01Three checks, and what each one is a property of
Normality, in most of the models a thesis uses, is a property of the residuals rather than of the raw variable. Students test the raw variable, find it skewed, panic, and transform something that never needed transforming. Ask what the model assumes to be normally distributed and the answer is almost always the leftover variation, not the measurement itself.
Homogeneity of variance is a property of the groups being compared: are they similarly spread, so that one mean is being estimated no less precisely than the other. Linearity is a property of the relationship the model has assumed exists. Naming the owner of each assumption is not pedantry. It tells you which object to plot, and half of all assumption confusion is people plotting the wrong object.
02What the software prints, and what to look at first
The formal tests return significance values, and their behaviour depends heavily on how much data you have. With a large sample they will flag departures far too small to affect anything. With a small sample they will happily miss departures that matter. So a formal test alone is a poor basis for a decision, in either direction.
Look at the picture as well. A normal probability plot of the residuals shows you the shape and where it deviates. A scatter of residuals against fitted values shows non-constant spread and curvature in a single image, which is why it is the one plot worth producing every time regardless of what the tests say. Whatever you conclude, conclude it from both and say which you relied on.
Take them in that order rather than the order the software prints them. Residuals against fitted values first: it carries two assumptions at once, and a bend in it sends you back to how the model was specified before any assumption is worth arguing about. The probability plot second, because a curve there is often the same fault seen from a different side rather than a separate one. Judging normality before you have looked at the spread is how a transformation gets applied to a model that needed another term in it instead.
03When one fails, four honest moves
A failed check is not a dead end and it is not a confession. It changes which procedure is appropriate or which caveat is required, and every one of the following is defensible in a chapter provided you say what you did and why.
- Use the variant of the procedure that does not assume it, where the software offers one
- Move to a rank-based equivalent, and state plainly that what is being compared has changed
- Transform the variable, then interpret on the transformed scale and report that you did so
- Report the violation, keep the analysis, and carry the consequence into your limitations
- Not on this list: deleting the cases that spoil the plot without a rule stated in advance
- Also not on this list: running the analysis several ways and reporting only the version that survived
04Independence, which nothing prints
This is the one that gets work sent back at a defence, and no procedure will warn you about it. Measurements taken twice from the same person are not independent. Students inside one classroom are not independent of each other. Patients on one unit share a great deal that has nothing to do with your variables. A survey handed out by a manager to their own team is not a sample of individuals in any useful sense.
Clustered observations analysed as though they were independent produce an interval that is too narrow and a significance value that is too generous, which means the result overstates its own certainty. That is a design property, so you find it by describing how the data were collected, not by running anything. Do that description early, in writing, before the analysis is chosen.
If clustering exists, either the model accounts for it or the write-up names it as a limitation in specific terms rather than general ones. Where the design and the analysis have to be reconciled before a committee sees them, that reconciliation is what a statistics specialist is for.
FAQQuestions to control
Do I have to report assumption checks in the results chapter?
Report them, briefly, before the result they qualify. A sentence naming what you checked, how, and what you found is enough, with plots in an appendix if your programme wants them. Omitting the checks entirely reads as though they were never run, and a reader who suspects that will treat every number after it with the same suspicion.
Is a non-parametric test always the safe choice?
It is safer against one specific problem and it is not free. Rank-based procedures answer a slightly different question, usually about distributions or medians rather than means, so your write-up has to change wording to match. They also tend to detect less when the original assumption was actually fine. Choose one because a check failed, not as a precaution against checking.
My sample is small. Does that change what I should check?
It changes what you should trust. Formal tests lose the ability to detect real departures at small sizes, so the plots carry more of the weight and your judgement carries the rest. It also raises the value of saying so: a limitations paragraph that names the sample size and the resulting uncertainty about assumptions is stronger than a formal test reported as though it settled the matter.
Can I just remove outliers so the assumptions hold?
Only under a rule you set before looking, and only with the removals reported and the analysis shown both ways. Points removed because they were inconvenient change the result in a direction you chose, which is the thing peer review exists to catch. An extreme value is often the most interesting observation in the set, and it usually deserves a sentence rather than a deletion.
