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This might be a simple question, but just wanna make sure I am on the right track: I need to compare two samples using a 1-sided t-test. The sample sizes in the two groups are unequal, but it should be ok. The crucial part is that, the data in both groups are not independent. To be more specific, I have 4 vectors, v1, v2, v3 and v4, and I calculate the pairwise angles: v12, v13, v14, v23, v24, v34. Then I put them into two groups:

Group1: v12, v34
Group2: v13, v14, v23, v24

Then I compare group 1 and 2 using one-sided two-sample t-test. This violates the independent samples assumption of t-test, right? I am considering a solution of using permutation test in the coin R package (one_way() function). Is that the correct way to do?

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  • $\begingroup$ What hypothesis do you mean to test with the t-test? $\endgroup$
    – IMA
    Commented Jul 23, 2013 at 13:08
  • $\begingroup$ @IMA: to test (alternative H1:) mean of Group1 is smaller than mean of Group2 $\endgroup$
    – alittleboy
    Commented Jul 23, 2013 at 13:16
  • $\begingroup$ Do you really, actually only have four vectors, or is this a simplification (ie, how large are the groups in reality?) $\endgroup$
    – IMA
    Commented Jul 23, 2013 at 13:24
  • $\begingroup$ @IMA: it's only a simplification: in reality, I have 56 data points in Group1 and 64 data points in Group2. BTW, they are not necessarily normally distributed, which is another issue I concern... $\endgroup$
    – alittleboy
    Commented Jul 23, 2013 at 13:26
  • $\begingroup$ Is your basic data an N x 16 matrix whose columns you split into two sets, say 1..8 and 9..16, and are you asking if the average within-set angle is greater that the average between-set angle? If so then what is N? Are the N rows independent samples from the same 16-variate distribution? What is the sample space? Are you really interested in the angles, as opposed to their cosines or some other function? $\endgroup$
    – Ray Koopman
    Commented Jul 24, 2013 at 4:17

1 Answer 1

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Thank you for clarifying. It seems that the columns, not the rows, are the random elements. In the absence of any information about what the within-set or between-set distribution of angles might be in the ~15K-dimensional space (or even with that information), a permutation test is quite reasonable. Note that 16_choose_8 = 12870, so you can easily get the exact distribution.

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  • $\begingroup$ thanks! I think the number of sample within group is 56, and between group is 64, for a total of choose(16,2)=120 data points. still, a permutation test is reasonable, i guess :) $\endgroup$
    – alittleboy
    Commented Jul 25, 2013 at 11:45
  • $\begingroup$ Sorry, I should have been more explicit. You need to look at all possible splits of the 16 columns into 2 sets of 8. For each split, you get the difference between the within-set mean and the between-set mean. You want to know where the within-between mean difference for your actual split comes in that distribution of all possible mean differences. (Since the within-between comparison doesn't care which is Set 1 and which is Set 2, you need to look at only 15_choose_7 = 6435 splits, with one of the columns always being in Set 1.) $\endgroup$
    – Ray Koopman
    Commented Jul 25, 2013 at 13:27

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