I have a ratio variable - women's income : men's income, women in the numerator and men in the denominator. This variable is intended to measure income disparity between women and men. In a logistic regression model - a one unit increase in women's income relative to men's means that income disparity is decreasing rather than increasing, correct?
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1$\begingroup$ Suppose the ratio starts at 1:1 and then increases to 2:1. Wouldn't that be an increase in disparity? The difficulty here is that the ratio does not measure disparity: disparity is a deviation between the ratio and the ideal value of 1:1. It sounds like you may need to reconsider how you define and quantify disparity in your model before you go any further. $\endgroup$– whuber ♦Commented May 30, 2019 at 16:26
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The struggle to interpret ratios evidences their many problems when applied in regression analysis. It is worth repeating, verbatim, the abstract from Kronmal's 1993 paper "Spurious Correlation and the Fallacy of the Ratio Standard Revisited ".
In it he says:
Spurious correlation refers to the correlation between indices that have a common component. A 'per ratio' standard is based on a biological measurement adjusted for some physical measurement by division. Renowned statisticians and biologists (Pearson, Neyman and Tanner) have warned about the problems in interpretation that ratios cause. This warning has been largely ignored. The consequences of using a single ratio as either the dependent or one of the independent variables in a multiple-regression analysis are described. It is shown that the use of ratios in regression analyses can lead to incorrect or misleading inferences. A recommendation is made that the use of ratios in regression analyses be avoided.
"Disparity" is a declaration of injustice. A (non-1) ratio is not a disparity per se. Suppose for instance all men worked 50% FTE and all women worked 100% FTE yet the ratio is 0.5. Is that a disparity?
And what is the basis of men's income being the denominator? Women's income is as much a variable as men's income, and yet men's income is being treated as a standardization factor without any account of variability (measurement error). As the paper shows, even adjusting for matching factors does not resolve the issue. The proposed approaches are in increasing order of complexity.
Adjust for men's and women's income (presuming some appropriately paired type of design was even available in the first place), and their interaction. Consider a log transform of the outcome. Observe the direction and trend in the interaction parameter
Fit a fully stratified model with flexible adjustment (splines, polynomial, or other) adjustment for male/female income, possible confounders (education, age, tenure) and observe/comment on trends.
Fit a structural model including possible mediators like number of children, marital status, education, etc. and show post-estimates of predicted outcomes for male and female outcomes.
