Variability in the Log Domain and Limitations to Its Approximation by the Normal Distribution

Analyzing lognormal data: A nonmathematical practical guide

Lognormal distributions are pervasive in pharmacology and elsewhere in biomedical science, arising naturally when biological effects multiply rather than add. Despite their ubiquity in pharmacological parameters (eg, EC50, IC50, Kd, and Km), lognormal distributions are often overlooked or misunderstood, leading to flawed data analysis. This largely nonmathematical review explains why lognormal distributions are common, how to recognize them, and how to analyze them appropriately. We show that many measured variables are lognormal. So are many derived parameters, particularly those defined as ratios of lognormal variables. Through examples and simulations accessible to working scientists, we demonstrate how misidentifying lognormal distributions as normal leads to reduced statistical power, unnecessarily large sample sizes, false identification of outliers, and inappropriate reporting of effects as differences rather than ratios. We challenge the common practice of using normality tests to decide how to analyze data, showing that many data sets pass both normality and lognormality tests, especially with small sample sizes. Instead, we advocate for assuming lognormality based on the nature of the variable. This review provides practical guidance on recognizing and presenting lognormal data, and comparing data sets sampled from lognormal distributions. Based on Monte Carlo simulations, we recommend the lognormal Welch's t test or nonparametric Brunner-Munzel test for comparing 2 unpaired groups, the lognormal ratio paired t test for paired comparisons, and lognormal ANOVA for ≥3 groups. By recognizing and properly handling lognormal distributions, pharmacologists can design more efficient experiments, obtain more reliable statistical inferences, and communicate their results more effectively. SIGNIFICANCE STATEMENT: Lognormal distributions are common in pharmacology and many scientific fields, but they are often misunderstood or overlooked. This review provides a detailed guide to recognizing and analyzing lognormal data, aiming to help pharmacologists perform more appropriate and more powerful statistical analyses, draw more meaningful conclusions from their data, and communicate their results more effectively.

The Pulvinus Is the Weak Point for Stem Lodging Resistance in Ripe Barley

Stem lodging is a serious problem for the ripe barley crop because it can reduce grain yield and quality. Although biometrical traits (stem diameter and wall thickness) and mechanical properties (stiffness and strength of the culm) have an obvious role in determining lodging resistance, they have only a partial capability to predict lodging resistance. We, therefore, investigated how factors like stem wetting and the point of application of the bending force affect the assessment of these traits. A three-point bending test using a height gauge can provide measures of bending strength (BS), material strength (σb), modulus of elasticity (E), and stiffness (EI). Since the first two parameters are of greatest interest, a quick manual method for measuring them is proposed. We used it specifically to compare the results of tests made by loading the bending force either on the node or the internode. It was shown that the pulvinus (which forms a complex with the node) is the weak point for mechanical resistance to bending in ripe barley stems, as a drop in BS between -31% and -41% (depending on whether the stems were dry or wet) was observed when the loading force was applied on the node/pulvinus complex with respect to the internode. We also found that, overall, BS plummeted -62% with respect to dry stems when the stems were wetted. This was due to an equivalent (-62%) plunge in σb. Similar drops in BS (-64%) and σb (-68%) following wetting were measured with the height gauge. Wetting, therefore, greatly lowers the mechanical resistance of stems. Moreover, the existence of a weak point-i.e., the pulvinus-in mature barley stems is an important feature that must be considered when evaluating the lodging-related characteristics of this crop. These findings improve our understanding of the mechanical properties of barley stems and, thus, our capability to identify genotypes with better lodging resistance.

Bayesian estimation in veterinary pharmacology: A conceptual and practical introduction

Sophisticated mathematical and computational tools have become widespread and important in veterinary pharmacology. Although the theoretical basis and practical applications of these have been widely explored in the literature, statistical inference in the context of these models has received less attention. Optimization methods, often with frequentist statistical inference, have been predominant. In contrast, Bayesian statistics have not been widely applied, but offer both practical utility and arguably greater interpretability. Veterinary pharmacology applications are generally well supported by relevant prior information, from either existing substantive knowledge, or an understanding of study and model design. This facilitates practical implementation of Bayesian analyses that can take advantage of this knowledge. This essay will explore the specification of Bayesian models relevant to veterinary pharmacology, including demonstration of prior selection, and illustrate the capability of these models to generate practically useful statistics, including uncertainty statements, that are difficult or impossible to obtain otherwise. Case studies using simulated data will describe applications in clinical trials, pharmacodynamics, and pharmacokinetics, all including multilevel modeling. This content may serve as a suitable starting point for researchers in veterinary pharmacology and related disciplines considering Bayesian estimation for their applied work.

Pharmacogenetic Testing or Therapeutic Drug Monitoring: A Quantitative Framework

BACKGROUND

Pharmacogenetic profiling and therapeutic drug monitoring (TDM) have both been proposed to manage inter-individual variability (IIV) in drug exposure. However, determining the most effective approach for estimating exposure for a particular drug remains a challenge. This study aimed to quantitatively assess the circumstances in which pharmacogenetic profiling may outperform TDM in estimating drug exposure, under three sources of variability (IIV, inter-occasion variability [IOV], and residual unexplained variability [RUV]).

METHODS

Pharmacokinetic models were selected from the literature corresponding to drugs for which pharmacogenetic profiling and TDM are both clinically considered approaches for dose individualization. The models were used to simulate relevant drug exposures (trough concentration or area under the curve [AUC]) under varying degrees of IIV, IOV, and RUV.

RESULTS

Six drug cases were selected from the literature. Model-based simulations demonstrated that the percentage of patients for whom pharmacogenetic exposure prediction is superior to TDM differs for each drug case: tacrolimus (11.0%), tamoxifen (12.7%), efavirenz (49.2%), vincristine (49.6%), risperidone (48.1%), and 5-fluorouracil (5-FU) (100%). Generally, in the presence of higher unexplained IIV in combination with lower RUV and IOV, exposure was best estimated by TDM, whereas, under lower unexplained IIV in combination with higher IOV or RUV, pharmacogenetic profiling was preferred.

CONCLUSIONS

For the drugs with relatively low RUV and IOV (e.g., tamoxifen and tacrolimus), TDM estimated true exposure the best. Conversely, for drugs with similar or lower unexplained IIV (e.g., efavirenz or 5-FU, respectively) combined with relatively high RUV, pharmacogenetic profiling provided the most accurate estimate for most patients. However, genotype prevalence and the relative influence of genotypes on the PK, as well as the ability of TDM to accurately estimate AUC with a limited number of samples, had an impact. The results could be used to support clinical decision making when considering other factors, such as the probability for severe side effects.

Tacrolimus population pharmacokinetics in adult heart transplant patients

INTRODUCTION

Tacrolimus is an immunosuppressant largely used in heart transplantation. However, the calculation of its exposure based on the area under the curve (AUC) requires the use of a population pharmacokinetic (PK) model. The aims of this work were (i) to develop a population PK model for tacrolimus in heart transplant patients, (ii) to derive a maximum a posteriori Bayesian estimator (MAP-BE) based on a limited sampling strategy (LSS) and (iii) to estimate probabilities of target attainment (PTAs) for AUC and trough concentration (C0).

MATERIAL AND METHODS

Forty-seven PK profiles (546 concentrations) of 18 heart transplant patients of the Pharmacocinétique des Immunosuppresseurs chez les patients GREffés Cardiaques study receiving tacrolimus (Prograf®) were included. The database was split into a development (80%) and a validation (20%) set. PK parameters were estimated in MONOLIX® and based on this model a Bayesian estimator using an LSS was built. Simulations were performed to calculate the PTA for AUC and C0.

RESULTS

The best model to describe the tacrolimus PK was a two-compartment model with a transit absorption and a linear elimination. Only the CYP3A5 covariate was kept in the final model. The derived MAP-BE based on the LSS (0-1-2 h postdose) yielded an AUC bias ± SD = 2.7 ± 10.2% and an imprecision of 9.9% in comparison to the reference AUC calculated using the trapezoidal rule. PTAs based on AUC or C0 allowed new recommendations to be proposed for starting doses (0.11 mg·kg-1 ·12 h-1 for the CYP3A5 nonexpressor and 0.22 mg·kg1 ·12 h-1 for the CYP3A5 expressor).

CONCLUSION

The MAP-BE developed should facilitate estimation of tacrolimus AUC in heart transplant patients.

A two-sample approach to retrograde extrapolation of blood THC concentrations - Is it feasible?

BACKGROUND

Retrograde extrapolation of drug concentrations in blood can be relevant in cases of drug-impaired driving and is regularly used in forensic toxicology in Norway. Δ9-tetrahydrocannabinol (THC) has complex, multi-compartmental pharmacokinetics, which makes retrograde extrapolation of blood THC concentrations problematic. In the present study, we evaluated an approach to retrograde extrapolation in which momentary rates of decrease of THC were estimated from two consecutive blood samples in apprehended drivers.

MATERIAL AND METHODS

Data were collected from apprehended drivers in Norway 2000-2020. We included 548 cases in which THC was detected in two consecutive blood samples collected ≥ 20 min apart. THC concentrations were measured by GC-MS and UHPLC-MS/MS. In each case, THC concentrations and the time between the two sampling points (Δt) were used to estimate the rate constant k. The relationship between THC concentration and k was modelled by linear regression.

RESULTS

The median Δt was 31 min (interquartile range, IQR = 9). The median blood THC concentration was 2.4 μg/L (IQR = 3.4) at the first sampling point and 2.3 μg/L (IQR =3.1) at the second. The concentration decreased in 62% and increased in 38% of all cases. However, considering measurement uncertainty, the changes were not statistically significant in 87% of cases. The mean of k was 0.12 h-1, corresponding to an apparent t1/2 of 6.0 h. The t1/2 predicted from linear regression of k against THC concentration ranged from 0.93 to 13 h for the highest and lowest concentrations observed (36 and 0.63 μg/L, respectively). The time from driving to blood collection had a median of 1.7 h (IQR = 1.5), and did not correlate with k.

CONCLUSIONS

The apparent t1/2 of THC calculated from the mean of k was 6.0 h, which is shorter than the terminal elimination t1/2 suggested in previous population studies. This indicates that blood samples were often taken during the late distribution phase of THC. Because Δt was short relative to the rates of decrease expected in the late distribution and elimination phases, the underlying true concentration changes related to in vivo pharmacokinetics were small and masked by the relatively larger "false" changes introduced by random analytical and pre-analytical error. Therefore, individual values of k calculated from only two blood samples taken a short time apart are unreliable, and a two-sample approach to retrograde extrapolation of THC cannot be recommended.

mvLognCorrEst: an R package for sampling from multivariate lognormal distributions and estimating correlations from uncomplete correlation matrix

BACKGROUND AND OBJECTIVE

Pharmacometrics (PMX) is a quantitative discipline which supports decision-making processes in all stages of drug development. PMX leverages Modeling and Simulations (M&S), which represents a powerful tool to characterize and predict the behavior and the effect of a drug. M&S-based methods, such as Sensitivity Analysis (SA) and Global Sensitivity Analysis (GSA), are gaining interest in PMX as they allow the evaluation of model-informed inference quality. Simulations should be correctly designed to obtain reliable results. Neglecting correlations between model parameters can significantly alter the results of simulations. However, the introduction of a correlation structure between model parameters can cause some issues. Sampling from a multivariate lognormal distribution, which is the typically distribution assumed for PMX model parameters, is not straightforward when a correlation structure is introduced. Indeed, correlations need to respect some constraints which depend by the CVs (i.e., coefficients of variation) of lognormal variables. In addition, when correlation matrices have some unspecified values, they should be properly fixed preserving the positive semi-definiteness of the correlation structure. In this paper, we present mvLognCorrEst, an R package developed to address these issues.

METHODS

The proposed sampling strategy was based on reconducting the extraction from the multivariate lognormal distribution of interest to the underlying Normal distribution. However, with high lognormal CVs, a positive semi-definite Normal covariance matrix cannot be obtained due to the violation of some theoretical constraints. In these cases, the Normal covariance matrix was approximated to its nearest positive definite matrix using Frobenius norm as matrix distance. For the estimation of unknown correlations terms, the graph theory was used to represent the correlation structure as weighed undirected graph. Plausible value ranges for the unspecified correlations were derived considering the paths between variables. Then, their estimation was performed by solving a constrained optimization problem.

RESULTS

Package functions are presented and applied on a real case study, that is the GSA of a PMX model that has been recently developed to support preclinical oncological studies.

CONCLUSIONS

mvLognCorrEst package is an R tool to support simulation-based analysis for which sampling from multivariate lognormal distributions with correlated variables and/or estimation of partially defined correlation matrix are required.