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If my difference scores are not normally distributed - and I want to do a parametric paired t-test - do I:

  1. log transform the the original scores and perform a paired t-test on these scores
  2. log transform the difference scores and do a one sample t-test against a test value of 0
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For a paired test, what is relevant is the distribution of the difference scores, not the individual distributions. Even if the original scores have a nonnormal distribution (difficult to say with only $n=17$), the difference might be normal (or at least symmetric). So I would have first a look at the qqplot of the differences against a normal distribution. Then, if necessary, transform the difference score. Or use a nonparametric procedure. More discussion in the related post at Skewness transformation for one but not the other variable?

An alternative could be a permutation test (permuting the signs of the absolute differences), which do not depend on distribution assumptions. For further discussion of alternatives see Best practice when analysing pre-post treatment-control designs

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  • $\begingroup$ Not sure OP's question was completely answered, Kjetil. Can he do either option 1 or 2? I personally would find it easier to interpret option 1 and we can also get a correlation score. $\endgroup$ May 6, 2021 at 2:19
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    $\begingroup$ @ScottEdwards2000 The first sentence in this answer is clear: it depends on the distribution of the differences (as well as on your probability model for the data). Either option is possible (although it would be rare for option (2) to be appropriate). +1. $\endgroup$
    – whuber
    Apr 7, 2023 at 12:53
  • $\begingroup$ @whuber thanks for the help but still not sure I get it - the answer says what matters is the difference scores distribution, which to me would indicate the 2nd option would make more sense because those are the scores we are normalizing. But you are saying that it would be rare for that to be appropriate? thx $\endgroup$ Oct 1, 2023 at 17:48
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    $\begingroup$ @ScottEdwards2000 I hope the issue might become obvious when you contemplate the possibility that some of the differences will be positive and some negative: that precludes simple, natural transformations like the logarithm altogether, even when the original data might themselves be positive and when their individual logarithms might be nicely distributed (e.g., normally distributed). $\endgroup$
    – whuber
    Oct 1, 2023 at 18:31
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For dependent observations, transform the individual data NOT the differences. Then you can back-transform the mean difference to give the ratio of the geometric means. If the paired data are before and after measurements, this is interpreted as the ratio of the two geometric mean measurements.


Reference: Oxford Handbook of Medical Statistics 2nd edition, page(332)

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    $\begingroup$ Although a handbook might assert this -- and it's generally good advice for beginners -- the nuanced answer originally posted here explains why either option might be appropriate depending on the data and how to begin examining the data. $\endgroup$
    – whuber
    Apr 7, 2023 at 12:54
  • $\begingroup$ Thank you @whuber $\endgroup$
    – Abo7aneen
    Apr 7, 2023 at 16:31

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