Mixing of variability in the log-domain with normal-domain variability components may lead to biased estimation
Jeroen Elassaiss-Schaap (1,2), Lisa Martial (1)
(1) PD-value, The Netherlands (2) Sytems Pharmacology and Pharmacy, LACDR, the Netherlands
Introduction:A formal comparison of properties of the normal and the log-normal distribution has been available since 2020 [1]. The log-normal distribution has tails that can be represented by those of normal distributions up to lognormal standard deviations of about 0.25 to 0.67 depending on the choice of criteria. At higher values, the relative position of mean, median and mode of the lognormal distribution change. It has not been established whether, and if, how, these properties influence estimation of distributional variances in mixed-effects frameworks. To study such properties in isolation, a simulation study was set up utilizing one of the simplest models: estimation of a population mean (expectation value), variability and residual variability.
Objectives:
Compare estimation outcomes applying and mixing log-normal and normal distributions at the population and residual level under various levels of inter-individual variability
Methods: Richly sampled distributions around a population mean of 8 were simulated for 8 different levels of inter-individual variability (IIV) and 6 levels for residual variability (res), with 10 samples per individual. Where needed, log-sd values were converted into normal-domain sd values using the equality [1]:
sd = exp(0.5*log-sd^2)*√(exp(log-sd^2)-1)
Simulations were conducted in R 3.6.3, and IIV values were always simulated using a log-normal distribution whereas res values were simulated using either a log-normal or a proportional normal-domain distribution. Negative values arising from proportional samples were removed without replacement.
Models were estimated in NONMEM 7.4.4 in $PRED using the following settings (4 combinations):
Either normal or log-normal specification of thetas: THETA(1)*EXP(ETA(1)) or EXP(THETA(1)+ETA(1)), and
Either proportional residuals or log-normal residuals, the latter through a transform-both-sides approach;
using the following estimation methods:
FOCE-I, LAPL-I, and SAEM
Each simulation consisted of 100 individuals with 10 generated samples each. The quality of re-estimation was judged using the ratio of estimated divided by true simulated value.
Results: Initial re-estimation runs demonstrated zero gradient occurrences when the initial theta was set to the simulation value, and was therefore set to a 10-fold lower value, while boundary tests were switched off for all subsequent runs. The majority (>95%) of subsequent runs were successful.
The estimated-to-true ratios were not overly sensitive to the magnitude of either IIV or res, although deviations increased more than proportionally with higher variability. As a general rule, estimation of the population mean was most sensitive, followed by that of IIV while estimation of res was least sensitive. However, the impact of mixing variability domains was high regardless of the magnitude of IIV and res, i.e. deviations were already large at lowest simulated values. The mean deviated more than 4-fold from estimated even in the rich sampling scenario applied, in the case were the theta was estimated linear and the sigma (res) log-linear, or vice versa. Estimates under matching distributions for theta and sigma were close to the truth. Results were very similar under LAPL-I, where some partial improvements were noted under SAEM, for 1 out 12 scenarios. An exploration of simulations with two simple 2-compartmental PK models with absorption revealed that a similar sensitivity did not occur under rich sampling with IIV on clearance or Frel, while some sensitivity was observed with a turnover model with IIV on baseline. Further evaluations were performed in NONMEM, where re-estimation of the matching, well-performing, models was performed with 100 replicates. Bias was low at any SD, but the RMSE of estimation was noticeably larger in the linear case, increasing with IIV-sd to about 150%.
Conclusion: Estimation in NONMEM is very sensitive to cross-specification of theta-level and sigma-level distribution, e.g. normal for theta and log-normal for sigma, even under rich-data situations for a simple model, with 4-fold or higher biases. The impact is consistent with the importance of residual-error gradients in estimation across estimation methods. It seems therefore advisable to execute prudence when applying such cross-specification in daily practice, although no impact was observed in the context of compartmental PK models.
References:
[1] Elassaiss-Schaap J, Duisters K. Variability in the Log Domain and Limitations to Its Approximation by the Normal Distribution. CPT Pharmacometrics Syst Pharmacol. 2020 May; 9(5): 245–257