Hi,
I am analyzing temporal trends in parameters like the autocorrelation and the variance. I know that, albeit they are calculated for interval scaled variables, correlations are not interval scaled but ordinal scaled. How about variances? It is proper to treat them as interval scaled?
My first idea was that it could be difficult, because of the squaring down during the calculation. If this would be the problematic point, could it be helpful to use the SD instead of the variance?
I appreciate any comments (and sources)!
Brian, I don't think that the reference to "measurement scales" is of any help in your context. You may find this paper enlightning:
Thomas, M. A., Mathematization, Not Measurement: A Critique of Stevens’ Scales of Measurement (October 3, 2014). Available at SSRN: http://ssrn.com/abstract=2412765 or http://dx.doi.org/10.2139/ssrn.2412765
As a bottom line:
Maths (an stats) do not deal with "scales of measurement". Numbers are numbers, and "permissible operations" are given by set theory in general and by measure theory in particular. If may be that a measure-theoretic definition of a particular variable (like the SD) is not known and can not easily be based on appropriate axiomization, but operations with sensible results may be seen as empirical indications for the underlying properties. The statistical operations are based on assumptions that follow only from the empirical properties of the data and are not "permissible" or "forbidden" according to a "scale". These assumptions do not require "meaningful interpretations of differences" (whatever this is) - they require that some assumptions about (distributional and other) properties of the data are plausible.
This is all nicely worked out for "probabilities". Here Kolmogorov's axioms give the measure-theoretic definition of the probability space with all its features, and "permissible" algebraic and statistical operations follow from that. There is no question if probabilities are interval scaled or ratio scaled... probabilities are measure-theoretically defined within a probability space (Omega, F, P).
What may be much more interesting that the question if 4 SD units are representing "practically twice" the variation than 2 SD units is the question for the probability distribution of SD values (see http://mathworld.wolfram.com/StandardDeviationDistribution.html) and if this tells us something relevant w.r.t. your problem (you have not mentioned what you actually want to do, though).
Why do you think a correlation coefficient is only ordinaly scaled? Would your answer change if you apply Fisher transformation? What does this mean in the first place?
The same holds for variance and sd. A phrase like "this sd is twice as high as that sd" is meaningful, i.e. it is a ratio scale. If you feel uncomfortable because both are restricted to be positive, please bear in mind, that this is also true for your weight, heigt, age etc.
https://en.wikipedia.org/wiki/Fisher_transformation
Hi Eik,
you are right, to transform the correlation coefficient into Fisher's Z would create an approximately interval scaled variable (I did not have this in mind!). The correlation coefficient r per se is not interval scaled and it would be a mistake to treat it like that.
Concerning the variance/ SD I am not shure! The phrase you quoted may seem to be meaningful (sounds like that), but is a SD of 4 really twice as much as a SD of 2 (for example)? I have to think about it and maybe to check it by creating some small example datasets and estimating the variances/ SDs, because your reasoning sounds good to me but does not prove this assumption, I guess.
Thank you very much for your ideas!
Brian, I don't think that the reference to "measurement scales" is of any help in your context. You may find this paper enlightning:
Thomas, M. A., Mathematization, Not Measurement: A Critique of Stevens’ Scales of Measurement (October 3, 2014). Available at SSRN: http://ssrn.com/abstract=2412765 or http://dx.doi.org/10.2139/ssrn.2412765
As a bottom line:
Maths (an stats) do not deal with "scales of measurement". Numbers are numbers, and "permissible operations" are given by set theory in general and by measure theory in particular. If may be that a measure-theoretic definition of a particular variable (like the SD) is not known and can not easily be based on appropriate axiomization, but operations with sensible results may be seen as empirical indications for the underlying properties. The statistical operations are based on assumptions that follow only from the empirical properties of the data and are not "permissible" or "forbidden" according to a "scale". These assumptions do not require "meaningful interpretations of differences" (whatever this is) - they require that some assumptions about (distributional and other) properties of the data are plausible.
This is all nicely worked out for "probabilities". Here Kolmogorov's axioms give the measure-theoretic definition of the probability space with all its features, and "permissible" algebraic and statistical operations follow from that. There is no question if probabilities are interval scaled or ratio scaled... probabilities are measure-theoretically defined within a probability space (Omega, F, P).
What may be much more interesting that the question if 4 SD units are representing "practically twice" the variation than 2 SD units is the question for the probability distribution of SD values (see http://mathworld.wolfram.com/StandardDeviationDistribution.html) and if this tells us something relevant w.r.t. your problem (you have not mentioned what you actually want to do, though).
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