**Rapid sample size calculations for a defined likelihood ratio test-based power in mixed effects models**

Camille Vong, Martin Bergstrand, Mats O. Karlsson

Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden

**Objectives:** Efficient power calculation methods have previously been suggested for Wald test based inference in mixed effects models (1) but for Likelihood ratio test (LRT) based hypothesis testing, the only available alternative has been to perform computer-intensive multiple simulations and re-estimations (2). For correct power calculations, a type 1 error assessment to calibrate the significance criterion is often needed for small sample sizes, due to a difference between the actual and the nominal (chi squared) significance criteria(3). The proposed method is based on the use of individual Objective Function Values (iOFV) and aims to provide a fast and accurate prediction of the power and sample size relationship without any need for adjustment of the significance criterion.

**Methods:** The principle of the iOFV sampling method is as follows: (i) a large dataset (e.g. 1000 individuals) is simulated with a full model and subsequently the full and reduced models are re-estimated with this data set, (ii) iOFVs are extracted and for each subject the difference in iOFV between the full and reduced models is computed (ΔiOFV), (iii) ΔiOFVs are sampled according to the design for which power is to be calculated and a starting sample size (N), (iv) the ΔiOFVs sum for each sample is calculated (∑ΔiOFVs), (v) steps iii and iv are repeated many times, (vi) the percentage of ∑ΔiOFVs greater than the significance criterion (e.g. 3.84 for one degree of freedom and α=0.05) is taken as the power for sample size N, (vii) steps iii-vi are repeated with increasing N to provide the power at all sample sizes of interest. The power versus sample size relationship established via the iOFV method was compared to traditional assessment of model-based power (200 simulated datasets) for a selection of sample sizes. Two examples were investigated, a one-compartment IV-Bolus PK model with sex as a covariate on CL (3) and a more complex FPG-HbA1c model with a drug effect on kout for FPG (4).

**Results:** Power generated for both models displayed concordance between the suggested iOFV method and the nominal power. For 90% power, the difference in required sample size was in all investigated cases less than 10%. To maintain a 5% type 1 error a significance criteria calibration at each sample size was needed for the PK model example and the traditional method but not for power assessment with the iOFV sampling method. In both cases, the iOFV method was able to estimate the entire power vs. sample size relationship in less than 1% of the time required to estimate the power at a single sample size with the traditional method.

**Conclusions:** The suggested method provides a fast and still accurate prediction of the power and sample size relationship for likelihood ratio test based hypothesis testing in mixed effects models. The iOFV sampling method is general and mimics more closely than Wald-test based methods the hypothesis tests that are typically used to establish significance.

**References:**

[1] Ogungbenro K, Aarons L, Graham G. Sample size calculations based on generalized estimating equations for population pharmacokinetic experiments. J Biopharm Stat2006;16(2):135-50.

[2] Ette EI, Roy A. Designing population pharmacokinetic studies for efficient parameter estimation. . In: Ette EI, Williams PJ, editors. Pharmacometrics: the Science of Quantitative Pharmacology. Hoboken: John Wiley & Sons; 2007. p. 303-44.

[3] Wahlby U, Jonsson EN, Karlsson MO. Assessment of actual significance levels for covariate effects in NONMEM. J Pharmacokinet Pharmacodyn2001 Jun;28(3):231-52.

[4] Hamren B, Bjork E, Sunzel M, Karlsson M. Models for plasma glucose, HbA1c, and hemoglobin interrelationships in patients with type 2 diabetes following tesaglitazar treatment. Clin Pharmacol Ther2008 Aug;84(2):228-35.