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I am a Phd Student in experimental psychology and I try hard to improve my skills and knowledge about how to analyze my data.

Until my 5th year in Psychology, I thought that the regression-like models (e.g., ANOVA) assume the following things:

  • normality of the data
  • variance homogeneity for the data and so on

My undergraduate courses lead me to believe that the assumptions were about the data. However in my 5th year, some of my instructors underlined the fact that the assumptions are about the error (estimated by the residuals) and not the raw data.

Recently I was talking about the assumptions question with some of my colleagues who also admited that they discovered the importance of checking the assumptions on the residual only in their last years of university.

If I understand well, the regression-like models make assumptions on the error. Thus it makes sense to check the assumptions on the residuals. If so, why some people check the assumptions on the raw data? Is it because such checking procedure approximate what we would obtain by checking the residual?

I would be very intersting in a disccussion about this issue with some people who have more accurate knowledge than my colleagues and I. I thank you in advance for your answers.

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Basically, you are on the right track. You will find a discussion about the aspect of normality in Normality of dependent variable = normality of residuals?

Some assumptions of the classic linear model are indeed about errors (using residuals as realizations of them):

  • Are they uncorrelated? (Relevant for inference and optimality of the OLS-estimators)
  • Do they have equal variance? (Relevant for inference and optimality of the OLS-estimators)
  • Are they centered around 0? (Key assumption to get unbiased estimators and predictions)
  • If the sample is very small: are they normal or at least symmetrically distributed? (Relevant for inference)

Other conditions are about "raw data":

  • Are there no gross outliers in regressors? (High leverage observations can destroy the whole model)
  • No perfect multicollinearity? (Would cause computational problems, at least in some software packages)

Now, your undergrad teacher might be correct as well:

  • Maybe you were focusing on univariate tests like the one-sample t-test. There, the assumptions are about the raw data.
  • If the $R^2$ is quite low and the response variable looks everything but normal, then the same will most likely also be true for the residuals.
  • How would you check homoscedasticity etc. based on raw data? Maybe you misunderstood him or her.
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  • $\begingroup$ Ok thank you very much for your answer and for the link which is very useful. Some of my colleagues and I belived until recently that the raw data should have equal variances. As you said we maybe missed something in our courses. In some book we can read the following : $\endgroup$ – Psychokwak Nov 26 '13 at 8:12
  • $\begingroup$ "Most common statistical procedures make two assumptions that are relevant to this topic: (a) an assumption that the variables (or their error terms, more technically) are normally distributed, and (b) an assumption of equality of variance (homoscedasticity or homogeneity of variance), meaning that the variance of the variable remains constant over the observed range of some other variable." Does it means that when one talks about "variable" he or she systematically talks about "their error terms"? If so I am ok with that but without explicit mention it is far from obvious (at least for me). $\endgroup$ – Psychokwak Nov 26 '13 at 8:15
  • $\begingroup$ Finally, I have a last question about your answers. If t-test and ANOVA are particular cases of the regression, why the assumptions are about the data in a one-sample t-test? Thanks again for your useful answer. $\endgroup$ – Psychokwak Nov 26 '13 at 8:18
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    $\begingroup$ To answer your last comment: The one-sample t-test can also be viewed as a special case of regression. The model simply consists of the intercept (=mean) and the error term, i.e. the response is a shifted error. Since shifts are irrelevant for any assumption, it's equivalent to talk about data or residuals. $\endgroup$ – Michael M Nov 26 '13 at 11:17
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I find the differentiation between the residuals and the raw data unhelpful since both refer more to your actual sample and not the underlying population distribution. It's better to think of as some requirement being "in-group requirements" and others "between group assumptions".

For example, variance homonenity is a "between-group assumption" since it say that the within group variance is the same for all groups.

Normality is a "within group" assumption which requires that within each group y is distributed normally.

Note that having normality over your entire raw y usually means you have no effect - look at the distribution of gender without differentiating between females and males. It will not be normally distributed, because of the strong gender effect. But within each gender it holds quite well.

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    $\begingroup$ Thanks for your answer too. It is an interesting way to see the question. I had never think about normality in such a way (i.e., "that having normality over [the] entire raw y usually means [we] have no effect"). $\endgroup$ – Psychokwak Nov 26 '13 at 8:20

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