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I'm conducting a project where I determine which factors (3 in total) contribute to the accuracy in which one would hit a dartboard. Each factor has two levels, low and high. For example, one of the factors is distance and the lowest distance (closest to the dart board) is 2 meters while the highest distance is 4 meters. Another factor I'm using is the left and right hand to throw.

My problem is, how do I measure the response? should I indicate on the dart board two areas where one is considered a bulls-eye (success) and the other a failure? Or should I measure the distance in centimeters between the bulls-eye and the dart to determine which dart has come closest?

which is the appropriate method?

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If I were to go with measuring the distance from the center then that would mean the closer to the center the better the accuracy is, is it acceptable to state that a lower distance from the center means a better accuracy in the experiment? – silent Jun 3 '12 at 4:00
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Well, with regard to the research question - does distance matter, or does it only matter whether or not you can hit the bullseye? – Macro Jun 3 '12 at 4:43
The reason I asked was I can't hit a bulls-eye with my right hand even after 50tries lol. Is it valid to use the distance? – silent Jun 3 '12 at 4:49
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That depends on the research question. Are you interested in seeing how often you hit the bullseye or how far you get from the bullseye on average? In either case, if you just measured the distance it would leave your options open. – Macro Jun 3 '12 at 4:53
@Macro's comments are very useful. As for me, I'm developing an allergy to "should" questions. (Though one of these days I'll ask one on this site and have to eat my words.) I'd rather see people thinking in terms of "if I take this action, what are the likely consequences? What about that action?" And then they can judge how well their goals will be met by each course of action. – rolando2 Jun 3 '12 at 18:23

2 Answers

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I would go for the distance and then use a linear model to describe this score. The other option would be a logistic regression approach, but if you have mostly "failures" as you point out in the comments, this will not be very informative.

The only drawback with the distance is that it is not well behaved in terms of distribution. It is certainly not Gaussian, and probably the variance will change with the different conditions (example: right hand versus left hand). So you might already think of a transform to regularize it, perhaps the log of the inverse distance or something like this.

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When you say a linear model to describe the score, would it be suitable to use a scale from 1-10, where 10 describes a bulls-eye. – silent Jun 3 '12 at 12:10
I had more in mind the real distance in centimeters or inches. And then taking $\log (1/d)$ assuming that $d$ is never 0. – gui11aume Jun 3 '12 at 13:48
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I agree that distance is your best option. The use of a cutoff throws away information about the accuracy. So accuracy change with distance from the target is better assessed with a continuously changing measure. If the shooter is unbiased (i.e. his aim is on center and the hits scatter around the center then if the x an y coordinates on the target are normally distributed and independent with equal variance normalized coordinates would have a Rayleigh distribution for their distance. This is the square root of a chi square and will be skewed. As gu11aume points out for linear regression you might want to apply a transformation to the distance to make it symmetric so that least squares regression could then be applied. Handedness could be used as an indicator variable in the model.

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would a scale from 1-10 be good, where a 10 would be the highest accuracy – silent Jun 3 '12 at 12:30
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Well that is changing a continuous variable into a categorical one it still throws out information in the distance measure. It is better than binary but not as good as using the distance itself. The skewness can be dealt with, As gui11aume suggested a log transformation might work or you could try to find a best Box-Cox power transformation. – Michael Chernick Jun 3 '12 at 12:39

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