An evaluation of extended vs weighted least squares for parameter estimation in physiological modeling

Weighted least squares (WLS) is the technique of choice for parameter estimation from noisy data in physiological modeling. WLS can be derived from maximum likelihood theory, provided that the measurement error variance is known and independent of the model parameters and the weights are calculated as the inverse of the measurement error variance. However, using measured values in lieu of predicted values to quantify the measurement error variance is approximately valid only when the noise in the data is relatively low. This practice may thus introduce sampling variation in the resulting estimates, as weights can be seriously mis-specified. To avoid this, extended least squares (ELS) has been used, especially in pharmacokinetics. ELS uses an augmented objective function where the measurement error variance depends explicitly on the model parameters. Although it is more complex, ELS accounts for the Gaussian maximum likelihood statistical model of the data better than WLS, yet its usage is not as widespread. The use of ELS in high data noise situations will result in more accurate parameter estimates than WLS (when the underlying model is correct). To support this claim, we have undertaken a simulation study using four different models with varying amounts of noise in the data and further assuming that the measurement error standard deviation is proportional to the model prediction. We also motivate this in terms of maximum likelihood and comment on the practical consequences of using WLS and ELS as well as give practical guidelines for choosing one method over the other.