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My question: What is the most appropriate way to analyse cases in which two variables and their ratio are all believed to influence some outcome? I would ideally like guidance in a regression framework, but am open to other approaches as well.


Background: In the fields of oceanography & limnology, the Redfield ratio is the ratio of nitrogen to phosphorus (or N:P) in water. This ratio is believed to have important implications for the biology of aquatic systems and so it is frequently used as an independent variable in a regression framework. The absolute values of the numerator and denominator (N & P) are believed to strongly influence these systems as well, distinct from their ratio.

However, in most analyses, the ratio is the only one of the three included as an independent variable; the numerator and denominator (N & P) are excluded because of worries about multicollinearity.

I believe this approach is flawed, and would like to know 1) if it's possible to construct a linear model that allows us to rigorously estimate the distinct effects of numerator, denominator & the ratio, and 2) if so, how to go about it.

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  • $\begingroup$ What is "some outcome" ? Is it a continuous variable or binomially scaled ? What is your linear model ? $\endgroup$
    – user10619
    Aug 24, 2017 at 14:14
  • $\begingroup$ @subhashc.davar A continuous variable. As for the model structure, that is part of my question above; how do we construct one in a meaningful & appropriate manner? The details don't matter a great deal here, I'm after some insight into the statistics/experimental design. $\endgroup$
    – mkt
    Aug 29, 2017 at 11:04
  • $\begingroup$ To me, postulating a correct model depends on logic and one should not apply a particular statistical model without valid reasons. Further ,I do not think that variables with absolute values and ratio led variables can be linearly combined without approprlate transformations/ rescaling etc. $\endgroup$
    – user10619
    Aug 30, 2017 at 9:20

1 Answer 1

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Among N, P, and their ratio there are only 2 functionally independent predictors. So you could, for example, examine the ratio and one of either N and P (as a measure of overall concentration) as 2 predictors in a regression.

My hunch is that these analyses would be best done on a logarithmic scale for the predictors, as your N and P values are strictly positive and measurement errors in practice often tend to be proportional to measured values. As the log of the ratio is simply the difference between the logs of the 2 values, you could even choose to do the regression initially with log(N) and log(P) as predictors and extract the ratio as a contrast thereafter. Again, however, you will only have 2 independent predictors among the 3 of N, P, and their ratio.

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  • $\begingroup$ Thanks, I appreciate this contribution. I think your intuition is correct about measurement errors in N and P on a linear scale. Their biological effects are (I believe) linear on a log scale too, which is an additional reason to log the data. $\endgroup$
    – mkt
    Jul 13, 2017 at 11:44

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