Multicollinearity in Path Analysis: A Simple Method to Reduce Its Effects

Abstract

The multicollinearity in path analysis was investigated in different scenarios. A biometrical approach identified the multicollinearity‐generating traits. Data derived from averages overestimated the correlation coefficients. The use of all sampled observations increased the accuracy in path analysis. A simple sample tracking method that reduces multicollinearity is proposed. Some data arrangement methods often used may mask correlation coefficients among explanatory traits, increasing multicollinearity in multiple regression analysis. This study was performed to determine if the harmful effects of multicollinearity might be reduced in the estimation of the X′X correlation matrix among explanatory traits. For this, data on 45 treatments (15 maize [Zea mays L.] hybrids sown in three places) were used. Three path analysis methods (traditional, with k inclusion, and traditional with trait exclusion) were tested in two scenarios: with X′X matrix estimated with all sampled observations (ASO, n = 900) and with the X′X matrix estimated with the average values of each plot (AVP, n = 180). The condition number (CN) was reduced from 3395 to 2004 when the matrix was estimated with all observations. On average, the factors that inflate the variance of regression coefficients were increased by 61% in the AVP scenario. The addition of the k coefficient reduced the CN to 85.40 and 51.17 for the ASO and AVP scenarios, respectively. Exclusion of multicollinearity‐generating traits was more effective in the ASO than the AVP scenario, resulting in CNs of 29.62 and 63.66, respectively. The largest coefficient of determination (0.977) and the smallest noise (0.150) were obtained in the ASO scenario after the exclusion of the multicollinearity‐generating traits. The use of all sampled observations does not mask the individual variances and reduces the magnitude of the correlations among explanatory traits in 90% of cases, improving the accuracy of biological studies involving path analysis.

Mean Performance and Stability in Multi‐Environment Trials II: Selection Based on Multiple Traits

“…Using the EFA, it was possible to determine how many factors exist (in other words, in how many latent variables the original set of variables could be reduced), the relationship between the factors and how the variables were associated with these factors (Ullman, 2006). Finally, the estimation of final factor scores allowed dealing with the multicollinearity, a systemic issue in multivariate analyses (Olivoto et al, 2017), incorporating in the new first latent variables the original structure of the data, thus leading to dimensional reduction.…”

Multivariate analysis revealed genetic divergence and promising traits for indirect selection in black oat

“…To proceed with the path analysis, the multicollinearity of the correlation matrix among the 14-explanatory traits (put here the explanatory traits) was initially diagnosed by the condition number (CN), given by the ratio between the largest and smallest eigenvalues of explanatory traits correlation matrix. We decided to exclude the variables causing severe multicollinearity problems from the analysis, as suggested by Olivoto et al (2017). After solving possible problems with multicollinearity, a phenotypic path analysis was performed considering GY as the dependent trait and those variables that remained after multicollinearity adjustment as explanatory traits.…”

“…When evaluating the multicollinearity of the phenotypic correlation matrix among the explanatory traits, we observed severe multicollinearity effects caused by the traits SL, SW, TGW, FGC, SHL, RL. Thus, these traits were removed from path analysis to circumvent the harmful effects of multicollinearity (Olivoto et al, 2017).…”

Path Analysis Applied to Evaluation of Biomass Estimates in Subtropical Forests at Brazil

“…To better understand the factors affecting the correlations between variables, Wright (1921) proposed a method called path analysis, which deconstructs the simple correlations between direct and indirect effects of independent variables on the dependent variable. This method was originally used in plants by Dewey and Lu (1959), and was subsequently applied in the context of various agricultural crops (OLIVOTO et al, 2017). However, forested areas are rarely studied (RESENDE et al, 2016) and no researches are found in the Brazilian literature available for modeling forest biomass.…”

“…In the prediction of direct and indirect effects of a set of variables on the dependent variable, it is necessary to estimate the path coefficients that are obtained by linear regression, in which the variables were previously standardized (OLIVOTO et al, 2017). The estimate of coefficients can be adversely influenced by the multicollinearity effects among the variables involved.…”