Indirect selection for multiple technological and nutritional traits in common bean cultivars under different degrees of multicollinearity

ABSTRACT

When implementing path analysis between technological and nutritional traits in common bean without considering the degree of multicollinearity, it is important to assess whether there will be errors in indirect selection. This study proposed: to assess path analysis under different degrees of multicollinearity for technological and nutritional traits in common bean; and to recognize the degree of multicollinearity that allows an accurate interpretation of direct and indirect effects of these traits for the indirect selection of fast-cooking. Then, 10 technological traits and seven nutritional traits were measured in 25 common bean cultivars obtained across four experiments. Path analysis was carried out using different degrees of multicollinearity: severe (including all traits), moderate to strong (excluding three traits), and weak (excluding four traits). The magnitude and algebraic sign of direct and indirect effects between technological and nutritional traits varied when path analysis was performed using different degrees of multicollinearity. With severe multicollinearity (condition number = 6,798.28), regression coefficient values were excessively high and in an unfavorable direction for selection. Under weak multicollinearity (condition number = 66.21), the value and sign of the generated correlation coefficients were more in line with the biological phenomenon studied, allowing for an accurate interpretation of the direct and indirect effects of technological and nutritional traits on cooking time. Indirect selection based on the lowest values of mass of 100 grains and calcium concentration is recommended in this study for the selection of fast-cooking common bean cultivars.

Key words
Phaseolus vulgaris L.; condition number; trait exclusion; path analysis; fast-cooking

INTRODUCTION

In the development of new common bean (Phaseolus vulgaris L.) cultivars, apart from aiming for high grain yield, resistance to biotic and abiotic stresses, and other desirable agronomic traits, it is imperative to ensure technological and nutritional quality. This is due to the increasing significance of grain traits such as color, size, cooking time, and nutritional value for consumers when purchasing common bean.

Mineral concentration in common bean grains exhibits quantitative inheritance, meaning its regulation by multiple genes, susceptibility to environmental influences, and low heritability (Ribeiro et al. 2019, Ribeiro and Mezzomo 2020). This complex nature of mineral concentration complicates the selection process, prolonging the release of new biofortified common bean cultivars. Furthermore, assessing mineral concentration involves the acquisition of chemical products controlled by the military and federal law enforcement agencies. These reagents are relatively costly, and the acid digestion process generates chemical residues that are environmentally polluting and harmful to health. Consequently, indirect selection through traits that are less complex, display higher heritability, require simpler and quicker measurement, and pose reduced risks of chemical contamination is promising for common bean breeding programs.

The study of simple correlations has enabled the identification of promising technological and nutritional traits for indirect selection in common bean (Steckling et al. 2017, Ribeiro and Kläsener 2020, Ribeiro et al. 2021a, 2021b, Ribeiro and Maziero 2023). Nevertheless, while correlation coefficients are highly useful to quantify and indicate the direction of influence of factors affecting complex traits, they fall short in determining the exact relative importance of direct and indirect effects (Cruz and Regazzi 1997). This decomposition of direct and indirect effects of explanatory variables on the main variable is achieved through path analysis.

Path analysis has proven to be an effective tool for indirect selection targeting high grain yield in common bean (Coimbra et al. 1999, Cabral et al. 2011). However, its application towards indirect selection for technological (Ribeiro et al. 2000, Santos et al. 2016) and nutritional (Zilio et al. 2017) traits remains underexplored by common bean breeding programs. No studies have been found in the literature examining the direct and indirect effects of technological and nutritional traits on cooking time in common bean. Additionally, no record was found on whether errors may occur in indirect selection if path analysis is implemented without considering the effects of multicollinearity between agronomic, technological and/or nutritional traits in common bean.

In this context, it is necessary to ascertain the degree of multicollinearity to be employed in path analysis that provides correlation coefficients whose value and sign align with the biological phenomenon evaluated. The hypothesis is that without multicollinearity a more accurate interpretation of the direct and indirect effects of technological and nutritional attributes on cooking time in common bean will enable. As a result, more coherent and reliable correlation estimates will be generated, ultimately leading to greater gains in indirect selection for technological and nutritional traits in common bean.

In this way, common bean breeding programs are expected to achieve greater efficiency in indirect selection for grain traits that decisively influence consumer choices. The primary objective of this study was to analyze the direct and indirect effects of technological and nutritional traits on cooking time in common bean under varying degrees of multicollinearity. Additionally, the study aimed to determine the degree of multicollinearity that enables the accurate interpretation of path analysis results and to recommend promising traits for the indirect selection of fast-cooking common bean cultivars.

MATERIALS AND METHODS

Description of experiments

Four experiments were conducted in Santa Maria, Rio Grande do Sul, Brazil (29º42’ S latitude, 53º49’ W longitude, and 95 m altitude). Three of these were established during the rainy season (sowing in October) in the years 2019, 2020, and 2021, and one during the dry season of 2021 (sowing in February). According to the Köppen’s classification, the climate in the region is humid subtropical (Alvares et al. 2013).

The soil in the experimental area is classified as a typic alitic Argisol, Hapludalf. Soil analysis revealed the following chemical composition before sowing in the 2019 experiment: pH (H₂O) = 6.7; organic matter = 2%; K = 60 mg·dm^-3; P = 12.3 mg·dm^-3; Ca = 5.6 cmol·dm^-3; Mg = 3 cmol·dm^-3; and Zn = 1 mg·dm^-3. New soil analyses were conducted in the subsequent years to adjust the fertilizer requirements for each experiment. For all experiments, the soil was prepared through conventional cultivation methods, which involved one plowing and two harrowing operations, thus ensuring uniformity across the area for sowing and seedling emergence.

This study employed a randomized-block experimental design with three repetitions. The experimental unit was constituted by four 4-m-long rows spaced 0.5-m apart, resulting in a total area of 8 m². However, the usable plot area comprised solely the two central rows (4 m²), to prevent the intermingling of cultivars. The evaluated treatments were 25 common bean cultivars (Table 1). These cultivars represent a sample of the cultivars that were released by public breeding programs in Brazil and were registered for cultivation in Rio Grande do Sul, with some also registered for other states (Brasil 2023). Hence, the cultivars selected for this study are representative of the technological advancements achieved by the main common bean breeding programs in Brazil.

Table 1
Common bean cultivars evaluated, gene pool, grain type, and breeding program.

Cultivar	Gene Pool1	Grain Type2	Breeding Program
1. BRSMG Realce	Andean	cranberry	EMBRAPA
2. Iraí	Andean	cranberry	SAPDR
3. IPR Uirapurú	Mesoamerican	black	IDR
4. IAC Netuno	Mesoamerican	black	IAC
5. BRS Esplendor	Mesoamerican	black	EMBRAPA
6. IPR Tuiuiú	Mesoamerican	black	IDR
7. BRS Valente	Mesoamerican	black	EMBRAPA
8. BRS Esteio	Mesoamerican	black	EMBRAPA
9. Fepagro 26	Mesoamerican	black	SAPDR
10. Guapo Brilhante	Mesoamerican	black	EMBRAPA
11. IPR Graúna	Mesoamerican	black	IDR
12. BRS Expedito	Mesoamerican	black	EMBRAPA
13. IPR Tiziu	Mesoamerican	black	IDR
14. Fepagro Triunfo	Mesoamerican	black	SAPDR
15. BRS Campeiro	Mesoamerican	black	EMBRAPA
16. IPR Siriri	Mesoamerican	carioca	IDR
17. IPR Tangará	Mesoamerican	carioca	IDR
18. IAC Milênio	Mesoamerican	carioca	IAC
19. BRSMG Pioneiro	Mesoamerican	carioca	EMBRAPA
20. Fepagro Garapiá	Mesoamerican	carioca	SAPDR
21. Pérola	Mesoamerican	carioca	EMBRAPA
22. IAC Imperador	Mesoamerican	carioca	IAC
23. IPR Juriti	Mesoamerican	carioca	IDR
24. SCS 205 Riqueza	Mesoamerican	carioca	EPAGRI
25. BRS Estilo	Mesoamerican	carioca	EMBRAPA

• ¹
  Andean:
• large-sized grains (mass of 100 grains > 40 g); Mesoamerican: grain size varied from small (mass of 100 grains < 25 g) to medium (mass of 100 grains of 25 to 40 g);
• ²
  cranberry (cream seed coat with red streaks);
• black (black seed coat); and carioca (beige seed coat with brown streaks); EMBRAPA: Brazilian Agricultural Research Corporation; SAPDR: Secretariat of Agriculture, Livestock, and Rural Development of the State of Rio Grande do Sul; IDR: Rural Development Institute of Paraná; IAC: Agronomic Institute of Campinas; EPAGRI: Agricultural Research and Rural Extension Corporation of Santa Catarina.

Uniform and consistent management practices were applied across all experiments. The amount of fertilizers (05-20-20 formula and urea) employed in each experiment was based on the requirements indicated by annual soil analysis reports. Seed treatment involved the application of the fungicide Maxim (Fludioxonil and Metalaxil-M) and the insecticide Cruiser 350 FS (Thiamethoxam) at a rate of 200 mL·100 kg^-1 of seeds. Insect control measures were implemented during crop development with the application of Engeo^TM Pleno (Thiamethoxam and Lambda-cyhalothrin) at a rate of 125 mL·ha^-1. Weed management was addressed through a combination of mechanical (weeding) and chemical (herbicides) methods: Dual Gold (S-Metolachlor) and Basagran (Bentazone) at the rates of 1.25 and 1.50 L·ha^-1, respectively. Sprinkler irrigation was employed whenever periods of prolonged water deficit occurred.

Harvesting and determination of technological and nutritional traits

Harvesting occurred at the maturity stage, characterized by 90% of the plants in the usable plot area exhibiting dry pods and grains displaying the standard color of each cultivar. Plants were manually uprooted, labeled, and air-dried until processing. Manual grain threshing was carried out to prevent cross-contamination of cultivars and potential physical and mechanical grain damage. Subsequently, the grains were stored in paper bags, which were then wrapped in plastic bags and kept in a cold room (temperature of 5°C and relative humidity of 75%) until subsequent evaluations of technological traits and mineral concentration (Table 2).

Table 2
Description of the 17 technological and nutritional traits determined in 25 common bean cultivars in experiments conducted between 2019 and 2021.

Trait	Methodology description	Unit
1. Lightness (L* value)	Quantified with a portable colorimeter: ranges from 0 (black) to 100 (white)
2. Chromaticity a (a* value)	Analyzed with a portable colorimeter: ranges from -60 (green) to + 60 (red)	-
3. Chromaticity b (b* value)	Determined with a portable colorimeter: ranges from -60 (blue) to + 60 (yellow)	-
4. Grain length	Measured parallel to the hilum in 10 grains collected randomly	mm
5. Grain width	Length from the hilum to the opposite side in 10 grains collected randomly	mm
6. Grain thickness	Measurement perpendicular to the length and width of 10 grains collected randomly	mm
7. Mass of 100 grains	Average weight of three 100-grain samples collected randomly in each replication	g
8. Normal grains	Percentage of grains that absorbed water after 8 hours of soaking in distilled water	%
9. Water absorption	Calculated by the Eq.: [(weight of grains after soaking – weight of grains before soaking)/weight of grains before soaking] × 100	%
10. Cooking time	Evaluated in a Mattson cooker with 25 metal stems after 8 hours of soaking the grains in distilled water	min
11. Potassium concentration	Determined after nitric-perchloric digestion and flame photometer reading.	g·kg-1 dry matter (DM)
12. Phosphorus concentration	Analyzed after nitric-perchloric digestion and reading on an optical emission spectrophotometer	g·kg-1 DM
13. Calcium concentration	Measured after nitric-perchloric digestion and reading on an atomic absorption spectrophotometer	g·kg-1 DM
14. Magnesium concentration	Estimated after nitric-perchloric digestion and reading on an atomic absorption spectrophotometer	g·kg-1 DM
15. Iron concentration	Determined after nitric-perchloric digestion and reading on an atomic absorption spectrophotometer	mg·kg-1 DM
16. Zinc concentration	Assessed after nitric-perchloric digestion and reading on an atomicabsorption spectrophotometer	mg·kg-1 DM
17. Copper concentration	Quantified after nitric-perchloric digestion and reading on an atomic absorption spectrophotometer	mg·kg-1 DM

Statistical analyses

The data obtained in the four experiments were subjected to individual analysis of variance, combined analysis of variance, multicollinearity diagnostics, and path analysis, using Genes software (Cruz 2016). For all statistical analyses implemented, a significance level of 0.05 probability was adopted. The cooking time was converted to seconds. The normality assumption was not achieved neither for normal grain (%) nor for water absorption (%) data in most evaluated environments, when judged by symmetry, kurtosis and Lilliefors test. These data exhibited a Poisson distribution and the following transformation was applied, as recommended by Storck et al. (2000) (Eq. 1):

$Y_{ij}=\sqrt{}(x+0.5)$ (1)

in which: Y_ij: transformed variable; x: the trait value.

Analysis of variance was carried out individually, based on the obtained data for the technological and nutritional traits evaluated in each experiment. These analyses allowed us to determine whether the residual variances were homogeneous or heterogeneous, by the Hartley’s maximum F-test.

Combined analysis of variance was implemented by the least squares method, according to Eq. 2:

$Y_{ijk}=m+B/E_{jk}+C_i+E_j+C{E}_{ij}+E_{ijk}$ (2)

in which: Y_ijk = response variable referring to cultivar i, in block k and in environment j; m = overall mean; B/E_jk = block k in environment i; C_i = effect of cultivar i; E_j = effect of environment j; CE_ij = interaction between cultivar i and environment j; E_ijk = experimental error.

In this model, all effects were considered as random, except for the mean and the ‘cultivar’ source of variation, which were treated as fixed effects. The F test was executed to identify traits exhibiting significant effects related to cultivar, environment, and/or cultivar × environment interaction.

Multicollinearity diagnostics were carried out using the phenotypic correlation matrix generated from combined analysis of variance. The condition number (CN), which estimates the ratio between the highest and lowest eigenvalues of the correlation matrix, defined the degree of multicollinearity.

Path analysis was performed based on the phenotypic correlation matrix derived from combined analysis of variance, considering three degrees of multicollinearity: severe (CN ≥ 1000), moderate to strong (100 < CN < 1000), and weak (CN ≤ 100), proposed by Montgomery et al. (2012). In path analysis, cooking time served as the main variable, while the other traits were considered explanatory variables.

Multicollinearity diagnostics allowed us to identify the pair of traits displaying the highest correlation. The selection of which of these traits should be eliminated was based on detecting the trait with the greatest contribution to the last eigenvectors and which showed the highest variance inflation factor value (Cruz and Carneiro 2003). Multicollinearity diagnostics was repeated, and the need to eliminate traits was assessed again until reaching 100 < CN < 1000 (moderate to strong) and CN ≤ 100 (weak).

RESULTS AND DISCUSSION

Individual and combined analyses of variance

Uniform residual variances were obtained for all traits evaluated across the four experiments, except for normal grains, mass of 100 grains, and iron concentration. For these three traits, the ratio between the highest and lowest residual mean square exceeded 7, necessitating an adjustment in the degrees of freedom of the error and of the cultivar × environment interaction, as recommended by Cruz (2016). This procedure led to uniform residual variances for all technological and nutritional traits and allowed the execution of combined analysis of variance.

Cultivar and cultivar × environment interaction effects were not significant only for normal grains (Ribeiro et al. 2024), indicating a lack of genetic variability. Consequently, this trait was not included in the path analyses. However, 12 out of the 17 evaluated traits displayed a significant cultivar × environment interaction, demonstrating variation in technological and nutritional traits between common bean cultivars grown in different environments. The presence of a genotype × environment interaction has been previously described for multiple technological and nutritional traits (Steckling et al. 2017, Ribeiro and Kläsener 2020, Ribeiro et al. 2021b) determined in common bean lines. In our study, if the traits of technological grain quality and potassium, phosphorus, and calcium concentrations were used for indirect selection for fast cooking in common bean, the obtained outcomes are likely to depend on the growing environment.

Nonetheless, if indirect selection relies on distinct traits in each growing environment, this approach may yield limited selection gains, consequently constraining its use in common bean breeding programs. However, Pearson’s linear correlation analysis using data derived from three experiments revealed a high level of coincidence in the detection of significant correlations between technological and nutritional traits in common bean lines (Ribeiro and Maziero 2023). For the current study, path analysis of technological and nutritional traits was conducted using data from four common bean cultivar competition trials, which means more reliable correlation coefficients are expected.

Regarding the concentrations of magnesium, iron, zinc, and copper, only a significant cultivar effect was observed, signifying genetic variability for these mineral concentrations. Preliminary studies have also reported that common bean lines differ in the concentrations of magnesium, iron, zinc, and copper (Silva et al. 2012, Gouveia et al. 2014, McClean et al. 2017). Therefore, selection based on the concentrations of these minerals could be effective for indirect selection for fast cooking in common bean, regardless of the growing environment. Consequently, greater gains are expected with indirect selection.

Path analysis with severe multicollinearity

Path analysis with severe multicollinearity was implemented considering all evaluated traits, except for normal grains (Table 3, Fig. 1), due to the lack of genetic variability between the evaluated common bean cultivars. The mean value of normal grains was 97.27% (Ribeiro et al. 2024), indicating that the common bean cultivars had a low percentage of hard grains, i.e., grains that do not absorb water after soaking–also known as hardshell grains. Previous studies also found that common bean lines have a high percentage of normal grains (Ribeiro and Kläsener 2020, Ribeiro et al. 2021b, Kläsener et al. 2022). These findings highlight the success of breeding programs in selecting lines with a reduced percentage of hardshell grains, thus enhancing the cooking quality of new common bean cultivars.

Table 3
Path analysis under severe multicollinearity considering the direct (DE) and indirect (ID) effects obtained between the traits of luminosity (L*), chromaticity a* (a*), chromaticity b* (b*), grain length (length), grain width (width), grain thickness (thickness), mass of 100 grains (mass), water absorption (absorption), and concentrations of potassium (K), phosphorus (P), calcium (Ca), magnesium (Mg), iron (Fe), zinc (Zn), and copper (Cu) on cooking time (Ct) obtained in 25 common bean cultivars evaluated in four experiments carried out from 2019 to 2021.

L*		a*		b*		Length		Width		Thickness
DE on Ct	-2.41	DE on Ct	-1.00	DE on Ct	3.22	DE on Ct	0.14	DE on Ct	-0.17	DE on Ct	-0.40
ID		ID		ID		ID		ID		ID
a*	-0.98	L*	-2.35	L*	-2.41	L*	-1.33	L*	-1.00	L*	-1.34
b*	3.22	b*	3.13	a*	-0.98	a*	-0.70	a*	-0.52	a*	-0.63
Length	0.08	Length	0.10	Length	0.07	b*	1.73	b*	1.29	b*	1.76
Width	-0.07	Width	-0.09	Width	-0.07	Width	-0.13	Length	0.10	Length	0.11
Thickness	-0.22	Thickness	-0.25	Thickness	-0.22	Thickness	-0.34	Thickness	-0.24	Width	-0.10
Mass	0.79	Mass	0.97	Mass	0.76	Mass	1.44	Mass	1.16	Mass	1.33
Absorption	-0.05	Absorption	-0.05	Absorption	-0.05	Absorption	-0.04	Absorption	-0.01	Absorption	-0.05
K	0.05	K	0.07	K	0.05	K	0.10	K	0.09	K	0.13
P	-0.00	P	-0.01	P	-0.00	P	-0.03	P	-0.03	P	-0.02
Ca	-0.38	Ca	-0.41	Ca	-0.38	Ca	-0.39	Ca	-0.29	Ca	-0.32
Mg	0.02	Mg	0.03	Mg	0.02	Mg	0.04	Mg	0.04	Mg	0.02
Fe	-0.10	Fe	-0.10	Fe	-0.10	Fe	-0.05	Fe	0.02	Fe	-0.06
Zn	-0.09	Zn	-0.11	Zn	-0.09	Zn	-0.12	Zn	-0.15	Zn	-0.07
Cu	-0.04	Cu	-0.04	Cu	-0.04	Cu	-0.02	Cu	0.00	Cu	-0.02
Total	-0.20		-0.12		-0.21		0.30		0.29		0.33
Mass		Absorption		K		P		Ca		Mg
DE on Ct	1.47	DE on Ct	-0.16	DE on Ct	-0.20	DE on Ct	-0.11	DE on Ct	0.58	DE on Ct	-0.09
ID		ID		ID		ID		ID		ID
L*	-1.29	L*	-0.81	L*	0.65	L*	-0.11	L*	1.61	L*	0.47
a*	-0.66	a*	-0.30	a*	0.33	a*	-0.08	a*	0.72	a*	0.29
b*	1.67	b*	1.06	b*	-0.83	b*	0.11	b*	-2.10	b*	-0.57
Length	0.13	Length	0.03	Length	-0.07	Length	0.03	Length	-0.09	Length	-0.06
Width	-0.13	Width	-0.01	Width	0.07	Width	-0.05	Width	0.09	Width	0.08
Thickness	-0.36	Thickness	-0.13	Thickness	0.26	Thickness	-0.06	Thickness	0.22	Thickness	0.11
Absorption	-0.04	Mass	0.35	Mass	-0.91	Mass	0.33	Mass	-0.92	Mass	-0.60
K	0.12	K	0.01	Absorption	0.01	Absorption	-0.03	Absorption	0.02	Absorption	-0.04
P	-0.02	P	-0.02	P	-0.03	K	-0.06	K	-0.06	K	-0.08
Ca	-0.36	Ca	-0.06	Ca	0.18	Ca	-0.03	P	0.01	P	0.02
Mg	0.04	Mg	-0.02	Mg	-0.03	Mg	0.01	Mg	-0.05	Ca	0.30
Fe	-0.04	Fe	-0.06	Fe	0.00	Fe	-0.00	Fe	0.07	Fe	-0.02
Zn	-0.12	Zn	0.04	Zn	0.05	Zn	-0.04	Zn	0.09	Zn	0.11
Cu	-0.02	Cu	-0.00	Cu	0.01	Cu	0.02	Cu	0.02	Cu	0.00
Total	0.39		-0.09		-0.51		-0.06		0.19		0.06
Fe		Zn		Cu
DE on Ct	0.17	DE on Ct	-0.28	DE on Ct	0.08
ID		ID		ID
L*	1.34	L*	-0.79	L*	1.20
a*	0.55	a*	-0.40	a*	0.49
b*	-1.79	b*	1.02	b*	-1.64
Length	-0.04	Length	0.06	Length	-0.03
Width	-0.02	Width	-0.09	Width	-0.00
Thickness	0.14	Thickness	-0.10	Thickness	0.12
Mass	-0.37	Mass	0.61	Mass	-0.32
Absorption	0.05	Absorption	0.03	Absorption	0.01
K	-0.00	K	0.03	K	-0.03
P	0.00	P	-0.02	P	-0.03
Ca	0.25	Ca	-0.19	Ca	0.13
Mg	0.01	Mg	0.03	Mg	0.00
Zn	-0.05	Fe	0.03	Fe	0.11
Cu	0.05	Cu	0.01	Zn	-0.02
Total	0.28		-0.04		0.07
Condition number (CN)					6,798.28
Coefficient of determination (R2)					0.691
Residual variable effect					0.555

Figure 1
Causal diagram of path analysis under severe, moderate to strong, and weak multicollinearity considering the direct and indirect effects obtained between the technological traits and mineral concentration on cooking time obtained in 25 common bean cultivars evaluated in four experiments carried out from 2019 to 2021.

When path analysis was performed with all traits for which cultivar and/or cultivar × environment interaction effects were significant, a CN = 6,798.28 was observed (Ribeiro et al. 2024), characterizing severe multicollinearity as per the classes established by Montgomery et al. (2012). This severe multicollinearity was associated with a high coefficient of determination (R² = 0.691) (Table 3). Similarly, studies focusing on technological (Santos et al. 2016) and grain yield (Coimbra et al. 1999) traits in common bean lines also reported high R² values in path analysis. High R² values indicate that the set of explanatory variables analyzed is sufficient to explain variations in the main variable (Cruz and Carneiro 2003). Moreover, when a high R² value coincides with a zero residual variable effect, it implies that explanatory variables account for all the observed effects on the main variable (Coimbra et al. 1999, Santos et al. 2016). In the present study, the residual variable effect was 0.555, indicating that multicollinearity issues are partially obstructing the assessment of the influence of explanatory variables on the response of the main variable (cooking time).

Additionally, the major direct effects on cooking time in common bean were provided by the chromaticity b (b* value), followed by the lightness (L* value), mass of 100 grains, and the chromaticity a (a* value) (Table 3). The regression coefficients for these traits exhibited high magnitude, with some having an algebraic sign contrary to our initial expectations. Prior research employing Pearson’s linear correlation analysis, after excluding highly correlated traits that contributing to multicollinearity, did not found a correlation between cooking time and L* and a* values in common bean lines of varying grain colors originating from Mesoamerican and Andean gene pools (Ribeiro et al. 2021b, Kläsener et al. 2022). Therefore, when conducting path analysis with and without multicollinearity, changes may be present in both the magnitude and direction of the estimates of direct and indirect effects of technological and nutritional traits on cooking time in common bean. This, in turn, leads to errors in indirect selection.

This issue arises because, under severe multicollinearity, variances associated with path coefficients, which measure the direct effects of the explanatory variables on the main variable, may reach excessively high values, generating estimates that do not align with the studied biological phenomena and lack reliability (Cruz and Carneiro 2003). In our study, when path analysis for technological and nutritional traits in common bean was conducted with severe multicollinearity, several regression coefficients with unexpected magnitude and sign were generated (Table 3). Many of these outcomes could not be rationalized biologically and thus should not be employed for indirect selection aimed at reduced cooking time in common bean due to their low reliability.

Path analysis with moderate to strong and weak multicollinearity

To address the adverse effects of multicollinearity, we identified variables that were strongly associated and, consequently, responsible for the most significant multicollinearity issues. These variables were removed from the dataset before performing path analysis, a strategy proven effective in managing collinearity, as observed in the context of indirect selection for high oil content in soybean lines (Del Conte et al. 2020). Similarly, excluding highly correlated traits in Pearson’s linear correlation analysis was effective in mitigating the undesired effects of multicollinearity in the study of correlations between technological quality and nutritional traits in common bean (Ribeiro et al. 2021b, Kläsener et al. 2022). Our approach did not involve maintaining all analyzed variables and adding a constant (K value) to the diagonal of the matrix due to the difficulty in determining an appropriate value for this constant. Furthermore, the inclusion of a K value resulted in less reliable correlation coefficient estimates compared with those obtained after excluding highly correlated traits in path analysis of important agronomic traits for soybean breeding (Del Conte et al. 2020).

After removing the a* and b* values and grain length, we achieved a CN = 125.11, allowing us to perform path analysis under moderate to strong multicollinearity (Table 4, Fig. 1). In addition, path analysis was conducted under weak multicollinearity (CN = 66.21) after excluding the a* and b* values, grain length, and grain thickness (Table 5, Fig. 1).

Table 4
Path analysis under moderate to strong multicollinearity considering the direct (DE) and indirect (ID) effects obtained between the traits of luminosity (L*), grain width (width), grain thickness (thickness), mass of 100 grains (Mass), water absorption (absorption), and concentrations of potassium (K), phosphorus (P), calcium (Ca), magnesium (Mg), iron (Fe), zinc (Zn), and copper (Cu) on cooking time (Ct) obtained in 25 common bean cultivars evaluated in four experiments carried out from 2019 to 2021.

L*		Width		Thickness		Mass		Absorption		K
DE on Ct	-0.03	DE on Ct	-0.13	DE on Ct	-0.23	DE on Ct	1.20	DE on Ct	-0.18	DE on Ct	-0.21
ID		ID		ID		ID		ID		ID
Width	-0.06	L*	-0.01	L*	-0.02	L*	-0.02	L*	-0.01	L*	0.01
Thickness	-0.13	Thickness	-0.14	Width	-0.08	Width	-0.11	Width	-0.01	Width	0.06
Mass	0.64	Mass	0.94	Mass	1.08	Thickness	-0.21	Thickness	-0.07	Thickness	0.15
Absorption	-0.06	Absorption	-0.01	Absorption	-0.06	Absorption	-0.04	Mass	0.28	Mass	-0.74
K	0.06	K	0.09	K	0.13	K	0.13	K	0.01	Absorption	0.01
P	-0.00	P	-0.03	P	-0.01	P	-0.02	P	-0.02	P	-0.03
Ca	-0.34	Ca	-0.26	Ca	-0.28	Ca	-0.32	Ca	-0.06	Ca	0.16
Mg	-0.01	Mg	-0.02	Mg	-0.01	Mg	-0.01	Mg	0.01	Mg	0.01
Fe	-0.15	Fe	0.03	Fe	-0.09	Fe	-0.07	Fe	-0.09	Fe	0.01
Zn	-0.10	Zn	-0.17	Zn	-0.08	Zn	-0.13	Zn	0.05	Zn	0.05
Cu	-0.02	Cu	0.00	Cu	-0.01	Cu	-0.01	Cu	0.00	Cu	0.00
Total	-0.21		0.29		0.33		0.39		-0.09		-0.51
P		Ca		Mg		Fe		Zn		Cu
DE on Ct	-0.09	DE on Ct	0.52	DE on Ct	0.03	DE on Ct	0.27	DE on Ct	-0.32	DE on Ct	0.03
ID		ID		ID		ID		ID		ID
L*	-0.00	L*	0.02	L*	0.01	L*	0.02	L*	-0.01	L*	0.01
Width	-0.04	Width	0.07	Width	0.07	Width	-0.01	Width	-0.07	Width	-0.00
Thickness	-0.03	Thickness	0.13	Thickness	0.06	Thickness	0.08	Thickness	-0.06	Thickness	0.07
Mass	0.27	Mass	-0.75	Mass	-0.49	Mass	-0.30	Mass	0.49	Mass	-0.26
Absorption	-0.03	Absorption	0.02	Absorption	-0.04	Absorption	0.06	Absorption	0.03	Absorption	0.01
K	-0.06	K	-0.06	K	-0.08	K	-0.00	K	0.04	K	-0.03
Ca	-0.03	P	0.00	P	0.01	P	0.00	P	-0.01	P	-0.02
Mg	-0.00	Mg	0.02	Ca	0.27	Ca	0.22	Ca	-0.17	Ca	0.11
Fe	-0.00	Fe	0.12	Fe	-0.03	Mg	-0.00	Mg	-0.01	Mg	-0.00
Zn	-0.05	Zn	0.11	Zn	0.12	Zn	-0.06	Fe	0.05	Fe	0.17
Cu	0.01	Cu	0.01	Cu	-0.00	Cu	0.02	Cu	0.00	Zn	-0.03
Total	-0.06		0.19		-0.06		0.28		-0.05		0.07
Condition number (CN)					125.11
Coefficient of determination (R2)					0.672
Residual variable effect					0.572

Multicollinearity diagnostics revealed a high linear relationship (r ≥ 0.975) between L* and a*, as well as L* and b*. Similar correlation estimates were reported between these traits in common bean lines of different grain types (Ribeiro and Kläsener 2020, Ribeiro et al. 2021a), confirming the existence of interrelationships between traits that make up the color of common bean grains. Additionally, mass of 100 grains exhibited a high linear relationship (r ≥ 0.900) with grain length and thickness. Length and thickness were the grain dimensions most associated with mass of 100 grains across black, carioca, and other-colored common bean genotypes (Ribeiro et al. 2000), validating the presence of multicollinearity between these traits.

Highly correlated traits convey similar information and are, therefore, considered redundant in the selection process. Consequently, the elimination of traits such as a*, b*, grain length, and grain thickness in path analysis helped exclude these redundant variables, allowing breeding programs to concentrate efforts on the evaluation of the most promising traits for indirect selection. This, in turn, will reduce the workload and financial resources required in future studies to determine technological and nutritional traits in common bean while preserving the efficiency of indirect selection. In addition, because the traits of a*, b*, grain length, and grain thickness showed a variance inflation factor > 10, their removal was necessary before path analysis was implemented, as recommended by Cruz and Carneiro (2003) and Olivoto et al. (2016). Eliminating these four traits also contributed to obtaining correlation estimates with a magnitude consistent with the studied biological phenomenon.

In Pearson’s linear correlation analysis, the exclusion of highly correlated traits, traits with a greater weight in the last eigenvectors, and traits with higher variance inflation factors successfully resolved multicollinearity issues. This allowed promising traits to be identified for use in indirect selection for technological quality and mineral concentration in common bean lines (Ribeiro and Maziero 2023). However, whether this methodology, when applied to decomposing correlation coefficients into direct and indirect effects in path analysis, results in more coherent and reliable estimates is yet to be fully understood. In this case, the recognition of technological and nutritional traits with greater direct effects on cooking time in common bean could enhance the gains obtained with indirect selection.

When path analysis was performed under moderate to strong (excluding the traits of a*, b*, and grain length) and weak (excluding a*, b*, grain length, and grain thickness) multicollinearity, the R² values were 0.672 and of 0.666, respectively (Tables 4 and 5). These R² values were slightly lower than the value of 0.691 observed in path analysis with severe multicollinearity, that is, involving all traits (Table 3). However, using moderate to strong multicollinearity (Table 4) and weak multicollinearity (Table 5), there was a sharp reduction in CN to 125.11 and 66.21, respectively. As a consequence, the correlation coefficients obtained with moderate to strong or weak multicollinearity were similar in magnitude and sign, demonstrating biological significance and greater coherence (Tables 4 and 5) than those observed under severe multicollinearity (Table 3).

Table 5
Path analysis under weak multicollinearity considering the direct (DE) and indirect (ID) effects obtained between the traits of luminosity (L*), grain width (width), mass of 100 grains (Mass), water absorption (absorption), and concentrations of potassium (K), phosphorus (P), calcium (Ca), magnesium (Mg), iron (Fe), zinc (Zn), and copper (Cu) on cooking time (Ct) obtained in 25 common bean cultivars evaluated in four experiments carried out from 2019 to 2021.

L*		Width		Mass		Absorption		K		P
DE on Ct	-0.06	DE on Ct	-0.08	DE on Ct	0.99	DE on Ct	-0.19	DE on Ct	-0.15	DE on Ct	-0.12
ID		ID		ID		ID		ID		ID
Width	-0.03	L*	-0.02	L*	-0.03	L*	-0.02	L*	0.02	L*	-0.00
Mass	0.53	Mass	0.78	Width	-0.06	Width	-0.01	Width	0.04	Width	-0.02
Absorption	-0.06	Absorption	-0.01	Absorption	-0.04	Mass	0.23	Mass	-0.61	Mass	0.22
K	0.04	K	0.07	K	0.10	K	0.01	Absorption	0.01	Absorption	-0.04
P	-0.00	P	-0.04	P	-0.03	P	-0.02	P	-0.03	K	-0.04
Ca	-0.35	Ca	-0.26	Ca	-0.33	Ca	-0.06	Ca	0.16	Ca	-0.03
Mg	0.00	Mg	-0.01	Mg	-0.01	Mg	0.00	Mg	0.01	Mg	-0.00
Fe	-0.15	Fe	0.03	Fe	-0.07	Fe	-0.09	Fe	0.01	Fe	-0.00
Zn	-0.10	Zn	-0.16	Zn	-0.12	Zn	0.05	Zn	0.05	Zn	-0.04
Cu	-0.02	Cu	0.00	Cu	-0.01	Cu	0.00	Cu	0.01	Cu	0.01
Total	-0.20		0.29		0.39		-0.09		-0.51		-0.06
Ca		Mg		Fe		Zn		Cu
DE on Ct	0.52	DE on Ct	0.01	DE on Ct	0.26	DE on Ct	-0.30	DE on Ct	0.04
ID		ID		ID		ID		ID
L*	0.04	L*	0.01	L*	0.03	L*	-0.02	L*	0.03
Width	0.04	Width	0.04	Width	-0.01	Width	-0.04	Width	-0.00
Mass	-0.62	Mass	-0.40	Mass	-0.25	Mass	0.41	Mass	-0.22
Absorption	0.02	Absorption	-0.04	Absorption	0.06	Absorption	0.03	Absorption	0.01
K	-0.05	K	-0.06	K	-0.00	K	0.03	K	-0.02
P	0.00	P	0.02	P	0.00	P	-0.02	P	-0.03
Mg	0.01	Ca	0.27	Ca	0.22	Ca	-0.17	Ca	0.11
Fe	0.11	Fe	-0.03	Mg	-0.00	Mg	-0.01	Mg	0.00
Zn	0.10	Zn	0.12	Zn	-0.06	Fe	0.05	Fe	0.16
Cu	0.01	Cu	-0.00	Cu	0.02	Cu	0.00	Zn	-0.02
Total	0.19		-0.06		0.29		-0.04		0.07
Condition number (CN)					66.21
Coefficient of determination (R2)					0.666
Residual variable effect					0.578

Thus, correlation coefficients obtained using weak multicollinearity (CN = 66.21) (Table 5) allowed a better understanding of the causes involved in the association between technological and nutritional traits in common bean. This was attributed to the exclusion of all highly correlated traits, traits with a greater weight in the last eigenvectors, and traits with a variance inflation factor > 10 prior to executing path analysis (Table 5). This approach allowed a considerable reduction of the undesirable effects of multicollinearity in path analysis. Therefore, the set of explanatory variables analyzed to carry out path analysis was sufficient to explain the variations in the main variable. Consequently, the use of weak multicollinearity made it possible to accurately interpret the direct and indirect effects of technological and nutritional traits on cooking time in common bean.

Promising traits for indirect selection for fast cooking in common bean

The greatest positive direct effects on cooking time were recorded for mass of 100 grains (0.99) and calcium concentration (0.52) (Table 5). Therefore, common bean cultivars with a higher mass of 100 grains and a higher calcium concentration exhibited longer cooking times. Mass of 100 grains and calcium concentration showed a high positive correlation with cooking time in Mesoamerican common bean lines (Ribeiro and Kläsener 2020). However, calcium concentration did not correlate with cooking time when Mesoamerican and Andean common bean lines were evaluated (Rivera et al. 2018), and no direct effect was found between mass of 100 grains and cooking time in Mesoamerican common bean lines (Santos et al. 2016). The differences found between the described correlation coefficients in these studies can be explained by the genetic variability of the germplasm, number of growing environments, variations in environmental conditions, management practices, and whether the path and simple correlation analyses were carried out with or without multicollinearity.

Indirect effects via mass of 100 grains were mostly negative and of low magnitude. Mass of 100 grains had the greatest negative indirect effect on calcium concentration (-0.33), showing that selection for increased mass of 100 grains contributes to a decrease in calcium concentration in common bean. Previous studies also found a negative correlation between mass of 100 grains and calcium concentration in Andean common bean lines (Ribeiro et al. 2021a) and in Mesoamerican and Andean common bean accessions (Rivera et al. 2018). Approximately 95% of calcium in common bean is concentrated in the seed coat, which represents 8.2 to 10.7% of the total seed dry matter (Ribeiro et al. 2012). Therefore, a higher mass of 100 grains corresponds to lower calcium content in common bean seeds. This finding indicates that Mesoamerican common bean cultivars (small to medium sized grains) may have a greater potential to benefit from calcium biofortification.

Indirect effects via calcium concentration were considered negligible for all technological and nutritional traits, except for mass of 100 grains. Therefore, calcium concentration had the greatest indirect negative effect on mass of 100 grains (-0.62) (Table 5). This result suggests the feasibility of indirect selection based on mass of 100 grains and/or calcium concentration to achieve improvements in cooking time in common bean. However, as mass of 100 grains also showed an indirect negative effect on calcium concentration on cooking time in this research, it is necessary to find a balance in reducing the mass of 100 grains and calcium concentration aiming the development of fast-cooking common bean cultivars.

The other technological and nutritional traits exhibited low direct effects on cooking time. Traits with a low direct effect on main variable are not recommended for use in indirect selection (Cruz and Regazzi 1997). In this case, the auxiliary trait is not the major determinant of changes in the main variable, which results in unsatisfactory gains in indirect selection for fast cooking in common bean.

In the present study, only mass of 100 grains and calcium concentration displayed high direct effects in a favorable direction for selection aiming at fast cooking. Therefore, for the evaluated common bean cultivars, selection for the lowest values of mass of 100 grains and calcium concentration provided reduction in cooking time without compromising the nutritional value of the grains. Indirect selection based on the lowest values of mass of 100 grains and calcium concentration is thus recommended for selecting fast-cooking common bean cultivars.

CONCLUSION

The magnitude and algebraic sign of direct and indirect effects of technological and nutritional traits in common bean vary when path analysis is conducted under different degrees of multicollinearity.

Under weak multicollinearity (condition number = 66.21), correlation coefficients exhibit values and signs that align more coherently with the biological phenomena under investigation. This, in turn, allows for an accurate interpretation of the direct and indirect effects of technological and nutritional traits on cooking time in common bean.

Indirect selection based on the lowest values of mass of 100 grains and calcium concentration is recommended in this study for the selection of fast-cooking common bean cultivars.

ACKNOWLEDGMENTS

To the Coordenação de Aperfeicoamento de Pessoal de Nível Superior and to the Conselho Nacional de Desenvolvimento Científico e Tecnológico for financial support and scholarships.

• How to cite: Ribeiro, N. D., Andrade, F. F. and Maziero, S. M. (2025). Indirect selection for multiple technological and nutritional traits in common bean cultivars under different degrees of multicollinearity. Bragantia, 84, e20230283. https://doi.org/10.1590/1678-4499.20230283

• FUNDING

  Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
  Grant No: 88881.844984/2023-01
  Conselho Nacional de Desenvolvimento Científico e Tecnológico
  Grant No: 302167/2019-6

DATA AVAILABILITY STATEMENT

The data are available in https://doi.org/10.5281/zenodo.13910430

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Section Editor:

Luciana Carlini-Garcia https://orcid.org/0000-0002-8098-4227

Publication Dates

• Publication in this collection
  22 Nov 2024
• Date of issue
  2025

History

• Received
  06 Dec 2023
• Accepted
  15 July 2024