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On the Mathematics site an OP who is just learning statistics gave his description of the difference between linear regression and ANOVA and asked if his interpretation was correct. I responded that linear regression considers how a set of covariates relate to a response in a functional form (could have added "that is linear in the parameters") whereas ANOVA categorizes the response into a class or classes of group(s) and tests for a difference between group means. A member downvoted my answer saying that ANOVA can include continuous predictors as well. His own answer indicated that he was considering the term ANOVA to mean the testing of significant terms from the decomposition of variance in the general linear model. We both gave descriptions of linear regression that agreed.

My question is: "What do you think is the best answer? His answer? My answer? An explanation providing the two meanings of ANOVA? Something else?

Link: http://math.stackexchange.com/questions/183704

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Can you provide the link to the discussion? Also we have an anova tag, but I don't have sufficient rep to suggest/create a synonym with your created analysis-of-variance tag, could you create the synonym? – Andy W Aug 18 '12 at 13:28
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Also there is a related discussion in a prior question on the site (although not a duplicate), Why is ANOVA taught / used as if it is a different research methodology compared to linear regression? – Andy W Aug 18 '12 at 13:29
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@smillig The discussion was not about computer routines. We just discussed SAS in my case and R in his as support to our positions. It is common to refer to the variance decomposition table in the general linear model as the ANOVA table. But ANOVA is usually differentiated from regression in the way I stated. Thanks for linking the other post. – Michael Chernick Aug 18 '12 at 13:59
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I can't understand why nobody uses the purpose argument, i.e. that linear regression is a modelling method and ANOVA is a hypothesis testing framework. I know that they can be extended in a way they overlap, but the canonical formulations are perfectly separated. – mbq♦ Aug 18 '12 at 14:26
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@MichaelChernick I know the discussion wasn't about computer routines, but since the other commentator was referring to R functions, I thought the documentation for what each of those things do might shed light on where each of you is coming from (which, as I think mbq is alluding to in his comment, is probably behind the differences between the answers on the original post). I think your answer is correct and definitely didn't deserve to be downvoted. – smillig Aug 18 '12 at 15:15
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ANOVA and Linear regression are twin princesses grown in different castles. Please see the book of Andy Field: Discovering statistics using SPSS. He has a very nice explanation including the evolution in time of this two. Anyway put bluntly: they are very similar and developed in parallel for a certain period of time by different scientific communities. NB: Of course the comparison is software independent.

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