The Lasso – A Novel Method for Predictive-Covariate Modelling of Population Pharmacokinetics/Pharmacodynamics

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Objectives: To implement a new method (the LASSO) for covariate selection within NONMEM and to compare this method to the commonly used stepwise-covariate modelling (SCM).

Introduction and Theory: Identification and quantification of covariate relations is an important part of population pharmacokinetic/pharmacodynamic (pop PK/PD) modelling. The covariate model is often built with stepwise-covariate-selection procedures such as SCM (1). However, the problems of using stepwise regression and other subset selection methods have been debated for decades and in the area of pop PK/PD it has recently been shown that these selection methods often are inappropriate for small- to moderately-sized datasets (< 100 individuals), where the covariate model may even reduce the predictive performance (2).

One method for obtaining a more predictive model in linear regression is “Ridge regression”. This method mainly shrinks the covariate coefficients and does not provide the big picture of which covariates are important by selecting among the many potential covariate relations one may wish to investigate. The LASSO is a similar and increasingly popular method which performs both selection and shrinkage of the covariate coefficients simultaneously.(3)

The lasso-estimate of the covariate model is the maximum-likelihood estimate subject to the restriction: absolute-sum of the covariate coefficients ≤ t. The t-value will determine the amount of shrinkage and the model size. An optimal t-value can be estimated using cross validation.

Method: A method for performing the LASSO in NONMEM was implemented in Perl using the Perl-speaks-NONMEM-software package (4,5). The implemented method was applied to a real PK dataset (6) investigating 20 parameter-covariate relations and the procedure was compared to the SCM (forward selection at p < 0.05). The comparison was made on run times, which covariates were selected and the magnitude of the covariate coefficients.

Results: The LASSO method required only half the run time for the SCM. The LASSO retained the two most important covariates selected by the SCM. As expected the retained covariate coefficients were shrunken when using the LASSO method.

Conclusion: In this example the LASSO selected the two most important covariates, as identified by the SCM. The LASSO required only half the computer-run time and shrunk the two selected coefficients compared to the SCM. We believe this shrinkage is necessary to improve the predictive performance of the model considering that 20 covariate relations have been investigated in a dataset with only 64 individuals. However, an evaluation on external data must be performed to assess predictive performance.

References:
[1] Jonsson EN and Karlsson MO. Automated Covariate Model Building Within NONMEM. Pharm Res, 1998. 15(9): p. 1463-8.
[2] Ribbing J. and Jonsson EN. Power, Selection Bias and Predictive Performance of the Population Pharmacokinetic Covariate Model. J Pharmacokinet Pharmacodyn, 2004. 31(2): p. 109-34.
[3] Tibshirani R. Regression Shrinkage and Selection via the LASSO. Journal of the Royal Statistical Society, Series B. 1996. 58(1): p. 267-288
[4] Lindbom L, Ribbing J, Jonsson EN. Perl-speaks-NONMEM (PsN)–a Perl module for NONMEM related programming. Comput Methods Programs Biomed. 2004 Aug;75(2): p. 85-94.
[5] Lindbom L, Pihlgren P, Jonsson EN. PsN-Toolkit A collection of computer intensive statistical methods for non-linear mixed effect modeling using NONMEM. Submitted to Comput Methods Programs Biomed, March 2005.
[6] Karlsson MO and Sheiner LB. The Importance of Modeling Interoccasion Variability in Population Pharmacokinetic Analyses. J Pharmacokinet Biopharm, 1993. 21(6): p. 735-50.