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So, I have two scales, A and B, both ranging from 0 to 10. I would like to combine these into one as follows:

At one endpoint, high values on A and high values on B At the other endpoint, high values on A and low values on B

How should I proceed?

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    $\begingroup$ How do the scales differ? Are they like systolic vs diastolic blood pressure, i.e., different metrics of the same thing? $\endgroup$ – Mike Hunter Oct 30 '15 at 17:22
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    $\begingroup$ What do you want to do w/ low values on A? $\endgroup$ – gung - Reinstate Monica Oct 30 '15 at 17:27
  • $\begingroup$ Why wouldn't PCA work? It would automatically scale the metrics in the desired directions. $\endgroup$ – Mike Hunter Nov 3 '15 at 14:32
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Why would you want to do this? Presumably, the two things measure different quantities. This provides evidence that perhaps some combination of them is not necessarily meaningful. For example, suppose the distance from my house to office measured in feet and the distance from my office to my grocer measured in miles. If these locations are collinear, increases in one will be offset in another, but their combination (i.e., sum) is only meaningful if they're expressed on a common scale.

For feet and miles, the common scale is explicitly defined. I infer that since you're asking the question, you don't how to determine the common scale.

One way to infer a common scale is to do a regression. If we're predicting height from age and gender, in a dimensional analysis sense, the units of the coefficients are the unit conversions from age units and gender units into height units. This is not without problems, though. The coefficient estimates are only consistent if the model is correctly specified, i.e. it includes all relevant terms. Clearly age and gender alone are not the only determinants of height, so the coefficients of this model will be wrong.

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