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Is there ever a compelling reason to conduct multiple hypothesis tests versus a multiple regression when comparing a response in two treatment groups among many (possibly thousands or millions) explanatory variables?

For example, say we have the alternatives:

  • Multiple comparisons of the response $y$ in treatment group $x_{trmt}=A$ vs. $x_{trmt}=B$ on many variables $x_1,x_2,...,x_p$ (controlling for the familywise error rate or false discovery rate with Bonferroni, Benjamini–Hochberg procedure, etc.), or
  • Multiple regression of $y$ on $x_{trmt},x_1,x_2,...,x_p$ with interaction terms $x_{trmt}x_1,x_{trmt}x_2,...,x_{trmt}x_p$.

Especially when considering strong positive or negative correlations between the explanatory variables may exist, a penalized regression such as a lasso or ridge regression would seem a more powerful alternative to multiple comparison techniques that avoids issues with interpreting $p$-values. The lasso regression can also handle data where the number of explanatory variables exceeds the number of subjects in the dataset.

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