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A note on confidence intervals with extended least squares parameter estimates


It has previously been shown that the extended least squares (ELS) method for fitting pharmacokinetic models behaves better than other methods when there is possible heteroscedasticity (unequal error variance) in the data. Confidence intervals for pharmacokinetic parameters, at the target confidence level of 95%, computed in simulations with several pharmacokinetic and error variance models, using a theoretically reasonable approximation to the asymptotic covariance matrix of the ELS parameter estimator, are found to include the true parameter values considerably less than 95% of the time. Intervals with the ordinary least squares method perform better. Two adjustments to the ELS confidence intervals, taken together, result in better performance. These are: (i) apply a bias correction to the ELS estimate of variance, which results in wider confidence intervals, and (ii) use confidence intervals with a target level of 99% to obtain confidence intervals with actual level closer to 95%. Kineticists wishing to use the ELS method may wish to use these adjustments.

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  1. 1.

    C. C. Peck, L. B. Sheiner, and A. Nichols. The problem of choosing weights in nonlinear regression analysis of pharmacokinetic data.Drug Metab. Rev. 15:113–150 (1984).

  2. 2.

    C. C. Peck, S. L. Beal, L. B. Sheiner, and A. Nichols. Extended least squares nonlinear regression: A possible solution to the choice of weights problem in analysis of individual pharmacokinetic data.J. Pharmacokin. Biopharm. 12:545–558 (1984).

  3. 3.

    L. B. Sheiner and S. L. Beal. Pharmacokinetic parameter estimates from several least squares procedures: Superiority of extended least squares.J. Pharmacokin. Biopharm. 14:185–201 (1985).

  4. 4.

    S. L. Beal. Asymptotic properties of optimization estimators for the independent not identically distributed case with application to extended least squares estimators. Technical Report, Division of Clinical Pharmacology, University of California, San Francisco (1984).

  5. 5.

    G. E. P. Box and M. E. Muller. A note on the generation of random normal deviates.Ann. Math. Stat. 29:610–611 (1958).

  6. 6.

    P. Lewis, A. Goodman, and J. Miller. A pseudorandom number generator for the system 360.IBM Syst. J. 8:135–146 (1969).

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Correspondence to Lewis B. Sheiner.

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Sheiner, L.B., Beal, S.L. A note on confidence intervals with extended least squares parameter estimates. Journal of Pharmacokinetics and Biopharmaceutics 15, 93–98 (1987).

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Key words

  • Extended least squares
  • ordinary least squares
  • confidence intervals