Nonlinear Mixed Effects Model And Its Application In Forestry

Nonlinear mixed effects models (NLMEMs) are built on the regression function, whichrelying on the nonlinear relationship of fixed effects parameters and random effects parameters.They are modern statistical methods to analyze longitudinal data, multilevel data and repeatedsurvey data. They also can be considered as powerful statistical tools, which not only reflectthe overall variation trend of population, but also describe individual differences. In recentlyyears, NLMEMs are getting much attention from the scholars in various disciplines, such asmedicine, engineering, agriculture and forestry and so on. After30years' development,several scholars have proposed the single level NLMEMs and the multi-level NLMEMs, andtypical statistical software such as SAS and S-Plus were developed to estimate them. Inpractice, we found two main problems in existing NLMEMs and estimation methods. One isthat the main software such as SAS and S-Plus may not always reach convergence, especiallywhen the model contains many estimable parameters. The other is that the current NLMEMsmay not cover all types of random effects combinations (such as interaction that is commonlyused in forestry), which limits application of NLMEMs. The objectives of this paper aretherefore to (1) propose an algorithm with good convergence to estimate single level andmulti-level NLMEMs,(2) put forward a unified expression that includes all types of randomeffects of NLMEMs and corresponding parameter estimation method,(3) complete and realizethe coding for the above two algorithm in ForStat software,(4) and use our programs toresolve a practical problem in forestry, which is not resolved yet.The study has successfully achieved the above four objectives. Specific contents of thepaper are as follows:1) Based on the first-order conditional expectation linearization–expectation maximation (FOCE-EM), it draws calculation formulas and processes of singlelevel and multi-level NLMEMs.2) The study proposed a normal standard expression forNLNEMs, which contains all types of random effects, and put forward the relevant parameterestimation method, naming linearization approximation-sequential quadratic programming algorithm.3) We found the mixed procedure in SAS may not guarantee the variance ofrandom effects parameters as non-negative definite, so we used one specific case to verify thisdrawback. Therefore, we proposed the linearization approximation-sequential quadraticprogramming algorithm to estimate the parameters in the NLMEMs standard expression, andwe present ten common variance types of constraint conditions to guarantee each variancepositive definite or half positive definite.4) The standard expression of this model can dealwith classification (namely quantification problem) of both fixed effects and random effectsparameters. We also pointed out some defects of classification differential method in nlmefunction of S-Plus, and put forward the solutions such as linearization approximation-sequential quadratic programming algorithm to overcome this drawback5) It is the first timethat we used two factors (stand density and site index class) NLMEMs within interactions toanalyze the height–diameter model for Larix olgensis. We also further analyzed therelationship of random effects and altitude based on the two factors NLMEMs.Based on the analysis and results from the study, the following conclusions can be drawn:1) the proposed normal NLMEMs standard expression contain various types of NLMEMs(random effects is normal distribution), such as single level NLMEMs, grading multi-levelNLMEMs, multi-factor NLMEMs with main effects, multi-factor NLMEMs with main effectsand interaction effects, and general NLMEMs that combined several types of random effects..We also spread the normal NLMEMs standard expression to those NLMEMs that thevariances of random effects are related to some factors (also called as group variable). Ascompared with the traditional models, the proposed models are more general and useful.2)When calculating single level or grading multi-level NLMEMs, the FOCE-EM algorithm hasa similar computational accuracy with the LB algorithm of SAS and S-Plus. However,FOCE-EM algorithm in theory ensures the linear step convergence, therefore, the convergenceof this algorithm is significantly better than LB algorithm.3) The study proposed linearizationapproximation-sequential quadratic programming algorithm to estimate the general types ofNLMEMs. It is necessary to choose this algorithm to guarantee the random effects variance asnon-negative definite.4) The computational speed of FOCE-EM algorithm is more quickly than linearization approximation-sequential quadratic programming algorithm. Therefore, it isbetter to use the FOCE-EM algorithm to analyze single level and multi-level NLMEMs. Forother types of NLMEMs, linearization approximation-sequential quadratic programmingalgorithm would be better.5) Through using the two factor NLMEMs to analyze the height-diameter model for Larix olgensis, it can be drawn that the prediction precision can beobviously increased by considering the interactions of stand density and site index class, and ifthe altitude was considered as group variable in the mixed model, the prediction precisionwould be further improved.