Number of trials to estimate the condition number in rye traits<sup/>

Neu, Ismael Mario Márcio; Cargnelutti Filho, Alberto; Silveira, Daniela Lixinski; Pezzini, Rafael Vieira; Bandeira, Cirineu Tolfo

doi:10.5935/1806-6690.20230046

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Scientific Article • Rev. Ciênc. Agron. 54 • 2023 • https://doi.org/10.5935/1806-6690.20230046 copy

Number of trials to estimate the condition number in rye traits¹ 1 Paper extracted from the thesis of the first author, presented to the graduate program in Agronomy of the Universidade Federal de Santa Maria (UFSM), Santa Maria-RS, Brasil. Study financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Multicollinearity must be diagnosed in multivariate analyses. Among the indicators, the condition number can be used to quantify the degree of multicollinearity. Hence, this study sought to determine the number of measurements (trials) necessary to estimate the number of condition in linear correlation matrices between rye traits. Five uniformity trials were carried out with ‘BRS Progresso’ rye, and eight morphological traits and eight productive traits were evaluated, forming two groups. In each group of traits, six cases (combinations of traits) were planned and the multicollinearity diagnosis was performed. Repeatability analyses were performed using the following methods: analysis of variance, principal component analysis, and structural analysis, and the number of measurements (trials) was determined for different levels of precision. A higher condition number of repeatability coefficients was obtained by the principal component methods (based on correlation and variance and covariance matrices) and structural analysis based on the variance and covariance matrix. A greater number of measurements (trials) is necessary to estimate the number of conditions in productive traits compared to morphological ones. One trial is enough to efficiently estimate the condition number with a minimum accuracy of 80% in morphological and productive traits of rye, whereas at least three trials are required for 95% accuracy.

Keywords:
Secale cereale L; Repeatability analysis; Multicollinearity; Experimental planning

Case¹	NCT²	No. of Trials	No. of Comb.³	Minimum	Mean	Median	Maximum	Range	PCWM (%)⁴
				------------------------------ Morphological traits ------------------------------
2	28	5	140	1.0	2.61	1.45	16.6	15.6	100.00
3	56	5	280	1.1	11.07	3.20	377.1	376.0	98.21
4	70	5	350	1.5	37.30	6.43	853.0	851.6	92.00
5	56	5	280	2.7	100.20	22.52	965.2	962.5	76.79
6	28	5	140	3.6	223.81	94.05	1043.7	1,040.1	50.71
7	8	5	40	75.6	426.78	352.37	1067.5	991.9	20.00
				-------------------------------- Productive traits ---------------------------------
2	28	5	140	1.0	5.85	2.25	78.3	77.3	100.00
3	56	5	280	1.5	18.95	6.61	170.3	168.9	97.14
4	70	5	350	2.6	50.57	27.67	519.5	516.9	85.43
5	56	5	280	7.1	120.00	69.54	723.0	716.0	49.29
6	28	5	140	23.6	251.10	198.46	1008.5	984.9	24.29
7	8	5	40	144.7	468.25	384.32	1170.8	1,026.2	0.00

Case¹	ANOVA²		PCR		PCS		SAR		SAS
Case¹	MORP	PROD	MORP	PROD	MORP	PROD	MORP	PROD	MORP	PROD
rc - repeatability coefficient
2	0.923	0.817	0.948	0.885	0.954	0.872	0.948	0.882	0.923	0.817
3	0.859	0.849	0.983	0.879	0.991	0.873	0.983	0.877	0.859	0.849
4	0.838	0.885	0.979	0.917	0.988	0.914	0.979	0.916	0.838	0.885
5	0.825	0.880	0.980	0.933	0.989	0.932	0.980	0.933	0.825	0.880
6	0.812	0.831	0.983	0.937	0.992	0.942	0.983	0.937	0.812	0.831
7	0.792	0.707	0.988	0.961	0.994	0.969	0.988	0.961	0.792	0.707
Mean	0.842 a³	0.828 a	0.977 a	0.918 b	0.985 a	0.917 b	0.977 a	0.918 b	0.842 a	0.828 a
R² - coefficient of determination
2	0.984	0.957	0.989	0.975	0.991	0.972	0.989	0.974	0.984	0.957
3	0.968	0.966	0.997	0.973	0.998	0.972	0.997	0.973	0.968	0.966
4	0.963	0.975	0.996	0.982	0.998	0.981	0.996	0.982	0.963	0.975
5	0.959	0.973	0.996	0.986	0.998	0.986	0.996	0.986	0.959	0.973
6	0.956	0.961	0.997	0.987	0.998	0.988	0.997	0.987	0.956	0.961
7	0.950	0.924	0.998	0.992	0.999	0.994	0.998	0.992	0.950	0.924
Mean	0.963 a	0.959 a	0.996 a	0.982 b	0.997 a	0.982 b	0.996 a	0.982 b	0.963 a	0.959 a

Case¹	Morphological traits					Productive traits
	---------------------------------------------- Coefficient of determination (R²) ---------------------------------------------
	0.80	0.85	0.90	0.95	0.99	0.80	0.85	0.90	0.95	0.99
	----------------------------------------------- ANOVA - Analysis of variance -----------------------------------------------
2	0.3	0.5	0.8	1.6	8.3	0.9	1.3	2.0	4.3	22.2
3	0.7	0.9	1.5	3.1	16.3	0.7	1.0	1.6	3.4	17.6
4	0.8	1.1	1.7	3.7	19.1	0.5	0.7	1.2	2.5	12.9
5	0.8	1.2	1.9	4.0	21.0	0.5	0.8	1.2	2.6	13.5
6	0.9	1.3	2.1	4.4	22.9	0.8	1.1	1.8	3.9	20.1
7	1.1	1.5	2.4	5.0	26.0	1.7	2.3	3.7	7.9	40.9
Mean	0.76 a²	1.08 a	1.72 a	3.63 a	18.92 a	0.86 a	1.21 a	1.93 a	4.07 a	21.21 a
	------------------------------ PCR - Principal components based on the correlation matrix ------------------------------
2	0.2	0.3	0.5	1.0	5.4	0.5	0.7	1.2	2.5	12.9
3	0.1	0.1	0.2	0.3	1.7	0.6	0.8	1.2	2.6	13.7
4	0.1	0.1	0.2	0.4	2.1	0.4	0.5	0.8	1.7	9.0
5	0.1	0.1	0.2	0.4	2.0	0.3	0.4	0.6	1.4	7.1
6	0.1	0.1	0.2	0.3	1.7	0.3	0.4	0.6	1.3	6.7
7	0.0	0.1	0.1	0.2	1.2	0.2	0.2	0.4	0.8	4.0
Mean	0.10 b	0.13 b	0.21 b	0.45 b	2.35 b	0.36 a	0.51 a	0.81 a	1.71 a	8.90 a
	------------------------------ PCS – Principal components based on the covariance matrix ------------------------------
2	0.2	0.3	0.4	0.9	4.7	0.6	0.8	1.3	2.8	14.5
3	0.0	0.1	0.1	0.2	0.9	0.6	0.8	1.3	2.8	14.4
4	0.0	0.1	0.1	0.2	1.2	0.4	0.5	0.8	1.8	9.3
5	0.0	0.1	0.1	0.2	1.1	0.3	0.4	0.7	1.4	7.2
6	0.0	0.0	0.1	0.2	0.8	0.2	0.3	0.6	1.2	6.1
7	0.0	0.0	0.1	0.1	0.6	0.1	0.2	0.3	0.6	3.1
Mean	0.07 b	0.09 b	0.14 b	0.30 b	1.56 b	0.37 a	0.52 a	0.83 a	1.75 a	9.10 a
	------------------------------- SAR – Structural analysis based on the correlation matrix --------------------------------
2	0.2	0.3	0.5	1.0	5.5	0.5	0.8	1.2	2.5	13.2
3	0.1	0.1	0.2	0.3	1.7	0.6	0.8	1.3	2.7	13.9
4	0.1	0.1	0.2	0.4	2.1	0.4	0.5	0.8	1.7	9.1
5	0.1	0.1	0.2	0.4	2.0	0.3	0.4	0.6	1.4	7.1
6	0.1	0.1	0.2	0.3	1.7	0.3	0.4	0.6	1.3	6.7
7	0.0	0.1	0.1	0.2	1.2	0.2	0.2	0.4	0.8	4.0
Mean	0.10 b	0.13 b	0.21 b	0.45 b	2.36 b	0.36 a	0.51 a	0.82 a	1.73 a	9.00 a
	-------------------------------- SAS - Structural analysis based on the covariance matrix --------------------------------
2	0.3	0.5	0.8	1.6	8.3	0.9	1.3	2.0	4.3	22.2
3	0.7	0.9	1.5	3.1	16.3	0.7	1.0	1.6	3.4	17.6
4	0.8	1.1	1.7	3.7	19.1	0.5	0.7	1.2	2.5	12.9
5	0.8	1.2	1.9	4.0	21.0	0.5	0.8	1.2	2.6	13.5
6	0.9	1.3	2.1	4.4	22.9	0.8	1.1	1.8	3.9	20.1
7	1.1	1.5	2.4	5.0	26.0	1.7	2.3	3.7	7.9	40.9
Mean	0.76 a	1.08 a	1.72 a	3.63 a	18.92 a	0.86 a	1.21 a	1.93 a	4.07 a	21.21 a

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