# Looking at residuals vs. residual percentages

Suppose I fit a linear regression to some data (say, weight vs. height), and all the standard linear regression assumptions are satisfied (in particular, the data is homoscedastic). For example, here's a random figure pulled from amstat.org that looks like it satisfies what I'm thinking of:

Now I'm doing some exploratory data analysis, so I want to look at examples where the linear regression is particularly off; that is, I want to sort all the individuals by how badly the regression predicts their weight from their height (so that, say, I can look at further details, like whether people whose weight the model underpredicts tend to eat a lot of junk food). My question is:

• Should I sort all the individuals by their raw residuals?
• Or should I sort all the individuals by the residuals as a percentage of the prediction, i.e., by residual weight / predicted weight?

On the one hand, it seems like sorting by raw residuals might be the way to go, since standard linear regression errors are based off the squared residuals, and not the residual percentages. On the other hand, someone who weighs 70kg when their predicted weight is 50kg seems much more of an outlier than someone who weighs 120kg when their predicted weight is 100kg.

Is it just a matter of preference or the particular model at hand?

• Just out of curiousity, do the two methods lead to a much different ordering? – Dominic Comtois Jul 18 '11 at 6:22
• Do you have a rough noise model, e.g. $\sigma(x)$ increasing with x, with scaled residuals $(\frac{y - ax}{\sigma(x)})^2$ roughly flat ? – denis Jul 18 '11 at 8:41
• Ratios are indeed better indicators than absolute differences in this case @dominic yes there should be a different ordering, since in mentioned example both residuals are 20, but relative errors are $2/7 > 2/12$. @raegtin it seems you just want to define what are the outliers in the regression model, isn't it? – Dmitrij Celov Jul 18 '11 at 11:40
• @Denis: I'm assuming that $\sigma(x)$ is constant. @Dmitrij: yep, I basically just want to find the outliers. From a "minimize sum of squares" point of view, it seems like I shouldn't be scaling residuals to find outliers, but intuitively it seems like I should. @dominic: yep, like Dmitrij said, the two methods should lead to a much different ordering (especially if there's a large range in the x-axis). – raegtin Jul 18 '11 at 20:14
• Maybe part of the problem is that I'm surprised many models seem to have constant variance. For example, I'm surprised in the example above that there's not a larger variance in weight as height increases. – raegtin Jul 18 '11 at 20:17