Wojciech Krzyzanski , William J Jusko
University at Buffalo, Buffalo, USA
Objectives: Circadian rhythms determined by the sleep/wake 24 h cycle are common features in longitudinal measurements of variety of pharmacodynamic responses. Cosinor analysis is a classic approach towards detection and quantification biological of rhythms [1]. A cosinor model consists of a single or multiple cosine functions of varying frequencies that can be parameterized in terms of amplitudes and acrophases or decomposed to a linear combinations of cosines and sines (harmonics). The latter representation constitutes a linear model. Both regression and mixed effects techniques have been employed in the cosinor analysis for parameter estimation [2]. Determination of the optimal number of harmonics is subject to the model selection techniques. The likelihood ratio test (LRT) is commonly used for mixed effect models. The objective of this project was to develop a robust approach for selection of the number of harmonics in the mixed effects cosinor model. The approach was tested on previously published data of cortisol plasma concentrations versus time in adolescent subjects [3].
Methods: The mixed effects cosinor model consisted of multiple harmonics as fixed effects, diagonal covariance matrix for between subject variability, and homoscedastic residual error. The model was implemented as a new R function lmecosinor based on the existing R function lme for linear mixed effects models [4]. The lmecosinor outputs the maximum likelihood parameter estimates of the cosinor model as well as the value of the objective function. The R functions lrtest and anova [5] were used to perform LRT and F-tests for model comparison. Selection of harmonics was done according to the forward inclusion followed by the backward elimination approach. For forward inclusion we started with the 0th harmonic and compared to the cosinor model with 0th and ith harmonic Model(0,i), where i =1,2,…, 6. The best two-harmonic model was chosen based on the lowest p-value of the test. In the next step Model(0,i1) was compared to Model(0,i1,j), where j =1,2,…, 7, and j ≠i1. The process continues until all p-values for the test comparing Model(0,i1,…,ik) with Model(0,i1,…,ik,j) were greater than 0.05. We considered three scenarios: A) Each model had the diagonal covariance matrix and LRT was performed; B) Each model did not have covariance matrix (naïve pooled data) and LRT was performed; C) Each model did not have a covariance matrix and F-test was performed. The backward elimination step started with the final model Model(0,i1,…,ik) with the covariance matrix and the entries of the covariance matrix were fixed to 0 to obtain a reduced model. The models were compared using LRT. The process stopped when all p-values were less than 0.05. Time courses of log-transformed cortisol plasma concentrations in adolescent subjects (N=18) sampled at 30 min intervals over 24-48 h periods served as a case study [3]. Performance of the selected final model was assessed by goodness of fit diagnostics, visual predictive checks, and inspection of standard errors.
Results: The sequence of selected models was same for all scenarios: Model(0,1) → Model(0,1,2) → Model(0,1,2,4). Whereas Model(0,1,2,4) was final for scenarios B and C, for scenario A forward selection continued to Model(0,1,2,4,3) with p=4.1E-7 when it was discontinued. The backward elimination resulted in all entries of the covariance matrix for Model(0,1,2,4) to be significant (p<0.00012). This model estimate of the typical value of the mesor of cortisol plasma concentration was 71.ng/mL and the primary peak time was 7.56 h (7:34 AM). The diagnostic plots did not reveal any flaws of performance and relative standard errors of all parameter estimates were less than 51% except for B4 = -0.08 (%RSE=236) due to a small value relative to other parameters
Conclusions: Cosinor analysis of circadian population data can be performed using linear mixed-effects modeling. The new R function lmecosinor adequately employs the R lme function for cosinor analysis. LRT is “anti-conservative” for selection of harmonics for the mixed-effects cosinor model. LRT applied to naïve-pooled data results in a robust selection of the best harmonics. The F-test and LRT applied to naïve-pooled data yield very similar p-values. The best performing mixed-effects model for the cortisol data consists of 24, 12, and 6 hour rhythms.
References:
[1] Cornelissen G (2014) Cosinor-based rhythmometry. Theoretical Biology and Medical Modelling 11:16
[2] Mikulich SK et al. (2003) Comparing linear and nonlinear mixed model approaches to cosinor analysis. Statist Med 22:3195–3211
[3] Jusko WJ, Slaunwhite WR, Aceto T (1975) Partial pharmacodynamic model for the circadian- episodic secretion of cortisol in man. J The Journal of Clinical Endocrinology and Metabolism 40:278-289
[4] R Package ‘nlme’.https://svn.r-project.org/R-packages/trunk/nlme
[5] R Package ‘lmtest’ https://cran.r-project.org/web/packages/lmtest/index.html
Reference: PAGE () Abstr 9339 [www.page-meeting.org/?abstract=9339]
Poster: Methodology - New Modelling Approaches