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Let's say I'm measuring viral load post-infection. The two groups I have are vaccinated and unvaccinated.

We expect the distribution of the unvaccinated cohort's viral load to resemble a normal distribution centered around a certain positive number. We expect the vaccinated group's viral loads to be centered near 0, which means that we won't see a symmetric bell curve.

The t-test is robust for bell-shaped distributions as I understand it, so is there a better/more specific test one should use in this scenario? Or would we need to rely on a non-parametric test like Mann-Whitney U test even though we have an idea of the distribution? Thank you.

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What you are saying is that you have two samples with very different distributions and you want to compare the mean values (which I agree would be a statistic of major interest for your example). I assume that the sample sizes are not 'small'. Here are three possible analyses:

  • DO NOTHING: If the sample is 'very large' then put your trust in the Central Limit Theorem and perform a two-sample t-test (Fagerland 2012). This could at least be a useful check for the next two analyses.
  • EASY: Two-sample t-test with Welch-Sattherwaite approximate degrees of freedom (which allows for unequal variances).
  • MORE ROBUST: Non-parametric bootstrap resampling to estimate a confidence interval for the difference of means, the ratio of means or some other informative measure of effect size. Note that Permutation testing can't be used because the two samples are not identically distributed.

Beware that quite often the Mann-Whitney U-test provides only some vague inference that there is some difference in the distributions of the two samples (first and foremost, it's not a comparison of means) and then the P-values are too small in large samples. Do NOT use the Mann-Whitney U-test for P-value hacking.

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