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I have a dataset that essentially amounts to a sample of K unique elements from an ordered sequence of length N. I believe that elements are more likely to be sampled from certain portions of the sequence (e.g. earlier elements are more likely to be selected), so my null hypothesis is that all elements are equally likely to be sampled. Equivalently, one could imagine an ordered sequence of K ones and (N-K) zeros, and I want to ask whether the ones appear to be uniformly distributed throughout the sequence or whether they vary in density along the sequence. What is a good statistical test to use for this question?

N is approximately 20000, and K is anywhere from 3000 to 10000.

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I finally figured out that if I restrict myself to asking whether the ones are distributed, on average, closer to the beginning or end of the sequence (as opposed to the more general question of any non-uniformity), then I can get my answer with a Mann-Whitney U test on the indices of the ones and zeros.

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    $\begingroup$ You might want to consider a much more informative approach than that, such as viewing a plot of the empirical cumulative distribution (which would amount to a QQ plot). Any pattern of deviations from the expected linear behavior not only would reveal a violation of the null hypothesis, but would also indicate how the null is violated. A Kolmogorov-Smirnov statistic could be used as a formal test. $\endgroup$ – whuber Oct 31 '14 at 16:43
  • $\begingroup$ Ooh, yes, a QQ plot would be a good idea. However, the original reason I asked this question is because my collaborator showed an informative plot that clearly indicated a robust trend, but her committee refused to believe her unless we agreed to become enablers for their crippling addiction to p-values. Of course, the p-values from the Mann-Whitney test (and Kendall's tau test too) for these datasets turned out to be less than the smallest representable double-precision floating point value (~10^-300), as expected. $\endgroup$ – Ryan C. Thompson Oct 31 '14 at 19:57

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