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When Predictors Sum to a Constant: Trade-Off Effect Analysis Using a Regression Model Based on Isometric Log-Ratio Transformation
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DOI：10.1037/met0000668.png)
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Abstract
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The standard regression model is not feasible when the sum of predictors is a constant, which is a common occurrence in proportional data or ipsative data. Davison et al. (2022) described a set of reduced-rank regression models in which each regression coefficient can be interpreted as a predictor trade-off effect. However, the assumption of linearity and symmetry in their method is too rigid, and the compositional nature of the predictors should not be disregarded. In this article, from the perspective of compositional data, a new method named isometric-log-ratio-transformed trade-off effect analysis (ITEA) is proposed. The predictors are transformed into isometric log-ratio coordinates using a planned sequential binary partition, and trade-off effects are then estimated using a regression model with isometric log-ratio coordinates. Instead of directly relying on regression coefficients, the trade-off effect is defined as the difference in the dependent variable before and after the trade-off, from which the 95% confidence interval can be further derived. Moreover, the main results of the ITEA are not affected by the variation in orthonormal bases. Applying the ITEA to the data in Davison et al.'s (2022) study yields more flexible and interpretable results of trade-off effects. We also provide an empirical example of a forced-choice questionnaire to verify the validity of the ITEA, with visualization attempts of trade-off effects under different conditions. Usefulness, suitable applications, and potential extensions are discussed.
Keywords:
trade-off effect
regression
compositional data
isometric log-ratio transformation
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