Biomarker ratios: does it matter if the biomarker is designated as the numerator or the denominator?
Yue Ma, Kao Lee Yang, Barbara B Bendlin- Psychiatry and Mental health
- Cellular and Molecular Neuroscience
- Geriatrics and Gerontology
- Neurology (clinical)
- Developmental Neuroscience
- Health Policy
- Epidemiology
Abstract
Background
Ratios of two variables are commonly used in research, such as biomarker ratios, body mass index (BMI), and waist‐to‐hip ratio. Although there are conventions about which variable serves as the numerator and which serves as the denominator in a ratio, these conventions are not always followed. Furthermore, new biomarkers may not have established conventions, leading to difficulties in comparing findings from studies that calculate ratios reciprocally. Taking the logarithm of a ratio equals to the difference between the logarithms of the numerator versus the denominator, transforming it into a linear function of the two terms, and thus may provide a linkage between analyses that employ reciprocal ratios.
Method
For linear and logistic regressions, we mathematically derived the relations in parameter estimates and test statistics between models with log transformed reciprocal ratios in the predictor and/or outcome. Two real data examples (n = 305; National Alzheimer’s Coordinating Center) were analyzed, including linear regressions of CSF P‐tau/Aβ1‐42 ratio on BMI, and logistic regressions of cognitive impairment on P‐tau/Aβ1‐42.
Result
Under log transformation, taking the reciprocal of the predictor flips the sign of the regression coefficient and the test statistic for the predictor only. Whereas, taking the reciprocal of the outcome flips the sign of the coefficient and the test for all variables, including the intercept, the predictor, and all other covariates. In both situations, the magnitudes of coefficients and tests, standard errors, and p‐values all remain the same. For logistic regressions, when the sign of a coefficient is flipped, the corresponding odds ratio becomes reciprocal. In any case, the estimated relationship between the predictor and the outcome is essentially unchanged.
Conclusion
A model including log transformed ratio variables can be easily converted to the model with reciprocal ratios by flipping the signs of regression coefficients, allowing the comparison of findings from studies using reciprocal ratios. However, reciprocal and log transformations introduce difficulties in result interpretation. More meaningful interpretations could be achieved by converting ratio variables back to their original untransformed scales, plotting predicted outcome values against the predictor, and reporting predicted values at clinical cutoffs or key statistical points capturing the data distribution.