Nonlinear mixed effect models to describe the dry matter accumulation in garlic plants

Teixeira, Filipe Ribeiro Formiga; Nascimento, Moysés; Cecon, Paulo Roberto; Silva, Anderson Rodrigo da

doi:10.1590/0103-8478cr20230471

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CROP PRODUCTION • Cienc. Rural 54 (12) • 2024 • https://doi.org/10.1590/0103-8478cr20230471 copy

Nonlinear mixed effect models to describe the dry matter accumulation in garlic plants

Modelos não lineares mistos na descrição do acúmulo de matéria seca em plantas de alho

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT:

Given the importance of describing the accumulation of total dry matter in garlic accessions and the advantage of Nonlinear Mixed Effect Models (NLME) in this process, the present work research compared four nonlinear equations (Gompertz, Logistic, Richards, and von Bertalanffy) in the fit of accumulation of total dry matter per plant of 30 garlic accessions. The objective was also to identify the best accessions according to each growth parameter by estimating the random effects around the mean through the best among the models. The analysis was carried out using the R software. The best model was the Logistic according to the criteria used for comparison (AIC, BIC,

R_{a j .}^{2}

, MSE and MAE), presenting estimates closer to the actual observed values. According to the random effects estimated by this model, which represent deviations from the mean, the accessions that showed the highest asymptotic weight were 4505, 4826 and 4500, while accessions 4826, 4837 and 4491 took longer to reach the inflection point of the curve. The NLME approach used one fit per equation to obtain information on all individuals in the sample, efficiently adjusting the accumulated total dry matter and identifying the best accessions according to the estimated random effects of its parameters.

Key words:
Allium sativum; fixed effects; growth; random effects; statistical modeling

Model	--------------------------------------------------------------Parameters--------------------------------------------------------------
	${\hat{β}}_{1}$	${\hat{β}}_{2}$	${\hat{β}}_{3}$	${\hat{β}}_{4}$
Gompertz	23.69 (0.9069)	98.90 (0.5913)	0.06 (0.0027)	-
Logistic	23.17 (0.8632)	106.56 (0.6931)	0.09 (0.0025)	-
Richards	22.57 (0.8238)	115.90 (5.6231)	0.21 (0.1598)	4.16 (3.7557)
von Bertalanffy	26.60 (1.1627)	10.28 (1.6764)	0.04 (0.0019)	-

Universidade Federal de Santa Maria Universidade Federal de Santa Maria, Centro de Ciências Rurais , 97105-900 Santa Maria RS Brazil , Tel.: +55 55 3220-8698 , Fax: +55 55 3220-8695 - Santa Maria - RS - Brazil
E-mail: cienciarural@mail.ufsm.br

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[1] E-mail: filiperfteixeira@ufpi.edu.br. ^*Corresponding author

DAP	Average ± SD	Median	Minimum	Maximum
60	0.97 ± 0.30	0.93	0.45	1.60
90	4.44 ± 0.96	4.34	2.86	6.72
120	17.70 ± 3.59	17.40	11.80	28.80
150	22.60 ± 4.70	22.10	13.00	36.10

Reference	-----------------------------------------Equation^*-----------------------------------------
Gompertz	$y_{i j} = ϕ_{1 i} e x p \{- e x p [- ϕ_{3 i} (x_{i j} - ϕ_{2 i})]\} + ε_{i j}$
Logistic	$y_{i j} = ϕ_{1 i} {\{1 + e x p [- ϕ_{3 i} (x_{i j} - ϕ_{2 i})]\}}^{- 1} + ε_{i j}$
Richards	$y_{i j} = ϕ_{1 i} {\{1 + ϕ_{4 i} e x p [- ϕ_{3 i} (x_{i j} - ϕ_{2 i})]\}}^{- \frac{1}{ϕ_{4 i}}} + ε_{i j}$
von Bertalanffy	$y_{i j} = ϕ_{1 i} {[1 - ϕ_{2 i} e x p (- ϕ_{3 i} x_{i j})]}^{3} + ε_{i j}$
Linear Mixed Effect (LME) Model	$y_{i j} = ϕ_{1 i} {+ ϕ_{2 i} x}_{i j} + ε_{i j}$

Model	AIC	BIC	$R_{a d j .}^{2}$	MAE	MSE
Gompertz	491.53	511.05	0.9873	0.83	1.11
Logistic	472.94	492.45	0.9931	0.61	0.59
Richards	472.74	497.83	0.9902	0.64	0.86
von Bertalanffy	529.39	548.91	0.9805	1.10	1.70