How do I formally test outliers from a linear regression?

Question

Hello (and thanks for reading this).

I have a set of data that looks at the area of remaining retina against age. I have three points of data per patient. We know that over time, the area of remaining retina decreases exponentially. I have done a linear regression to model loss of area over time (see graph). X axis = age, Y axis = retinal area (log scale).

In this data there are some patients that are outliers because of their genetic mutation. I am interested in doing a formal test to see whether because of this mutation, they have a significantly larger area of retina remaining for their age, compared to the rest of the cohort. I have read about various tests (Grubbs, Dixon etc) but they don't seem to apply here for bivariate data. I am aware that I can identify outliers by visualising them, but I want a formal test with a p value if possible to satisfy reviewers who have requested it.

Please see the attached graph to demonstrate what the data looks like. The red and green dots are the patients I am interested in testing. Any suggestions on how to analyse this would be most appreciated. I am using Prism 7 for analysis. I can't seem to find an answer for this question on stack anywhere. As you can probably tell, my stats experience is fairly limited. Thanks for your help.

Answer 1

I don't know how big your measurement error for the area is, but presumably it's small (looking at how consistent the data is within person). Looking at the graph suggests to me that you blatantly have people for whom your model really does not fit/that are outliers relative to the current model. It's very easy to see, because you conveniently have multiple records per person.

You could look at the residuals per person and do a stratified test of normality or look at how unlikely each record was to arise under your model (but the real problem with the current model is not taking person into account). You'd clearly see that the residuals are not normal for each person. You should look at it by person, because that matches the sampling. Doing formal tests for outliers or normality has lots of problem (see many other questions), but here it's glaringly obvious that the model had problems.

A model that might do better might be a model with a random subject effect on the intercept and possibly also on the slope.