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Please tell me what the actual assumptions of a two way anova are. I read somewhere that this being similar to multiple regression, the only assumption is that the residuals are to be normally distributed with equal variance. But then a web search gave the following assumption "Two-way anova, like all anovas, assumes that the observations within each cell are normally distributed and have equal variances". I am confused. Please help out.

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  • $\begingroup$ What exactly are you confused about? The two descriptions sound the same to me. $\endgroup$ – gung - Reinstate Monica Aug 4 '14 at 2:30
  • $\begingroup$ If the model is "true," the statements are equivalent: $\varepsilon_i\sim N(0,\sigma^2)\Longleftrightarrow y_i\sim N(x_i^T\beta,\sigma^2)$, since $x_i^T\beta=\mathbb{E}[y_i|x_i]$. I'm not sure if I got that precisely right, but that's the idea at least. This is why people use the term "variance" so loosely when they talk about regression, because it doesn't matter which one they're referring to. $\endgroup$ – shadowtalker Aug 4 '14 at 3:27
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There are a number of assumptions that apply to two-way anova if you're performing inference; different people will place more or fewer things into the assumptions or into the model or leave them as obvious/implied, but a basic list for the most common tests/intervals is probably:

  1. independence. The observations are assumed to be independent.

  2. equal variance. The observations (equivalently, the errors) are assumed to have equal variance.

  3. normality. The errors (not the residuals!) are assumed to be normal. Equivalently the observations within each combination of factors are normal. However, since we don't have access to the errors, we examine this assumption by reference to the residuals, which (if the other assumptions hold) will approximate them.

  4. The values of the factors are known/fixed.

  5. The form of the model for the mean is correct.

The last two are often omitted from the list but they're still there.

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