Not sure about the interpretation of this residual plot

I'm analyzing a residual plot of the residuals vs the fitted values.

I'm not quite sure how to interpret this plot since there looks like there is a pattern and the average is not actually zero. There are diagonal straight parallel lines. I'm working with several ordinal IV's.

Any ideas?

• Looks like your dependent/explained/left-hand-side/$y$-variable has only 5 categories. Is that correct? Oct 20 '14 at 14:01
• Correct. @MaartenBuis Oct 20 '14 at 14:03
• If there is a pattern in your residual plot it means there is some underlying factor that affects the dependent variable that isn't being taken into account in your regression. If you find that variable and include it in your model the pattern should disappear
– Mike
Oct 20 '14 at 14:04
• The average of the residuals is easily calculated with any decent software (looks like Stata in your case, but any software should support that). Oct 20 '14 at 14:16
• If you told me an individual would be on the lowest Likert band, I would guess their residual was negative. This seems to be an exploitable bias. Oct 20 '14 at 16:16

You are using a Likert scale or similarly defined ordinal dependent (and independent) variable and modeling it in the same way as a continuous variable. The debate of whether Likert scale can be modeled as a continuous numeric variable is a bit murky and can be summarized here:

http://www.theanalysisfactor.com/can-likert-scale-data-ever-be-continuous/

In practice, I tend to not have a problem with treating ordinal independent variables as numeric variables if there are at least five values and you are somewhat confident they are equally spaced on the spectrum of possible values -- that is, a difference between any two consecutive is approximately of equal magnitude. This practice is not grounded in theory, but experience has shown me that results tend to not be badly biased.

If this work is intended to be published, I suggest you either carefully justify your decision to use this method or look into alternative methods. EDIT: As suggested by @NickCox in the comments, ordered logit (or probit) regression would be a suitable alternative to deal with the ordinal response variable.

One consequence of treating the variables as continuous is that the residual plot can appear as a series of parallel lines, as your plot shows. This makes the typical pattern detection of the residual plot more difficult since the points clearly will not be randomly scattered.

As far as your plot is concerned, I believe it is acceptable. There is no clear curvature of the plot, and no fanning (i.e. no heteroscedasticity). They appear (to me) to form a roughly horizontal band if you take the density of points into account when forming the band. I would suggest plotting the absolute value of the residuals to help assess heteroscedasticity further.

• Ordered logit (probit) is an alternative: there is no obvious need to jump all the way to nonparametric. Oct 20 '14 at 14:14
• How about the fact that the obvious omitted variable causes variance at the tails to be twice what it is in the middle? Not trying to be antagonistic, but if you don't see a problem with it I'm interested as to why. Oct 20 '14 at 14:15
• @ssdecontrol How do you reach that conclusion? Oct 20 '14 at 14:17
• @underminer Yes, but also a discrete response variable, as can be inferred from the pattern; in any case that has been confirmed by the OP in comments. Oct 20 '14 at 14:30
• @underminer Not a mistake on your part, as OP could helpfully have been clearer. Oct 20 '14 at 14:51