Mixed-effects beta regression for modeling continuous bounded outcome scores using NONMEM when data are not on the boundaries

Abstract

Beta regression models have been recommended for continuous bounded outcome scores that are often collected in clinical studies. Implementing beta regression in NONMEM presents difficulties since it does not provide gamma functions required by the beta distribution density function. The objective of the study was to implement mixed-effects beta regression models in NONMEM using Nemes’ approximation to the gamma function and to evaluate the performance of the NONMEM implementation of mixed-effects beta regression in comparison to the commonly used SAS approach. Monte Carlo simulations were conducted to simulate continuous outcomes within an interval of (0, 70) based on a beta regression model in the context of Alzheimer’s disease. Six samples per subject over a 3 years period were simulated at 0, 0.5, 1, 1.5, 2, and 3 years. One thousand trials were simulated and each trial had 250 subjects. The simulation–reestimation exercise indicated that the NONMEM implementation using Laplace and Nemes’ approximations provided only slightly higher bias and relative RMSE (RRMSE) compared to the commonly used SAS approach with adaptive Gaussian quadrature and built-in gamma functions, i.e., the difference in bias and RRMSE for fixed-effect parameters, random effects on intercept, and the precision parameter were <1–3 %, while the difference in the random effects on the slope was <3–7 % under the studied simulation conditions. The mixed-effect beta regression model described the disease progression for the cognitive component of the Alzheimer’s disease assessment scale from the Alzheimer’s Disease Neuroimaging Initiative study. In conclusion, with Nemes’ approximation of the gamma function, NONMEM provided comparable estimates to those from SAS for both fixed and random-effect parameters. In addition, the NONMEM run time for the mixed beta regression models appeared to be much shorter compared to SAS, i.e., 1–2 versus 20–40 s for the model and data used in the manuscript.

Appendix

A sample NONMEM code for the mixed effects beta regression model for the ADNI data is presented.

Code	Comments
$PROB beta regression	; Data specification
$INPUT $DATA ADNI.csv
$PRED	; Mean model
B0 = THETA(1) + ETA(1) B1 = THETA(2) + ETA(2)	; Define parameters for baseline and slope
LINP = B0 + B1 * TIME MU = EXP(LINP)/(1 + EXP(LINP))	; Linear predictor on logit scale ; Conditional mean by anti-logit transforming the linear predictor
LTAU = THETA(3) TAU = EXP(LTAU)	; precision parameter
	; specify the log likelihood based on the density function for beta distribution (Eq. 2 )
X1 = TAU X2 = MU * TAU X3 = (1−MU) * TAU
LG1 = 0.5 * (LOG(2 * 3.1415) − LOG(X1)) + X1 * (LOG(X1) − 1) + (5/4) * X1 * (LOG (1 + (1/(15 * X1 * 2)))) LG2 = 0.5 * (LOG(2  * 3.1415) − LOG(X2)) + X2 * (LOG(X2) − 1) + (5/4) * X2 * (LOG (1 + (1/(15 * X2 * 2)))) LG3 = 0.5 * (LOG(2 * 3.1415) − LOG(X3)) + X3 * (LOG(X3) − 1) + (5/4) * X3 * (LOG (1 +  (1/(15 * X3 * 2))))	; Approximation of the log(gamma) function (Eq. 4 ) ; The first part of the log likelihood, 0.5 * (LOG(2 * 3.1415), can be omitted if computation of full likelihood is not required. ; Log Likelihood of the beta distribution (Eq. 2 )
LOGL = LG1 − LG2 − LG3 + (MU * TAU−1) * LOG(DV) + ((1−MU) * TAU−1) * LOG(1−DV) Y = −2 * LOGL
SOR = (DV − MU)/SQRT(MU * (1−MU)/(1 + TAU))	; Pearson residuals (standardized ordinary residuals)
$THETA	; specify initial values
$OMEGA
$EST MAX = 9999 PRINT = 5 METHOD = COND − 2LOGLIK LAPLACIAN NOABORT $COV PRINT = E MATRIX = R $TABLE	; estimation step