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I wonder if anyone has any links or advice on specifying a ratio of two explanatory variables in a linear regression? That is, specifying the two independent variables plus their ratio. We have data where the ratio term seems to be significant. Are there any known issues regarding the functional form or any other issues (e.g., collinearity)?

I meant we are considering a model $y = ax_1 + bx_2 + cx_1/x_2$, where $y$, $x_1$, $x_2$ are all continuous and wondered if this was reasonable, or if there were some good references on assessing appropriate functional form of multiple covariates.

I think we were "inspired" to try a ratio of covariates as we saw a similar approach taken in this paper.

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    $\begingroup$ Your question seems rather vague. There may or may not be issues, depending on the circumstances. $\endgroup$ – Glen_b -Reinstate Monica Aug 22 '14 at 10:05
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    $\begingroup$ Gravitational force is inversely proportional to squared distance, but proportional to masses. If your experiment was about gravitational force and your variables were mass and distance then you would find strong causal relationships between both, but their ratio would be more informative. $\endgroup$ – EngrStudent Aug 22 '14 at 12:20
  • $\begingroup$ In any event, be particularly careful if there are any small values of x2. $\endgroup$ – EdM Aug 22 '14 at 13:41
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    $\begingroup$ Could you tell us a little about your motivation for using $x_1/x_2$ in this model? For instance, is it suggested by theory, or did it emerge from a principled exploratory data analysis, or did it perhaps turn up after trying lots of different models? $\endgroup$ – whuber Aug 22 '14 at 14:20
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The "ratios" in the paper you cite were determined (according to the "Experimental procedures") as the difference between two measurements (cycles-to-threshold in polymerase chain reaction, PCR) that are related to logarithms of the underlying variables (amounts of messenger RNA, mRNA, for each of two gene transcripts). Since log(x1/x2)= log(x1) - log(x2), so you only have 2 linearly independent variables in this scale among x1, x2, and the ratio.

Log-transformed measurements of things like mRNA are often better behaved in statistical analyses than their linear-scale values. If applicable to your study, try regression using log(x1) and log(x2) as independent variables. If their ratio is "really" the important variable, then the regression coefficients will be close to equal in absolute magnitude and opposite in sign.

And if you are getting inspiration from that paper, also get inspired by the multi-stage discovery and validation process the authors used: discovery of candidates by microarray analysis of thousands of genes, technical validation of candidates by a different technology (PCR), and biological validation by manipulating the expression levels of the 2 genes used to form the "ratio" and finding results consistent with predictions. And even with such effort, the authors might today be expected to perform more thorough statistical validation of their model than they did for this study a decade ago.

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One reference that would be good to read and consider is:

Spurious Correlation and the Fallacy of the Ratio Standard Revisited, Richard Kronmal, Journal of the Royal Statistical Society. Series A, Vol. 156, No. 3 (1993), 379-392.

This brings up some of the situations that can occur with using ratios.

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Your ratio term, $x_1/x_2$ can be understood as an interaction between $x_1$ and a transformed version of your second variable $x_2\rightarrow \frac 1 {x_2}$. I don't know what your variables are, or how much data you have, but I might consider fitting a full model $y = ax_1 + bx_2 + c1/x_2 + dx_1x_2 + ex_1/x_2$, and then conducting a nested model test. You will probably also want to try assessing the $R^2$ shrinkage for the two models via cross-validation. The nested model test that would only drop your ratio term would be the same as the test of that term that already came with your output, but you may want to cross validate a model without the ratio term as well.

Unless there are strong prior theoretical reasons for only including certain transformations or interaction terms, or the transformation are just for meeting model assumptions, it is generally not recommended to omit lower level terms.

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