et al, 2011) Dataset: Rich simulated dataset with five levels of BLQ data (1, 2.5, 5, 7.5, and 10%) Structural model: One-and two-compartment models with first-order absorption and first-order elimination (PK) Estimation methods: Beal methods 1 and 3 Specificity: Low percentages of BLQ data (Senn et al, 2012) Dataset: Rich simulated and real datasets. This is a theoretical study on continuous proportions of BLQ data Structural model: Analysis of a sample on a single unknown mean rather than measurements over time Estimation methods: MLE with either truncated or censored samples Specificity: No comparison with omission or substitution methods MLE, maximum-likelihood estimation, LOQ, limit of quantification; BLQ, below the lower quantification limit; PK, pharmacokinetic study; PD, pharmacodynamic study; CV, coefficient of variation.…”