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In a recent publication, the authors report the following transformation when aggregating across three different scales:

Cognitive style level was used as a control variable and captured as (sqrt(OV) + sqrt(SV) + sqrt(V))^2 which is similar to, but a more robust measure than, the sum of the means of the three cognitive styles, particularly when team size varies. It captures both the level and the range of the member cognitive styles in the team (Matousek 2002, Rudin 1987).

The referenced publications are both lengthy real analysis textbooks, which don't provide a clear intuition for why this is valuable. What is the intuition behind this? Are there more robust transformations?

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    $\begingroup$ $\left( |x_1|^p + |x_2|^p + \ldots + |x_n|^p \right)^{1/p}$ for $p \geq 1$ is known as the p-norm. For $p=2$, it is the classic Euclidean norm. Your case though is $p = \frac{1}{2}$ which makes the measure concave (rather than convex) and hence isn't a norm (because it doesn't satisfy the triangle inequality: the norm of a sum is less than the sum of the norms). This isn't my field and don't know why a concave measure would be desired. $\endgroup$ – Matthew Gunn Dec 4 '18 at 1:10
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    $\begingroup$ The threads at stats.stackexchange.com/questions/3682 and stats.stackexchange.com/questions/244202 will illuminate this. @Matthew being a norm is not important. This is a multiple of a "root mean;" it lies on a continuum between the arithmetic mean ($p=1$), geometric mean ($p=0$), and harmonic mean ($p=-1$). You can see how closely it is tied to the Box-Cox family of transformations. $\endgroup$ – whuber Dec 4 '18 at 1:14

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