Nonlinear regression models for estimating linseed growth, with proposals for data collection

Nonlinear regression models represent an alternative way to describe plant growth. In this study, we aimed to model the growth of linseed using four methods for data collection (longitudinal, mean, random, and cross-sectional) and fitting the logistic and Von Bertalanffy nonlinear regression models. The data came from experiments conducted between 2014 and 2020 in the municipality of Curitibanos, Santa Catarina, Brazil. The study had a randomized block design, with experimental units consisting of six lines, 5.0 m long and 3.0 m wide, containing the varieties and cultivars of linseed with four replicates. We performed weekly assessments of the number of secondary stems and plant height and measured total dry mass fortnightly. After tabulation, the data were analyzed using the four methods, and the logistic and Von Bertalanffy models were fitted. The logistic model for the plant height variable exhibited the best performance using the longitudinal, mean, and cross-sectional methods. It was an alternative approach that reduced the time and labor required to conduct the experiment.

Keywords:
logistic; Von Bertalanffy; Linum usitatissimum; nonlinear regression

Authorship

Mariane Peripolli

Departamento de Fitotecnia, Universidade Federal de Santa Maria, Av. Roraima, 1000, 97105-900, Santa Maria, Rio Grande do Sul, Brazil. Universidade Federal de Santa MariaBrazilSanta Maria, Rio Grande do Sul, BrazilDepartamento de Fitotecnia, Universidade Federal de Santa Maria, Av. Roraima, 1000, 97105-900, Santa Maria, Rio Grande do Sul, Brazil.

Alessandro Dal'Col Lúcio

Darlei Michalski Lambrecht ^** Author for correspondence. E-mail: darleilambrecht@gmail.com

https://orcid.org/0000-0002-1376-3504

Jaqueline Sgarbossa

Lana Bruna de Oliveira Engers

Sidnei José Lopes

Leosane Cristina Bosco

Departamento de Agricultura, Biodiversidade e Florestas, Universidade Federal de Santa Catarina, Curitibanos, Santa Catarina, Brazil.Universidade Federal de Santa CatarinaBrazilCuritibanos, Santa Catarina, BrazilDepartamento de Agricultura, Biodiversidade e Florestas, Universidade Federal de Santa Catarina, Curitibanos, Santa Catarina, Brazil.

Dislaine Becker

* Author for correspondence. E-mail: darleilambrecht@gmail.com

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (3)
Tables (4)

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Figure 1
Confidence intervals for parameters and critical points of the logistic nonlinear model for the variable plant height (cm) for data collection methods: longitudinal (L), mean (M), random (R), and cross-sectional (C) for linseed in the cultivar Aguará INTA, 2016 (April 26th, 2016). β₁ (represents the height), β₂ (represents the time of growth), β₃ (represents the growth rate), XMAP (maximum acceleration point), XIP (inflection point), XMDP (maximum deceleration point), XADP (asymptotic deceleration point), and concentration (XMDP-XMAP).

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Figure 2
Confidence intervals for parameters and critical points of the Von Bertalanffy nonlinear model for the variable plant height (cm) for data collection methods: longitudinal (L), mean (M), random (R), and cross-sectional (C) for linseed in the cultivar Aguará INTA, 2016 (April 26th, 2016). β₁ (represents the height), β₂ (represents the time of growth), β₃ (represents the growth rate), XMAP (maximum acceleration point), XIP (inflection point), XMDP (maximum deceleration point), XADP (asymptotic deceleration point), and concentration (XMDP-XMAP).

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Figure 3
Growth curves for the variable plant height (cm) for linseed in the cultivar Aguará INTA, logistic nonlinear regression Model (A) and its critical points (C), Von Bertalanffy nonlinear regression Model (B), and its critical points (D), 2016 (April 26^th, 2016).

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Table 1
Experiment features: crops, sowing dates, linseed varieties (brown and golden) and cultivars (Aguará INTA and Caburé INTA), variables evaluated in the 2014/2015 to 2020/2021 crops.

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Table 2
P value for the tests of normality (Shapiro-Wilk - SW), heteroscedasticity (Breusch Pagan - BP), and error independence (Durbin Watson - DW) for the nonlinear logistic and Von Bertalanffy models for the four data collection methods.

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Table 3
Adjustment quality indices: fitted coefficient of determination (R² _aj), random standard error (RSE), adjustment standard deviation (ASD), Akaike information criterion (AIC), and Bayesian information criterion (BIC) for the nonlinear logistic and Von Bertalanffy models for the four data collection methods for linseed plant height of cultivar Aguará INTA, 2016 (April 26^th, 2016).

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Table 4
Percentage of adjustments for intrinsic (c ^I ) and parametric (c ^θ ) nonlinearity in the nonlinear logistic and Von Bertalanffy models in the cultivars Aguará INTA and Caburé INTA and golden and brown varieties for the four data collection methods concerning the variables plant height, number of secondary stems, and total dry mass for linseed.

Figure 1 Confidence intervals for parameters and critical points of the logistic nonlinear model for the variable plant height (cm) for data collection methods: longitudinal (L), mean (M), random (R), and cross-sectional (C) for linseed in the cultivar Aguará INTA, 2016 (April 26th, 2016). β₁ (represents the height), β₂ (represents the time of growth), β₃ (represents the growth rate), XMAP (maximum acceleration point), XIP (inflection point), XMDP (maximum deceleration point), XADP (asymptotic deceleration point), and concentration (XMDP-XMAP).

Figure 2 Confidence intervals for parameters and critical points of the Von Bertalanffy nonlinear model for the variable plant height (cm) for data collection methods: longitudinal (L), mean (M), random (R), and cross-sectional (C) for linseed in the cultivar Aguará INTA, 2016 (April 26th, 2016). β₁ (represents the height), β₂ (represents the time of growth), β₃ (represents the growth rate), XMAP (maximum acceleration point), XIP (inflection point), XMDP (maximum deceleration point), XADP (asymptotic deceleration point), and concentration (XMDP-XMAP).

Figure 3 Growth curves for the variable plant height (cm) for linseed in the cultivar Aguará INTA, logistic nonlinear regression Model (A) and its critical points (C), Von Bertalanffy nonlinear regression Model (B), and its critical points (D), 2016 (April 26^th, 2016).

Table 1 Experiment features: crops, sowing dates, linseed varieties (brown and golden) and cultivars (Aguará INTA and Caburé INTA), variables evaluated in the 2014/2015 to 2020/2021 crops.

Crops	Seeding	Harvest	Genotypes	Variables
2014	August 14^th, 2014	January 05th,2015 (Golden) December 17th, 2014 (Brown)	Golden var. Brown var.	Plant height and total dry mass
2015	July 23^rd, 2015	December 9th, 2015	Golden var. Brown var.	Plant height and total dry mass
2016	April 26^th, 2016	November 9th, 2016	Aguará INTA cv. Caburé INTA cv. Golden var.	Plant height and secondary stems
2016	May 20^th, 2016	December 6th, 2016	Aguará INTA cv. Caburé INTA cv. Golden var.	Plant height, secondary stems, and total dry mass
2018	April 13^th, 2018	November 20th, 2018 October 22nd, 2018 (Golden)	Aguará INTA cv. Caburé INTA cv. Golden var. Brown var.	Plant height and number of secondary stems
2018	May 24^th, 2018	November 24th, 2018	Aguará INTA cv. Caburé INTA cv. Golden var. Brown var.	Plant height and number of secondary stems
2019	May 27^th, 2019	December 22nd, 2019	Brown var.	Total dry mass
2020	September 6^th, 2020	December 30th, 2020	Brown var.	Plant height and number of secondary stems

Table 2 P value for the tests of normality (Shapiro-Wilk - SW), heteroscedasticity (Breusch Pagan - BP), and error independence (Durbin Watson - DW) for the nonlinear logistic and Von Bertalanffy models for the four data collection methods.

	Logistic			Von Bertalanffy
Method	SW	BP	DW	SW	BP	DW
Longitudinal	2.36 x 10^-5	1.42 x 10^-3	0.596	6.97 x 10^-5	0.002807	0
Mean	0.049479	0.587301	0.054	0.499341	0.928462	0
Random	0.002611	0.389177	0.886	0.020393	0.492409	0.164
Cross-sectional	0.885824	0.075044	0.412	0.988261	0.431512	0.004

Table 3 Adjustment quality indices: fitted coefficient of determination (R² _aj), random standard error (RSE), adjustment standard deviation (ASD), Akaike information criterion (AIC), and Bayesian information criterion (BIC) for the nonlinear logistic and Von Bertalanffy models for the four data collection methods for linseed plant height of cultivar Aguará INTA, 2016 (April 26^th, 2016).

	Logistic
Method	R² _aj	RSE	RSD	AIC	BIC
Longitudinal	0.945126	6.849924	3.954806	1743.434	1757.677
Mean	0.998158	1.384269	0.799208	49.93614	52.19594
Random	0.945314	9.216176	5.320962	99.22663	100.0864
Cross-sectional	0.994246	2.439279	1.408318	64.66593	66.92573
	Von Bertalanffy
Method	R² _aj	RSE	RSD	AIC	BIC
Longitudinal	0.938026	7.27964	4.202902	1775.073	1789.316
Mean	0.990659	3.117422	1.799845	71.04384	73.30363
Random	0.932642	8.75237	5.053183	97.8841	100.1439
Cross-sectional	0.988687	3.420312	1.974718	73.45469	75.71449

Table 4 Percentage of adjustments for intrinsic (c ^I ) and parametric (c ^θ ) nonlinearity in the nonlinear logistic and Von Bertalanffy models in the cultivars Aguará INTA and Caburé INTA and golden and brown varieties for the four data collection methods concerning the variables plant height, number of secondary stems, and total dry mass for linseed.

Plant height
	Logistic				Von Bertalanffy
Method	Aguara	Cabure	Golden	Brown	Aguara	Cabure	Golden	Brown
Longitudinal	100%	100%	100%	100%	100%	100%	100%	60%
Mean	100%	100%	100%	100%	100%	40%	50%	40%
Random	100%	80%	100%	20%	20%	0%	0%	0%
Cross-sectional	60%	80%	100%	100%	80%	40%	16.67%	0%
Total dry mass					Number of secondary stems
	Logistic				Logistic
Method	Aguara	Cabure	Golden	Brown	Aguara	Cabure	Golden	Brown
Longitudinal	50%	33.33%	50%	0%	50%	50%	100%	100%
Mean	100%	66.67%	25%	0%	50%	50%*	100%	100%
Random	50%	0%	0%	0%	0%	0%	0%	100%
Cross-sectional	50%	0%	0%	0%	0%	0%	0%	100%

*Von Bertalanffy model with 33.33%.

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Nonlinear regression models for estimating linseed growth, with proposals for data collection

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[1] * Author for correspondence. E-mail: darleilambrecht@gmail.com