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Estimating Bias In Parameters For Some NONMEM Models For Ordered Categorical Data

Siv J÷nsson and Mats O. Karlsson

Uppsala University, Sweden

Introduction Side effect data are commonly reported as ordered categorical data e.g. none, mild, moderate and severe. Oftentimes only a relative minor fraction of the patients experience a side effect, which may affect the estimation properties of the methods used. The present study aimed at investigating the bias in parameter estimates for models for ordered categorical data using NONMEM (1).

Methods A population logistic regression model for ordered categorical data was used for simulation and estimation within NONMEM. The model predicts for each individual observation, Yit, the probability of having a score that is greater than or equal to a given score m = 0, 1, 2, 3 and has the general structure

Pr (Yit >= m|ETA) = exp(INT + ETA )/[1 + exp(INT + ETA)]

where INT is the sum of THj (j=1 to m). ETA denotes the individual random effect which is assumed to be a symmetrically distributed, zero-mean random variable with a variance of OM2.

Bias in the population estimates was studied based on Monte Carlo simulated data sets (n=100), each data set comprising 1000 patients with 4 observations each. The model given above was fitted to each simulated data set, followed by simulations of new data based on the parameter estimates.

Results and conclusions When nominal parameter estimates were chosen so that relatively even distributions between responses were obtained, NONMEM performed well and simulations under the estimated model parameters mimicked the real data. However, in cases where only a minor fraction of the observations was non-zero, one or more of the population estimates were highly biased, resulting in that simulations under the estimated model parameters did not reflect the observed data, see one example below. The bias appears to be linked to a non-normal distribution of estimated ETAs and increases with increasing value of OM2.






Nominal value

- 8.25

- 2.43

- 2.86


Estimated value

mean (range)

- 9.28

(-9.71, -8.60)

- 2.86

(-3.38, -2.46)

- 3.30

(-4.08, -2.55)


(83, 141)


Fraction of observations = 0

Fraction of observations = 1

Fraction of observations = 2

Fraction of observations = 3

Simulations based on nominal values

mean (range)


(0.87, 0.91)


(0.041, 0.073)


(0.024, 0.046)


(0.011, 0.029)

Simulations based on estimated values

mean (range)


(0.81, 0.85)


(0.052, 0.077)


(0.035, 0.060)


(0.043, 0.077)

1. Beal SL, Sheiner LB. NONMEM users guides. NONMEM Project Group. San Francisco: University of California at San Francisco; 1998.