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I'm measuring the error produced by operators executing planar cuts under different guiding systems. I define a target plane and then measure the euclidean distance from uniformly sampled points in the target plane to the executed plane. Since the target plane is my coordinate system reference, I define it as the plane $xz$ in $\mathbb{R}^3$. The executed plane $E$ is defined as $ax + by + cz + d=0$ with $b \neq 0$ and, given that I'm sampling points from $xz$ and projecting them into $E$, then $b$ could never be $0$. Then $E$ could be rewriten as $y = -\frac{a}{b}x - -\frac{c}{b}z -\frac{d}{b}$ with $x \sim \mathcal{U}(x_{min}, x_{max})$ and $z \sim \mathcal{U}(z_{min}, z_{max})$. Of course, the cutting tool also produces some (maybe normal) noise that affects $y$. The random variable $y$ is not normally distributed (I've tested it and I've plotted some qq-plots of it).

I have 2 different guiding systems and I have to test if they are significantly different wrt the above described measurement method. Which method do you recommend me to use to test my hypothesis?

In yellow one of the methods, in green the other one. I'm trying to show that the green method is better

Thanks!

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    $\begingroup$ What hypothesis? $\endgroup$ – Glen_b -Reinstate Monica May 15 '14 at 19:01
  • $\begingroup$ @Glen_b Presumably that the executed plane coincides with the target plane. Federico, what can you tell us about the nature of the discrepancy? You have mentioned measurement error of the distances, but could you tell us more specifically how the guiding system might fail to miss the target plane? For instance, perhaps it is designed in a way that can create only a translational error (causing the two planes to be parallel) or maybe only a rotational error (guaranteeing the planes will intersect). Perhaps (alternatively) they are constrained to have a fixed intersection? $\endgroup$ – whuber May 15 '14 at 19:06
  • $\begingroup$ @whuber but then wouldn't the population value of $b$ be 0? $\endgroup$ – Glen_b -Reinstate Monica May 15 '14 at 19:08
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    $\begingroup$ @whuber actually the options you give (translational error, rotational error, intersection or not) may all appear at the same time (except for a rotation that makes the executed plane perpendicular to xz; in that case b = 0 and the method does not work). This complexity is why we are trying to assess the effectiveness of the guiding system by measuring the error represented by the random variable y. The guiding system is just helping the human operator, that's why we want to know which one has a better effect on the outcome. Thanks! $\endgroup$ – Federico May 15 '14 at 19:57
  • $\begingroup$ @Glen_b the hyphotesis that the guiding systems produce different outcomes, in a statistically significant way; then I think we would be able to keep the one with the smallest dispersion and whose expectation is closest to 0 (the result of an ideal guiding system would be one in which y is always zero, that is, plane $E$ is plane xz). Thanks $\endgroup$ – Federico May 15 '14 at 20:02
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You suggest the possibility of a Kruskal-Wallis test, and it's quite possible that it's suitable.

Indeed, the criterion you mention in comments - "closer to zero more often" - directly suggests doing a Kruskal Wallis on the absolute deviation from zero. The two-sample version (a Wilcoxon-Mann-Whitney test) in fact has a direct interpretation in a "smaller more often" sense -- indeed in that case you can scale the U-statistic to an estimate of that probability (by dividing the U statistic by the total number of pairwise comparisons - i.e. its maximum possible value).

If your interest is instead more in some measure of location shift, then with an additional assumption (that the distributions are identical in shape, aside from possible location shift), the Kruskal-Wallis may still be a good choice. Further, in the two-sample case, there's a location-shift estimate that's readily available (one can also obtain acceptance regions - akin to confidence ellipsoids in ANOVA - for the whole set of location-shifts)

Those two possible ways to view the Wilcoxon-Mann-Whitney are discussed in more detail here.

The Kruskal-Wallis is similar, but there's not an exact correspondence because the K-W statistic can be broken up into transitive (WMW-type) and non-transitive differences (See the discussion in item (3) in this answer); usually the second component is relatively small, but sometimes can be important.

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