This issue seems to rear its ugly head all the time, and I'm trying to decapitate it for my own understanding of statistics (and sanity!).
The assumptions of general linear models (t-test, ANOVA, regression etc.) include the "assumption of normality", but I have found this is rarely described clearly.
I often come across statistics textbooks / manuals / etc. simply stating that the "assumption of normality" applies to each group (i.e., categorical X variables), and we should we examining departures from normality for each group.
does the assumption refer to the values of Y or the residuals of Y?
for a particular group, is it possible to have a strongly non-normal distribution of Y values (e.g., skewed) BUT an approximately (or at least more normal) distribution of residuals of Y?
Other sources describe that the assumption pertains to the residuals of the model (in cases where there are groups, e.g. t-tests / ANOVA), and we should be examining departures of normality of these residuals (i.e., only one Q-Q plot/test to run).
does normality of residuals for the model imply normality of residuals for the groups? In other words, should we just examine the model residuals (contrary to instructions in many texts)?
To put this in a context, consider this hypothetical example:
- I want to compare tree height (Y) between two populations (X).
- In one population the distribution of Y is strongly right-skewed (i.e., most trees short, very few tall), while the other is virtually normal
- Height is higher overall in the normally distributed population (suggesting there may be a 'real' difference).
- Transformation of the data does not substantially improve the distribution of the first population.
Firstly, is it valid to compare the groups given the radically different height distributions?
How do I approach the "assumption of normality" here? Recall height in one population is not normally distributed. Do I examine residuals for both populations separately OR residuals for the model (t-test)?
Please refer to questions by number in replies, experience has shown me people get lost or sidetracked easily (especially me!). Keep in mind I am not a statistician; though I have a reasonably conceptual (i.e., not technical!) understanding of statistics.
P.S., I have searched the archives and read the following threads which have not cemented my understanding:
- ANOVA assumption normality/normal distribution of residuals
- Normality of residuals vs sample data; what about t-tests?
- Is normality testing 'essentially useless'?
- Testing normality
- Assessing normality of distribution
- What tests do I use to confirm that residuals are normally distributed?
- What to do when Kolmogorov-Smirnov test is significant for residuals of parametric test but skewness and kurtosis look normal?