I am working with some data that vary along an arbitrary scale of -1 to 1. More specifically, this scale refers to a difference between two scores (with 0 meaning no difference, 1 being the maximum possible difference when A>B, and -1 being the maximum possible difference when B>A)

I think it could be useful to express the observed scores as POMPs (Percent Of Maximum Possible score; Cohen, P., Cohen, J., Aiken, L. S., & West, S. G. (1999). The problem of units and the circumstance for POMP. Multivariate behavioral research, 34(3), 315-346). These scores are easy to calculate through the provided formula

100*[(observed-minimum)/(maximum-minimum)], where maximum/ minimum refers to the boundaries of the scale.

However, I am uncertain whether I should set minimum score = 0 (so score=0 retains the original meaning of no difference and the sign indicates the sign of the difference) or, to the contrary, I should calculate the POMP score considering minimum= -1 (in which case, 0 scores would become 50% of the maximum possible difference).

I thank you in advance for any possible answer

  • $\begingroup$ Could you elaborate on why it might be useful to re-express your data and why you chose this particular method? $\endgroup$ – whuber Jan 7 at 14:13
  • $\begingroup$ As the article of Cohen et al. (1999) highlights, POMP scores might be especially useful when the original scale is arbitrary and they ease comparisons with alternative measures of the same construct (which can use a different scales). In this regard, and as the same article illustrates, POMP scores are sample independent, hence displaying several advantages over other possible transformations (e.g. Z and T scores),. Moreover, POMP scores preserve test statistics and statistical power, and convey immediate meaning. $\endgroup$ – cs- Jan 7 at 14:55
  • $\begingroup$ This sounds like a mere linear transformation of the data. Is there anything more to it than that? $\endgroup$ – whuber Jan 7 at 15:01
  • $\begingroup$ You are right. This is "just" a linear transformation of the data. However, it provides some gains at the interpretability and comparability levels without affecting the results of statistical comparisons or statistical power. Thus, for example, another researcher may construct a different instrument to measure the same construct but using a, let's say, -3 to 3 scale. POMP scores would be comparable, while the original scores would be not. Moreover, by describing the effect as % of maximum possible helps to figure out the size of the effect in a scale without meaningful units $\endgroup$ – cs- Jan 7 at 15:10
  • $\begingroup$ Since both the alternatives mentioned in your post are such linear transformations, the only basis for choosing among them would be "interpretability"--there appears to be no possibility of a statistical answer. However, because you present your situation abstractly, you haven't supplied any information about what interpretability might mean in this instance. That makes it hard to answer your question. $\endgroup$ – whuber Jan 7 at 15:19

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