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I was taught in applied stats to verify the assumption of normality on my variables separated by group. For example, if I'm studying a continuous variable like weight (g) in two groups of subjects: A and B, and I'll use a t-test to test a hypothesis of the difference in means, I'd proceed to use plots and summaries (kurtosis, skewness) or even the least favored (Shapiro-Wilk, Kolmogorov-Smirnov, etc.) to assess normality.

If I do this and found one of the groups to deviate significantly from normality, I'd think of another approach to analyze the data or keep with normality after approximating through transformations. However, if I instead input the same data in a linear regression model with 'group' as an independent variable, I then reassess the distribution of my residuals, the normality assumption now holds.

  1. I know that the residuals are combined, but what are the fundamental differences between the two approaches?
  2. Wouldn't be wiser to use the last one?
  3. Why was this taught this way, is there a more appropriate way?

I also learned that several statistical tests have underlying linear models, so hence the question.

BioLeal
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    The assumption is about the residuals, but there's also an assumption (usually) that the residuals are independently and identically distributed (iid). If you have one group with the residuals strictly on the left side of some Gaussian and another group with the residuals strictly on the right side of that same Gaussian, you have normal residuals when you combine all of the residuals, but you (badly) violate the iid assumption. – Dave Mar 24 '21 at 14:49
  • I found an answer here: https://stats.stackexchange.com/questions/60410/normality-of-dependent-variable-normality-of-residuals – BioLeal Mar 26 '21 at 10:16

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